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PT). present the mechanical properties of LG-SW after multiple heating-cooling cycles obtained in this experiment and the corresponding degeneration factors for grades 1670 and 1960, respectively. The degeneration factors are also plotted in as the function of heating-cooling cycle index.Results show that although little differences were presented in (a), (b), and (c) due to normal test errors, generally speaking, all the elastic modulus, yield strength, ultimate strength, and ductility of both Grade 1670 and Grade 1960 LG-SW after cyclic heating-cooling were almost the same as those after only one heating-cooling cycle regardless of the temperature to which they had been exposed. This finding indicates that the effects of cyclic heating-cooling on the mechanical properties of both lower- and higher-grade LG-SW are insignificant. Hence, they can be ignored in the assessment of the post-fire mechanical properties of LG-SW. The effects of cyclic heating-cooling were also not considered in the predictive equations proposed in , most of the current studies on post-fire mechanical properties focused on structural steels of varying manufacturing processes (hot-rolled or cold-formed) and grades (mild or high-strength), and reliable predictive models have been developed based on extensive experimental studies. Moreover, in British Standard 5950-8 (2003) Annex B The comparison of the post-fire elastic moduli obtained in this study for LG-SW with those of Grade 1570 SW-PCS, Q235, Q345, S460, S690, G300, and G500 in the existing studies is shown in . The residual factors of LG-SW demonstrate trends that differ significantly from those of Grade 1570 SW-PCS and various kinds of structural steels. The elastic moduli of all four grades of LG-SW remained almost unchanged after cooling from temperatures up to 1000 °C, whereas different degrees of reduction are observed for Grade 1570 SW-PCS and structural steels cooled down from certain elevated temperatures.The post-fire yield strengths obtained in this study for LG-SW are compared with those of Grade 1570 SW-PCS, Q235, Q345, S460, S690, G300, and G500, as well as with the recommendation on the post-fire reuse of structural steels in BS 5950-8, as presented in (b). The yield strengths used for comparison are defined based on the 0.2% proof stress method. The yield strength of LG-SW decreased much earlier and faster with respect to exposure temperature than that of most structural steels but more slowly than that of Grade 1570 SW-PCS. In addition, the recommendations in BS 5950 for all S235, S275, and S355 obviously cannot ensure safe reuse of LG-SW when the exposure temperature exceeds 600 °C. Thus, such recommendations are unsuitable for evaluating the post-fire yield strengths of LG-SW.The post-fire ultimate strengths derived in this study for LG-SW are compared with those of Grade 1570 SW-PCS, Q235, Q345, S460, S690, G300, and G500, as well as with the recommendation in BS 5950-8, as shown in (c). Similar to the comparison of yield strength, the ultimate strength of LG-SW showed much more significant decreasing trends than those of various structural steels, whereas such trends are more moderate than that of Grade 1570 SW-PCS. Furthermore, the recommendations in BS 5950 for all S235, S275, and S355 steels are dangerous for post-fire evaluation of ultimate strength of LG-SW.(d) presents the comparison of the post-fire ductility obtained in this study for LG-SW with those of Grade 1570 SW-PCS, Q235, and Q345. The results for S460, S690, G300, and G500 are not included because the specific values of fracture strains are not available in the corresponding literatures Generally, the variations in the mechanical properties of LG-SW with exposure temperature are significantly different from those of steel wires used for prestressed concrete structure and various kinds of structural steels available in previous literatures because of the considerable differences in both chemical compositions and manufacturing processes. Hence, the outcomes of generated from these materials should not be applied to LG-SW directly. Moreover, the recommendations in BS 5950-8 for the post-fire reuse of steels are obviously unsafe for the evaluation of strength of LG-SW; hence, they also cannot be adopted.The experimental results demonstrate that the mechanical properties of LG-SW markedly changed after exposure to elevated temperatures. Therefore, predictive equations that are sufficiently accurate for practical use and easy to master by engineers to evaluate the post-fire performances of LG-SW are significant. However, no design guides and studies in previous literatures on the post-fire mechanical properties of LG-SW are available; moreover, as discussed in , the predictive models proposed based on the results of various kinds of structural steels and steel wires used for prestressed concrete structure are inappropriate for post-fire assessment of LG-SW. Therefore, new predictive equations that are simple in form but high in precision are proposed in this section to estimate the post-fire mechanical properties of LG-SW of varying grades on the basis of the experimental results. Given that the exposure temperature is the main factor causing the deterioration of mechanical properties, the predictive equations were developed as a function of the highest exposure temperature T, where the significant influence of different cooling methods was also incorporated.The elastic moduli of all grades of LG-SW remained almost unchanged after exposure to various elevated temperatures up to 1000 °C for both air cooling and water cooling. Hence, for practical use, a conservative percentage, i.e., 95%, of the original elastic modulus is recommended in Eq. to evaluate the post-fire elastic modulus residual factors of all four grades of LG-SW regardless of exposure temperature and cooling method. compare the elastic modulus residual factors predicted by Eq. with the experimental results. The predictions are all lower than but approximate to the minimum value of the experimental data.The yield strength residual factors of all four grades of LG-SW cooled by air presented three main regions with respect to exposure temperature. Hence, piecewise equations that contain three equations were developed to ensure accuracy. For water-cooled LG-SW, an exposure temperature of 700 °C is defined as critical failure temperature [] because the LG-SW after water cooling from higher temperatures show complete brittleness with nearly no ductility and strength thus cannot be reused. Hence, predictive equations for water cooling condition were developed only in the range of 20 °C–700 °C, which is below the critical failure temperature. In addition, considering the influence of steel grades on the change of yield strength was insignificant, the same equations were developed for all four grades of LG-SW cooled by same cooling methods.For air cooling, all grades of LG-SW can be calculated as follows:fy,PTfy=120°C≤T≤400°C1.221+1.39×10−4T−1.78×10−6T2400°C<T≤750°C−3.535+8.44×10−3T−4.43×10−6T2750°C<T≤1000°CFor water cooling, all grades of LG-SW can be calculated as follows:fy,PTfy=120°C<T≤400°C1.700−1.73×10−3T400°C<T≤700°C compares the post-fire yield strength residual factors of LG-SW predicted by Eqs. with the experimental results. A good agreement is observed between the predictions and test results.Similar to the post-fire yield strength, the same piecewise equations were developed for all four grades of LG-SW to predict the ultimate strength residual factors of LG-SW because of the minimal difference caused by various steel grades; whereas the influence of different cooling methods was considered.For air cooling, all grades of LG-SW can be calculated as follows:fu,PTfu=120°C≤T≤400°C1.911−2.50×10−3T+5.78×10−7T2400°C<T≤750°C−5.950+1.40×10−2T−7.46×10−6T2750°C<T≤1000°CFor water cooling, although LG-SW still maintained at least 30% of their original ultimate strength at the exposure temperature of 750 °C, they cannot be reused after fire events because of significant loss of ductility. Hence, the equations were also proposed only in the range of 20 °C–700 °C, which is lower than the critical failure temperature. All grades of LG-SW can be calculated as follows:fu,PTfu=120°C<T≤400°C1.764−1.90×10−3T400°C<T≤700°C shows a comparison of the post-fire ultimate strength residual factors predicted by Eqs. with the experimental results, and a good agreement can be observed between them.Compared with elastic modulus and strength, the variations in the ductility of LG-SW cooled by air differed greatly with steel wire grades. Therefore, piecewise predictive equations were developed for the lower- and higher-grade LG-SW separately under air cooling condition. As for the water cooling method, the same equations were proposed for all grades of LG-SW because of the insignificant influence of steel wire grades.For air cooling, lower-grade 1670 and 1770 LG-SW can be calculated as follows:δu,PTδu=120°C≤T≤500°C5.325−1.84×10−2T+1.95×10−5T2500°C<T≤750°C5.725−7.26×10−3T+2.62×10−7T2750°C<T≤1000°CHigher-grade 1860 and 1960 LG-SW can be calculated as follows:δu,PTδu=1.001−1.39×10−4T+5.02×10−7T220°C≤T≤600°C20.526−6.91×10−2T+6.13×10−5T2600°C<T≤750°C143.472−0.436×10−2T+4.46×10−4T2−1.52×10−7T3750°C<T≤1000°CFor water cooling, the equations were also proposed in the region lower than the critical failure temperature. All grades of LG-SW can be calculated as follows:δu,PTδu=1.001−8.83×10−5T20°C≤T≤500°C1.038−2.09×10−3T+1.72×10−5T2−5.02×10−8T3+4.66×10−11T4500°C<T≤700°C shows that the post-fire ductility residual factors predicted by Eqs. agree well with the experimental results.This paper presents a detailed experimental study on the post-fire mechanical properties of low-relaxation hot-dip galvanized prestressed steel wires (LG-SW) of various grades, i.e., 1670, 1770, 1860, and 1960. LG-SW is the basic component material of high-strength steel cables that are widely used as load-bearing members in prestressed steel structures. The specimens were initially heated to 13 preselected elevated temperatures up to 1000 °C and subsequently cooled down to ambient temperature by air or by water spraying. Tensile coupon tests were subsequently conducted on the specimens at ambient temperature. The associated mechanical properties, such as stress–strain curves, elastic moduli, yield strengths, ultimate strengths, and ductility, were obtained. Additional tests were also conducted to explore the effects of cyclic heating-cooling on the post-fire mechanical properties. Test results showed that the post-fire mechanical properties of LG-SW were significantly influenced by the exposure temperatures and the cooling methods, whereas the influences of steel wire grades and cyclic heating-cooling were limited. LG-SW of varying grades can regain their original elastic moduli after exposure to elevated temperatures up to 1000 °C. Their yield and ultimate strengths remained unchanged after exposure to temperatures up to 400 °C, whereas significant changes occurred thereafter. Under air cooling condition, the post-fire strength of LG-SW rapidly decreased after exposure to temperatures exceeding 400 °C and reached the lowest value at the exposure temperature of 750 °C, with a reduction of 60% and 70% for yield and ultimate strength, respectively. As the exposure temperature continued to increase, an obvious rebound in both the yield and ultimate strength was observed up to 1000 °C because of the formation of martensite crystalline structures. With regard to ductility, air-cooled LG-SW showed a notable increase in ductility after exposure to temperatures up to 750 °C while a sharp reduction followed. By contrast, when water cooling method adopted, all the yield and ultimate strengths, and ductility of LG-SW followed a similar change trend as those of the air-cooled specimens until 700 °C; thereafter, however, steel wires lost nearly all their strength and ductility because of the significant embrittling effect of water cooling. Hence, 700 °C was defined as the critical failure temperature for water-cooled LG-SW, and great caution must be exercised to avoid the use of LG-SW that was cooled from temperatures higher than 700 °C by water spraying. Neither the current design guide nor previous literatures have provided suitable recommendations for the post-fire mechanical properties of LG-SW. Hence, predictive equations were developed for air and water cooling separately to accurately evaluate the post-fire elastic moduli, yield strengths, ultimate strengths, and ductility of LG-SW. The application of the outcomes from this study will lead to the appropriate assessment of the mechanical performance and safe reuse of the high-strength steel cables composed of LG-SW after fire events.Mechanistic contribution of the interplay between microstructure and plastic deformation in hot-rolled Fe–11Mn–2/4Al–0.2C steelWe elucidate here the mechanistic contribution of the interplay between microstructural constituents and plastic deformation behavior in hot-rolled Fe–0.2C–11Mn–2Al (2Al steel) and Fe–0.2C–11Mn–4Al (4Al steel) transformation induced plasticity (TRIP) steels. The steel containing lower aluminum content (2 wt%) was characterized by excellent combination of tensile elongation (TE) of 31.4%, ultimate tensile strength (UTS) of 1407 MPa, and UTS×TE of 44.2 GPa%, where the ultrahigh strength is attributed to the cumulative contribution of the newly transformed martensite with high microhardness and Portevin-Le Chatelier (PLC) effect. In contrast, the higher aluminum (4 wt%) containing steel indicated lower UTS of <1200 MPa, but higher TE of 34–40%, which resulted from the TRIP effect and the cooperative deformation of α-ferrite and δ-ferrite.The increasing demand for energy conservation, environmental protection, and reduced weight of automobiles has led to significant interest in the development of new advanced high-strength steels. Medium Mn-content (5–12%) transformation-induce plasticity (TRIP) steels with high-austenite (20–80%) content are potential candidates for automotive applications where the TRIP effect can be completely utilized It is suggested that superior mechanical properties can be obtained with increase in Mn and C content, which increases the volume fraction of retained austenite. For example, Luo et al. In recent studies, Al was added in medium Mn-content TRIP steels to optimize austenite stability by suppressing cementite formation. Furthermore, Al in TRIP steels facilitated the presence of α-ferrite and δ-ferrite and contributed to excellent tensile properties The objective of the study described here is aimed at studying the relationship between microstructure and properties in hot-rolled Fe–11Mn–2/4Al–0.2C steels, and demonstrate the potential of proposed alloy design-processing relationship in obtaining desired mechanical properties.The chemical composition of the two experimental steels had a nominal composition (in wt%) of Fe–11Mn–0.2C–2Al (2Al steel) and Fe–11Mn–0.2C–4Al (4Al steel), and the actual chemical analyses of the two experimental steels is presented in . The ingots were cast using a vacuum furnace and heated at 1200 °C for 2 h and hot forged to rods of section size ~100×30 mm2, followed by air cooling to room temperature. Subsequently, the rods were soaked at 1200 °C for 2 h, and hot rolled to 4 mm thickness in the temperature range of 1150–850 °C, and finally air cooled to ambient temperature.An innovative heat treatment, referred as “austenite reversed transformation” (ART), was applied to medium-Mn steels Tensile specimens of dimensions 12.5 mm width and gage length of 50 mm were machined from the heat-treated sheets with the tensile axis parallel to the prior rolling direction. Tensile tests were carried out at room temperature using a universal testing machine (SANSCMT5000) at a constant crosshead speed of 3 mm min−1. The samples were etched with 25% sodium bisulfite solution. Microstructural examination was carried out using scanning electron microscope (SEM). Microhardness measurements were carried out using Vicker hardness tester. The selected pressure was 50 kgf, and the holding time was 10 s. The volume fraction of austenite was determined by X-ray diffraction (XRD) with CuKα radiation using direct comparison method where Iγ is the integrated intensity of austenite and Iα is the integrated intensity of α-phase. is representative microstructure of the as-hot-rolled steels. The microstructural constituents consisted of martensite and austenite. The streaks in the hot-rolled 4Al steel spreading out as layers during hot-rolling are δ-ferrite, and were retained during the following heat treatment. is the SEM micrographs of the hot rolled 2Al steel heated at different temperatures. a and b describe the microstructure of the samples quenched at 650 °C and 700 °C, respectively, followed by tempering at 200 °C. The microstructural constituents consisted of acicular α-ferrite and austenite as the dominant phase. When the samples were quenched at 750 °C and 800 °C, respectively, as shown in c and d, the microstructure comprised of martensite and austenite. For clarity, a mixture of acicular α-ferrite and austenite is marked with a rectangle in a, and is illustrated at high-magnification in the inset of the micrograph, and the magnified region of martensite and austenite is presented in the inset of shows the microstructure of heat treated 4Al steel. Because of a higher Al concentration, δ-ferrite was developed. As shown in a and b, the microstructural constituents of the samples quenched at 750 °C and 800 °C, followed by tempering at 200 °C consisted of δ-ferrite, α-ferrite and austenite. While the samples quenched at 850 °C and 900 °C, respectively, the microstructure consisted of δ-ferrite, austenite and martensite (The variation in the volume fraction of austenite obtained from XRD is summarized in . The accuracy of XRD measurements of austenite fraction is ~1%. 2Al steel maintained a high-austenite fraction of ~80% in the temperature range of 600–700 °C, followed by drastic decrease to ~25%, when quenching was carried out in the temperature range of 750–850 °C because of martensitic transformation. Similar trend in the austenite fraction was observed in 4Al steel. It is obvious that 4Al steel has lower austenite content than 2Al steel. demonstrates the engineering strain–stress plots of 2Al steel and 4Al steel. The mechanical properties of these two steels were summarized in . In 4Al steel, the ultimate tensile strength (UTS) increased continuously with increase in temperature up to 850 °C, whereas total elongation (TE) decreased with increase in temperature. Compared to 4Al steel, 2Al steel had a significantly higher UTS of 1370–1450 MPa than 4Al steel (735–1200 MPa). However, it is interesting that 2Al steel with a higher austenite content () had a lower TE of 22–32% than 4Al steel (34–40%), when 2Al and 4Al samples were quenched from 600 to 700 °C and 700 to 800 °C, respectively. In 4Al steel, the product of UTS and TE attained a maximum of 37.4 GPa% at 800 °C. In contrast, 2Al steel attained 44.2 GPa% at 650 °C.The underlying reason for the variation in tensile properties in 4Al steel can be related to the fraction and stability of austenite, as demonstrated in our previous work where σα, σγ, σm are the flow stresses and ƒα, ƒγ, ƒm, are volume fraction of ferrite, austenite and martensite, respectively. However, the rule is not applicable in the case of 2Al steel. For 2Al steel, there are substantial differences in constituents of the microstructure between the samples quenched from 600 to 700 °C and the samples quenched from 750 to 850 °C, however, the variation in UTS in these samples is unconventional.), the samples quenched from 600 to 700 °C were composed of similar austenite content (~80%) and α-ferrite, and the transformation ratio of austenite increased with increase in quenching temperature. Thus, the corresponding UTS increased from 1370 MPa to 1452 MPa. As regards, the samples quenched from 750 to 850 °C, the microstructural constituents consisted of martensite and austenite (~25%). Based on the rule of mixtures, higher marteniste fraction should contribute to higher UTS. However, the samples quenched from 750 to 850 °C with higher martensite content are not characterized by higher UTS. Thus, to understand the paradox, the microhardness of austenite and martensite was measured., in the case of 4Al samples, quenched from 800 °C and 900 °C (referred as 4Al-800 and 4Al-900), the Vickers hardness of the newly generated martensite, namely, transformed martensite, evolved during tensile tests was close to the original martensite formed during quenching. Thus, the rule of mixture was applicable for 4Al steel. In contrast, for the 2Al samples, quenched from 650 °C and 750 °C (labeled as 2Al-650 and 2Al-750), the Vickers hardness of the newly generated martensite was more than 1.5 times than the original martensite. Hence, we propose use of a modified rule of mixtures:where σnm is the flow stresses and ƒnm is volume fraction of newly generated martensite. It is estimated that σnm (2381 MPa) is greater than σm (1451 MPa). Combining XRD results and Eq. , we deduced that the superior UTS in 2Al samples quenched from 600 to 700 °C resulted from the transformed martensite, which provided a contribution of σnm to UTS (TRIP effect).b), two interesting phenomena were observed. First, 2Al-650 sample was comprised of 84 vol% austenite, and after tensile deformation to fracture, 59% of the austenite was transformed to martensite. In contrast, 2Al-600 sample and 2Al-700 sample comprised of 80 and 83 vol% austenite, respectively. The austenite transformation ratio of the two samples was 46% and 82%, respectively. Interestingly, the corresponding TE for the two samples was similar (2Al-600: 22.5%, 2Al-700: 23.3), but were below the value of 31.4% obtained for 2Al-650 sample. Second, 2Al-700 sample had a significantly higher austenite content than 2Al-750 sample (28 vol%), but the corresponding TE for the two samples was similar (2Al-750: 22.7%). The underlying reason for the two phenomena can be elucidated by studying the work hardening behavior.a and b, the work-hardening behavior of 2Al-600 sample and 2Al-650 sample exhibited three stages of work hardening rate (WH) evolution: in stage 1, WH rapidly decreases; in stage 2, WH increases (in the range of 4500–7500 MPa for 2Al-650 sample); and in stage 3, WH decreases with marked fluctuations. In the case of 2Al-700 sample, the WH of stage 2 decreases slowly from 11000 to 8100 MPa which is much higher than the value of other samples. The majority of studies reported in the literature , ƒγ0, ƒγ and k are the initial austenite fraction, the austenite fraction at strain ε, and the mechanical stability of austenite, respectively. A higher value of k corresponds to lower austenite stability. Austenite in 2Al-700 sample with a low-stability (k=7.4) quickly transformed during the early stages of deformation leading to high-WH and low-ductility. 2Al-600 sample (k=2.7) and 2Al-650 sample (k=2.8) with similar austenite stability correspond to similar WH. However, as mentioned above, 2Al-650 sample had superior ductility than 2Al-600 sample. Thus, besides austenite stability, there are other factors influencing ductility, and assuming that it is connected with the jerky behavior in WH.These fluctuations in stage 3 are directly connected to the observed serrations in the stress–strain plots. As shown in a, the big fluctuations were mixed with small serrations. Thus, the serrations can be divided into two types: dense and sparse. In the most commonly accepted models for dense serrations, the behavior was related to Portevin-Le Chatelier (PLC) effect ), PLC effect played a dominant role in 2Al-600 sample, while discontinuous TRIP effect played a leading role in 2Al-650 sample, and contributed to higher ductility. Despite far lower austenite fraction, 2Al-750 sample (k=5.1) had a higher austenite stability than 2Al-700 sample (k=7.4) and without PLC effect, which led to similar ductility in the two samples. Thus, it is deduced that the PLC effect is detrimental to ductility. In conclusion, the ductility was influenced by TRIP effect and PLC effect. Furthermore, TRIP effect is primarily influenced by austenite stability rather than austenite fraction., 2Al steel had a significantly higher UTS than 4Al steel, while 4Al steel had a superior ductility than 2Al steel. In order to find out the reason for the difference in the tensile properties between the two steels, the microstructure and deformation behavior in 2Al steel and 4Al steel were studied.Based on the aforementioned discussion, it is inferred that 2Al steel with higher UTS is consistent with the two aspects. As shown in , the microhardness of austenite and the newly generated martensite in 2Al samples was higher than in 4Al samples. According to the Eq. , we can deduce a higher UTS in 2Al steel. On the other hand, when comparing strain–stress plots of two steels in , it is obvious that PLC effect only existed in 2Al steel, and it was reported that PLC effect help in improving the tensile strength Let us compare 2Al-650 sample and 4Al-800 sample. Comparing b, the microstructure of 2Al-650 sample is composed of austenite and acicular α-ferrite, while 4Al-800 sample was composed of austenite, lath-type α-ferrite and δ-ferrite. Comparing b, it was noted that the thickness of α-ferrite in 4Al-800 sample was larger than 2Al-650 sample. On fracture, as marked in a and b, it is obvious that both α-ferrite and δ-ferrite were considerably squeezed by the transformed martensite because of volume expansion. Thus, the cooperative deformation of α-ferrite and δ-ferrite led to superior ductility. In contrast, the thickness of acicular α-ferrite in 2Al-650 sample was extremely fine, which could not effectively accommodate the volume expansion of transformed martensite leading to high internal stress. Thus, the newly generated martensite had significantly higher microhardness than the original martensite (Microstructural evolution, mechanical properties, and work hardening behavior of 2Al steel and 4Al steel were studied. The main conclusions are as follows:2Al-650 sample exhibited best combination of mechanical properties, and was characterized by excellent combination of TE of 31.4%, UTS of 1407 MPa, and UTS×TE of 44.2 GPa%, which were significantly superior to a number of hot-rolled medium Mn-content TRIP steels at the ultrahigh-strength level.The ductility of 2Al steel was influenced by TRIP effect and PLC effect. PLC effect was proven to be detrimental to ductility. TRIP effect, which primarily depends on austenite stability, played a positive role in ductility. 4Al steel with superior ductility contributed not only to TRIP effect, but also to the cooperative deformation of α-ferrite and δ-ferrite.The rule of mixtures was modified to describe the variation in UTS in 2Al steel. The ultrahigh-strength in 2Al steel was a cumulative contribution of the newly transformed martensite with high-microhardness and PLC effect. In contrast, PLC effect was absent in 4Al steel. Moreover, the microhardness of newly developed martensite was significantly lower than 2Al steel, leading to a lower UTS in 4Al steel.In 4Al steel, the cooperative deformation of α-ferrite and δ-ferrite absorbed the volume expansion of transformed martensite such that the newly generated martensite had microhardness similar to the original martensite. In contrast, acicular α-ferrite in 2Al steel could not effectively accommodate the volume expansion of transformed martensite leading to high internal stress. Thus, the newly generated martensite had a higher microhardness.Dynamics behavior of rotating bladed discs: A finite element formulation for the study of second and higher order harmonicsAnnular finite elements for the computation of second and higher order harmonics modes of bladed rotating discs are developed. The elements take into account gyroscopic effect and stiffening due to centrifugal and thermal stresses (the latter not present in arrays of blades). The displacement field is expressed by a truncated Fourier series along the angle and by polynomial shape functions in the radial direction. This paper is the generalization of a previous study limited to zero- and first-order harmonics and deals only with second and higher order harmonics modes that are uncoupled from the modes involving the behavior of the rotor as a whole. Several cases have been studied to verify the accuracy of the disc and array of blades elements.Many rotors, like those of turbomachinery, can be modeled as composed of shafts, discs and arrays of blades. In elementary rotordynamics, the bladed discs are assumed to be rigid bodies, contributing to the inertia of the rotor but not to its compliance. However, there are cases like, for instance, circular saws The blades can be modeled as an array of beams attached to the outer edge of the disc and this has been successfully done for studying the interaction between discs and blades, but this approach does not allowed to predict the local modes of the blades Dynamic instability of rotating discs in contact with stationary pads have been shown for circular saws and for disc brakes; the latter case is studied using the finite element method by Kang It is important to study the higher order modes of the disc and array of blades even if they are not coupled to the shaft dynamics since they can be excited at a resonance and compromise the safety of the rotor system. The aim of the present paper is to generalize the finite element formulation taking into account both the gyroscopic effect and centrifugal stiffening to study the flexural behavior in the flexible discs and array of blades which have already been introduced in The basic approach is what usually defined as 112 dimensional approach, i.e. the shaft is modeled as a beam (one-dimensional solid) and the discs and the arrays of blades are assumed to be annular elements with displacements developed in Fourier series along the angle.The discs are connected to the shaft and a shaft–disc transition element is used as interface, while blades are assumed to be attached to the outer diameter of the discs and the interface is simulated by a disc–array of blades transition element. The element matrices are developed by a Lagrange approach and written in the complex coordinate described in Refs. The element is modeled as a two-dimensional annular object, with all properties concentrated at its mid-plane. In addition, it is assumed to be perfectly balanced, i.e. its center of mass lies in its geometrical center and its principal axis of inertia coincides with its rotation axis. However, a static and couple unbalance can be added into the equations of motion later.The deformation of the bladed disc is made by a rigid body configuration deviation and a compliant body deformation. During the deformation, the mid-plane of the disc is assumed to maintain the same orientation in space of a rigid body attached to the relevant shaft cross-section, while the disc exits its mid-plane owing to its flexibility.The generalized coordinates can be defined with reference to the frames shown in . Frame OXYZ is the inertial frame with the origin at O and Z-axis coinciding with the rotor rotation axis in its undeformed position; frame Ox*y*Z is the rotating frame, axes x* and y* rotate in the XY-plane at a rotating speed ω (at constant speed condition it is rotated by angle ωt); frame CX′Y′Z′ has its original point C located at the center of the shaft in the disc attachment cross-section, but its axes remain parallel to axes X,Y,Z. The deformed position of a generic point P on the disc can be defined by the following rotations:Rotate the axes of CX′Y′Z′ frame about the X′‐axis by an angle Φx until Y′‐axis enters the mid-plane of the disc in its deformed configuration; let the axes so obtained be yx and zx and the rotation matrix expressing the coordinates of P in CX′Y′Z′ frame from those referred to CX′yxzx be R1.Rotate frame CX′yxzx about yx-axis by an angle Φy until X′‐axis enters the mid-plane of the disc in its deformed configuration; let the axes so obtained be xy and zy and the rotation matrix be R2.Rotate frame Cxyyxzy in xyyx plane through an angle Φz+ωt, define the so-obtained frame as Cxyz, which is fixed to the mid-plane of the disc and will be referred to as the rotating and whirling frame, and the rotation matrix refer to the rotating about z-axis can be split into two terms R3 (rotation Φz) and R4 (rotation ωt). The expressions for the above-mentioned rotating matrices are defined in Refs. Let u, v and w be, respectively, the radial, tangential and axial displacement components of a generic point whose undeformed position P0 is defined by radius r and angle θ in the reference frame Cxyz defined above that follows both the rotation and the deformation of the shaft. The position of point P after deformation iswhere Rk are rotation matrices mentioned above and they are functions of the angles characterizing the rigid body displacement and of the rotation ωt.For second and higher order harmonics, the displacement field is uncoupled with the flexural behavior of the rotor and thus rigid body motions need not be included in the formulation. Consequently, the deformation of point P in the inertial reference OXYZ (shown in ) related to the i th-order harmonics can be expressed by neglecting the displacement (C−O¯) in Eq. An annular finite element with a thickness varying linearly with the radius is shown in . A non-dimensional radial coordinate χ that goes from 0 at the inner radius to 1 at the outer one is defined. The thickness h(r) and the non-dimensional radius χ are thus defined asThe displacement field within the element can be expressed as a trigonometrical series in the angular position θ: where n is the number of harmonic terms. The coefficients of the harmonic terms contributing to the in-plane displacement are uic, uis, vic and vis, while those contributing to the out-of-plane displacement are wic and wis.It is already noted the dynamic behavior of the zero- and first-order harmonics was studied in Ref. implies a decoupling between each harmonic. The displacement field of each harmonic can thus be expressed asui(χ,θ,t)=uiccosiθ+uissiniθvi(χ,θ,t)=viccosiθ+vissiniθwi(χ,θ,t)=wiccosiθ+wissiniθfori≥2.As usual in the circular plate's theory, the coefficients of the harmonic terms are functions both of the non-dimensional radius and of timeuic(χ,t),uis(χ,t),vic(χ,t)vis(χ,t),wis(χ,t),wic(χ,t)fori≥2.As usual in the FEM, the matrices of the shape functions are introduced, and the dependence of the displacements on the radius and the time can be expressed as uic(χ,t)=nu(χ)qux(t),uis(χ,t)=nu(χ)quy(t),vic(χ,t)=nv(χ)qvx(t),vis(χ,t)=nv(χ)qvy(t),wic(χ,t)=nw(χ)qwx(t),wis(χ,t)=nw(χ)qwy(t).The shape functions for the in-plane displacements u and v are assumed to be linear; while cubic ones are adopted for the out-of plane displacement w. In terms of the non-dimensional coordinate χ they arenu(χ)=nv(χ)=[1−χχ],nw(χ)=[AriBΔrCr0DΔr],where A,B,C,D are shape functions similar to those used in the “simple Timoshenko beam” as reported in Ref. The generalized coordinates used to express the deformation of the disc deflection field are thus qwx={φy1βvy1φy2βvy2}T,qwy=−{φx1βvx1φx2βvx2}T.Two nodes are defined on each disc element when dealing with second and higher order harmonics: nodes 1 and 2, which are located at the inner and the outer radius of the element, respectively. The total number of the real degrees of freedom of the element is 16; nodes 1 and 2 have 8 each. If complex coordinates are introduced into the equations of motion, 8 complex flexural degrees of freedom will result.The generalized coordinates used to approximate the displacements of nodes 1 and 2 describe the deflections of the element from its rigid configurations. The equations of motion of the element are obtained from the expressions of the kinetic and the potential energies and following a Lagrangian approach.The kinetic energy is computed by neglecting the terms related to the deviation of the rigid configurations under the assumption that the thickness of the disc is small compared to the radial dimensions and the shear deformation is negligible.Pi denotes the displacement of a point in the mid-plane (z=0) of the element relative to the initial reference frame, the kinetic energy can be written aswhere ρ is the density of the material of the disc element.The harmonic terms are composed of two uncoupled parts, the in-plane displacement u, v and the out-of-plane term w. So the kinetic energy splits into two independent contributions:The in-plane and out-of plane contributions to the kinetic energy can be expressed as equations of the shape function and element degrees of freedom, respectively, asTinp,i=12[ω2(quxTminp,iqux+quyTminp,iquy+qvxTminp,iqvx+qvyTminp,iqvy−2quxTminp,iqvx−2quyTminp,iqvy)+2ω(quyTminp,iq˙ux−qvyTminp,iq˙ux−quxTminp,iq˙uy+qvxTminp,iq˙uy−quyTminp,iq˙vx+qvyTminp,iq˙vx+quxTminp,iq˙vy−qvxTminp,iq˙vy)+q˙uxTminp,iq˙ux+q˙uyTminp,iq˙uy+q˙vxTminp,iq˙vx+q˙vyTminp,iq˙vy],Toutp,i=12[ω2(qwxTmoutp,iqwx+qwyTmoutp,iqwy)+ω(2qwyTmoutp,iq˙wx−2qwxTmoutp,iq˙wy)+q˙wxTmoutp,iq˙wx+q˙wyTmoutp,iq˙wy].Matrices minp,i and moutp,i are given by the integrals: minp,i=2π∫rir0ρrhnuTnudr=2π∫rir0ρrhnvTnvdr,Contributions to the potential energy due both to the elastic stresses in the material (Ue,i) and to the ‘geometric effect’ (Ug,i) have been considered, as explained in Ref. If the thickness of the disc is small, it can be considered as a rotating Kirchhoff plate and transverse shear deformations can be neglected. The expression of elastic energy of i th-order harmonics due to bending isUe,i=12∫rir0∫02πεiTDεirdrdθ=Ueinp,i+Ueoutp,i.The generalized strain vector is obtained from the displacement field within the disc; it can be split into the two contributions due to in-plane and out-of-plane deformationεinp,i=∂ui∂ruir+1r∂vi∂θ1r∂ui∂θ+∂vi∂r−vir,εoutp,i=∂2wi∂r2−1r(∂wi∂r+1r∂2wi∂θ2)2(1r2∂wi∂θ−1r∂2wi∂θ∂r).Matrix D is used to represent the element elastic properties as functions of the disc thickness and material characteristics; it can be split into two separate terms, too. Both can be computed under the assumption that the sections perpendicular to the radial directions remain plane after deformation and the effect of shear deformation is negligible:account for the in-plane and out-of-plane elastic behavior, E is Young's modulus, andBy substituting the discretized displacement equations and shape functions into Eq. , the elastic potential energy can be expressed as Ueinp,i=12(quxTkuu,iqux+quyTkuu,iquy+qvxTkvv,iqvx+qvyTkvv,iqvy+quxTkuv,iqvx+quyTkuv,iqvy),kuu,i=π∫rir0einp2r2{[2+i2(1−υ)]nuTnu+2υr[nuTn′u+(n′u)Tnu]+2r2(n′u)Tn′u}rdr,kvv,i=π∫rir0einp2r2{[2i2+(1−υ)]nuTnu−r(1−υ)[nuTn′u+(n′u)Tnu]+r2(1−υ)(n′u)Tn′u}rdr,kuv,i=iπ∫rir0einp2r2[(3−υ)nuTnu−r(1−3υ)(n′u)Tn′u]rdr,kww,i=π∫rir0eoutpr4{[i4+2i2(1−υ)]nwTnw−r[i2+2i2(1−υ)][nwTn′w+(n′w)Tnw]+r2[1+2i2(1−υ)](n′w)Tn′w−υi2r2[nwTn″w+(n″w)Tnw]−υr3[(n′w)Tn″w+(n″w)Tn′w]+r4(n″w)Tn″w}rdr.The geometric potential energy Ug,i is generated by the effect of centrifugal stiffening and thermal stresses in the disc. The restoring force due to centrifugal and thermal stresses is very important if the disc is like a membrane, i.e. has a low bending stiffness. This contribution can be expressed in the formUg,i=12∫rir0∫02πσr∂wi∂r2+σc1r∂wi∂θ2+σr∂vi∂r2+σc1r∂ui∂θ2rhdrdθ.The radial and circumferential stresses σr and σc can be evaluated by using, for example, the Manson method Integrating the equations, the geometric potential energy is obtained: Uginp,i=12(quxTkgu,iqux+quyTkgu,iquy+qvxTkgv,iqvx+qvyTkgv,iqvy),Ugoutp,i=12(qwxTkgwc,iqwx+qwyTkgwc,iqwy+qwxTkgwr,iqwx+qwyTkgwr,iqwy).The stiffness matrices are given by the integrals of the shape functions kgv,i=π∫rir0σrhr(n′v)Tn′vdr,kgu,i=i2π∫rir0σchrnuTnudr,kgwr,i=π∫rir0σrhr(n′w)Tn′wdr,kgwc,i=i2π∫rir0σchrnwTnwdr.Second and higher order harmonics are uncoupled from the flexural behavior of rotor. If no external force acts on the element, the equations of motion for the second and higher order harmonics have the same form as in Ref. Minp,iQ¨inp,i−iωGinp,iQ˙inp,i+(Kinp,i+ω2Kωinp,i−ω2Mniinp,i)Qinp,i=0,Moutp,iQ¨outp,i−iωGoutp,iQ˙outp,i+(Koutp,i+ω2Kωoutp,i−ω2Mnioutp,i)Qoutp,i=0.The in-plane and out-of-plane motions can be assembled in vectors asQinp,i=qux+iquyqvx+iqvy(4×1),Qoutp,i={qwx+iqwy}(4×1).The element mass, gyroscopic, centrifugal stiffening, thermal gradients and stiffness matrices are obtained from the element kinetic and potential energies through Lagrange's equations Ginp,i=2iminp,i−minp,i−minp,iminp,i,Goutp,i=2imoutp,i,Mniinp,i=i2minp,i−minp,i−minp,iminp,i,Mnioutp,i=i2moutp,i.The stresses in the radial and tangential directions can be computed from centrifugal and thermal loadings, the latter is independent of the rotational speed. Thus the stress field can be written as where σr and σc are the stresses in radial and circumferential direction of the disc, σrω and σcω are the centrifugal stresses due to a unit rotational speed, ω=1rad/s, while σrT and σcT indicate the stress field due to the thermal gradients. In this case, the stiffness matrices can be split as The stiffness matrices can be obtained by taking into account the arrangement of vector Kinp,i=kuu,i+kguT,i−kuvT,i−kuvT,iTkvv,i+kgvT,i,Kωinp,i=kguω,i00kgvω,i,Koutp,i=kww,i+kgwcT,i+kgwrT,i,Kωoutp,i=kgwcω,i+kgwrω,i.The elements are implemented with a numerical integration routine based on a four point Gauss procedure since the expressions of the matrix terms are difficult to achieve by analytical integrations.Beam and disc elements cannot be directly linked to the present annular disc element and the compatibility of the displacement at the disc–shaft interface must be insured. A suitable transition element has thus been developed which is provided with two nodes, the first located at the inner radius of the transition element (outer radius of the beam) describes the interface between the beam and the disc, and the second located at the outer radius of the element. The two nodes have 2 complex degrees of freedom and 4 complex degrees of freedom for flexural behavior, respectively.The matrices for the shaft–disc transition element have been obtained from those described above for the disc element by just deleting all rows and columns linked with the degrees of freedom at node 1. This corresponds to constraining the displacements of the point at the inner radius of the element as a rigid body motion.The main assumptions to analyze the array of blades are that all blades are equal, are aligned along the radial direction and their shear center coincides with the center of mass of each section. The number of blades must be 3 or larger, so that the assumption that the array is axisymmetric is satisfied A typical cross-section of a blade perpendicular to the radial direction is shown in . G is the mass center of the cross-section while T is the shear center of it.The array is modeled as a two-dimensional object, all its properties concentrated in the mid-plane of the blade, like for the disc element. The deformation of the array is referred to the same reference plane of the disc that is a plane perpendicular to the inflected deformation of the shaft passing through the disc–shaft attachment. The array exits this reference plane owing to its flexibility.Let uj, vj, and wj be, respectively, the radial, tangential and axial displacement components of a point Pj of a section of the jth blade taken at a radius r. Pj is the coordinate in an inertial reference of point P expressed as(Pj−O¯)=(C−O¯)+∏k=14Rk({r00}T+{ujvjwj}T),where C is the coordinate of the shaft–disc attachment point, Rk are the rotation matrices as a function of angle of the rigid body motion as reported in Ref. The second and higher order harmonics are uncoupled with the flexural behavior of the rotor as a whole while being important in the study of the local modes of the bladed disc. Rigid body motions can thus be neglected since they are uncoupled and do not enter the formulation of the present model. According to these indications, the deformation in the initial frame of point can be represented as the same form as Eq. To define the shape functions approximating the deformations of the array of blades, the latter has been subdivided into annular elements. A non-dimensional radius χ has been defined in the same way as seen for the disc element. A, I2 and I3 are the area of the cross-section of each blade and its area moments of inertia about the principal inertia axis (u2, u3 in ) of the cross-section. They are, together with angle ψ, linear functions of the non-dimensional radius χ. The displacements uj, vj, wj are then approximated by means of a truncated Fourier's series in the angular coordinate θj: uj(χ,θj,z,t)=u0+∑i=1n(uiccosiθj+uissiniθj),vj(χ,θj,z,t)=v0+∑i=1n(viccosiθj+vissiniθj),wj(χ,θj,z,t)=w0+∑i=1n(wiccosiθj+wissiniθj).The coefficient of the various harmonics displacement uic,s and vic,s refer to the in-plane displacement while wic,s are related to the out-of-plane displacement.The dynamic behavior of the zero- and first-order harmonics of array of blades element have already been studied in Ref. uj,i(χ,θj,t)=uiccosiθj+uissiniθjvj,i(χ,θj,t)=viccosiθj+vissiniθjwj,i(χ,θj,t)=wiccosiθj+wissiniθjfori≥2.The use of shape functions to eliminate the displacement dependence on θj of the displacement field u, v, and w lead to the unknown functions in the same form as Eq. , which also can be approximated by shape functions written as Eq. The shape functions for the in-plane radial and out-of-plane deformations are the same as those used in the disc element, expressed in Eq. , while the in-plane circumferential deflection of the blade element is coupled with the out-of-plane deformation and can be expressed with the same cubic polynomial shape functions as the out-of-plane shape functions.The generalized coordinates used to express the deflections of the array of blades coupled to the flexural behavior are qvx=−{vx1βwx1vx2βwx2}T,qvy={vy1βwy1vy2βwy2}T,qwx={wx1βvx1wx2βwx2}T,qwy=−{wy1βvy1wy2βvy2}T.Two nodes are defined on the element, one at the inner radius of the element and the other at the outer radius. The total number of the real degrees of freedom of the element is 20; nodes 1 and 2 have 10 each. If complex coordinates are introduced into the equations of motion, 10 complex flexural degrees of freedom will be used.The generalized coordinates used to approximate the displacements describe the deflections of the element from the rigid-body configuration. The equations of motion of the element have been written in the same way as already seen for the disc element.The kinetic energy has been computed taking into account the contributions due to the rotational inertia of the array while the contributions of the rotational inertia the cross-section of a single blade is neglected when considering the deviations from the rigid-body configuration.Let Pj,i denoting the displacement of the mass center of the jth blade at the radius r, relative to the inertial reference. The kinetic energy iswhere ρ is the density of the material of the blades and A(r) is the cross-section of the blade at radius r.In differentiating with respect to time, angle θj must be considered as a function of time. Owing to the orthogonality of the harmonic functions, a decoupling between the modes of the various orders occurs. The kinetic energy can be split into in-plane and out-of-plane contributions:The in-plane and out-of plane contributions to the kinetic energy can be expressed as functions of the shape function and element degrees of freedom asTinp,i=12[ω2(quxTminp1,iqux+quyTminp1,iquy+qvxTminp2,iqvx+qvyTminp2,iqvy−2quxTminp3,iqvx−2quyTminp3,iqvy)+ω(quyTminp1,iq˙ux−qvyTminp3,iq˙ux−quxTminp1,iq˙uy+qvxTminp3,iq˙uy−quyTminp3,iq˙vx+qvyTminp2,iq˙vx+quxTminp3,iq˙vy−qvxTminp3,iq˙vy)+q˙uxTminp1,iq˙ux+q˙uyTminp1,iq˙uy+q˙vxTminp2,iq˙vx+q˙vyTminp2,iq˙vy],Toutp,i=12[ω2(qwxTmoutp,iqwx+qwyTmoutp,iqwy)+ω(2qwyTmoutp,iq˙wx−2qwxTmoutp,iq˙wy)+q˙wxTmoutp,iq˙wx+q˙wyTmoutp,iq˙wy].Matrices minp,i and moutp,i are given by the integrals: minp1,i=N2∫rir0ρAnuTnudr,minp2,i=N2∫rir0ρAnvTnvdr,minp3,i=N2∫rir0ρAnuTnvdr,moutp,i=N2∫rir0ρAnwTnwdr.Shear deformation in the blade is neglected, since each single blade is modeled as an Euler–Bernoulli's beam. The elastic energy is thus related to the radial extension (axial deformation for the blade) and flexural deflections:Ue,i=12∑j=1N∫rir0EAΔr2(s′1,i)2+1Δr4(I2(s″2,i)2+I3(s″3,i)2)dr.The prime indicates the partial derivative related to the radial coordinates r and E is Young's modulus. The displacements s along the inertial axes are linked to the axial, tangential, radial directions by angle ψwhere angle ψ is a function of the radial coordinate only. Owing to the orthogonally of trigonometric functions, the in-plane and out-of-plane displacements are decoupled (s″2,i)2=(v″j,i)Tv″j,i2cos2ψ+(w″j,i)Tw″j,isin2ψ,(s″3,i)2=(v″j,i)Tv″j,isin2ψ+(w″j,i)Tw″j,icos2ψ.Prime indicates the partial derivative of the displacements u, v, w. The contributions to the elastic potential energy are expressed in terms of element generalized coordinates as Ueinp,i=12(quxTkeinp1,iqux+quyTkeinp1,iquy+qvxTkeinp2,iqvx+qvyTkeinp2,iqvy),Ueoutp,i=12(qwxTkeoutp,iqwx+qwyTkeoutp,iqwy).The stiffness matrices are obtained from the shape functions by the following integrals: keinp1,i=N2Δr2∫rir0EA(n′u)Tn′udr,keinp2,i=N2Δr4∫rir0EIw(n″v)Tn″vdr,where Iv and Iw are the area moments of inertia of the cross-section in circumferential and axial direction v and w in The geometric potential energy is caused by the centrifugal forces Fr,i. Assuming that the blades are free to expand radially at their tips, the thermal effect do not induce any radial load along the axis of blades and force Fr,i can be expressed asThe restoring force due to centrifugal stress is very important in the case of rotating pendulum. The geometric contribution to the potential energy can be expressed asUg,i=12Δr2∑i=1N∫rir0Fr,i(r)[v′2j,i+w′2j,i]dr.The terms in the potential energy can be split into two independent contributionsSubstituting the shape functions and displacement equations into Eq. Uginp,i=12(qvxTkginpω,iqvx+qvyTkginpω,iqvy),Ugoutp,i=12(qwxTkgoutpω,iqwx+qwyTkgoutpω,iqwy).The stiffness matrices are given by the integrals of the shape functions: The equation of motion for the second and higher order harmonics of the array of blades element is the same as the equations describing the array of blades element in Ref. Qinp,i=qux+iquyqvx+iqvy(6×1),Qoutp,i={qwx+iqwy}(4×1)The element mass, gyroscopic, centrifugal and thermal stiffening and stiffness matrices are obtained by using Lagrange's equations Minp,i=minp1,i00minp2,i,Moutp,i=moutp,i,Ginp,i=2iminp1,i−minp3,i−minp3,iminp2,i,Goutp,i=2imoutp,i,Mniinp,i=2i2minp1,i−minp3,i−minp3,iminp2,i,Mnioutp,i=i2moutp,i.The stresses in the radial and tangential directions can be computed from centrifugal and thermal loading. The stiffness matrices can be written as Kinp,i=keinp1,i00keinp2,i,Kωinp,i=000kginpω,i,Koutp,i=kww,i+kgoutp,i,Kωoutp,i=kgoutpω,i.The elements are implemented following the same numerical integration procedures as the disc element.Even if the displacements within the array are very similar to those of the disc element, different kinds of shape functions are used for the tangential displacement fields (linear functions for the disc element; cubic functions for the array of blades) and thus a disc–blades transition element is needed. Two suitable transition elements should then be developed to insure the compatibility of the displacement fields at the shaft–array of blades (if the blades are connected directly to the shaft) and the disc–array of blades (if a disc is interposed between the shaft and the blades) interface. Due to the fact that the blades are seldom connected to the shaft directly, only the disc–array of blades transition element has been developed.The disc–array of blades transition element is provided with a node 1 contains 4 complex degrees of freedoms for the flexural behavior which located at the inner radius of the transition element describes the interface between the disc and array of blades, while a node 2 which has 5 complex degrees of freedom for flexural behavior located at the outer radius of the transition element. Their matrices have been obtained from those derived for the array of blades element by constraining the rotation about the tangential directions at node 1.A rotating membrane is a very thin disc with negligible flexural stiffness which constitutes a good test case for verifying the mass, gyroscopic and centrifugal stiffening matrices of the element in out-of-plane modes. In an analytical point of view The test has been performed using the following data: thickness: 1×10−7 |
m; outer diameter: 1 m; inner diameter: 0 m; Young's modulus: 1×10−5 |
N/m2; Poisson's ratio: 0.3; density: 7800 kg/m3. Two models, respectively with four and nine disc element and one transition element, have been built and each model has been studied by using both the whole set of generalized coordinates and performing Guyan reduction, which only consider the out-of-plane degrees of freedom as the master ones. The Campbell diagram is obtained by repeating the computations at three different values of the spin speed, namely 0, 0.5 and 1 Hz. The results are compared with the analytical results obtained by Southwell From the table it follows that the present element performs quite well, proved that a sufficient number of elements are used. Comparing the results of the two models with 5 elements and 10 elements, it is clear that the accuracy of the numerical results increases by increasing the number of elements in the model: the numerical results tend to the analytical solution Another model has also been built using ANSYS 11.0 code with the same geometric and material parameters in previously developed FEM and analytical model. Both the centrifugal stiffening (angular velocity inertia) and gyroscopic effect (Coriolis force) are considered in the ANSYS model. The membrane is meshed with 1078 4-node shell element, SHELL181. The Campbell diagram is obtained by repeating the computations at five different values of the spin speed, namely 0, 0.25, 0.5, 0.75 and 1 Hz. The results of the second- and third-order harmonics are compared in The figure shows that the natural frequencies of the flexural behavior at standstill are vanishingly small, the first natural frequency for the second-order harmonics, as an example, is 2.1×10−21 |
Hz at standstill (present FEM model) and 5.7×10−6 |
Hz (ANSYS model). The natural frequencies increase linearly with speed according to the Campbell diagram, which is predicted by the theory. But the present FEM model matches very well the analytical result as already demonstrated in , while the ANSYS model under-estimates the whirl frequencies of not only the forward but also the backward mode, and the errors increase with the increasing speed, which indicates that compared to the analytical results and the finite element model, the conventional FEM codes yield poor results when gyroscopic terms are accounted for. Furthermore, the present FEM model requires a much smaller number of element than the ANSYS model (10 VS 1078), while providing a much better accuracy.Consider a constant thickness steel disc with the following geometrical and material data: thickness 5.94 mm; outer diameter 1220 mm; inner diameter 76.2 mm; Young's modulus 2.1×1011 |
N/m2; Poisson's ratio 0.3 and density 7800 kg/m3. As no analytical solutions are available, the comparison is made by using the results obtained using the ANSYS 11.0 code, with 1642 4-node shell elements, SHELL181. The first 30 frequencies for null angular speed are obtained by using a Block Lanczos technique. The present finite element model is built using one shaft–disc transition element and five disc elements. The solution is performed both by resorting and not resorting to Guyan reduction, which consider the out-of-plane degrees of freedom as the master ones. Only the first three natural frequencies for the second- and third-order harmonics are computed which are reported in . The first three mode shapes for the second-order harmonics disc element and ANSYS model are plotted using non-dimensional deformation in From all the data above it follows that at null angular speed whatever the reduction method is used or not, not only the frequencies but also the modes for the higher order harmonics in a case of constant thickness pierced non-rotating disc fits very well with the maximum error at about 0.5 percent.Now consider a steel disc whose thickness is variable in radial direction. The cross-section of the disc is illustrated in . Geometrical and material data are: outer diameter 640 mm; inner diameter 40 mm; Young's modulus 2.1×1011 |
N/m2; Poisson's ratio 0.3 and density 7800 kg/m3. A FEM model with one transition element and nine disc elements is built, and a comparison model has been set up using ANSYS 11.0 with 4684 eight node three-dimensional structural solid element, SOLID45. The first three frequencies at null angular speed of the second-order harmonics are compared in . The mode shapes are plotted using non-dimensional deformation in The precision obtained is remarkable. For the lowest three frequencies of the second-order harmonics element the maximum error is about 1 percent, which means for the disc if the thickness in radial direction is changed, the higher order harmonics element is also perfectly suitable. But if the thickness in radial direction is nonlinear, it is clear that more disc elements which are shorter in radial direction must be used to approximate the thickness changing. These tests on non-rotating systems show that the stiffness matrix, not tested in the example on the membrane, is essentially correct (or at least, yields the same results as commercial FEM codes).Consider the same steel disc studied in Example 4.3, but now taking into account rotation, with a speed range between 0 and 100 Hz. The mathematical model is the same as seen already, but here a further model in which the gyroscopic terms are neglected is studied. The results are compared with ANSYS 11.0 in . The frequencies computed using DYNROT and ANSYS codes are in accordance at standstill. When considering rotation, DYNROT model still agrees with ANSYS model if the gyroscopic effect is neglected. This means that in both models only centrifugal stiffening is taken into consideration.If all the contributions in DYNROT model are accounted for, ANSYS code under-estimates the whirl frequencies of both forward and backward mode and these errors increase with increasing speed. It proves that the conventional FEM codes make worse approximation and consequently reach worse accuracy than the present one when the gyroscopic effect is accounted for, even though the ANSYS model is meshed with much more elements than the present DYNROT model (4684 Vs 10).A rotating pendulum is a limit case for testing the inertial and centrifugal stiffening matrices of the present array of blades element. The natural frequencies of the rotating pendulum are related to the spinning speed by the relationshipswhere l is the length of the pendulum and r is the radius of the disc.The disc at which the pendulum is attached is modeled using two beam elements for the shaft constrained at both ends by rigid supports, one shaft–disc transition element, one disc element, and all these elements being very stiff. A disc–array of blades transition element and nine array of blades elements are added to model the pendulum. All geometrical and material data come from Ref. are 100 and 141.41 rad/s, respectively. The results for the second-order harmonic components, seen as the flexural oscillations of the row corresponding to the in-plane and out-of-plane oscillations of all pendulums, have been reported in . Each oscillation mode generates two flexural modes, which are backward and forward, respectively.The natural frequencies are also computed at other values of the spin speed, showing a linear dependence as predicted by the theory. The present model yields results which are close to the correct ones.A finite element model with a shaft and a flexible disc and four prismatic blades are considered to verify the behavior at standstill. A model with a cylinder and four beams is also set up using ANSYS 11.0 which is meshed with 4054 eight node three-dimensional structural solid element, SOLID45, to compare the results. The geometric configuration of the rotor structure is given in . The material parameters are: density 7800 kg/m3, Poisson's ratio 0.3, Young's modulus 2.1×1011 |
N/m2.This model is only used to verify the modes related to second and higher order harmonics of the array of blades. The DYNROT model is made of two beam elements for the shaft, whose ends are constraint by rigid supports, one shaft–disc transition element, one disc element, a disc–array of blades transition element and nine array of blades elements. The computation is performed both by using all generalized coordinates and using Guyan reduction, which determines the out-of-plane degrees of freedoms as the master degrees of freedom. Only the first three natural frequencies for the second-order harmonics are computed and compared with the results reported in . The first three mode shapes for the second-order harmonics element and ANSYS model and the compare of mode shape using non-dimensional deformation of each blade are plotted in It indicates that whatever the reduction method is used or not, the frequencies for the second and higher order harmonics in a case of shaft with flexible disc and prismatic blades fit very well with a maximum error of no more than 2 percent. Also the mode shapes for higher order harmonics of DYNROT model are in good agreement with that of ANSYS though some of the blades have 180° phase difference. But as the disc has been completely constrained, the dynamics of each blade is decoupled from the others, which means the phase difference of some blades in DYNROT model relative to ANSYS modes is not relevant.Two finite elements aimed at modeling discs and arrays of blades for the studying of their flexural behavior have been developed. Both the displacement field within the disc element and array of blades are approximated by trigonometrical expansion along the tangential direction and a polynomial expansion along the radius. Only the second and higher order harmonics have been taken into account as they are uncoupled from the dynamic behavior of the rotor.The formulation for both elements have been obtained using complex coordinates following a Lagrangian approach which accounts for gyroscopic effects and stress stiffening. The elements have been implemented in the existing FEM code DYNROT. For disc element a constant stress contribution has been considered, for example due to the thermal stresses, and assumed as a part proportional to the square of the spin speed. However, for blade array elements, the constant stress contribution has not been considered as the blades are assumed to be unconstrained in the radial direction at their tip and thermal gradient is not considered in array of blades element.Since the modes of the disc here studied are uncoupled with the modes of the shaft, the disc element has two nodes, one at the inner radius of the element and the other at the outer radius. Also the array of blades element is provided with two nodes, being located at the element inner and outer radii. Two transition elements have been developed to connect the disc element with beam element used to model the shaft and array of blades element with disc element, respectively. It is assumed that the discs are attached to the outer radius of the shaft and blades are clamped to the disc at the inner radius of the array.A number of tests have been carried out to verify the accuracy of the two elements. For the disc element, a membrane has first been studied to compare the analytical results with the natural frequencies obtained by the current disc element model and ANSYS code.A very interesting result is that, while the present element formulation yields results that coincide with the analytical solutions, the results obtained through ANSYS are different, and in particular yield a value of the natural frequencies that is lower than the analytical one. Then several cases are used to compare the natural frequencies considering the different geometric structure of discs modeled using the present disc element and a commercial FEM code, ANSYS 11.0. The results show the disc element performs with a good accuracy, even when using a small number of degrees of freedom.Again, when the Coriolis terms are included the ANSYS model yields results that are different from the present ones while the results obtained neglecting Coriolis terms coincide. From the results obtained for the rotating membrane, apparently the formulation for the gyroscopic term in ANSYS yields a worse approximation than the present one.For the array of blades element, both the analytical results obtained using a rotating pendulum and the numerical results achieved by rotating untwisted blades connected to a rigid disc and a rigid shaft have been compared to those given by the array of blades element, showing in both cases a good agreement. In addition, the present finite element model requires a smaller number of degrees of freedom than conventional FEM models, while still preserving its accuracy and, in some cases, yielding much more accurate results.Fatigue, residual strength and non-destructive tests of an aging aircraft’s wing detailThis study concentrates on fatigue, residual strength and non-destructive tests of an aging aircraft’s wing detail of the Finnish Air Force’s Hawk Mk.51 jet trainer. The studied detail was the integral stiffener with a drain hole near the wing root. Fatigue tests were deemed necessary to verify experimentally the analytically observed short fatigue life, significant crack growth rates and eventually to re-assess the detail’s inspection period for the fleet jets. The results of the study have been utilized e.g. at the complementary type approval accomplished for the Finnish Air Force’s Hawk.The Finnish Air Force (FIAF) has been operating their Hawk Mk.51/Mk.51A jet trainer fleet since the 1980s/1993s. As the fleet gets older, the fatigue related issues have become more acute. This study concentrates on fatigue, residual strength and non-destructive tests of an aging aircraft’s wing detail. The studied detail was the integral stiffener with a drain hole near the wing root of the Hawk Mk.51. The Mk.51A is not considered in this study because it has the modified wing with removed stress raisers (e.g. drain holes), strengthened structural details and changed materials. The results of the analytical life estimation calculations are introduced, too. The results of the study have been utilized e.g. at the complementary type approval accomplished for the FIAF’s Hawk.Fatigue tests were deemed necessary to verify experimentally the analytically observed significant crack growth rate, high stress levels, short total life cycle and eventually to re-assess the detail’s inspection period for the fleet jets. Parallel to the above studies, the service inspections of the fleet aircraft resulted in a crack indication within the Mk.51 drain hole region. The crack indication was made before the certified safe life of the wing while no known repair method existed. According to the CSI (company structural instruction issued by the OEM) the wing with crack(s) in this location is not airworthy. Thus, the wing was removed from service.For fatigue test purposes described above, eight small-scale and six component test specimens were manufactured from a retired wing of a Hawk Mk.51 aircraft.Small-scale test specimens were primarily used for the purpose of material properties’ re-evaluation. To achieve an adequately comprehensive view of the desired material properties, both constant and variable amplitude tests were performed for small-scale specimens. In the presented study, no material tests or comparisons were carried out between aged and virgin aluminium.Only variable amplitude tests were performed for the component specimens. Component tests were desired to bring light for critical crack size, crack growth rate and eventually non-destructive inspection (NDI) interval. According to the CSI the NDI kick-off point, interval and method are presented in (FI = Fatigue Index, FH = Flight Hours). The calculation of the fatigue index is based on the average aircraft mass and the G level exceedances. All Hawk jet trainers have a G-counter which registers the exceedings of the G levels of the flights. 68 FI is the certified safe life for wing of the Hawk Mk.51.Non-destructive tests were also carried out to evaluate the initial, critical and smallest detected crack size.The test specimens were cut from two retired wings. Eight small-scale test specimens were cut from the right wing which had suffered 21.3 FI. One of the component specimens was cut out from the same wing as the small-scale specimens and five component specimens were cut out from another wing which had suffered 60.0 FI.The small-scale specimens were cut out from stringers 1–4 between ribs 1–3 (). The stringers, ribs and bottom surface of the wing form the integrated structure. The bottom surface side of the specimens was machined to constant dimensions. Finally the machined surfaces were polished. The holes were drilled for the specimens without any finishing treatment, thus corresponding better to real drain holes removed from sealing compound. The width, thickness, length and hole diameter of the specimens were about 20 mm, 6.7 mm, 280 mm and 5.1 mm, respectively.The component specimens represent drain holes where the rib 2 crossed the stingers 3–5 (). The nominal main dimensions of the component specimens are presented in b reveals the meaning of the different indexes of . The indexes xC and yC specify the location of the center of gravity.The material of the specimens is aluminium in accordance with the British standard BS 2L 93:1971 Two FIAF’s Hawk jet trainers have the permanent operational measurement loads (OLM) set-up since 2000 . The order of the flights in the spectrum was random. With the proper transfer function (from the OLM strain gauge location to the fatigue critical location of interest) including offset the strains/stresses of the test spectrum corresponded to a load level of the real drain hole.Six constant amplitude tests were performed using three load levels and two specimens were tested using spectrum load. The load level or the mean stress–stress range combination were selected for the constant amplitude tests so that the consumption of a lifetime was severe. In the spectrum test the stress level of the real drain hole was aspired to the vicinity of the specimen hole. So the prepared spectrum was scaled by a proper linear function and the peak-valley values were connected with each others using half sine wave. Naturally the flexibility of the specimens and the performance of the test machine were considered.The test facility was general purpose testing machine for the strength of materials – 100 kN INSTRON 8502.The component tests were performed using an individual 250 kN hydraulic cylinder under load (force) control with INSTRON 8500 digital controller (). The level of the cylinder force was controlled by strain gauges at the drain hole location: one on the stringer (R2) and the another on the bottom surface (R1) (). The R1 strain gauge was used to control the cylinder and the secondary bending was specified by the R2 strain gauge.The component specimens were asymmetric () and the bottom surfaces were slightly convex. For these reasons, special attention had to be paid to the load transfer to the specimens. To obtain the typical secondary bending level, the fastening had to be adjusted from specimen to specimen (b the extra fastening arrangement was created in order to achieve sufficient friction to the stringer and thus a proper secondary bending. Naturally the adjustment of the secondary bending caused different prestressing conditions. Because of this the stress level at the bottom of the drain hole (6 o’clock) (b) was aimed to be kept near the desired value and the possible errors were allowed to accrue at the top of the drain hole (12 o’clock).A visual and eddy current (ET) non-destructive testing methods were applied, for the two constant amplitude small-scale specimens (see The results of the constant amplitude tests are summarized in shows the specimen’s ID, minimum and maximum cylinder forces and its range as well as the hole peak stress range and the number of cycles to failure. The results of reveal that the scatter in the life data is insignificant.The results of two variable amplitude tests are presented in . The total number of flights of the spectrum is shown in columns corresponding to the presentation of Based on the test results the adjusted material data for the strain-life method are presented as L93_PFA#2 in . The data for the aluminium 2014-T6 from the database of the MSC.Fatigue-program the results of the NDI testing are presented. According to the indications of the average crack growth takes approximately 6000–6500 cycles when the corner crack grows from the size of 1 mm (a) up to 6.5 mm through the thickness. The previously performed crack growth calculation In five of the six component specimens cracks were detected but only one specimen fractured from the desired drain hole. On the other hand it must be noted that the residual strength was evaluated by two specimens when the appeared cracks achieved an appropriate size. concludes the test results. In this context a crack initiation means the moment when a crack or cracks were detected first time using the ET method. The corresponding crack size (ainit) is based on a visual observation, the ET method or estimation. the fatigue test results of the variable amplitude tests are summarized. The data of , excluding the results of the small-scale specimens (SSS#7 and SSS#8), include the life cycles prior to fatigue tests (i.e. the flight-induced FI consumption). The fatigue calculations performed for the stringer locations, from which the small-scale specimens were cut of, showed that the flight-induced FI consumption experienced prior to fatigue test cycling is negligible . On the other hand the crack growth scatter is significant from 7.6 FI to 46.1 FI. It is important to note that the test results are actual observations and they are not adjusted by any safety or scatter factors.Based on the probability analysis crack initiation probability should be 6.6% The results of the NDI tests also indicated quite wide scatter in crack growth ) which is believed to be caught with the commercially available ET method.The residual strength tests of the specimens STR4_r and STR5_aux indicated the margin of 1.2 and 1.1 ) in which case the a-dimension is through the thickness.The fatigue experiments of FIAF’s Hawk Mk5.1 jet trainer wing drain holes and their results are presented in this paper. The drain hole investigations comprised the small-scale and component fatigue tests. The scatter of the results of the small-scale test specimens was narrow. On the other hand the scatter of the results of the component tests was quite wide thus complicating the conclusions from the test results.The certified safe life for wing is 68 FI. The test results indicated that the probability of the crack initiation was 6.6% at the point of 68 FI. On the basis of the fatigue test results, quite comprehensive material properties knowledge was achieved. The tests strengthened the validity of the previous used fracture properties: crack growth rate, fracture toughness and critical crack size. Above all, the results indicate that previously performed fatigue calculation would be conservative.A reasonable estimate of the initial corner crack size was obtained based the test results, too. Overall the test results indicate that the NDI program presented in could be followed in future although the scatter of the results of the component tests was quite wide.Developing and distributing network based engineering solutionsThe need to have easy access to the solutions of a variety of frequently occurring engineering problems, has resulted in the development of a wealth of tools ranging from engineering handbooks to sophisticated desktop engineering software. In this paper, we examine the use of internet/intranet-based methods for providing the tools that are commonly needed by engineers in their daily work. Using the World Wide Web as the distribution medium and the Internet browser as the execution environment, Sun Microsystem's Java technology provides the foundation for the development of an Engineer's Tool Box (ETB) that provides a framework whereby independent engineering software tools are linked, managed, and accessed globally via the Internet. Individual applications are developed to address specific engineering problems using simple, straightforward interfaces and are linked into the distributed ETB framework to solve more complex problems. The capabilities and limitations of the Java platform for developing and supporting such Internet based distributed engineering software tools are discussed herein.The need to have easy access to the solutions to common engineering problems traditionally has been met with a wide range of reference handbooks, but more recently sophisticated engineering software packages have been developed that allow complex and diverse engineering problems to be efficiently modeled and solved. In the future computers will play an even greater role in day-to-day engineering activities. Faced with increasingly complex tasks and greater competition, engineering software tools must not only become more powerful, but must also be more cost-effective to deploy and maintain, simpler to use, and more flexible in operation, expandability, and customization. In addition, engineering software tools of the future must feature improved communication and collaboration capabilities to facilitate the distribution of information and data.As a communication and distribution medium, the Internet represents an amalgamation of many of the inventions of the past two centuries. Although the potential of this global network continues to outpace its supporting infrastructure, it has already received wide acceptance in the business community. The ability to reach customers and conduct electronic commerce has spurred the recent explosion in Internet hosts, along with initiatives to improve, regulate, and even tax the flow of information and business transactions. The World Wide Web holds the same potential for sparking a revolution in the scientific and engineering communities. The marriage of today's powerful monolithic desktop engineering applications to an open, extensible, Internet-based network of engineering tools and data formats (i.e. objects) promises to revolutionize the trading of technical data to the same degree that electronic commerce is transforming business.Using the World Wide Web as the distribution medium, and the Internet browser as the execution environment, Sun Microsystem's Java technology provides the first opportunity to harness recent broad advances in computer hardware and software to make such improvements to engineering tools a practical reality. As a modern, object-oriented, multithreaded programming language with integral networking and security features, Java simplifies the creation and distribution of platform-independent engineering tools. This inherent power and flexibility is an ideal combination for the development and implementation of a distributed “Engineer's Tool Box” where independent applications can be linked, managed, and accessed globally via the Internet. The individual applications are designed to address specific engineering problems using simple, straightforward interfaces, and are linked into the distributed tool box framework to solve more complex problems. The concept of an open, distributed, Internet-based framework of interoperable tools collected to form an “Engineer's Tool Box” is the subject of this paper.Sun Microsystem's Java technology is an Object Oriented Programming (OOP) computer language with syntax and concepts that are similar to those of the C++ language improved memory management, including automatic garbage collection;improved compiler and runtime error checking; |
However it is Java's execution environment, called the Java Virtual Machine (JVM), which makes the language truly unique. Java source code is compiled into platform-neutral “bytecode” that is interpreted and executed by the platform-specific JVM. As another layer of abstraction between the application source code and the underlying hardware, the JVM is essentially an operating system, with the unique advantage that it may be implemented on top of other operating systems. A schematic of Java's execution environment is shown in As an object-oriented programming language, Java programs consist of one or more “classes” that encapsulate individual functions (called “methods” in Java) and data “objects”. The Java application programmers interface (API) refers to a standard set of Java class libraries collected into groups called “packages” in the Java Development Kit (JDK). These classes can be imported and extended to form a framework for one of two types of Java programs: applications or applets. A Java application is a standalone program like a typical C++ application although it still requires a JVM to run. A Java applet is designed to be downloaded over a network and executed from within a web browser using the HTML 〈APPLET〉 tag. This tag calls the Java Applet Class Loader, which starts the applet as the controlling class of the browser environment.The two dominant browser packages, Netscape Communicator and Microsoft Internet Explorer, come bundled with their own proprietary JVMs to run Java applets. Although required under license by Sun Microsystems to conform to the standard Java APIs, neither JVM is fully compliant or complete, and some language features differ considerably in implementation. Other JVMs exist for various development packages (e.g. Symantec's Visual Café), operating systems (e.g. IBM's OS/2), and hardware platforms (e.g. Apple's Macintosh). Unfortunately, all of these environments have their own bugs, omissions, and inconsistencies, and the developer must recognize and compensate for these differences when preparing network-based applications such as are discussed herein. By limiting the scope of this initial study to JDK 1.1 tools and compliant browsers, we focus primarily on the fundamental feasibility issues of ETB.The idea of gathering solutions to engineering problems together for easy use is, of course, not new. Implementations range from engineering handbooks to computer-based collections of tools made easily accessible to the practicing engineer To explore solutions to these limitations, a prototype of an internet-based package of engineering tools collectively called the “Engineer's Tool Box” (ETB) was developed. The primary focus of the ETB design was to provide nontrivial solutions for practicing engineers using an open, extensible framework that provides built-in mechanisms for data transfer and communication between independent applications. By tying together purpose-built standalone engineering applications using the simple programming and interface standards developed and presented in this paper, the ETB framework allows any Java-fluent engineer or programmer to extend and leverage the capabilities of existing ETB components. To demonstrate these concepts, a preliminary core package of nine systems and structural and system analysis applications was developed and tied into the ETB framework. The menu structure of this preliminary version of the Engineer's Tool Box is shown in A number of decisions were made regarding the structure and standardization of the Engineer's Tool Box development before the first applications were coded The preliminary ETB applications were designed to be simple and interactive, such that their functions are obvious to the user. Each module consists basically of an input form, a calculation engine, and an output window for numerical and graphical results. Although most of the ETB's preliminary applets have relatively light computational requirements, two of the components required significantly complex and computationally intensive modules. These applets, the Dynamic Simulation applet and the General Beam Analysis applet, are particularly useful in showcasing the unique features of the Java language and evaluating the execution performance of Java bytecode. Currently concerns remain about Java's performance and whether data-intensive engineering applications (e.g. a full-featured finite element analysis program) can be effectively implemented via network applets. Experience gained in the development and implementation of the Engineer's Tool Box shows that such concerns are legitimate but surmountable. Java performance will be discussed further in following sections.Applications in the ETB package are intended to be straightforward, purpose-built tools that manipulate well-defined input data in some prescribed manner to create structured output data objects such that individual modules can be linked together for enhanced functionality. Although this definition clearly emphasizes the building-block nature of ETB applets, it should not be interpreted as a limit to the size or complexity of the modules that the ETB framework can accommodate.To demonstrate the suitability of Java for developing useful network-based analysis tools, nine typical engineering calculation applications were developed during this study. These ranged from simple applets in which a few formulas are evaluated from data supplied by the user using on-screen dialog boxes to applications with significant algorithmic and computational components. An example of the former is an applet that calculates the stresses in a Thick-Walled Cylinder using geometry, loading, and material data the user supplies by filling in forms.An example of an applet with significant algorithmic and computational requirements is the Dynamic Simulation module in which the user defines an arbitrary dynamic system by entering its state-space equations into an ETB text editor dialog box. The equations are parsed by the applet, and a symbolic-numerical Runge–Kutta integration process with error estimation and step-size control computes the response over the user specified time interval.The implementation of the Thick-Walled Cylinder applet is fairly obvious and is not discussed further here except to show its screen image in , but the Dynamic Simulation applet is described in some detail in the sections to follow.The Dynamic Simulation applet implements a fourth-order parameterized Runge–Kutta numerical integration algorithm to solve a set of differential equations of the form x′=f(x,t). This algorithm uses fifth-order approximations to calculate both the relative and absolute error in the fourth-order solution and adapts the time step to minimize this error. A more thorough discussion of the relevant theory can be found in Refs. In some respects, the Dynamic Simulation applet is the most complex of the Engineer's Tool Box modules and requires the highest degree of consistency among Java Virtual Machines to operate correctly. presents a list of Java-specific capabilities that were utilized to form the end result. The graphical input interface for the Dynamic Simulation applet is shown in The primary considerations in the development of the Dynamic Simulation applet were flexibility and ease of use. Toward this end, a design decision was made to implement a recursive input parser to allow the user to define any dynamic system and transient load using the standard state-space convention and enter these equations into the input editor using a simple syntax consisting of familiar mathematical operators and functions. While the implementation of the input parser greatly increases the complexity of the Dynamic Simulation applet, the user is not limited to a set of predefined dynamic systems or transient forces. Parsing errors and the status of the simulation are shown in a separate text area, allowing the users to easily access input errors and data. Plotted output is provided in a separate frame, shown in , to allow the user a number of display and scaling options in a window that can be easily resized.The Dynamic Simulation applet's numerical integration method is contained in the abstract class and supplies the implementation-specific methods described in In Java, an object of an abstract class cannot be instantiated. An abstract class only defines an interface or implementation to be inherited by objects of its subclasses. This premise is used in the Dynamic Simulation to separate explicitly the numerical integration method in from the implementation-specific methods defined in the class. This adds to the extensibility of the routine, allowing future programmers to easily modify or replace the user-supplied variable and function definitions without altering the integration algorithm.While the flexibility of the Dynamic Simulation applet's text editors and the plot frame are sufficient for accessing and visualizing the simulation data, the user must be given the ability to save and recall the simulation input and store the output data for archival or post-processing. The Java language provides a number of powerful file-processing features that allow access to both the local and remote file systems. Java views files as a sequential stream of bytes, referred to more simply as “streams”. Using Java's built-in stream classes to manage file and network access, the Dynamic Simulation applet allows the user to save and recall the simulation input, and, via the plot frame, to save the output data in tabular format.Currently, the Java Virtual Machines in the Netscape and Microsoft browsers have security restrictions that limit file access to only the local server where the Java applets reside, and only when there is no active network connection. These limitations are required to protect the local system from malicious applets, particularly those originating from remote systems. Such limitations can be overcome using current digital encryption technologies to digitally “sign” an applet. Such digital signatures allow the end user to identify the source of the applet and verify that the binary has not been altered.Although the Netscape, Microsoft, and Sun JVMs allow signed applets to obtain special access to the local system's resources (files, printers, etc.), their authorization procedures and digital signature formats are not compatible with one another. This forces the programmer to write separate code for each. Currently, the Dynamic Simulation applet only supports the Netscape Capabilities API. The classes in this API modify and extend the Java security manager to control an applet's access to a local machine's resources or a remote network connection. Assuming the applet is signed by an authorized entity, the Netscape browser responds to local machine access requests with the dialog box requesting user permission to continue. If the user grants the privilege request, the Java applet will be able to access the local hard drive and write to any file. Given the potential risks to files and data stored on the local machine, such privilege requests must be considered very carefully by the end user.Because the Engineer's Tool Box is based upon the premise of an open, extensible framework, built-in mechanisms for data transfer and communication between independent applications must be included in its foundations. The passing of data between individual applications, hereafter referred to as “interapplet communication”, is handled automatically and remains transparent to the user through the implementation of the techniques described below.To understand the requirements that guided the design of the ETB's underlying framework, consider the Uniform Beam Frequencies applet shown in . This analysis tool allows the user to provide geometric and material properties of a uniform beam as input and calculates the beam bending frequencies. This tool is of limited value if the appropriate input parameters for real-world beam section geometry and material properties can only be supplied by keyboard inputs.Thus the Uniform Beam Frequencies was designed to allow the user to import the appropriate geometry and material input data from other sources within the ETB framework. In the design the name, location, or inner workings of the source of the data need not be known to the receiving applet (the beam frequencies module in this case). Any Java engineering tool could be linked into the ETB framework and provide the required input data.The utility and flexibility of this open architecture can be demonstrated by using the Beam Section Properties applet shown in and the Material Properties applet shown in to transfer geometry and material properties to the Uniform Beam Frequencies applet. As seen in , the Uniform Beam Frequencies applet includes buttons to allow the user to load the supporting applets and link them into the ETB framework. Once complete, these applets can pass data objects directly to the Uniform Beam Frequencies applet, requiring only a single input command from the user. It is important to stress that the Beam Section Properties and Material Properties applets are and remain completely independent; they do not require the Uniform Beam Frequencies applet to run. Although the Uniform Beam Frequencies applet provides a hard-coded mechanism to load and link these applets, any applet linked into the framework could send the appropriate data objects to the Uniform Beam Frequencies applet.Interapplet communication in the Engineer's Tool Box is achieved using an object-oriented design derived from the Multi-Panel Applet Design (MPAD) concept In MPAD, an overseeing “panel manager” is created to dynamically load individual Java applets, called “panels” hereafter, and links them together using global communication and data structures. The panel manager requires that the Java applets contain a number of common interface methods to operate correctly within the MPAD framework. These required interface methods are included in the Engineer's Tool Box applets by extending the class, which extends the Java Applet class. This ensures that the applet can be run both independently (i.e. as an applet) and as part of the larger MPAD framework (i.e. as a panel).The Engineer's Tool Box panel manager is implemented in the class. It extends the Java Applet class, and is always the default applet called by the HTML 〈APPLET〉 tag. Because it only operates in the background, the class name be specified at runtime using an HTML 〈PARAM〉 tag named “ dynamically loads and creates a new instance of this default class and then displays it directly to the screen as if the object were itself the Java applet being executed. Additional -derived classes can be loaded and instantiated by the is primarily managed through the use of three interface methods: The ETB's interapplet communication framework differs considerably from the original MPAD design, which relied on a global data structure to exchange data between applets. In the ETB's framework, a hierarchical communication structure is maintained between individual panels. Each panel automatically contains a reference to its “parent” panel, and data can be passed using Java's standard event-driven communication model. A simple illustration of the ETB's communication framework is shown in As discussed previously, the Uniform Beam Frequencies applet uses the data objects from the Beam Section Properties and Material Properties applets. Upon detecting objects of these types, the applet post-processes the data and accesses only those items that affect the calculation of the beam natural frequencies (e.g. areas, moments of inertia, Young's Modulus, etc.). In the case of the object, the user must be given the opportunity to select the desired moment of inertia (i.e. Iyy or Izz) to be used in the calculation. This is handled by opening a dialog box, shown in , that contains two mutually exclusive check boxes.The process described above is not just a minor detail in the operation of the Uniform Beam Frequencies applet. It is an important example of the standard convention used by the Engineer's Tool Box applets for the dynamic filtering of transferred data. The standard convention is that the upstream applet (i.e. the applet receiving the data) is responsible for data filtering, and that the downstream applet makes no attempt to filter data based on knowledge of the upstream applet. The rationale for this standard is simple. Individual applets in the Engineer's Tool Box should act as independent entities that output standardized data objects. As a part of a distributed computing environment, no assumptions should be made about that environment, including the identification of recognizable classes.Although relatively new, Java is a remarkably complete and capable language for the development of a wide variety of software. However, like any innovation, Java suffers from a number of limitations due to design deficiencies linked to its immaturity. Unfortunately, evaluations of the capabilities and limitations of the Java language are often subjective in the highly competitive computer software industry. Much of the benefit of the initial ETB development effort was evaluating the current applicability of the Java language to solving technical problems. Through this research, three potential risks or limitations were identified in using Java for these technical tasks: JVM incompatibility, execution performance, and security procedures for local machine access.A number of JVM inconsistencies relating to operating system functions were also discovered. Many problems were related to how or when the JVM called a frame's method in response to an operating system function such as window resizing, iconification, or shuffling. These inconsistencies required some compromises to force the frame to automatically redraw the plot whenever the window was brought into view. For example, to ensure that the plot is redrawn whenever the output frame is selected, resized, or deiconified, the events to trigger the redraw. While this worked well in almost all cases, a side-effect was discovered in the JVM implementation in Netscape Navigator 4.0. The Netscape JVM sends a event whenever the mouse pointer exits a list box, causing a complete redraw of the plot and a noticeable flicker, particularly on slower machines.Another compromise due to incompatible JVM behavior was required to implement a “show plot” button on the Dynamic Simulation applet that would bring the plot window into view under any circumstance. The JDK 1.1 documentation states that a call to a frame's method will accomplish this relatively minor task. To achieve consistent results across all the JVM implementations required that the frame's methods be called in succession. Obviously, this solution can cause noticeable delays on slower machines. Such compromises should be eliminated as the Java language and its virtual machines mature.Java source code is compiled into bytecode that must be interpreted by the Java Virtual Machine and translated into native computer instructions. Initially Java earned a reputation for poor execution performance because bytecode interpreters operate many times slower than comparative native code created by compiled languages (e.g. C++, Pascal, etc.). However, this perception is misleading since most JVMs now use a just-in-time compiler (JIT) to dramatically speed Java execution times A JIT compiler converts bytecode into native code on the fly, storing the native code so that repeated method invocations do not need to be compiled again. Disregarding the conversion process, the Java code should operate just as fast as code written in other languages that is complied and executed. Moreover, there are some instances where JIT-compiled code could execute faster by optimizing its operation based on the runtime environment No attempt was made to compare the performance of the Engineer's Tool Box applets running on different machines, operating systems, or virtual machines since such comparisons are difficult to prepare and often misleading. However, the Dynamic Simulation applet was used to give an indication of relative performance by comparing solution times for the numerical integration algorithm in the class and a similar routine in a commercially available desktop software package (MATLAB). The problem to be solved is a third-order example system with step input taken from Ref. The above system equations were integrated from 0 to 2 s with convergence tolerances set such that a total of 8000 time steps was required. The time-history simulation was performed using the Microsoft Windows version of Matlab 5.0’s function and the dynamic simulation applet running under the Microsoft Internet Explorer 4.0 browser which employs a just-in-time compiler (JIT). Elapsed wall-clock times were noted for the two simulation exercises. This comparison is not exactly scientific in nature since the Java implementation has not been optimized for best execution times and the desktop software routines do not exactly match the code ran about three times faster, the total required end-user effort (engineering time) in setting up and solving the problem was about the same in both cases.The power and flexibility of network based, and in particular Internet based, distributed applications come with significant security issues. The distributed framework must provide adequate protection of the end user's physical and intellectual property from theft or destruction by malicious programs without impeding usability. While no system with external access is impenetrable, the goals in providing adequate security to minimize potential cost of a security breach;to maximize the difficulty of potential attacks.To adequately meet these objectives in today's sensitive business environment, both Netscape and Microsoft have implemented severe restrictions on the ability of Java applets to access local resources (e.g. hard drives, printers, etc.) from their web browsers. To gain access to such resources, an applet must be digitally signed using a purchased certificate from an independent “Certificate Authority”, CA, that verifies the identities of the individuals or organizations requesting the certificate. This procedure is complicated by the fact that Netscape and Microsoft do not support the each other's certificate format, and neither support Sun's Java applet-signing standard. Due to the extensive integral security features built into the maturing Java specification and the current lack of an adopted standard for digitally signing software, Java's security issues were among the most difficult to address in the ETB effort. This is not a unique conclusion. PC Magazine observes that “the lack of a consistent mechanism for accessing local files is one major impediment to Java's success as a replacement for traditional productivity applications on the desktop” For this preliminary research, the Netscape Capabilities API In Netscape Navigator 4.0, an unsigned applet cannot be authorized to access protected system or network resources. An applet signed by an authorized principal with a set authorization level for a given target must specifically request privileges to gain such access. The authorization of requested privileges is handled by the Netscape Capabilities PrivilegeManager class. Unlike other security manager implementations, the PrivilegeManager class allows for detailed control of access privileges. Examples of supported privileges are shown in Adding the PrivilegeManager to the existing Dynamic Simulation code was simple. For example, to request privileges to write to the local hard drive, the following code was required:Assuming the applet is signed by an authorized CA, the Netscape browser responds with a dialog box asking for the user's permission to continue. If the user grants the privilege request, the Java applet will be able to access the local hard drive and write to any file. Given the potential risks to files and data stored on the local machine, such privilege requests must be considered very carefully by the end user.The development and implementation of the ETB modules and framework described in this paper demonstrates the viability of network-based methods for distributing nontrivial engineering solutions involving moderately intensive computational tasks. The Java 1.1 platform provides an accessible foundation for applying state of the art computing concepts to develop and distribute engineering software tools. While performance, security, and JVM compatibility issues were encountered, most problems were attributable to the relative immaturity of Java support in current browsers. Work-around solutions were found for most implementation problems and also many limitations discovered in the Java 1.1 class library.The work described herein is the first step in providing a robust web-based distributed environment that provides access to engineering solution modules and facilitates data collaboration. Toward that end, a client-side computational model using Java applets was shown to be a viable alternative to monolithic, server-based applications with web connectivity. This approach has recognized savings potential, especially for large organizations (e.g. corporate intranets), when compared with multiple installations over a network with many nodes.However, the use of Java applets for client-side computing remains a controversial subject in the development community. Much of the initial hype surrounding the capabilities of Java applets has been tempered with the realities of browser incompatibility and security issues. And while the ETB project has overcome numerous obstacles presented by its applet-based design, many improvements and extensions to this work have been identified to take advantage of the unique capabilities of the existing ETB system.One significant capability that is being studied is the ability to access ETB modules and data objects from remote locations via the internet. The Java 1.1 class libraries include the fundamental tools for providing such access using modern programming concepts such as reflection, serialization (a.k.a. persistence), sockets, and remote object invocation (e.g. RMI In addition, the incorporation of Java beans Java is a rapidly maturing platform. The current Java 1.2 class libraries offer significant improvements to the GUI and Security APIs. Unfortunately, inconsistencies remain in the evolving Java landscape. Most significantly, the Java 1.2 class libraries included with Microsoft's (increasingly industry-standard) Internet Explorer 5.0 are not compatible with Sun's JDK 1.2 specification. The current version of the ETB is based solely on the Java 1.1 class libraries to maximize compatibility with existing web browsers. Sun's Java Plug-in initiative The interested reader can try current versions of the ETB modules by connecting to http://mae.uta.edu/~lawrence/toolbox.Experimental study of fire-resistant steel H-columns at elevated temperatureThis paper presents the experimental studies of axially loaded fire-resistant steel columns under elevated temperature. With the advancement of metal production, fire-resistant steel with enhanced mechanical properties at elevated temperatures has been developed recently. However, extensive research work is needed in order for the application of fire-resistant steel in building structures. In this study, a series of fire-resistant steel columns was loaded to their ultimate states at specified temperature. The effects of width–thickness ratios, slenderness ratios and residual stress on the performance of fire-resistant steel H-columns are examined. Based on this study, it is found that the section property of fire-resistant H-columns should be at least a non-compact section in order to prevent local buckling. Column strength is sensitive to slenderness ratio at elevated temperature. The strength of a slender column decreased sharply especially for temperatures above 600 ∘C. It is also found that the failure mode of steel columns changed from inelastic global buckling at room temperature to local buckling at elevated temperature, due to the release of residual stress in fire. An analytical model is proposed which is able to predict the behavior of fire-resistant steel H-columns under elevated temperature. Design guidelines are also proposed for the design of fire-resistant steel columns in fire conditions.The strength of a steel column is usually governed by the stability due to local buckling or global buckling. Local buckling is a phenomenon of plate failure. The buckling strength of a plate is affected by the Young modulus, E, Poisson’s ratio, υ, and width–thickness ratio of the section, b/t. The critical stress of a plate subjected to compression can be determined by Eq. The value of Poisson’s ratio, υ, and Young’s modulus, E, are normally treated as of constant value at room temperature. However, Young’s modulus deteriorates greatly at elevated temperature. The Young’s modulus decreases rapidly at temperatures exceeding 500 ∘C (). At 600 ∘C, the Young’s modulus drops to approximately half of its value at room temperature and this affects the strength of steel plates greatly.The global behavior of steel columns is even more complicated. The factors, such as the yield strength, residual stress, initial crookedness and column slenderness, are all connected to the global buckling strength of steel columns at room temperature. With the existence of residual stress, the strength of inelastic columns is greatly reduced. At room temperature, the value Cc(=2π2E/Fy) is generally adopted as the mean to determine the slenderness ratio to distinguish inelastic global buckling and elastic global buckling of steel columns. An inelastic column is the column with slenderness ratio less than Cc, and its behavior can be determined by an inelastic buckling curve, Pcr=Py[1−Py4π2EI(KL)2] (). A column with slenderness ratio greater than Cc is classified as an elastic column. The global behavior of elastic columns can be determined based on the Euler formula, Pcr=π2EI(KL)2. Because thermal effect greatly reduces the Young’s modulus and the yield strength of steels, the resistance of steel columns must be deteriorated accordingly.To conquer the deterioration of steel columns in fire, fire-protection materials are normally applied on the steel members to isolate the heat transfer. The wet sprayed fire-resistive material and the dry type fire-proofing material are the common fire-protection materials adopted in steel construction. However, the traditional fire-proofing materials may not function well in the fire event. This is because the fire-protection material might be split due to the softening and distortion of steel members in fire. A new type of high-performance steel with an improved fire resistance characteristic of steel itself has been developed to enhance the fire resistant properties of steel structures. With additional chemical compositions, such as micro-alloying molybdenum (Mo) (), fire-resistant steels have been proven to perform well at elevated temperature, despite retaining the same mechanical properties as those of the conventional structural steel at room temperature. compares the deterioration of the Young’s modulus and yield strength of conventional structural steel, SM490, and fire-resistant steel, FR490, in fire. FR490 is the same steel grade as that of SM490. The nominal yield stress of these two steels is 343 MPa at room temperature. As shown in , both the Young’s modulus and the yield strength of FR490 steel is higher than those of SM490 steel at elevated temperature. The yield strength of FR490 is up to 40% higher than the strength of SM490. At 600 ∘C, FR 490 retains at least 2/3 of the nominal yield stress of room temperature, which is a value of 216 MPa. However, it should be noted that the Young’s modulus of FR490 is even smaller than that of SM490 at 700 ∘C. The deterioration of the Young’s modulus of FR490 is greater than that of SM490 when temperature is above 600 ∘C. Designers should pay attention to the deterioration of strength if the fire-resistant steel members are subjected to temperatures higher than 600 ∘C.Research related to fire-resistant steel has been carried out and fire-resistant steels have been put into practice in Japan Two series of column specimens, stub columns (Specimens No. 1–12) and columns with longer length (Specimens No. 13–32) were designed in this study. The basic idea of a stub column test is to design columns with limited length in order to include the same initial residual stress pattern as the original steel members but to exclude the possibility of failure due to overall buckling The purpose of long columns is aiming at examining the overall behavior of steel columns at elevated temperature. The slenderness ratios and residual stress are the two main factors affecting the global behaviors of steel columns. To avoid occurrence of local buckling before global buckling and to minimize the interaction of local buckling and global buckling, columns with lengths longer than stub columns are designed as plastic sections, based on the results obtained from stub column tests. To diminish the secondary effect on column strength, column specimens were carefully fabricated to fulfill the construction criteria of L/1000The steady state method was adopted in order to directly derive the load–displacement behavior of steel columns at specified temperatures. That is, the specimens were heated up to the specific temperature and then axial loads were applied. The current applicant of fire-resistant steel is set up at the temperature of 600 ∘C; therefore, the temperature selected in this study is from room temperature up to 600 ∘C. After a column specimen reached its ultimate load, the test was stopped when the strength decreased to 60% of the ultimate load. During the heating process, the specimen expanded freely without any restraint to its thermal expansion. To monitor the temperature distribution of the column specimens and to ensure the temperature of the specimens remains steady at the desired level, thermocouples were installed on each specimen, following BS476 regulations An identical specimen was also tested at room temperature as comparison. During the loading process at room temperature, strain gages were installed symmetrically across the specimens not only to monitor the stress–strain variation but also to ensure the compressive force was centrally loaded without eccentricity. The lateral dial gages are also installed to monitor the onset of the local buckling of stub columns and to determine the effective length of longer columns. Dial gages were instrumented longitudinally both at room temperature and at elevated temperature to ensure the load was centrally applied and to calculate the axial displacement of the specimens under load. The instrumentation of the specimens is shown inStructural behavior of steel columns in fire can be evaluated from the load–displacement behavior of the experimental results, as shown in . In these figures, the load ratio is the column strength to the nominal yield strength (Py,n) of room temperature. The nominal yield strength is determined based on the nominal yield stress of fire-resistant steel at room temperature, which is 343 MPa in this study. Basically, stub columns failed because of local buckling; while flexural buckling about the weak axis is the failure mode of most of the long columns. Behavior of steel columns in fire is discussed in the following. shows the behavior of stub columns in fire. With the same width–thickness ratios, the strength and stiffness of columns decrease with temperature. This primarily results from the deterioration of yield strength and Young’s modulus at elevated temperature. At the same temperature level, strength also decreases with the increasing width–thickness ratios. The strength of non-compact sections decreases more rapidly than compact and plastic sections, and provides less ductility accordingly. The load ratios are greater than unity in the cases of temperatures less than 600 ∘C. This means that columns with non-compact sections or better can reach nominal yield strength unless the temperature exceeds 600 ∘C. At 600 ∘C, strengths retain at least 60% of the nominal yield strength, which is the allowable design strength at room temperature.It is also found that the load ratios of ultimate strength (Pu) to yield strength of elevated temperature (Py,T) are from 1.11 up to 1.67 for compact sections or plastic sections ( summarizes the experimental results of stub columns, in which Pu/Py,T is the ratio of ultimate load to yield load at elevated temperature. At elevated temperature, columns of non-compact sections or better cannot only reach their yield load (Py,T) but also get into the strain hardening stage. There is no premature yielding induced by the residual stress, and both the strength and the ductility of columns are enhanced at elevated temperature. Column behavior is mainly governed by the yield stress at specified temperature. Because the yield stress of fire-resistant steel remains more than 60% of the value of room temperature at 600 ∘C, the deterioration of columns is not significant for plastic and compact sections. However, strength reduction of non-compact sections is greater. A non-compact section loses its resistance soon after reaching the yield strength.The columns were designed as plastic sections to exclude the onset of local buckling before global failure. The performance of long columns is listed in , including the load ratios, (Pu/Py,n), ductility and failure modes. The load ratio, Pu/Py,n, is the ultimate load at elevated temperature to the nominal yield load of room temperature. With increasing column slenderness and temperature, the strength and ductility of columns reduce greatly. At 600 ∘C, inelastic columns (slenderness ratios of 29, 60, and 80) retain about 60% of the nominal yield strength, while elastic columns (slenderness ratios of 108 and 120) buckled suddenly and their strength is less than 50% of the ambient strength. The ductility is defined as the ultimate displacement (Δu) to the displacement of the specimen yields (Δy). Ductility is greater than 5 for columns with slenderness of 29. And it is up to more than 10 for stub columns of plastic section. However, ductility decreases to 2–3 for columns of slenderness ratio 60. The ductility is even smaller for elastic columns. The greater the column slenderness is, the less the ductility. It is also found that ductility increases at a temperature of 600 ∘C. These are the basic mechanical properties of steel at elevated temperature. The elongation of the steels increases significantly as temperature reaches 600 ∘C (). This results in the larger ductility of columns at higher temperature. shows the effect of the width–thickness ratio on the performance of steel columns at elevated temperature, in which “RT” represents room temperature. At the same temperature level, column strength is not sensitive to the width–thickness ratio. However, reduction of column strength due to the thermal effect is significant. In current design practice, steel sections are classified as non-compact, compact, or plastic section according to the width–thickness ratio of steel plates. With the proper width–thickness ratio, the steel column is able to possess certain degrees of ductility and to exclude brittle failure induced by local buckling. Based on the experimental results, it was found that the criteria developed at room temperature to prevent premature failure of local buckling can be adopted for fire-resistant steel sections at elevated temperature. Because of the increase of the elongation and the softening of steel in fire (), the ductility of steel columns increases accordingly. Brittle failure due to local buckling for columns made of fire-resistant steel can be prevented either at room temperature or at elevated temperature if it is a compact section or better. No extra effort is needed in determining the width–thickness ratio of steel members if fire-resistant steel is used for building structures even under fire conditions. The effect of the width–thickness ratio on the behavior of fire-resistant steel columns is examined in detail and resulting criteria are proposed in Ref. The effect of slenderness ratios on the column strength at different temperatures are shown in . Only the columns with plastic sections are considered here to avoid the interaction between local buckling and global buckling. Based on the experimental results, it is found that the column strength decreases with increasing temperature and slenderness ratio. Column strengths are sensitive to column slenderness at temperatures less than 600 ∘C. At temperatures reaching 600 ∘C, strength decreases gradually until a slenderness ratio of 120. The deterioration of inelastic columns (slenderness ratios <108) is within 20% when the temperature level is less than 500 ∘C. It retains at least 60% of the nominal yield strength at 600 ∘C. However, the strength of elastic columns (slenderness ratios of 108 and 120) drops to 26% of the yield strength. That is, strength of a fire-resistant steel column is greater than 60% of the nominal yield strength at 600 ∘C, if its slenderness ratio is less than 80. Moreover, for the majority of steel columns in steel buildings, with slenderness ratios less than 50, columns retain at least 70% of the nominal yield strength. Furthermore, the variations of column strength are small for temperature levels between 400 and 500 ∘C. While the difference is obvious as temperatures reach 600 ∘C, this is the natural behavior of the fire-resistant steel, when the deterioration of stress and stiffness is small at 400 and 500 ∘C.It was also found that in addition to the slenderness ratio, column strength also depended on the failure mode. Higher strength can be obtained if local buckling can be defended until reaching yield load, such as columns with slenderness ratios of 18 and 29. On the contrary, column strength is reduced if it is governed by global buckling. lists the corresponding failure modes of the column specimens in fire. Local buckling, inelastic global buckling and elastic global buckling are the three failure modes of steel columns, as summarized in . The failure modes of steel columns depend on the slenderness ratios as well as the temperature levels.It is also noticed that the dominant failure modes of identical steel columns with slenderness ratios of 29 changed from inelastic global buckling at room temperature to local buckling at elevated temperature. Steel columns reach their yield strength at elevated temperature before the onset of local buckling. Unlike the behaviour of steel columns at room temperature, premature failure due to the existence of residual stresses no longer exists in the fire condition. Columns with intermediate length are able to reach the yield strength at elevated temperature. Because residual stress is the main factor to reduce column strength to reach their yield strength, it is believed that the change of failure mode from global buckling at room temperature to local buckling is due to the limited effect from the residual stress in fire.Residual stress is induced into steel members during the fabrication process such as cold forming and welding. With the existence of residual stress, the ultimate strength of inelastic steel columns is greatly reduced at room temperature. Extensive research related to the residual stress in steel members has been conducted and conclusive results have been established accordingly Based on the research conducted by Chen Numerical analyses were carried out to analyze the non-linear behavior of column behavior at elevated temperature and the results are compared with that derived from experimental works. Columns were meshed as a three dimensional model by four-node shell elements with six degrees of freedom at each node . The stress–strain curves derived from the tensile coupon test at elevated temperature were adopted in the numerical modes. Residual stress is assumed to be distributed as a Lehigh pattern with maximum compression of 0.3Fy at room temperature; while residual stress at elevated temperature is neglected. is the comparison between the numerical analysis and experimental results. The mean value of the predicted strength to the experimental strength is 1.02 with a standard deviation 0.09. The numerical model is able to predict the performance of fire-resistant steel columns with reasonable accuracy. Therefore, parametric studies were conducted to determine the load reduction factor of columns with varied slenderness ratios at elevated temperature.In the current steel structures design, fire protection materials are normally adopted to enhance the fire resistance of structures. The conventional construction cost of fire protection materials can add up to 30% of the cost of bare steel structures. If the fire resistance of steel members can be improved, the construction cost can be greatly reduced. compared the effect of the slenderness ratio on column strength for columns made of conventional steel and fire-resistant steel From the experimental results, it has been found that width–thickness criteria for compact and plastic sections are more rigorous at room temperature than that at elevated temperature. The criteria for compact sections and plastic sections of conventional steel can be applied for fire-resistant steel. The numerical model is able to predict the structural behavior in fire under the assumption of no residual stress at elevated temperature. Parametric studies were carried out for the nonlinear finite element analysis and resulting reduction factors of column strength at elevated temperature are proposed and listed in . The reduction factor is defined as the ultimate strength at a specified temperature to the nominal yield strength at room temperature. Fire-resistant steel columns can be designed according to the critical temperature and the corresponding strength needed in the fire event. For example, the ultimate strength of a fire-resistant steel column with slenderness ratio of 29 retains 75% of the nominal yield strength at 600 ∘C (). The fire-protection material can be released in the case the fire load is less than 75% of the nominal yield strength and the temperature is not higher than 600 ∘C. The slenderness ratio of 29 is the widely used column slenderness in building structures in seismic areas.The experimental studies of the fire-resistant steel columns loaded in fire were presented and discussed in this paper. These columns include stub columns, inelastic columns and elastic columns. Based on the experimental results of this study, it is found that compressive strength can be greatly improved by using fire-resistant steel. Strength of fire-resistant steel is not sensitive to the width–thickness ratio at elevated temperature. Stub columns with non-compact sections are capable of reaching the yield strength before undergoing local buckling at both room temperature and elevated temperature. However, ductility is not ensured for non-compact sections. On the other hand, compact and plastic sections perform ductile both at room temperature and at elevated temperature.At elevated temperature, the slenderness ratio governs the strength of steel columns. At 600 ∘C, fire-resistant steel columns retain more than 60% of the nominal yield strength for the majority of steel columns used in steel buildings. Due to the limited effect of residual stress in fire, the failure mode of steel columns changed from inelastic global buckling at room temperature to local buckling at elevated temperature. At elevated temperature, strength of fire-resistant steel columns can be calculated and designed according to the guidelines proposed in this study.Finite element modeling for chemical mechanical polishing process under different back pressuresIn this paper, the revolutions of wafer and pad were considered the same and the force forms including the pressure exerted on the top of wafer surface and the carrier back pressure were axisymmetric distributed, a 2D axisymmetric quasi-static model for chemical mechanical polishing process (CMP) was first established. Based on the principle of minimum total potential energy, a 2D axisymmetric quasi-static finite element model with a carrier back pressure compensation for CMP was then established. In this model, the four-layer structures including wafer carrier, carrier film, wafer and pad are involved. The effect of a given carrier back pressure on the stress components and von Mises stress on wafer surface was analyzed and the effect of different carrier back pressures on the von Mises stress and nonuniformity on wafer surface was investigated. The findings indicated that the axial stress was the dominant factor to the von Mises stress distribution on wafer surface. Because that the back pressure had the maximum affect on the axial stress and it made the axial stress increased along the −z direction. Thus, while applying a back pressure, the von Mises stress distribution increased. In addition, the changes of back pressure had the trend to be proportional to the von Mises stress variation and to be inversely proportional to the nonuniformity variation. The result showed obviously that during the CMP process, it could achieve the purpose to improve the planarization of wafer surface by compensating the different carrier back pressures.The mechanism of chemical mechanical polishing (CMP) consists of wafer carrier, carrier film, pad and platen as shown in . The carrier is attached to the wafer back by means of vacuum. The wafer surface, i.e. the IC part to be planarized, is placed on the platen with one or more layers of pads. Slurry is sprayed continuously through a tube and uniformly scattered on the pad. The wafer is placed between the carrier film and pad. The relative motion generated by the carrier and platen brings the wafer in contact with particles in the slurry, which generates the multiple actions including the mechanical friction, chemical reaction and removal of chemical solvent to accomplish the highly efficient material removal. For the future semiconductor industry, based on the enhancement of precision and capabilities of electronic devices and the increase in storage space and memory capacity, it is inevitable that the size of wafer must be enlarged. It needs the more strict requirements of nonuniformity on wafer surface. Therefore, the global planarization technology becomes increasingly important and CMP is the most important method in this field. In addition, the usage of the carrier back pressure compensation is regarded as one of research topics to improve the nonuniformity of wafer surface. is very complicated intrinsically to not understand very clearly yet and extremely difficult to analyze its polishing mechanism. Therefore, it is necessary to simplify its model. Runnels and Renteln The objective of this article was to develop a 2D axisymmetric quasi-static finite element model with carrier back pressure compensation for CMP. In this model, the effect of a given carrier back pressure on the stress components and von Mises stress distribution on wafer surface was analyzed and the effect of different carrier back pressures on von Mises stress and nonuniformity on wafer surface was then investigated.CMP is used mainly for the material removal of wafer surface. The material removal rate (MRR) during CMP process can be considered as a function of the applied normal pressure and the relative velocity. It is usually expressed by Preston’s equation where P is the normal pressure, V the relative velocity and Cp a Preston’s constant., the normal pressure, P can be controlled and the relative velocity, V→ for point A on oxide film in V→=V→w-V→p=R→w×ω→w-R→p×ω→p=R→w×(ω→w-ω→p)-R→wp×ω→pwhere V→w and V→p are the absolute velocities of point A on the oxide film and on pad, respectively; R→w and R→p the distances from point A to the oxide film center, O′ and to the pad center, O, respectively.If both the wafer revolution and pad revolution assume to be the same, i.e., ω→w=ω→p, the relative velocity of point A on the wafer to the pad, V→ in can be simplified as -R→wp×ω→p. It is obviously a constant value and it results in a constant shear stress to be uniformly distributed on the wafer surface–pad interface, therefore, the effect of shear stress can be neglected and a quasi-static model is established. In addition, since the force form includes two sources, one is the down pressure exerted on the top surface of the carrier and the other is the back pressure of carrier, both of them are axisymmetrically distributed. The axisymmetric geometry of pad can be achieved by assuming that it possesses a huge smooth surface. The CMP model in this paper can thus be simplified into a 2D axisymmetric quasi-static model as shown in Based on the principle of minimum potential energy and Hooke’s law where [K] is elastic stiffness matrix, {δ} the nodal displacement vector and {Q} the nodal force vector.[K]=∑allelements∫∫∫V[B]T[De][B]dV,[Q]=∑surface∫∫∫V[N]T{Fb}dV+∫∫S[N]T{Td}dSwhere [B] is strain–displacement matrix, [De] the elastic stress–strain relation matrix, [N] the shape function matrix, {Fb} the body force, {Td} the surface traction, V the volume, S the prescribed surface exerted by traction.When the carrier attaches to the wafer back by means of vacuum and exerts a back pressure as shown in , the contact equation concerning to the pressure distribution between the wafer surface and pad can be assumed as where pb is the pressure distribution, p the back pressure, r a given point on wafer and R the radius of wafer.While a 2D surface load is distributed to surface nodes, the nodal force can be expressed as {Qs}=∫Sf[N]T{prpz}dSf=∫sΓ[Li00LiLj00LjLk00Lk]{prpz}2πrdsΓwhere {Qs} is the nodal force vector on surface, Sf the surface exerted by force, Li, Lj and Lk the natural coordinates, sΓ the distance between the two nodes that define the edge under consideration and pr, pz the pressure exerted on surface., if the carrier load is exerted on the edge ij of a triangular element on carrier surface along the axial direction, i.e., sΓ=sij, Lk=0, pr=0 and pz is a given pressure. Besides, if the back pressure of carrier is exerted on the edge ij of a triangular element on carrier film surface along the axial direction, i.e., sΓ=sij, Lk=0, pr=0 and pz=pb. is divided into a total of 6800 triangular elements and 3661 nodes. The basic assumptions are: (1) surfaces of the carrier, carrier film, wafer and pad are smooth; (2) materials including the carrier, carrier film, wafer and pad are all isotropic; (3) all materials are tightly stacked; (4) since the contact surface between the wafer and pad is fully smooth, only a normal pressure is considered while the back pressure of carrier is applied. Besides, the boundary conditions are: (1) bottom surface of the pad sustains a fixed support, while nodes at the bottom are subject to complete limitation in all directions; (2) left side is a symmetric boundary condition and enjoys a roller support, while nodes at this side are r direction limitation and able to move freely in z direction; (3) a uniformly distributed down pressure exerted on the top surface of the carrier and a back pressure of the carrier are considered.von Mises proposed in 1913 that yielding occurs when a combination of stresses, i.e., von Mises stress exceeds the yield strength of material. The von Mises stress applied in the CMP model developed in this paper can be simplified as σ¯=12[(σrr-σzz)2+(σzz-σθθ)2+(σrr-σθθ)2+6τrz2]1/2where σ¯ is the von Mises stress; σrr, σθθ, σzz and τrz the radial, hoop, axial and shear stresses, respectively.Under the condition that the effect of slurry was ignored and the material properties and geometries for carrier, carrier film, wafer and pad listed in (a), the scale in y axis of this part is magnified and plotted in (a) and (b), it shows that near the wafer center, von Mises stress distribution was almost uniform, then increased gradually with a small amount. However, near the wafer edge, it would decrease in a large range. Finally, it would increase dramatically and peak significantly at the edge. This result was similar to that of Srinivasa-Murthy et al. is an experimental diagram for material removal rate variations by using two different carrier films are oxide-polishing results obtained with two carrier films, R200T3 and DF200, which have elastic modulus of 0.69 and 0.407 MPa, respectively. It shows that there is an obvious variation in material removal rate from the average values at the edge for two films. While comparing between , although the simulated conditions in this paper are different from that of , the trend of von Mises stress distribution profile in (a) and (b) is similar to that of removal rate profile, i.e., the characteristics of the curves in match qualitatively. It verifies that the proposed CMP model is acceptable. shows the distributions stress components including axial, hoop, radial and shear stress on wafer surface without back pressure compensation. It is found from that the von Mises stress is an effective stress, whose factor includes the above four stress components. Since only a down pressure of 0.0689655 MPa is applied on the top surface of the carrier along the axial direction, i.e., −z direction, it induces that the axial stress has a negative value. Besides, by comparing these four stress components on wafer center and their maximum on wafer surface listed in , it is found that the magnitude of axial stress, σzz is significantly higher than those of the other three components and its value is at least approximately ten times higher than those of the other three stresses. Therefore, it is obvious that the main contribution to the calculated von Mises stress is the axial stress, i.e., the magnitude and profile of the von Mises stress correlate well with the axial stress component. shows the distributions of stress components on wafer surface with a given back pressure of 0.0137931 MPa (2 psi). Since the direction of carrier back pressure is the same as that of the down pressure applied on the top surface of the carrier, that is, all of them are along the axial direction, it causes that the axial stress along −z direction increases as shown in . The axial stress on wafer center and its maximum without back pressure compensation and with a back pressure of 0.0137931 MPa, and their absolute increase amounts are all listed in . As a result, it induces that the von Mises stress increases as shown in and its increase amount is similar to that of the axial stress along −z direction, but the former is somewhat smaller than the latter due to the slight variations of the radial stress, hoop stress and shear stress as listed in In addition, six kinds of back pressures ranging from 0.0 to 0.03448276 MPa were used in order to investigate the effects of different back pressures on von Mises stress and nonuniformity of wafer surface and according to the definition of nonuniformity of wafer surface by Wang et al. where R is the nonuniformity of wafer surface, σmax the maximum von Mises stress and σc that on wafer center. shows the larger the back pressure of carrier, the higher the values including the von Mises stress on wafer center, σc, and the maximum von Mises stress, σmax, but the lower the nonuniformity, R of wafer surface.Furthermore, in order to observe and interpret more deeply why the values including σc and σmax increase, but R decreases with the increase of back pressure and since the units of σc and σmax are different from that of R, it is no way to compare among them, we thus define three dimensionless quantities, that is, σc ratio, σmax ratio and R ratio under the condition of no back pressure compensation as a basis. These three ratios mean that the values of σc, σmax and R are divided by those under the condition of no back pressure compensation. All of them were listed in . Three dimensionless curves including σc ratio, σmax ratio and R ratio under the condition of different back pressures were involved in , it indicated that the σc ratio curve is steeper than that of the σmax ratio curve while the back pressure increases gradually, therefore, the R ratio curve would decline with a negative slope from the definition of nonuniformity in . From the above observations and explanations, the nonuniformity decreases gradually with the increase of back pressure as shown in , it indicates that during the CMP process, it can achieve the purpose to improve the planarization of wafer surface by compensating the different back pressures of carrier and the higher the back pressure, the better the improvements for wafer planarization.From the simulation and analysis of the CMP model developed in this paper, the major findings are summarized as below: A 2D axisymmetric quasi-static finite element model with a carrier back pressure compensation for CMP composed of four-layer structures of wafer carrier, carrier film, wafer and pad is developed.The von Mises stress increases due to the increase of the axial stress along −z direction while the back pressure is applied and its increase amount is similar to that of the axial stress along −z direction, but the former is somewhat smaller than the latter due to the slight variations of the radial stress, hoop stress and shear stress.The larger the back pressure of carrier, the higher the values including the von Mises stress on wafer center and the maximum von Mises stress, but the lower the nonuniformity of wafer surface. Therefore, during the CMP process, it can achieve the purpose to improve the planarization of wafer surface by compensating the different back pressures of carrier and the higher the back pressure, the better the improvements for wafer planarization.Spark plasma sintering of Stellite®-6 superalloyThis paper aims at studying microstructure and mechanical properties of spark plasma sintered (SPSed) Stellite®-6 cobalt-based superalloy. SPS is a sintering technique, based on a relatively fast resistance heating using a pulsed current. Fast sintering process, associated with minimum grain growth, results in excellent mechanical properties. Samples were sintered at temperatures ranging from 950 to 1100 °C. Microstructure of samples were studied using scanning electron microscope (SEM), energy-dispersive X-ray spectroscope (EDS), X-Ray diffraction (XRD), and optical microscope. Hardness, impact test, as well as room and high temperature compression tests were used to evaluate the effects of sintering temperature and duration on the mechanical properties of SPSed samples. Results show that optimum mechanical properties can be obtained after sintering at 1050 °C for 10 min. The correlation between sintering parameters, microstructure, and mechanical properties are discussed.Superalloys are high performance strategic alloys that exhibit superior oxidation resistance, excellent high temperature erosion-corrosion resistance, and very good high temperature mechanical properties []. These alloys are widely used in different high temperature industrial applications. Amongst different grades of superalloys, Stellite is a cobalt-based grade, that shows optimum combination of wear resistance, oxidation resistance, and mechanical properties. More importantly, alloys in this grade have proven to maintain their properties in extreme temperature conditions. Stellite is essentially a Co-based alloy that mainly contains alloying elements, such as chromium (Cr), tungsten (W), and carbon (C). This alloy owes its excellent mechanical properties to solid solution strengthening, mostly achieved by dissolution of Cr in the matrix. Cr as the main alloying element also reacts with C to form complex and inter-dendritic carbides. Chromium carbide particles enhance the erosion resistance of the alloy as well as its high temperature strength. The other alloying elements, W, is also a strong carbide forming element. The distribution, morphology, and size of carbide particles greatly influences the mechanical properties of the alloy. That is why controlling processing conditions during manufacturing plays a prominent role in final characteristic of the alloy []. Depending on the composition and the microstructure, Stellite alloys can be used in different applications, including machine parts, gas turbines, hardfacing, valve seats, implants, and industrial saws []. Different petroleum, gas, and pharmaceutical industries benefit from Stellite alloys []. Also, different grades of Stellite alloys are widely used for repair purposes, i.e. in repair welding of turbine blades and nozzles.Different manufacturing methods are employed to make components from Stellite alloys among which casting, welding, and powder metallurgy routes are most widely used. The latter has the advantage of being performed in solid state, inferring that there is no need to deal with typical casting and solidification problems such as segregation, porosity, coarse grain structure, dendritic structure, and interconnected brittle eutectic carbide network between the dendrites []. In addition to that, powder metallurgy has more controllability over the microstructure-properties relationships. Numerous researchers have studied processing parameters-microstructure-properties relationship of Stellite alloy, fabricated by hot isostatic pressing (HIP) and powder injection molding (PIM) []. But, to our knowledge, there are very limited studies on the spark plasma sintering (SPS) of this alloy []. SPS has recently emerged as a powder metallurgy technique, with high-speed compaction characteristic. Overall it is a fast, near net-shape, low cost, and flexible method []. SPS essentially consists of high temperature pressing (20–100 MPa) of powders in a graphite die under simultaneous flow of current pulses. As mentioned before, it is considered as a fast P/M technique, which gives the possibility of consolidation with minimum grain growth, which in turn results in good mechanical properties []. In comparison to other more conventional sintering techniques, SPS has advantages of being conducted at comparatively lower sintering temperatures with relatively short sintering time. SPSed parts exhibit higher sintered densities, limited grain growth and minimal material loss during sintering, thereby making this technique promising with great potential []. In SPS technology, raw or mechanically alloyed ceramic, metallic, functional, oxide, and composite powders can be consolidated []. Rarely is there any comprehensive study, which addresses the correlation between the SPS parameters and the microstructure and mechanical properties of SPSed alloy. This study aims at optimization of SPS parameter to achieve highest possible mechanical properties. The experimental studies in this investigation mainly focus on the effects of SPS processing parameters (i.e., sintering temperature and sintering time) on the microstructure and mechanical properties of Stellite®-6 alloy. This specific alloy is the most widely used alloy in Stellite grade, with its application mostly being in P/M-made parts.Cobalt-based Stellite®-6 superalloy powder with the average particle size of 35 μm was used in this study. depicts SEM images of the powder, showing that particles have spherical morphologies with a very narrow particle size distribution, which is typical of atomized powders. Dendritic structure can be seen on particles in , which is a result of casting and solidification during manufacturing. gives the chemical composition of the alloy.All experiments were performed by the SPS machine KPF vacuum technology. The powder is poured in a graphite die with a cylindrical cavity with diameter 1.5 cm. The radial punches' surface and the inner surface of the die were shielded with a graphite foil (0.2 mm in thickness) to avoid sticking of the SPSed part to the die. An axial pressure of 50 MPa was applied throughout the heating stage under a controlled Ar atmosphere. Effects of sintering time on the microstructure of the SPSed parts were investigated by changing sintering time at 1050 °C. To study the effects of sintering temperature on the microstructure and mechanical properties of SPSed parts, the specimens were heated to 950, 1000, 1050, 1075 and 1100 °C at a rate of 150 °C/min and held at these temperatures for 10 min and cooled down at SPS machine. After removing the sintered specimens from the graphite die, the samples were cut, ground, polished, and etched for metallographic examination. The etchant for this alloy was 200 ml HCl (32%), 5 gr FeCl3 and 5 ml nitric acid (65%) solution. The microstructure of specimens was examined by scanning electron microscopy (SEM), connected to an energy dispersive X-ray spectroscopy (EDS). X-Ray Diffractometer (XRD) analyses were carried out on initial powders as well as sintered samples, using Cu K-alpha radiation for phase identification. Grain size was estimated from XRD patterns, using Scherrer formula [The densities of the sintered samples were measured according to the Archimedes' principle. Compression tests were performed on cylindrical samples of 5 mm diameter with approximate height of 7.5 mm at room temperature and at 650 °C. For high temperature compression test samples were pre-heated at 650 °C for 15 min. The hardness measurements were conducted using Vickers hardness through the application of 100 gf load for 10 s (Micro hardness tester/MICROMET-S101 made in Mitutoyo Japan). Sub-size Charpy-U notch samples for impact toughness testing were prepared according to STP1418 standard []. The samples of 4 mm × 3 mm × 27 mm having 1 mm depth with a 60° notch angle and 0.25 notch tip radius were machined with electrodischarge machining technique from sintered samples. shows the effects of sintering time on the size and morphology of porosities in the sample, SPSed at 1050 °C for 2, 5, 10, and 15 min. Porosity percentage and density are key factors, affecting the mechanical properties of metallic P/Med alloys. It is a basic fact that the higher the sintering temperature, the higher is the final density of the consolidated part. Higher density means lower internal microstructural discontinuities and defects, with both having negative implications for the mechanical properties and the integrity of P/Med powders. In order to evaluate the effects of sintering time on the densification behavior of Stellite-6, SPSed specimens were polished and studied with optical microscope. As can be seen in , after 2 minutes, particles are at the early stage of sintering, with particles being connected. Further holding up to 5 min is accompanied with the formation of diffusion necks between particles. After 10 minutes of holding, there is no sign of isolated rounded particles. Some isolated porosities are still left in the microstructure though. Further increase in holding time up to 15 min does not show any significant change in the area fraction of porosities, implying that 10 min is the optimum holding time. Shorter holding times result in the formation of a microstructure in which porosity is the dominant feature and longer holding time does not deliver a microstructure with a reduced porosity. Longer holding time increases the possibility of grain growth and carbide coarsening, with both having negative implications for the mechanical properties of the alloy. shows the comparison between carbide size in samples, sintered for 10 and 15 minutes. Carbides are clearly coarser in the latter. Controlling grain/carbide size is vital when it comes to controlling mechanical and physical properties of materials. Microstructures with finer carbides generally improve the creep, tensile, and fatigue properties of high temperature alloys in severe service conditions. Increasing sintering temperature and holding time, if not properly well planned, result in coarse grain/carbide microstructures [In order to study the influence of sintering temperature on the porosity percentage of SPSed specimens, samples were SPSed at 950, 1000, 1050, 1075, and 1100 °C. Results showed that SPS at 1075 and 1100 °C is associated with the partial melting of consolidated samples. So, it appears that sintering should be conducted below 1075 °C. shows the effects of sintering temperature on the porosity area fraction of sintered samples. Results show that 10 minutes of sintering at 950 and 1000 °C is not enough to make solid bulk, with minimum interconnected porosities in the microstructure. One can see that an increase in the sintering temperature from 1000 to 1050 °C leads to a significant reduction of porosities in the SPSed sample. Sintering at 950 °C and 1000 °C result in the average porosity percentage 22.5 and 16%, while at 1050 °C, this is less than 3%. The presence of these porosities in the microstructure can obviously negatively affect mechanical properties of the samples. Based on the obtained results, sintering at 1050 °C for 10 minutes gives the best outcome.XRD patterns of initial powder and samples sintered at temperatures 950 and 1050 °C are shown in . The peaks in all three patterns are at similar degrees, inferring that no new phase (including intermetallic compounds) is formed during sintering (at least not within the detection limit of XRD). Stellite microstructure comprises a Co-based matrix with M23C6, M7C3, M6C and WC carbides []. No WC, M6C, and M7C3 were detected by XRD, implying that the weight percentages of these phases are not enough to be detected by XRD. The majority of carbide phases in the microstructure belongs to M23C6. It is also noticeable that sintering has resulted in an overall reduction of peak intensities and peak broadening. This is comparatively more pronounced for the sample, sintered at 1050 °C. Both peak broadening and intensity reduction are indications of grain fragmentation/grain refinement and accumulation of lattice strain during sintering. Samples SPSed at 950 and 1050 °C have crystallite size of 39 and 31 nm respectively, whereas this is 65 nm in as-received powders.Typical microstructures of Stellite®-6 alloy SPSed at 1050 °C, are presented in . As can be seen, Stellite alloys have a Co-based matrix, essentially composed of intermingled complex dispersion of carbides. Carbides, present in the Stellite alloy, are reportedly mostly M23C6 and M7C3, with small amounts of M6C and WC []. The latter is known to be a high-temperature carbide, while other carbides are more important for low and intermediate temperatures. Overall, intermingled complex dispersion of carbides (see ) can enhance mechanical properties and improve wear/erosion resistance. EDS analyses of marked spots (given in . EDS spectrum of spot 1 shows high concentration of Co, showing that spot 1 is the matrix. A high amount of Cr is detected in point 2, showing that this is a chromium-rich carbide. These chromium-rich carbides are mostly M23C6 carbides []. The third analysis (spot 3) shows that the white phase is W-rich carbide. shows elemental mapping of alloying elements in the microstructure, which is in accordance with the presented XRD and EDS results.The corresponding room and high temperature (650 °C) engineering compression stress–strain curves for SPSed samples at 950, 1000 and 1050 °C with strain rate of 4.5 × 10−4/s are also shown in . There is a clear distinction between the mechanical properties of the sample, SPSed at 1050 °C and those of samples, sintered at lower temperatures (i.e. 950 and 1000 °C). The former shows remarkably better mechanical properties (yield strength as well as displacement). Samples, SPSed at 950 and 1000 °C, exhibit limited deformation before fracture, which obviously has to do with the fact that the sintering in these two temperatures is incomplete. Mechanical properties in structure with such high degrees of pore-connectivity are controlled by porosities rather than the material itself. On the contrary, mechanical properties of the sample, SPSed at 1050 °C, are controlled by complex primary and secondary chromium-rich M7C3 and M23C6 carbides as well as tungsten-rich carbides []. The same goes for the Vickers hardness values, where the hardness values of SPSed samples at 950, 1000, and 1050 °C are 116, 158, and 510 HV0.1 respectively. This can also be attributed to the reduced porosity with increased sintering temperature.The yield strength, and hardness values of SPSed Stellite®-6 alloy samples are compared with those of other manufacturing methods (see ). Interestingly, samples produced by SPS show the highest yield strength and hardness, without any need for additional heat treatment. Such perfect combination of properties can be attributed to the relatively fast synthesis in SPS, which eliminates the risk of carbides coarsening/grain growth. In addition to that, low levels of porosities and intermingled nature of carbides certainly have positive contributions to the obtained mechanical properties.Impact toughness test was also done on sintered specimens at room temperature. Results are shown in . Sample, sintered at 1050 °C for 10 min, showed the highest impact toughness value. But this is not a particularly high value. The microstructure of SPSed Stellite-6 essentially consists of an intermingled complex network of carbides in a Co-based matrix (note ). Carbides are inherently brittle and are known to be suitable positions for crack initiation and propagation []. Therefore, this relatively low value of impact toughness for optimum sample is not surprising. Also samples, sintered by SPS technique, have certain amount of porosity. Porosities can also act as preferred crack nucleation sites, negatively affecting impact toughness of sintered alloys. shows the fracture surface of samples, sintered at 950, 1000 and 1050 °C after compression test. Provided images are taken at low and high magnifications. The powdery nature of the fracture surface of samples, sintered at 950 and 1000 °C, is an indication that sintering at these two temperatures is incomplete. In the sample, sintered at 1000 °C, there is no sign of distinct powders. Interestingly, the fracture at the necks is typically a ductile fracture, confirmed by the presence of dimples on the fracture surface. By increasing sintering temperature to 1050 °C, there is no sign of isolated or interconnected particles at the fracture surface, which is in accordance with the comparatively lower porosity of this sample (see d). Contrary to samples, sintered at 950 and 1000 °C, in this case mechanical properties are controlled by the nature of material and not by the weak and poorly connected necks. shows the SEM fracture surface of specimens sintered at 1050 °C after fracture test. Microvoids and some porosities are evident in fracture surfaces (see red areas) that can certainly affect the impact toughness. As shown, the fracture mode is dominated by brittle fracture leading to low impact toughness. Even though some local ductile fracture areas exist on the fracture surface, the fracture is obviously dominated by cleavage fracture [This study aims at investigating the effects of spark plasma sintering (SPS) on the microstructure and mechanical properties of cobalt-based Stellite®-6 superalloy. Results showed that it was not possible to achieve a complete pore-free specimen with SPS. The minimum porosity (roughly 3%) and the highest densification, can be obtained after 10 min of sintering at 1050 °C. Higher holding time results in the coarsening of chromium carbides in the microstructure. Also, decreasing sintering temperature is associated with an incomplete sintering, such a way that isolated or poorly connected powders can be easily seen in the microstructure. EDS results showed that the main secondary phase is chromium-rich carbide. The XRD results also confirmed the presence of chromium carbides. It is also seen that sintering is associated with XRD peaks broadening, which is an indication of grain fragmentation and formation of smaller crystallites. Mechanical properties of the optimum specimen are much higher than those obtained by other P/M methods.The raw data required to reproduce these findings are available to download from . The processed data required to reproduce these findings are available to download from Microelectromechanical resonator manufactured using CMOS-MEMS techniqueThe fabrication of a microelectromechanical resonator using the commercial 0.35 μm complementary metal oxide semiconductor (CMOS) process and a post-process has been implemented. The resonator requires only one wet etching post-process. The suspended structures in the resonator consist of a membrane and four beams. The post-process utilizes an etchant to etch the sacrificial layer, and to release the suspended structures. Easy execution and low cost are the advantages of the post-process. The resonator comprises a driving part and a sensing part. The sensing part produces a change in capacitance when applying a driving voltage to the driving part in the resonator. A circuitry is used to convert the capacitance variation of the sensing part into the voltage output. Experimental results show that the resonant frequency of the resonator is about 39.5 MHz and the quality factor is 806.Micromechanical resonators are widely applied in filters and sensing elements due to their high-frequency sensitivity to physical parameters. For instance, Kobayashi et al. Recently, microelectromechanical system (MEMS) technology has become popular for the miniaturization of sensors The commercial CMOS process is usually employed to produce integrated circuits (ICs). Several studies have utilized the commercial CMOS process to fabricate various microdevices, such as micromirror arrays illustrates the micromechanical resonator that contains a driving part and a sensing part. The structure of the micromechanical resonator consists of a suspended membrane, four beams and two fixed electrodes. The suspended membrane is supported by beams. Two fixed electrodes under the suspended membrane are taken as the driving and sensing electrodes as shown in . The driving electrode is located under the sides of the membrane, and the sensing electrode is situated under the center of the membrane. The driving part is comprised of the membrane and the driving electrode and the sensing part are constructed by the membrane and the sensing electrode. When applying a driving voltage to the driving part in the resonator, the driving part produces an electrostatic force. The membrane is actuated by the electrostatic driving force. The gap between the membrane and the sensing electrode changes when the membrane displaces a distance, so that capacitance of the sensing part generates a variation. A differentiator circuit is employed to convert the capacitance variation of the sensing part into voltage output. Area of the membrane is 60×60 μm2. All beams are 20 μm long and 10 μm wide. Thickness of the membrane and the beams is approximately 1 μm. Driving electrode is 20 μm wide and 60 μm long, and the sensing electrode is 30 μm wide and 60 μm long. Gap between the membrane and the fixed electrodes is around 3.5 μm.The performance of the resonator depends upon dimensions and material properties of the resonator. Material properties consist of Young's modulus, Poisson's ratio and mass density. The finite element method (FEM) softwares, ANSYS and CoventorWare, are used to simulate the behavior of the resonator. The simulating procedure of the resonator consists of constructing the model, selecting element type, meshing the model, defining material properties, setting boundary conditions, applying loads and executing calculation. The model of the resonator is constructed in accordance with dimensions in . Triangular element is adopted to mesh the model. Material of the resonator is aluminum. Mass density of 2679 kg/m3, Young's modulus of 70 GPa and Poisson's ratio of 0.3 are adopted depicts the simulated results of the vibration for the micromechanical resonator. Results reveal that resonant frequency of the resonator is 39.7 MHz. shows the stress distribution of the resonator when a driving voltage of 60 V is applied. The maximum stress of 72 MPa located at the end of the beams is below the yield strength of aluminum, about 124 MPa. Thus, motion of the resonator can be operated in an elastic range under the driving voltage of 60 V.The micromechanical resonator is manufactured using the 0.35 μm CMOS process of Taiwan Semiconductor Manufacturing Company (TSMC). The resonator requires a post-process to release suspended structures after the CMOS process. The post-process of the resonator needs only one wet etching shows the process flow of the resonator. The schematic cross-sectional view of the resonator after the CMOS process is illustrated in . At this situation, the resonator contains a sacrificial layer and a structure layer. The sacrificial layer must be etched away to release the structure layer. The suspended structures of the resonator including the beams and membrane are the metal layer of the CMOS process. The oxide layer under the suspended structures, which is the sacrificial layer, has to be removed, so that suspended structures are released. The post-process is employed to remove the sacrificial layer. shows that wet etching with silox vapox III is employed to etch the sacrificial layer and release suspended structures. The silox vapox III (from Transene Company, Inc.) consists of ammonium fluoride, glacial acetic acid, aluminum corrosion inhibitor, surfactant and DI water. Etching rate is about 960 Å/min. shows the scanning electron microscope (SEM) image of the micromechanical resonator after the post-process. illustrates the experimental setup for measuring frequency response of the resonator. Power supply provided a dc bias voltage of 30 V and function generator (Aglient 4440A) supplied a swept sine voltage of 5 V to the driving part in the resonator to actuate the motion of the membrane. A spectrum analyzer (Aglient E4483c) was utilized to measure the output voltage of the sensing part and tested the frequency response of the resonator. shows the frequency response of the resonator. The experimental result revealed that resonant frequency of the resonator was approximately 39.5 MHz, which was in very good agreement with the simulated value (39.7 MHz). As shown in , the resonator had a 3 dB bandwidth (BW) of 49 kHz, so Q-factor of the resonator was 806. The resonator was manufactured using the CMOS-MEMS technique, and its fabrication was compatible with the CMOS process. Therefore, the resonator had a potential to integrate IC on a chip.The resonator presented by Beeby and White A micromechanical resonator has been fabricated using the commercial 0.35 μm CMOS process and post-process. The micromechanical resonator needed only one wet-etching post-process, which was easy to execute at low cost. The post-process employed the silox vapox III etchant to etch the sacrificial layer, and to release the suspended structures in the resonator. Area of the resonator was about 100×100 μm2. The resonator consisted of a driving part and a sensing part. When applying a driving voltage to the driving part in the resonator, the sensing part generated a change in capacitance. The capacitance variation of the sensing part was converted by the differentiator circuit into a voltage output. Driving voltage of the resonator was a DC voltage of 30 V with an AC voltage of 5 V. Experiments showed that resonant frequency of the resonator was 39.5 MHz and Q-factor was about 806. The micromechanical resonator had a potential to be integrated on a chip.Mechanical properties in long bones of rat osteopetrotic mutationsOsteopetrosis is a metabolic bone disease with increased skeletal density radiographically and increased risk of fracture. Experimental studies with rat osteopetrotic mutations have shown increased bone density and decreased bone strength. However, it is not known if this reduction in bone strength is only due to changes in structure and geometry or if the tissue properties of bone material itself are changed as well. We have evaluated bone tissue properties with nanoindentation in three osteopetrotic mutations in the rat (incisors-absent ia/ia, osteopetrosis op/op and toothless tl/tl) to test the hypothesis that reduced bone resorption in these mutations results in reduced tissue properties of bone material. No significant differences in elastic modulus or hardness were found between osteopetrotic mutants and their normal littermates (NLMs) in any of the three stocks. This indicates that the tissue properties of bone material are not changed significantly in osteopetrosis, even if the mechanical strength is decreased at the macroscopic level.Bone density is generally considered to be directly associated with mechanical competence of the skeleton and therapies for osteoporotic conditions concentrate on preserving and/or increasing bone density and bone mass. However, there are conditions, such as fluorosis (), in which increases in bone mass do not increase bone strength. Another condition, osteopetrosis, has been associated with fractures of long bones since it was first described almost a century ago (). The osteopetroses are a group of skeletal metabolic disorders of heterogenous etiology and varied severity that produce a generalized accumulation of skeletal mass due to reduced bone resorption (). Pathological fractures are common in children () with osteopetrosis in spite of increased bone mass and density (). Reports of short, thick, irregular collagen fibers and reduced mineral crystallinity in bones from osteopetrotic children () suggest that these are related to the increased susceptibility to fracture in osteopetrosis. There is a report of reduced bone and mineral density in bovine osteopetrosis (). Decreased elastic stiffness coefficient as measured with an ultrasonic method has also been reported in bovine (We have previously studied mechanical breaking force and mineral density in osteopetrotic mutations in the rat (). The peripheral quantitative computed tomography (pQCT) analysis showed that bone mineral density (BMD) and bone mineral content (BMC) were higher than or equal to normal littermates (NLMs) in all skeletal sites measured in the osteopetrotic mutants. However, the mechanical breaking force was equal to or lower than their NLMs in all sites. The cross-sectional structure of long bone shafts was markedly different in osteopetrotic mutants, having a thin cortex and a medullary area filled with primary trabecular bone (). These results indicated that osteopetrotic mutations in the rat increase bone density and decrease bone strength. However, it is not known if this reduction in bone strength is only due to changes in structure and geometry or if the tissue properties of bone are changed as well.Our objective was to measure bone tissue properties by nanoindentation in three different osteopetrotic mutations in the rat; incisors-absent (ia) () in which numbers of osteoclasts vary but bone resorption is reduced and bone mass is increased radiographically. In this study we tested the hypothesis that reduced bone resorption in these mutations results in reduced tissue properties of bone material.This study is based on bone samples used in another study (). The animals were maintained and used at the University of Massachusetts Medical Centre (UMMC) according to the recommendations in the Guide for the Care and Use of Laboratory Animals prepared by the Institute of Laboratory Animal Resources, National Research Council (DHHS publication NIH 86-23, 1985).The study was performed on 6 tl/tl, 3 op/op and 3 ia/ia rats and their normal littermates (age 2 weeks), which were obtained from breeding colonies at UMMC. Mutants were identified radiographically at birth by failure of development of marrow cavities in long bones. Normal homozygotes (+/+) and heterozygotes (+/−) are distinguishable only by breeding, and NLMs included both these genotypes. The same number of age-matched NLMs from each of the groups was used as controls. The animals were killed by CO2 suffocation, and the tibiae and femora were dissected out. The bones were stored at −20°C with soft tissue and thawed at room temperature just before testing the densitometric and mechanical properties of the bones as previously presented (). All tibias and totally eight femurs were available for the present study.The specimens were embedded without vacuum in epoxy resin at room temperature (EPO-THIN™ low viscosity epoxy, Buehler, Lake Bluff, IL, USA). The embedded samples were metallographically polished to produce the smooth surfaces needed for nanoindentation testing. After being ground using silicon carbide abrasive papers of decreasing grit size (600, 800, and 1200 grit) under deionized water, the specimens were polished on microcloths (TEXMET®, Buehler, Lake Bluff, IL) with successively finer grades of alumina powder, the finest being 0.05 μm grit. The last polishing step was on plain microcloth under deionized water, and the specimens were ultrasonically cleaned to remove surface debris.All experiments were performed using the TrioboIndenter (Hysitron Inc., Minneapolis, MN, USA). This fully automated hardness testing system makes small indentations at precise positions on a specimen surface while continuously monitoring the loads and displacements of the indenter. The specimens are held on an x–y table whose position relative to the microscope or the indenter is controlled with a resolution of 500 nm. The apparatus is enclosed in an insulated cabinet to provide thermal stability and suspended on an antivibration table to isolate it from external vibrations.) for determining the indentation modulus, hardness, and the indenter area function has been well documented, and will not be discussed at length. Measurements of load and displacement can be used to determine the contact stiffness. The reduced modulus can then be determined from the contact stiffness. The equations used to compute the hardness (H), and the reduced modulus (Er) arewhere Pmax is the peak displacement, Ac the contact area, and S the contact stiffness. The elastic modulus of specimen is derived fromwhere ν is Poisson's ratio. The elastic properties of the diamond indenter tip, νtip and Etip, are 0.07 and 1140 GPa, respectively.Fused silica, which exhibits elastic isotropy and has a relatively low modulus-to-hardness ratio, was used to calibrate the tip shape function. The elastic modulus of fused silica was calculated to be 71.2 GPa, which is similar to the known value, 72 GPa.Each nanoindentation test was conducted to a maximum load of 6 mN at a constant loading rate of 400 μN/s, including a preliminary thermal drift rate correction limited to a maximum of 0.05 nm/s. Indentation procedure included a linear loading part of 15 s, a holding period at maximum load of 10 s and a linear unloading part of 15 s. The 50–95% of the unloading curves was used for calculation of elastic properties. Sixteen indentations were made in each specimen in a longitudinal direction.The average of the 16 measurements was used as the value for each specimen. The values are given as mean±standard deviation (SD). The one-way analysis of variance (ANOVA) was used to compare the multiple study groups. Independent Student's t-test was used in pair-wise comparisons of parameters between mutants and normal littermates, stock by stock, and also by pooling the different stocks together. Values of p<0.05 were considered statistically significant.. The data for the mechanical breaking strength of whole bone, taken from the previous report (), are given for reference. ANOVA revealed no significant differences in Young's modulus or hardness between study groups. No differences were found in pair-wise comparisons between individual osteopetrotic mutants and their NLMs in any of the three stocks, and not even when the different stocks were pooled together. However, there was a slight trend to decreased strength properties in osteopetrotic mutants in all three stocks tested.The purpose of this study was to measure tissue properties of bone material in three osteopetrotic mutations in the rat to determine if the reduced mechanical strength of osteopetrotic bone is the result of intrinsic bone fragility. Our findings show that the tissue properties of bone material are not changed in osteopetrosis, even if the mechanical strength is decreased at the macroscopic level (). These results differ from the previous reports of decreased elastic stiffness coefficient in osteopetrosis (). These earlier studies were, however, performed indirectly by using ultrasound while we used direct mechanical measurement based on nanoindentation. Furthermore, the previous studies were performed in bovine () cortical bone from single individuals, while the present study was an experimental analysis from three to six rats in each of three different osteopetrotic mutations.The microstucture of human mature osteopetrotic bone differs from normal bone, lacking the typical secondary osteons (). We have speculated that the osteopetrotic rat bone is weaker because reduced bone remodeling permits the persistence of primary bone which has inherently less strength than secondary bone (). However, this hypothesis is not supported by the present study. We did not find any significant differences in the elastic modulus or hardness of bone material from the two groups. However, there was a trend to decreased strength properties in osteopetrotic mutants in all three stocks tested. This trend may be due to differences in bone composition. There is evidence that quality of mineral crystals and collagen are also affected in osteopetrosis (). The statistical power of the present study is highly limited due to the small number of samples in each study group. This is because of the restricted availability of the animals. The slight trend to reduced material properties in osteopetrotic mutants might have been clearer with a bigger sample size. However, the main result did not change when we pooled the different stocks together to compare the generic effect of an osteopetrotic mutation.The tissue properties of cortical bone material were tested in the present study. When considering the whole bone properties, the difference in macroscopic strength between various mutations might be affected partly by differences in the trabecular structure (). However, this does not account for the existing differences between mutations and their NLMs because in NLM diaphyses are mainly cortical with no trabecular structures (We studied bones only from 2-week old animals. Future studies should determine if there are compensatory mechanisms that develop in bones of low strength and low turnover. Furthermore, the effect on bone strength of exogenous colony stimulating factor 1 (CSF-1), which tends to normalize bone mass in tl rats (), needs to be evaluated. There is evidence that CSF-1 can normalize mineral crystallinity and collagen cross-links in tl/tl rats () but it is not known whether this also strengthens the biomechanical properties of bone.The purpose of this study was to test the hypothesis that reduced bone resorption in rat osteopetrotic mutations results in reduced material strength. The hypothesis was not supported by our findings which showed no differences in strength properties of bone in osteopetrosis, even when the mechanical strength is decreased at the macroscopic level. These findings may indicate that bone geometry and macroscopic structure are most important in predicting bone biomechanical competence in osteopetrosis.Starch media blast cleaning of artificially aged paint filmsStarch media blast cleaning is an environmentally benign coating removal process increasingly used as an alternative to traditional coating removal techniques. The main objective of this work was to provide a method for predicting the paint stripping rate of an aged paint system, through a knowledge of some fundamental physical properties of the paint/substrate system (e.g. hardness and modulus of elasticity). It was shown that aging of the aluminum panels painted with polyurethane significantly increased the hardness of the coatings, but did not have a significant effect on the coating modulus of elasticity. It was also shown that the coating became more erosion resistant as it aged, and that it was possible to predict the paint removal rate by in situ measurement of the coating dynamic hardness and coefficient of restitution. It was concluded that coating erosion resistance increased with increasing coefficient of restitution. The relative erosion resistance of the aged coatings was explained on the basis of differences in modulus of elasticity and dynamic hardness. As expected, for a constant modulus of elasticity, a higher dynamic hardness gives higher erosion resistance.Paint stripping and repainting of aircraft surfaces is required periodically during the lifetime of an aircraft. Traditionally, paint removal has been achieved with chemical stripping, however, the use of methylene chloride and phenol-based chemical strippers for aircraft paint removal generates large quantities of hazardous waste and creates health and safety problems for the operating personnel. One possible alternative to chemical stripping is starch media blast cleaning, which is increasingly being used to remove organic coatings from substrates. In this paint stripping method, a stream of starch media particles removes the coating by erosion Starch media blast cleaning is environmentally benign because the media is non-toxic and biodegradable. Due to its chemical inertness and erosion characteristics, starch media has been used for dry stripping of aircraft paint from aluminum alloys and polymer matrix composites. The process is also capable of selective stripping, removing only the top layer of a paint system while leaving the primer intact, thus reducing the amount of waste generated and the cost of repainting Paint stripping efficiency is a function of many factors, such as the coating thickness and the condition of the paint as a result of aging in a particular environment. For example, paint near a hot engine exhaust may be more difficult to remove than elsewhere. Efficient paint stripping requires the prediction of paint stripping rates at various places over the aircraft.The objective of this work was to provide a method for predicting the paint stripping rate of an aged paint system through a knowledge of some fundamental physical properties of the paint/substrate system. Measurements of Vickers hardness, coefficient of restitution, dynamic hardness, Young’s modulus and paint stripping rate were taken as a function of aging time.Starch media paint stripping technology appeared in the early 1990s. As is the case with erosion of organic coatings in general, publications addressing questions of a fundamental nature are difficult to find. Shipway and Hutchings All these investigations, however, were performed on relatively fresh paints, while in practice, it is usually an aged paint that is being stripped. The effect of aging on paint properties and starch media erosion has not been previously studied.During its service life, an aircraft is typically exposed to severe environmental effects including ultra-violet radiation at high altitudes, rapid and extreme temperature changes, and also driving rain and ice that may cause erosion and cracking of the paint.Many of the materials that come in contact with the body of an aircraft can also be damaging to paint films; in particular, hydraulic fluids, engine lubricants, brake fluids, and alcohol in de-icing fluids can soften and dissolve surface coatings, while burning fuels discharged through engine exhausts can have the reverse effect and harden coatings Below the glass transition temperature, organic coatings are in a glassy, non-equilibrium state. Physical aging is the term given to the changes in mechanical, thermal and dielectric properties that occur as the polymer approaches thermodynamic equilibrium at temperatures below the glass transition The behavior of paint films exposed to heat varies with the nature of the paint and the character of the exposure, particularly the temperature. Most coatings will yellow, and if exposure to heat is severe and prolonged, hardening will occur Polyurethane coatings are available either as two-component systems that are mixed shortly before use and cured by cross-linking, or as single-package materials that cure when exposed as a film to moisture, oxygen, heat, etc. There are different methods for artificially aging paint films. One commonly used standard employs a 7-day room-temperature cure, followed by a 100°C aging for 4 days Different paint aging processes can have a significant effect on the performance of dry starch media stripping. For example, comparing the above two accelerated aging methods, an aerospace paint aged at 100°C for 4 days is tougher to remove than the same paint subjected to 66°C for 7 days, and consequently the second aging regime makes it easier to selectively strip the paint leaving the epoxy primer intact The paint/substrate system used in this research was a typical aerospace system. The primer was a yellow two-component epoxy-polyamide (United States military MIL-P-23377) with a nominal thickness of 25 μm (1 mil). The topcoat was a matte gray two-component urethane (aliphatic iso-cyanate, United States military MIL-C-83286) with a thickness of 50–75 μm (2–3 mils). The density of the polyurethane topcoat was 1.27 g/ml±0.02. The substrate was AA2024-T3 clad aluminum pretreated with a chemical conversion coating. The panels were painted on both sides by the Canadian Department of National Defense and measured In order to accelerate the aging process and enhance reproducibility, painted aluminum panels were artificially aged in an oven at 100±3°C, for periods of 1, 2, 4 and 8 days. After each aging interval, the panels were removed and experiments were performed to measure Vickers hardness, dynamic hardness, coefficient of restitution, Young’s modulus and paint stripping rate with starch media blasting. Measurements were also made on a freshly painted panel before aging. These experiments and the results are discussed in the following sections. In addition, these properties were also measured on painted panels recovered from a relatively old Canadian military aircraft. Although the exact exposure history of these aircraft samples was not known, they were used to confirm that the artificial aging produced changes in paint properties that were approximately representative of actual exposure in the field.Hardness is a measure of a material’s resistance to indentation and it is, therefore, reasonable to assume that the response to the paint stripping process depends on the hardness of the paint film. This was investigated by comparing the hardness and stripping resistance of freshly painted panels with those that came from the field, and also with artificially aged panels. Vickers micro-indentation tests were performed on |
cm specimens cut from fresh and aged panels (three samples were cut from each panel). On each sample, 10 hardness readings were taken over scratch-free areas far from free edges with a minimum of 0.5 mm between indentations. A load of 20 g was applied for a period of 25 s, and the diagonals of the Vickers indentation were measured with a precision of 0.4 μm.Paint is a visco-elastic material and, therefore, tends to exhibit a certain amount of time-dependent recovery upon removal of the indenter load. The vast majority (90%) of the recovery occurs in the first few seconds after the load is removed, prior to optical measurement of the indentation shows the Vickers hardness of painted panels after different aging times in the oven. The average Vickers hardness of the fresh panel was 125 MPa, and it gradually increased to an average value of 158 MPa for a panel aged 8 days at 100°C. This increase in hardness with physical aging is expected. The Vickers hardness of the old field panels was in the range 140–150 MPa, which is comparable with 2 and 4 days aged panels.The dynamic hardness, defined as the instantaneous force resisting indentation during a collision divided by the instantaneous contact area, is an important property in determining the impact erosion behavior at high strain rates In the present case, the coefficient of restitution and dynamic hardness were measured by launching steel spheres (diameter 1.5±0.05 mm, mass=13.8±0.8 mg) at relatively low velocities (30–35 m/s), to limit the penetration depth to within the coating (∼40 μm). Incident and rebound velocities were measured using a video camera and strobe lights. The experimental apparatus is described in The coefficient of restitution e was calculated aswhere vr and vi are the rebound and incident velocities of the particles, respectively. In these experiments, the steel spheres were launched normal to the paint surface and did not suffer any plastic deformation, and thus e was largely determined by the amount of plastic deformation of the coating. Higher values of e indicated less coating damage, so that more of the incident kinetic energy was returned in the rebound. The coefficients of restitution of the panels are shown in . It is seen that e was approximately the same for all aged panels and the old field panel, and that the fresh panel had a significantly lower value. Thus e was sensitive to the initial effects of physical aging, but could discriminate among differing degrees of aging. The lower e found with fresh paint means that it would experience greater plastic (permanent) deformation upon impact and hence greater removal, a fact that was borne out in the stripping experiments.In order to examine the impact sites in more detail, three-dimensional profiles of the above described impact craters were obtained with an optical surface profilometer (WYKO). As can be seen in , the impact caused extrusion of the plastically deformed coating into raised edges around the crater. The dynamic hardness can be estimated by setting the incident kinetic energy to the work done in plastically deforming the coating where vi is the incident velocity, m the mass of the incident particle, and P(δ) is the load as a function of the indentation depth δ. If the effect of the pileup of the material adjacent to the crater edges is neglected, the indentation depth δ, and hence the force-depth relationship P(δ), can be expressed in terms of the contact radius a. Making the appropriate substitutions and rearranging results in the following expression for the dynamic hardness pdwhere R is the particle radius, and amax the maximum contact radius reached at the deepest penetration δmax (δmax is less than coating thickness).Upon impact, the coating below a particle will experience compressive plastic deformation in the direction of the impact and plastic radial expansion. On rebound, some of the compression in the direction of impact will be recovered due to elastic effects, but much less radial expansion will be recovered. The material surrounding the crater is thus left in a state of residual bi-axial compressive stress, which inhibits radial recovery within the crater. Furthermore, the elastic deformations up to the point of yield will be much greater in the direction of impact than in the radial direction . In this manner, measurements of the paint crater diameter can be used to estimate the dynamic hardness.For the present coating system, the dynamic hardness of the fresh and aged panels was calculated by measuring amax of the impact sites using the optical profilometer, and measuring incident velocity with the gas gun setup and image analysis software summarizes these results and also shows the coefficient of restitution and paint stripping rates (see below) corresponding to the same aging conditions. The differences between fresh panels and aged panels are statistically significant at a level of confidence greater than 95% (t-test). As will be discussed further in the following sections, it is seen that the paint stripping rate decreases with increasing dynamic hardness and coefficient of restitution.An ultrasonic time-of-flight method was used to measure the longitudinal sound wave velocity Vl, and Young’s modulus E, was then calculated fromwhere the paint density ρ=1.27±0.02 g/ml and Poisson’s ratio μ=0.378 |
cm) were cut from the fresh and aged panels, and the paint thickness was measured at five points. The same five locations were used for ultrasonic measurement of longitudinal sound wave velocity, and the average of these five measurements was used to calculate the Young’s modulus using shows the measured velocity and the calculated Young’s modulus.It was originally thought that the paint film would become stiffer as the paint aged. shows, however, that the modulus of elasticity of the paint film remained approximately constant for all aging times. Since physical aging produced significant changes in Vickers hardness, dynamic hardness and coefficient of restitution, it is concluded that physical aging affected plastic, but not elastic behavior.The painted panels were blasted with Envirostrip® 30/100 starch media (ADM/Ogilvie Inc.) under three controlled blasting conditions, and at various relative nozzle speeds. The blasting system allowed control of the media flow rate, air pressure, angle of attack, and table speed as described in shows the size distribution of the 30/100 media. Two approaches were used to measure the rate at which paint could be stripped from the various painted panels: the area of paint removed per unit time to a specified depth, and the paint thickness removed under fixed blasting conditions. In both cases, paint removal was measured over a 15 cm long strip of painted panel that had been exposed to the steady-state blast stream on a sliding table that moved before a fixed nozzle.The paint stripping rate was first determined as the maximum area of paint removed down to a specific layer (the primer in this case) per unit time, for a given set of blasting conditions (pressure, flow rate, nozzle angle, relative speed of nozzle and surface and stand-off distance). For the present paint system, the topcoat was gray and the primer was yellow, which made it relatively easy to identify topcoat removal or complete stripping of topcoat and primer. The experiments were performed on two separate batches of painted panels (each batch had been painted at the same time).The paint stripping rates psr, were calculated aswhere u was the speed of the nozzle relative to the surface (nozzle was stationary and table moved), and w the average width of the trace left by the abrasive stream on the panel. shows the paint stripping rates for topcoat removal of the panels blasted with Envirostrip® 30/100, at a media flow rate of 5.44 kg/min (12 lb/min), an air pressure of 207 kPa (30 psi), nozzle angle of attack 20°, nozzle-surface stand-off distance of 15 cm, and an average table speed of 1.26 m/min (4 ft/min). For this blasting condition, the average particle velocity was taken from previous measurement by Djurovic et al. The second measure of stripping resistance was obtained by recording the thickness of the paint on the aluminum substrates before and after exposure to the blasting stream. An eddy-current gage (DeFelsko Co., Positector model 6000), which had a resolution of ±1 μm was used to measure the paint thickness every 1.3 cm along the centerline of the blasting trace (total of 11 measurements). It is noted that the paint film thickness measurement with the eddy-current probe was a maximum value because the probe tip rested on asperity peaks. Nevertheless, the change in average peak height should be equal to the change in the average paint thickness.In order to quantify the blasting conditions, and to facilitate comparisons between them, the “work exposure” Wexp, was used to express the amount of energy to which a unit area of coating was exposed the average particle velocity (m/s) measured using a rotating disk device . The maximum overall uncertainty in calculating Wexp, due to measurement uncertainties in shows the coating thickness removal as a function of Wexp at two different incident angles for unaged paint. The results from In order to facilitate further comparisons with data available in The average particle velocities for calculating Wexp were taken from previous rotating disk measurements by Djurovic et al. gives the average topcoat thickness removed from panels under these blasting conditions as a function of Wexp. Note that the data at the three highest values of Wexp correspond to blasting condition C, an angle of attack of 20°, while all other data points correspond to conditions A and B, having a 45° angle of attack. For a given blasting condition (Wexp), the paint thickness removed tends to decrease as the paint ages. The thickness of coating removed increases with Wexp, before reaching a plateau. The minimum amount of coating removal occurred on a panel aged for 8 days, while the maximum amount was for a freshly painted panel. It is noted that there is some ambiguity in the interpretation of the curves of , because the starting thickness of the topcoat varied from panel to panel (50–75 μm) and over a single panel. Therefore, the plateau for a given panel age does not always correspond to complete topcoat removal to the primer layer. Supplementary visual observations of the stripped areas indicated that the plateau in the curve for the fresh panel corresponded to complete topcoat removal (with primer intact) at Wexp values greater than approximately 2 kJ/m2. Visual assessment of the 8-day aged panel showed that the topcoat was not completely removed at Wexp values as high as 6 kJ/m2, and that complete topcoat removal of the 1, 2 and 4-day aged panels corresponded to Wexp≥3.8 kJ/m2. The old panels from the field tended to behave as the 1- and 2-day aged panels. to show the amount of topcoat remaining on the differently aged panels as opposed to the amount removed. The value at Wexp=0 corresponds then to the average panel thickness of the topcoat before blasting. A remaining topcoat thickness of zero represents the nominal level of the epoxy primer, although there is some uncertainty (approximately ±10 μm) because of local thickness variations over the panel surface. For this reason, a negative remaining topcoat thickness may either simply reflect this uncertainty or indicate that some of the primer was removed. Note again that the data at the three highest values of Wexp correspond to blasting condition C, an angle of attack of 20°, while all other data points correspond to conditions A and B, having a 45° angle of attack. The underlying trend is that, for a given Wexp, the thickness of remaining topcoat increases as the paint ages, illustrating that the paint is more difficult to remove as the topcoat ages.In order to assess the reproducibility of the thickness removal results, a freshly painted panel and a panel aged 4 days were blasted with the same blasting parameters on two different dates. The reproducibility of the thickness removed was good, with a maximum difference of 10%.To determine variability in a single set of blasting experiments, a set of freshly painted and aged panels (0, 2, 4 and 8-days aged) were blasted three times, with a single set of blasting parameters. The standard deviation in the amount of coating thickness removal over each 15 cm long strip varied from 4 to 15% on a single panel.Artificially aging the aluminum panels painted with polyurethane significantly increased the Vickers and dynamic hardness, as well as the coefficient of restitution of the coatings, all of which are affected by plastic behavior, while aging did not have a significant effect on the coating’s modulus of elasticity. It was also observed that artificial physical aging, and aging in the field on an aircraft, decreased the psr as the coating became more erosion resistant. shows an idealized plot of the mean contact pressure pm versus the penetration depth δ, for a spherical particle penetrating two coatings having the same Young’s modulus E. It is assumed that the particle is not deformable (and thus much harder than the substrate), and that a fully plastic condition is reached very quickly in the impact process so that the elastic–plastic transition portion of the curve can be neglected illustrates that the particle with the higher dynamic hardness has a larger amount of recoverable elastic energy, causing a higher rebound velocity and a higher coefficient of restitution. This is consistent with the present data, as seen in , which shows that aged panels (which are harder than fresh panels, see end of ), have a higher coefficient of restitution than fresh panels.In order to measure the dynamic hardness of a paint film, the mass and velocity of the impact body must be carefully adjusted to control the depth of penetration. This is not practical for evaluation of stripping rate in the field. However, the coefficient of restitution e, can be conveniently measured using compact commercially available devices that propel a metal sphere through a guide tube toward the test surface. An induced current is generated within a coil encircling the guide tube, by the impacting and rebounding spheres, and the ratio of the voltages is used to calculate e. Although e is less sensitive to physical aging than dynamic hardness (), it is suggested that this simple measurement might be used to predict the relative paint stripping rates in various locations over an aircraft. Further experiments should be conducted to assess the sensitivity of e to the paint stripping resistance.It was shown that artificial aging of polyurethane paint significantly increases the Vickers and dynamic hardness of the coatings and the coefficient of restitution, while the process did not have a significant effect on the modulus of elasticity. The aging process decreased the paint stripping rate, i.e. the coating became more erosion resistant as it aged.The dynamic hardness, being a measurement of a coating’s resistance to indentation under dynamic conditions, is the dominant material property determining the erosion resistance of a coating. It was demonstrated that the paint stripping rate depends on either the coefficient of restitution or dynamic hardness of the paint film, and that, in principle, the coefficient of restitution (and dynamic hardness) of a paint film could be measured in situ using relatively simple equipment. This would make it possible to predict paint stripping rates at various locations over an aircraft or other structure.© 2020 Elsevier Ltd. All rights reserved.A modified yield function for modeling of the evolving yielding behavior and micro-mechanism in biaxial deformation of sheet metalsIn-depth understanding of the evolving plastic yielding behaviors and insight into their micro-scaled mechanisms are critical for fully exploiting of the formability of sheet metals, accurately forming of the needed shape and geometries, and precisely tailoring of the needed quality and property of the deformed parts. In this research, the in-plane yielding behaviors of dual-phase steel and aluminum alloy sheets were extensively investigated by biaxial tension experiments with the original and pre-strained specimens. It is found that the profile of the experimental plastic work contours changes with the increase of plastic deformation, no matter what the proportional or complex loading condition is. This indicates that the evolving yield behavior cannot be neglected. Based on the Yld2000-2d yield function, a modified yield function with introducing a variable exponent to represent the evolving yield behavior was proposed and then employed to model the evolving yielding of the given metallic sheets. To investigate the yielding micro-mechanisms, the simulated biaxial tension tests were conducted by using the established representative volume elements (RVEs) with a crystal plasticity model. The simulation results showed that the texture of the given sheet metals has a significant effect on the profile of the yield loci. Moreover, when the hard secondary phase is added into the polycrystalline aggregate, the optimum exponent of yield function for the given RVEs is increased, instead of decrease within a certain range of the plastic strain. The micro-mechanism of the evolving yielding behavior could be attributed to the ‘pinning’ effect of hard inclusions to the polycrystalline grains, i.e. the hardly-deformable particles strengthening the kinetic constraints to the polycrystalline matrix and further obstructing the rotation and plastic deformation of the neighboring grains. This research thus provides a comprehensive understanding of the effect of microscopic structure (crystal structure, texture and secondary hard phase) on the macroscopic plastic yielding behavior of metallic materials as well as a new high-fidelity modelling technique to describe the evolving yielding behavior phenomenologically, in such a way to support the application of FE simulation in sheet metal forming processes.To design and develop the light-weight and collision-safe automotive bodies, advanced high strength steels (AHSS) and light-weight alloys have been widely used in the contemporary automotive industries. Dual-phase (DP) steel is one of the promising AHSS materials, which has both high strength and good formability. Aluminum alloys, on the other hand, exhibits significant advantages with low density and have also been extensively utilized in aerospace and automotive industries for weight-reduction. Nevertheless, the mechanical responses of AHSS and aluminum alloy sheets are quite different compared with those of the conventional low carbon steel sheets. Therefore, high-fidelity modelling of the plastic yielding behavior has thus become an essential and tantalized issue as it critically affects the accurate determination of the deformation process and behavior of the given sheet metals.The phenomenological constitutive modelling has been widely used in sheet forming processes since it is generally focused on the macroscopic constitutive response of materials and can be implemented as time-efficient codes for finite element analysis. The yield criterion, which is a key component in the phenomenologically based plastic deformation model, defines a convex yield surface representing the boundary between pure elastic and elasto-plastic deformation. Moreover, it can also determine the direction of plastic flow when the associated flow law is utilized. Starting from the quadratic yield criterion proposed by von Mises, numerous kinds of yield criteria have been proposed to describe the yielding behaviors of different kinds of materials. For instance, firstly proposed an anisotropic yield criterion by introducing 6 anisotropic parameters into the original von Mises yield function. It has been verified that yield criterion is not capable in modelling the yielding behavior of FCC metals due to its quadratic form (). To overcome this problem, the non-quadratic isotropic yield function was proposed by . Based on the Hershey-Hosford yield function, further anisotropic extensions were made by , which added anisotropic parameters to the specific terms of the yield function; or by , etc., where the anisotropy was introduced by linearly transforming stress tensor with anisotropic tensor(s). Another approach to dealing with anisotropy is based on the stress invariants and the typical works were done by ) have also been extensively discussed. It should be noted that the validity and the micro-mechanism of these yield criteria have been examined either by , etc. with Taylor-Bishop-Hill (TBH) iso-strain crystal plasticity scheme (, etc. using full-field CPFEM/CPSM models. Both of which are limited to the mono-phase assumption. To the best of our current knowledge, few studies are focused on the effect of secondary phase, which is quite a common microstructure in many alloys, on the averaged yielding behaviors of polycrystalline aggregates with crystal plasticity approach.The hardening model is another component in the phenomenological plastic model, which describes the evolution of yield surface during the plastic deformation. Traditionally, the isotropic/kinematic hardening models are mostly used in constitutive modelling (), which assume the yield surface expands/translates in the stress space during the plastic deformation respectively. Nevertheless, some previous works revealed that the profile of the experimental plastic work contours for certain sheet metals would change when undergoing plastic deformation (). Generally, the anisotropic parameters of yield function are considered to be variable in order to describe this yield surface distortion behavior. The variable anisotropic parameters in these models are either determined by fitting the pre-calculated anisotropic parameters at equally spaced equivalent plastic strains (), or instantly computed by embedded numerical subroutine using multiple experimental tensile curves (). It should be mentioned that, for a general anisotropic yield function, the anisotropic parameters devote to calibrate the anisotropic plasticity, thus the above approach could only characterize the anisotropic part of plasticity evolution. As for the isotropic part of plasticity evolution, however, few studies have been conducted with a variable exponent (). Moreover, all these modelling techniques for distortional hardening are pure phenomenological, the micro-mechanisms of these techniques are not yet thoroughly explored and need to be investigated.From microscopic point of view, the plastic behavior of metal alloys is dominantly affected by several factors of material's microstructure, including phase composition (), preferential orientation of crystals (texture) (), etc. On the basis of crystal plasticity finite element method (CPFEM) and homogenization theories, the representative volume element (RVE) enables us to evaluate the statistically averaged mechanical behavior of the multi-phase metal alloys with hard secondary phase. For instance, studied the influence of martensite volume fraction and ferrite crystallographic orientation on the flow stress and ductile failure behaviors of DP steels with the CPFEM scheme. In addition, investigated the failure mechanisms of DP steel due to the heterogeneity of strain-stress partitioning between ferrite and martensite phases. established a multi-phase scheme with dislocation-based crystal plasticity model and successfully predicted the macroscopic tensile curves of DP800. modelled the martensitic transformation of AHSS with a rate-independent crystal plasticity model and predicted the FLC curves. Furthermore, other similar researches were also conducted by , etc. The above-mentioned studies are mainly focused on the relationship between the hard secondary phase and flow stress/fracture behaviors of the given materials, the effect of microstructure (especially the hard secondary phase) on the profile of the yield locus, however, has not yet been sufficiently and thoroughly investigated.In this research, the initial and subsequent experimental plastic work contours of DP steel and aluminum alloy sheets were determined by biaxial tension tests with original and pre-strained specimens. It indicated that the profile of the experimental plastic work contours would change when undergoing plastic deformation, no matter under proportional or complex loading conditions. Further analyses indicated that this kind of evolving yield behavior could be well modelled by introducing variable exponent to the yield function. Moreover, based on CPFEM/CPSM, the RVEs considering texture and hard secondary phase were established to investigate the micro-mechanism of this phenomenon. The proposed model was then validated by Nakazima bulging test and corresponding simulations. The present work bridges the microscopic crystal structure, texture and secondary hard phase with the macroscopic evolving plastic yielding behavior of metallic materials, and enlightens a new phenomenological modelling technique to enhance the accuracy in FE simulation of sheet metal forming processes.The materials used in this research are DP steel sheets with UTS of 590, 780, 980 MPa (named DP590, DP780 and DP980 respectively), and 6016 Al–Mg–Si alloy sheets with two different heat treatment processes: annealed state (6016-O) and solution heat treatment state (6016-T4P). The chemical compositions of the given DP steel and aluminum alloy sheets are listed in correspondingly. The martensite volume fraction of DP590, DP780 and DP980 is 10%, 23% and 38% accordingly, which has a significant correlation with the tensile strength of materials. As for 6016-T4P aluminum alloy, its heat treatment process consists of solution heat-treatment (SHT), quenching (with pre-aging) and followed by the naturally aging. During the aging process, the solid solute atoms tend to form clusters and then slowly evolve towards the so-called Guinier-Preston (GP) zones or nano-sized intermetallic precipitates, which consequently lead to a higher strength compared with the annealed state (The uniaxial tensile tests for each material along 0°, 45° and 90° to the rolling direction (RD) were conducted thrice with INSTRON tensile test machine, and the true stress-logarithmic plastic strain curves of the test samples are presented in lists the acquired mechanical properties of the test samples.In this research, a tailor-made hydraulic biaxial loading machine, detailedly illustrated by , was utilized for biaxial tensile tests. presents the geometry of the cruciform specimen for biaxial tensile tests according to the ISO-16842 (). The advantages of this kind of cruciform specimen with the slit arms have been detailedly illustrated by . Due to the small strain considered in this work, a pair of extensometers were used and placed on both sides of the specimen central area to measure the normal strain components, εx and εy properly.To determine the experimental plastic work contours of the given sheet metals, the biaxial tensile tests under proportional loading path were conducted with loading ratios (Fx:Fy) of 4:1, 4:2, 4:3, 4:4, 3:4, 2:4 and 1:4. Moreover, to examine the yielding and hardening behaviors under complex loading path, the pre-strain biaxial tension tests were also conducted, as presented in . In the first loading stage, pre-straining along RD (RD was defined before) and equi-biaxial direction (BD) were utilized. Afterwards, a series of proportional loading ratios were applied to these pre-strained specimens.The experimental uniaxial and biaxial tensile true stress-logarithmic plastic strain curves under the given proportional loading conditions for DP steel and 6016 alloy sheets are presented in , respectively. For the loading ratios of 4:0 and 0:4, uniaxial tensile tests with the standard specimens were conducted. It should be noted that there may exist fracture in the slit areas of the cruciform specimen before the central measured area could endure a relative larger plastic strain. Therefore, the limited strain range can be obtained by biaxial tensile tests with the slit arm cruciform specimen, compared with the standard uniaxial tensile test.In this research, the plastic work contour (), was employed to investigate the yield behavior of the given sheet metals. By utilizing the plastic work per unit volume principle (), the experimental plastic work contours under proportional loading conditions were determined and plotted in that the experimental plastic work contours expand in the stress space during the plastic deformation; while the profile evolution of the yield loci is non-neglectful. For DP780 and DP980, the profiles of the experimental plastic work contours vary from Mises-like to more Tresca-like with the increase of plastic dissipation. For 6016-O, however, inverse trend was observed. As for DP590 and 6016-T4, the profile evolution of the yield loci is less obvious. All these show that the distortional hardening needs to be considered in constitutive modelling of the sheet metals.The Yld2000-2d anisotropic yield function (), widely used for modelling anisotropy of the sheet metals, was utilized in this research to analyze the yielding behavior of the materials. It could be considered as the extended form of Hershey-Hosford yield function () with linearly transformed stress tensor. The plastic potential function of Yld2000-2d could be expressed as:{φ′=|X1′−X2′|Mφ″=|2X2″+X1″|M+|2X1″+X2″|Mand Xi′, Xj″ (i, j = 1, 2) are the principal values of the tensor X′ and X″:{Xi′=12(X11′+X22′±(X11′−X22′)2+4X′122)Xi″=12(X11″+X22″±(X11″−X22″)2+4X″122) are the Cauchy stress tensor σ linearly transformed by fourth-order tensors L′ and L″ respectively, i.e.:[L11′L12′L21′L22′L66′]=[2/300−1/3000−1/3002/30001][α1α2α7][L11″L12″L21″L22″L66″]=19[−228−201−4−4404−4−410−282−2000009][α3α4α5α6α8] presents the calculated Yld2000-2d yield loci with various values of exponent M (M ≥ 4). gives a clear intuition that the higher value of exponent M could result in a more Tresca-like yield locus; while the lower M value would lead to a more Mises-like yield locus (a general characteristic for the Hershey-Hosford derived yield functions). Therefore, variable exponents M was introduced to the Yld2000-2d yield function to characterize the evolving yield behaviors of the given sheet metals. Furthermore, (a), (c) also present the corresponding theoretical yield loci (marked as red lines) with the Yld2000-2d yield function, whose anisotropic coefficients were determined by using the mechanical properties at the corresponding equivalent plastic strain. As for the exponent M of the yield function, it was determined based on the L-M optimization algorithm for minimizing the deviation δ between the experimental yield points and theoretically predicted ones designated as:where σ1i and σ2i represent the stress component of the experimental yield point; di is the distance between the experimental yield point and theoretical one. (b), (d) give the optimal exponents of Yld2000-2d at different plastic deformation stages for given sheet metals. It could be observed that the optimal exponents Mopt for DP590, DP780, DP980 and 6016-T4P have a growing trend with the increase of ε‾p. As for 6016-O, however, an inverse trend exists and is observed.To assess the yielding behaviors under more general loading conditions, the pre-strain biaxial tension tests were carried out (see for details). Similarly, the subsequent experimental plastic work contours under complex loading paths could also be determined by plastic work per unit volume principle. It should be noted that the subsequent work contour strongly depends on the magnitude of the offset strain. The higher the magnitude of the offset strain, the more influence of the hardening effects caused by proportional loading on the subsequent yield locus. Therefore, the offset strain should be sufficiently small. In our present work, the offset strain was set to be 0.1%. present the subsequent experimental plastic work contours under complex loading paths for different sheet metals. It could be seen that, under the altered loading path, the subsequent experimental plastic work contours would not only expand, but also translate in the stress space compared with the initial experimental plastic work contours, indicating that the hardening behaviors of the given materials exhibit both the isotropic and kinematic hardening modes under altered loading path conditions.To evaluate the accuracy of different hardening modes clearly, the theoretical predicted subsequent yield loci with the isotropic (Iso), kinematic (Kinc) and mixed hardening modes under the pre-straining conditions were computed and presented in (a), (c). The theoretical subsequent yield loci of isotropic or kinematic hardening mode, as presented in , are simply the pure expansion or the translation of the initial yield loci in the stress space while passing through the yield point along the pre-straining direction. For the mixed hardening mode, however, both the expansion and translation phenomena of the yield surface exist, which needs more than one experimental yield point to distinguish the isotropic part from the kinematic one. Taking the RD pre-strain as an example, assuming that the back stress is X0=(XI,0,0) after pre-straining along RD, together with σY which calibrates the size of the yield surface. If the given subsequent experimental yield points (σI,0,0) and (0,σII,0) are along RD and TD respectively, the following equations should then be satisfied:where f(⋅) represents the Yld2000-2d yield function. Solving the nonlinear equations with numerical methods, the back stress component XI and the flow stress σY can be determined, and the theoretical yield loci predicted by the mixed hardening mode can then be computed. Moreover, the optimization was also conducted to determine the optimal exponent Mopt, as shown in , for the mixed hardening mode by minimizing the error between the experimental subsequent yield points and theoretical predicted ones., it is observed that the optimal exponent Mopt is increased with the equivalent plastic strain for DP590, DP780, DP980 and 6016-T4P; while it is decreased for 6016-O. All of these are consistent with the results under proportional loading conditions. Therefore, it can be concluded that the distortional hardening phenomenon need to be considered for these given sheet metals, no matter the proportional or complex loading condition is applied.Conventionally, the exponent M in the Yld2000-2d yield function is considered to be a constant generally associated with the metallic crystal structure, i.e. M = 6 for BCC metals and M = 8 for FCC metals (), which is consistent with many Hershey-Hosford derived anisotropic yield function (). However, for some sheet metals, alternative exponent values were also selected in the constitutive modelling (). Furthermore, it has been proven in Section that, by introducing rising and decreasing exponent to the yield function, we could better characterize the evolving yielding behavior of the given sheet metals, yet the micro-mechanism of this phenomenological modelling technique is still unclear.To reveal the mechanism of this phenomenon, the micromechanical model was introduced in this section based on the CPFEM. A continuum crystal plasticity model was utilized to model the slip deformation of the active slip systems and the rotation of the crystal lattice in a single grain (). The deformation gradient F of the crystalline material could be decomposed as:where Fe=Re⋅Ue is the elastic part of the deformation gradient denoting the stretching (Ue) and rotation (Re) of the crystal lattice; Fp represents the plastic deformation of material resulting from shearing along the active slip systems. The plastic deformation gradient Fp is further evolved as:where the plastic velocity gradient Lp is only related to the slipping rate of each (active) slip system and denoted as:Here the sum ranges over all (active) slip system; mα and nα are the unit vector along slip direction and the normal to the slip plane of the αth slip system in the intermediate reference configuration; mα⊗nα is the Schmid factor of the corresponding slip system. In this research, the rate-dependent crystalline plasticity scheme proposed by was utilized, in which the shearing rate of the αth slip system γ˙(α) could be expressed aswhere τ(α) is the resolved shear stress of the αth slip system; g(α) is the current strength of this slip system; γ˙0 is the reference strain rate; m is the rate sensitivity parameter.), the resolved shear stress on each slip system could be characterized as:where S is the second Piola-Kirchhoff (PK2) stress tensor. As for the strain hardening of the slip system, it could be characterized by the current strength g(α), whose evolution could be represented in a rate form in the following:where hαα (no sum on α) and hαβ(α≠β) are the self and latent slip hardening moduli, respectively. The slip hardening moduli is given by:Here, qαα=1 (no sum on α); qαβ(α≠β) is a material constant representing the interaction between slip systems. As for hβ, it is the hardening moduli on single slip system. Considering the prior researches of ) was chosen in this study, which is expressed as:where h0 is the initial hardening moduli; g(β) is the current strength of βth slip system; τs represents the saturated shear stress of the slip system; a is the fitting exponent.By programming the Abaqus Python script, five polycrystalline RVEs were generated with 8000 (20 × 20 × 20) C3D8R elements to get the averaged results, as presented in . The grain boundaries were generated according to the 3D-Voronoi algorithm with 50 randomly distributed seeds. For each grain, the random crystallographic orientation was assigned. Three independent elastic constants for monocrystalline, i.e. C11, C12 and C44, as presented in , were adopted due to the symmetry of cubic crystal. The aluminum is a typical FCC metals and has 12 slip systems ({111}⟨110⟩). As for the ferrite grain, 24 slip systems ({110}⟨111⟩ and {112}⟨111⟩) were considered for the BCC crystal structure. lists the parameters for each slip system in the ferrite and aluminum grains.To investigate the yielding behavior of the polycrystalline RVE, the boundary conditions presented in were applied. As the force with a constant loading rate was directly applied to the outer surfaces of the RVE model. The current CP model would sometimes encounter numerical non-convergence problem by using the Abaqus/Standard implicit solver with UMAT subroutine. Therefore, the CP model was re-programmed as VUMAT subroutine for Abaqus/Explicit solver, which was based on the UMAT previously developed by . Different from UMAT, VUMAT subroutine employs the Green-Naghdi stress rate instead of Jaumann stress rate, and the constitutive formulation is based on a corotational coordinate system, in which the rotation of the material is not needed to be considered. Therefore, the incremental rotation tensor ΔR in the finite deformation framework of VUMAT subroutine should be the identity tensor, instead of Hughes-Winget approach (Eq. Moreover, the Euler explicit integration algorithm was adopted in the VUMAT since the time increment in Abaqus/Explicit is sufficiently small (It is worth noting that the applied CP model is a viscoplastic model. Therefore, it is strain rate sensitive. According to Eq. , when the strain rate is changed, the strength and shear stress of each slip system will be changed proportionally. Therefore, the change of the strain rate would have an influence on the magnitude of the stress-strain curves, but would not change the profile of the yield loci, since all uniaxial/biaxial stress-strain curves are changed proportionally. presents the simulated uniaxial and biaxial tensile true stress-logarithmic strain curves under the proportional loading conditions for ferrite/aluminum polycrystalline RVE. Owing to the meticulously set boundary conditions, a large strain range could be achieved in simulation, which facilitates the investigation of yielding behavior in a wider strain range.Similar as before, by applying the plastic work per unit volume principle (), the simulated yield loci for ferrite and aluminum polycrystalline RVEs were determined and plotted in (a) and (c) respectively. What shown in (a) and (c) are the theoretical yield loci predicted by Yld2000-2d yield function, here the adopted anisotropic coefficients α1-α8 were determined based on the simulated material parameters at the corresponding equivalent plastic strain; and the optimal exponent Mopt, as shown in (b) and (d), was determined by the similar approach stated in Section (b) that, with the increase of plastic dissipation, the optimal exponent Mopt for ferrite polycrystalline is decreased significantly in the first place, and then goes into a steady zone at Mopt≈5.23 when ε‾p≥0.12, as presented (b). This phenomenon is caused owning to the rearrangement of the crystallographic orientation of the grain aggregate in plastic deformation, i.e. each grain in the polycrystalline tends to rotate so as to make active slip direction close to the maximum shear stress direction and thus facilitates the slipping of the active slip system, as presented in (a). In the ideal condition, the slip direction of the grain would finally be consistent with the shearing direction and Mopt would be reduced to approximately 4, which is identical to J2 plasticity theory (M = 4 for M>2.767 and M = 2 for 1≤M≤2.767 ()). However, due to the constraints of stress and strain compatibility across grain boundaries, the rotation of each grain is restricted and a final steady state is achieved when the interaction of grain rotation and grain boundary restriction realizes a balance.To evaluate the texture evolution (grain rotation) of the RVEs quantitatively, the following indicator is introduced:where θi represents the angle between the maximum shear stress direction and the closest active slip direction in each integration point, and n stands for the total number of RVE's integration points. Through programming the Abaqus Python script, the indicator Ave_Cosθ in each frame for pure ferrite RVEs can be calculated and presented in (b) that Ave_Cosθ does increase with the growth of the equivalent plastic strain, which proves the previous assumption that the grain rotation during plastic deformation generally brings about the active slip direction getting closer to the maximum shear stress direction. (d), the optimal exponent Mopt for aluminum polycrystalline is also decreased with the increase of the equivalent plastic strain, which is similar to the experimental results of 6016-O, as presented in (b). However, the decreasing speed is relatively smaller than that of ferrite polycrystalline, possibly due to the less slip systems in FCC metals., the optimum exponent Mopt in Yld2000-2d for pure ferrite polycrystalline is decreased with the increase of the equivalent plastic strain, which is contrary to the experimental results of DP steel sheets presented in Section . Considering the above analysis results, it is assumed that the martensite phase, the hard secondary phase in the ferrite matrix, would have a significant effect on the yielding behavior of the DP steel. To verify this assumption, a dual-phase RVE containing ferrite grains and martensite inclusions was modelled, in which the martensite elements were randomly positioned, as shown in . In this work, only the strength difference between polycrystalline matrix and secondary phase was considered. The simple J2 plastic model with Swift hardening was thus adopted for martensite phase according to the previous work of , in which the Swift hardening model could be expressed as:where σs is the initial yield stress; k is the Swift coefficient and n is the strain-hardening exponent. The material parameters for martensite phase, determined by Similarly, to investigate the micro-mechanism of plastic yielding of 6016-T4P, the RVE containing aluminum grains and secondary phase was generated, for which the generating procedures was the same as those of the dual-phase steel RVE. According to , the solid solute atoms (mainly Mg and Si) of 6016 aluminum alloys would finally form nano-sized intermetallic phase (mainly Mg2Si β-phase) during the naturally aging step. Since the Mg2Si is a brittle intermetallic compound, only the elastic properties () were considered in this study, as presented in Afterwards, the simulated yield loci for RVE containing ferrite grains and different volume fractions of martensite phase, were determined and presented in (a), (c), (e) and (g). The theoretical yield loci of Yld2000-2d yield function with the optimized exponent Mopt and anisotropic coefficients α1-α8 determined by simulated material parameters at corresponding deformation stages are also plotted in the figure. It is found from that the added hard inclusions of the martensite phase dramatically altered the yielding behavior of the RVE, i.e. the optimal exponent Mopt of the dual-phase RVE is increased with the plastic dissipation, which is similar to the experimental yielding behaviors of the DP steel sheets as shown in (b), (d), and (f). The micro-mechanism of this phenomenon might be attributed to the ‘pinning’ effect of the martensite inclusions to the ferrite grains, i.e. the hardly-deformable particles strengthening the kinetic constraints to the ferrite matrix, and further obstructing the rotation and plastic deformation of the neighboring grains (as presented in (f) and (h), when the volume fraction of martensite is increased to 30%–40%, the growth rate of Mopt vs. ε‾p would drop significantly and early ‘steady zone’ of Mopt would occur. It thus indicates that a strong ‘pinning’ effect of martensite particles would significantly block the rotation of the grains and lead to an early balance between grain rotating and stress/strain compatibility restrictions across the phases.To verify the proposed ‘pinning’ effect of martensite particles, the indicator Ave_Cosθ (defined in Eq. ) in each frame for DP (specifically ferrite+20% martensite) RVEs was calculated by the programmed Abaqus Python script and presented in (b). Also depicted in the figure is the Ave_Cosθ curves of pure ferrite RVEs for comparison. It could be observed from (b) that, with the increase of the equivalent plastic strain, the Ave_Cosθ curves of DP RVEs gradually run lower than the corresponding pure ferrite ones, proving that the martensite particles do have ‘pinning’ effect to the grain rotation.In addition, the simulation results of RVE containing aluminum grains and 1% volume fraction of Mg2Si are presented in that, the optimal exponent Mopt of the generated RVE is increased at the first place (ε‾p≤0.058), which is quite similar to the experimental yielding behavior of 6016-T4P presented in (d); when ε‾p>0.058, however, the optimal exponent Mopt of the generated RVE is gradually decreased with the increase plastic dissipation. This phenomenon might be attributed to the weak ‘pinning’ effect of the added Mg2Si secondary phase since the volume fraction of Mg2Si is sufficiently small and would not effectively obstruct the plastic deformation and rotation of the polycrystalline grains.It should be noted that the above work is more focused on revealing the effect of hard secondary phase on the averaged yielding behavior of the polycrystalline RVE in an ideal circumstance (without texture). The size, the geometric shape of the secondary phase, and the effect of the solid solute atoms on the polycrystalline grains, etc., however, have not yet been considered. When the volume fraction of the hard secondary phase is comparably small (e.g. DP590 and 6016-T4P), this simplified model could provide reasonable results. For the DP steel with a relatively large volume fraction of martensite (e.g. DP780 and DP980), the effects of solid solution atoms, the shape of the martensite phase and other related aspects are not neglectable. Due to the modelling complexity and significant computational cost, the size of Mg2Si inclusions and the shape of martensite have not yet been considered and thus our current work is in early stage. Advanced modelling techniques will be needed for further investigation to reveal the effects of these factors on macroscopic yielding behaviors. that the CPFEM simulations predicted von Mises type of the yield loci, which are quite different from the experimental ones of DP steel sheets presented in . This might owe to not considering specimens’ texture in the CP modelling procedures. In this section, the crystal plasticity spectral method (CPSM) RVE modelling technique was adopted to consider the texture of the specimens (). The modelling framework was based on the open-source code DAMASK with FFT-solver and its implemented phenomenological crystal plasticity model (stated in Eqs. ). As for the major input of the RVE modelling, the ODFs were obtained by EBSD texture analyses. According to the previous study of , the number of FFT points was set to be 323 in the RVE model. presents the calculated yield points considered texture of the given sheet metals. Also depicted were the previously calculated ones based on CPFEM with random crystallographic orientations (previously presented in ) and the experimental ones (previously presented in that, by considering specimen texture, the CPSM RVE models could enhance the fitting quality of the yield loci significantly for DP780, DP980 and 6016-O, indicating that the texture does have a non-neglectable influence on the yielding behaviors of sheet metals.Moreover, the optimal exponents Mopt of CPSM RVEs simulated yield loci were also determined (shown in ) by the similar approach stated in Section (a) (b) (c) and (e) that, the optimal exponents Mopt are decreasing, instead of increasing with plastic dissipation, which are not consistent with the experimental results of DP steel sheets (shown in (b), (d) and (f)) and 6016-T4P aluminum alloy sheets (shown in (d)). Considered the results in Section , it could be concluded that the textures of the given DP steels and 6016-T4P are not capable to block the grain rotation effectively, and the strong ‘pinning’ effect of secondary hard inclusions might have a greater impact on the evolving plasticity (i.e. increase of Mopt) of these materials.To validate the constitutive models for the given sheet metals, the Nakazima bulging test and corresponding FE simulations were conducted. The tool dimensions for the Nakazima bulging tests are marked in (a). To ensure the occurrence of crack at the top of the deformed specimen, the lubricating oil and plastic film were applied between the specimen and the punch. The corresponding FE model is presented in (b). Considering the symmetry in the forming system, only 1/4 of the specimen was modelled aiming at reducing the time consumption of the numerical computing. The Coulomb coefficient between the punch and specimen was set to be 0.01 (To precisely simulating plastic forming processes of the given sheet metals, the evolving yielding behavior should be considered. Previous studies often directly applied FE codes of the crystal plasticity models in simulation (). Nevertheless, this kind of approach is proven to be quite computationally inefficient. Phenomenological modelling is a practical way to characterize the evolving yielding behavior of sheet metals since it could be implemented as time-efficient codes for finite element analysis. In this section, the proposed Yld2000-Var, i.e. Yld2000-2d with a variable exponent, was implemented into Abaqus as VUMAT user subroutine via Next Increment Correct Error (‘NICE’) explicit integration algorithm (). The optimal exponents for given sheet metals at different plastic deformation stages, as presented in (b), (d), were fitted with the following bounded fitting function in Yld2000-VarMopt(ε‾p)=p1+p2⋅[exp(p3⋅ε‾p−p4)−1]/[exp(p3⋅ε‾p−p4)+1]where p1, p2, p3 and p4 are the fitting parameters, and the fitted results are listed in As for the eight anisotropic coefficients α1−α8 in Yld2000-Var, they were determined and updated by solving eight non-linear equations proposed by with Newton-Rapson iteration algorithm after each stress-updating step in VUMAT. This identification procedure needs instant material parameters at a certain deformation stage. The multiple experimental stress-equivalent plastic strain curves (including uniaxial tensile curves along 0°, 45° and 90° with respect to RD, as well as equi-biaxial tensile curve) and corresponding constant r-values () were thus needed to be input into VUMAT subroutine. The experimental stress-equivalent plastic strain curve along each direction could be fitted by the following extended Voce-type flow stress model (where A, B, C and n are the fitting parameters, and the fitted results are listed in . It is worth noting that the equivalent plastic strain ε‾p of the tensile curves along each direction is not simply the plastic part of the total strain, it was determined based on the plastic work per unit volume principle with the reference tensile curve σ−εp along RD (Moreover, three most commonly used models, i.e. Mises, Hill48 and Yld2000-2d, were also adopted in FE simulations for comparison. For the Mises and Hill48 models, they were already included in ABAQUS, thus only VUHARD subroutine was needed to be implemented into the extended Voce-type flow stress model presented in Eq. . As for the original Yld2000-2d model, another VUMAT subroutine was programmed with ‘NICE’ numerical integration algorithm (). The conventional fixed exponents were adopted for the original Yld2000-2d, i.e. M = 6 for BCC metals (DP steels) and M = 8 for FCC metals (aluminum alloys) (), and the corresponding fixed anisotropic coefficients for the given sheet metals were determined and presented in , it is clear that the proposed Yld2000-Var can well predict the force-displacement curves under Nakazima bulging process for nearly all given sheet metals. Other three models, however, would encounter relatively larger error in some cases. It could be concluded that, owing to the introduced variable exponent, the proposed Yd2000-Var model has considerable accuracy in modelling of the evolving yielding behaviors of the given materials.In this research, the experimental and crystal plasticity simulations were utilized to investigate the plastic yielding behavior of DP steel and 6016 aluminum alloy sheets. The main findings are summarized as follows:The initial and subsequent experimental plastic work contours were obtained by conducting original and pre-strain biaxial tension tests. Non-neglectable evolving yielding behavior was observed, i.e. the profile of the yield loci is changed with the increase of the equivalent plastic strain, no matter whether they were under proportional or complex loading conditions.Based on Yld2000-2d yield function, it was found that this kind of evolving yielding behavior could be well characterized by introducing a variable exponent to the yield function.Based on CPFEM/CPSM, the biaxial tensile simulations were performed on the established polycrystalline RVE. The simulated results showed that the texture of the given sheet metals affects the shape of the initial yield loci. Moreover, the hard secondary phase has a significant effect on the averaged yielding behavior of the RVE material, i.e. the optimum exponent Mopt of Yld2000-2d for pure ferrite/aluminum polycrystalline aggregate is decreased along with the undergoing plastic deformation (similar to the experimental results of 6016-O). When the hard secondary phase is added into the ferrite/aluminum polycrystalline aggregate, however, the optimum exponent Mopt of Yld2000-2d for the given RVEs is increased within a certain range of the equivalent plastic strain, which is similar to the experimental results of DP steels and 6016-T4P.The micro-mechanism of the evolving yielding behavior might be attributed to the ‘pinning’ effect of the hard inclusions to the polycrystalline grains, i.e. the hardly-deformable particles strengthening the kinetic constraints to the polycrystalline matrix, and further obstructing the rotation and plastic deformation of the neighboring grains.By analyzing the micro-mechanism of the metallic evolving yielding behavior, this study reinforced the theoretical basis of the proposed phenomenological modelling techniques, viz., introducing a variable exponent to yield function, especially in the large strain range where the conventional biaxial tensile test could not be done.The validity of the Yld2000-2d with a variable exponent was verified by Nakazima bulging test and the corresponding simulations and the modified yield criterion was shown to be efficient.We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.Since the exponent of the Yld2000-Var yield function depends on the plastic strain, the ‘NICE’ explicit integration algorithm needs the additional partial derivative ∂σ‾/∂M. The expression of partial derivative ∂σ‾/∂M is:∂σ‾∂M=σ‾MP1MlnP1σ‾+P2MlnP2σ‾+P3MlnP3σ‾P1M+P2M+P3Mand Xi', Xi" (i = 1, 2) are defined in Eq. self-propagating high-temperature synthesis (SHS) Dispersion of ultrafine SiC particles in molten Al-12Si alloy Jin-Ju PARK, Sang-Hoon LEE, Min-Ku LEE, Chang-Kyu RHEE Nuclear Materials Research Division, Korea Atomic Energy Research Institute, Daejeon 305-353, Korea Received 21 April 2010; accepted 10 September 2010 Abstract: The bulk Al-12 Si eutectic composites were fabricated through a conventional liquid metal casting route, especially with the help of ultrafine ceramic powders made by self-propagating high-temperature synthesis (SHS) process. The SHS powders were fabricated by the chemical reaction between micro-sized SiC and Al particles at very high combustion temperatures, producing the coarse Al particles (several tens of microns) containing ultrafine SiC ceramic particles. Microstructural observation revealed that the addition of ultrafine SiC particles has a crumbling tendency of Si eutectic phase. It is suggested that the casting method combined with SHS process is promising for fabricating the Al-based MMC with ultrafine ceramic particles. Key words: Al-12Si alloy; dispersion; self-propagating high-temperature synthesis (SHS); ultrafine ceramic; SiC 1 Introduction Aluminum-based metal matrix composites (MMCs) have been widely studied as an attractive choice for aerospace, automobile, and military applications due to their low density and superior specific properties including strength, stiffness and creep resistance[1−10]. Normally, to fabricate Al-based MMCs, micro-sized ceramic powders and fibers were used to improve the yield and ultimate strength of the metal. However, the ductility of the MMCs deteriorates, which limits a widespread use of the Al-based MMCs. It is of great interest to use nano-sized ceramic particles to strengthen the metal matrix, that is, metal matrix nano-composites (MMNCs), while the ductility of the matrix is retained. With nanoparticles reinforcement, especially high temperature creep resistance and better fatigue life could be achieved because thermally stable ceramic nanoparticles can maintain their properties at high temperatures. Currently, there are several fabrication methods of MMNCs, including mechanical alloying with high energy milling[5], ball milling[10], nano-sintering[11], vortex process[12], laser deposition, etc. Casting, as a liquid phase process for the fabrication of MMNCs, is capable of producing products with complex shapes. It will be attractive to produce as-cast lightweight bulk components of MMNCs with uniform reinforcement distribution and structural integrity. However, nano-sized ceramic particles present difficult problems. It is extremely difficult to obtain uniform dispersion of nano-sized ceramic particles in liquid metals due to poor compatibility as wettability and dispersability in the metal matrix, high viscosity, and a large surface-to-volume ratio. These problems easily induce agglomeration and clustering in the matrix. In this study, in order to solve these problems, the self-propagating high-temperature synthesis (SHS) process[13−17] is introduced to distribute and disperse ultrafine ceramic particles into Al alloy melts, thus making the production of cast high-performance Al matrix composite promising. The SHS process provides an economical and energy efficient route for the preparation of various hard ceramic particles, which can be subsequently incorporated in a metallic matrix. The aim of this study is to disperse the ultrafine SiC ceramic particles in Al-12Si alloy matrix through the use of a conventional casting method. For this purpose, the SHS process for the SiC ceramic and Al metal powders is utilized to promote the compatibility such as wettability and dispersability. The changes in the microstructure of the matrix are discussed in views of quantitative image analysis of the length of Si eutectic phase. Corresponding author: Jin-Ju PARK; E-mail: [email protected] Jin-Ju PARK, et al/Trans. Nonferrous Met. Soc. China 21(2011) s33−s36 s34 2 Experimental In the present work, the composite material was made from Al-12Si eutectic alloy by stir casting method. For the additive powders, the silicon carbide-aluminum (SiC-Al) powders were manufactured using combustion mode SHS and contained 15% SiC(mass fraction). The SHS process setup involved a burning velocity of 0.1−20 cm/s, combustion temperature of 2 300−3 800 K, heating rate of 10 3 −10 6 K/s, induction time for ignition of 0.2−1.2 s and ignition temperature of 800−1 200 K. After the SHS process, the surface and a cross-section of the SiC-Al composite powders were observed by using a scanning electron microscope (SEM). The X-ray diffraction (XRD) and energy dispersive spectroscopy (EDS) were also employed for the characterization of the SHS powders. The casting experimental setup consists of process and control parts. An electric resistance-heating unit was used to melt the alloy in a graphite crucible with size of 100 mm in diameter and 200 mm in height. The SHS SiC-Al composite powders were directly added into the melts with 5% during the process from the top of crucible. Since the SiC particles tended to settle at the bottom, the composite melt was stirred for 3 min at 700 r/min using a graphite impeller in order to produce a homogeneous distribution of SiC particles in the melt. The alloy melt was protected by argon atmosphere. This molten metal was then poured into a metal mold. The microstructural properties were observed via SEM with regard to the addition of the SHS SiC-Al composite powders. Thereafter, the SEM images of the silicon eutectic phase were analyzed quantitatively with respect to its length. 3 Results and discussion Figure 1 shows SEM image of surface morphology, XRD pattern and EDS mapping image of SHS SiC-Al powders. As shown in Fig.1(a), the SHS SiC-Al composite powders have an angular shape with particle sizes ranging from 50 μm to 100 μm. The XRD pattern reveals that the fabricated SHS composite powders consist of SiC and Al without any impurities. In addition, from the EDS mapping image shown in Fig.1(c), the SiC ceramic particles are well distributed in the Al powder matrix. In order to clarify the distribution of SiC particles in the Al matrix, a typical cross-sectional view of the SHS SiC-Al composite powders is shown in Fig.2. It can be seen that the SiC-Al composite powder consists of globular SiC particles represented as darker phase with size of 1−5 μm surrounded by the matrix of Al. Hence, it is confirmed that the SHS process can easily fabricate SiC-Al composite powders with very small and uniform size of SiC ceramic particles. Fig.1 SEM image (a) of surface morphology, XRD pattern (b) and EDS mapping image (c) of SHS SiC-Al powders Jin-Ju PARK, et al/Trans. Nonferrous Met. Soc. China 21(2011) s33−s36 s35 It is also found that the SiC particles in the composite powders are distributed inhomogeneously in the Al powder matrix. Figure 3 illustrates the microstructural change of Al-12Si matrix composite in the absence and presence of SiC ceramic particles. The microscopic examination of the specimen without SiC particles given in Fig.3(a) shows that the Al-12Si matrix is composed of the Al-Si eutectic phase, which consists of bright Al and dark Si phases in the microstructure image. As shown in Fig3(a), the eutectic Si phase is normally very long. However, Fig.2 SEM image of cross-section of SHS SiC-Al powders Fig.3 SEM images of Al-12Si alloy matrix in absence (a) and presence (b) of SHS SiC-Al powders in the presence of the SHS SiC-Al composite powders, the microstructure of the composite matrix changes remarkably. Although the main phase is the same as that of the matrix without SiC particles, the length of the eutectic Si phase is significantly reduced. Based on the microstructural change in the presence of SHS SiC-Al composite powders, it is realized that the addition of SiC particles has the tendency of crumbling eutectic Si phase. In order to confirm the effect of the addition of SiC particles on the microstructural change of the matrix, the quantitative image analysis is employed on the SEM image of Fig.3 as a function of the length of Si eutectic phase. Table 1 gives the values of minimum, maximum and average length of Si eutectic phase without and with SHS SiC-Al powder additive. When there are additive powders, the length of Si eutectic phase is shortened. For the better understanding, the results of the quantitative image analysis of the length of Si eutectic phase in view of its number and mass are presented in Table 2. When there are no additive powders, the length of Si phase ranged from 1 μm to 151 μm. However, in case of adding the powders, the distribution of the Si eutectic length is only within 80 μm. Figure 4 demonstrates the histogram of the distribution by the length of Si eutectic phase in views of its number and mass based on the results of Table 2. It can be easily seen that when the SHS SiC-Al powders are added into the matrix, the length of Si eutectic phase is shorter than that without additive powders and hence the length of Si eutectic phase is ranged from 1 μm to 40 μm. Table 1 Length of Si eutectic phase in Al-12Si alloy in absence and presence of SHS SiC-Al powders. Length of Si eutectic phase/μm SHS SiC-Al powders Minimum Maximum Average Absence (Fig.3(a)) 2.27 151.77 25.88 Absence (Fig.3(b)) 2.46 81.49 16.99 Table 2 Results of quantitative image analysis for change in length of Si eutectic phase in view of its number and mass Without SHS SiC-Al powders (Fig.3(a)) Without SHS SiC-Al powders (Fig.3(b)) Length/ μm Numbe r Ratio of number/% w/% Number Ratio of number/% w/% 0−20 77 52 13.5 96 71 34.5 20−40 42 28 29.5 30 22 39.5 40−60 21 13.5 31 7 5 19 60−80 5 3 9 3 2 7 80−100 3 2 9 0 0 0 100−120 1 0.5 3 0 0 0 120−140 1 0.5 2 0 0 0 140−160 1 0.5 3 0 0 0 Jin-Ju PARK, et al/Trans. Nonferrous Met. Soc. China 21(2011) s33−s36 s36 Fig.4 Histogram of distribution based on quantitative image analysis of Fig.3 with length of Si eutectic phase in matrix in views of its number (a) and mass (b) Hence, this study suggests that the addition of SHS processed powders can efficiently disperse ultrafine ceramic particles into a Al-based alloy through the conventional casting route. Further study is necessary to optimize the process and finally to improve the physical and mechanical properties. 4 Conclusions 1) Self-propagating high-temperature synthesis (SHS) process can easily fabricate SiC-Al composite powders contained ultrafine SiC ceramic particles in Al powder matrix. 2) In the presence of SHS SiC-Al composite powders, the length of the eutectic Si phase is significantly shortened. This implies that the addition of SiC particles has a tendency of crumbling eutectic Si phase. 3) Based on the results, ceramic dispersion strengthening composites with ultrafine ceramic particles using a liquid metal casting method is possible by the help of the addition of SHS processed powders. Acknowledgements This research was financially supported by the Korea Atomic Energy Research Institute (KAERI) R&D Program. References [1] ARSENAULT R J. The strengthening of aluminum alloy 6061 by fiber and platelet silicon carbide[J]. Materials Science and Engineering, 1984, 64: 171−181. [2] NARDONE V C, PREWO K M. On the strength of discontinuous silicon carbide reinforced aluminum composites[J]. Scripta Metallurgica, 1986, 29: 43−48. [3] IBRAHIM A, MOHAMED F A, LAVERNIA E J. Particulate reinforced metal matrix composites-A review[J]. Journal of Materials Science, 1991, 26: 1137−1156. [4] RAMAKRISHNAN N. An analytical study on strengthening of particulate reinforced metal matrix composites[J]. Acta Materialia, 1996, 44: 69−77. [5] KARANTZALIS A E, WYATT S, KENNEDY A R. The mechanical properties of Al-TiC metal matrix composites fabricated by a flux-casting technique[J]. Materials Science and Engineering A, 1997, 237: 200−206. [6] TONG X C, FANG H S. Al-TiC composites in situ-processed by ingot metallurgy and rapid solidification technology: Part II. Mechanical behavior[J]. Metallurgical and Materials Transactions A, 1998, 29: 893−902. [7] ZHANG B P, MASUMOTO H, SOMENO Y, GOTO T. Preparation of Au/SiO 2 nano-composite multilayers by helicon plasma sputtering and their optical properties[J]. Materials Transactions, 2002, 43: 2855−2859. [8] PENG L M. Creep of metal matrix composites reinforced by combining nano-sized dispersoids with micro-sized ceramic particulates or whiskers (review)[J]. International Journal of Materials and Product Technology, 2003, 18: 215−254. 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Further study is necessary to optimize the process and finally to improve the physical and mechanical properties. 4 Conclusions 1) Self-propagating high-temperature synthesis (SHS) process can easily fabricate SiC-Al composite powders contained ultrafine SiC ceramic particles in Al powder matrix. 2) In the presence of SHS SiC-Al composite powders, the length of the eutectic Si phase is significantly shortened. This implies that the addition of SiC particles has a tendency of crumbling eutectic Si phase. 3) Based on the results, ceramic dispersion strengthening composites with ultrafine ceramic particles using a liquid metal casting method is possible by the help of the addition of SHS processed powders. Acknowledgements This research was financially supported by the Korea Atomic Energy Research Institute (KAERI) R&D Program. References [1] ARSENAULT R J. The strengthening of aluminum alloy 6061 by fiber and platelet silicon carbide[J]. Materials Science and Engineering, 1984, 64: 171−181. [2] NARDONE V C, PREWO K M. On the strength of discontinuous silicon carbide reinforced aluminum composites[J]. Scripta Metallurgica, 1986, 29: 43−48. [3] IBRAHIM A, MOHAMED F A, LAVERNIA E J. Particulate reinforced metal matrix composites-A review[J]. Journal of Materials Science, 1991, 26: 1137−1156. [4] RAMAKRISHNAN N. An analytical study on strengthening of particulate reinforced metal matrix composites[J]. Acta Materialia, 1996, 44: 69−77. [5] KARANTZALIS A E, WYATT S, KENNEDY ADispersion of ultrafine SiC particles in molten Al-12Si alloyThe bulk Al-12 Si eutectic composites were fabricated through a conventional liquid metal casting route, especially with the help of ultrafine ceramic powders made by self-propagating high-temperature synthesis (SHS) process. The SHS powders were fabricated by the chemical reaction between micro-sized SiC and Al particles at very high combustion temperatures, producing the coarse Al particles (several tens of microns) containing ultrafine SiC ceramic particles. Microstructural observation revealed that the addition of ultrafine SiC particles has a crumbling tendency of Si eutectic phase. It is suggested that the casting method combined with SHS process is promising for fabricating the Al-based MMC with ultrafine ceramic particles.self-propagating high-temperature synthesis (SHS)Fluid systems above basement shear zones during inversion of pre-orogenic sedimentary basins (External Crystalline Massifs, Western Alps)In the inner part of the External Alps, inherited Liassic basins were buried and inverted during the Oligo-Miocene collisional phase of the Alpine orogeny. In northern Oisans, during crustal shortening, the basement was locally sheared while the cover was disharmonically folded above the main basement shear zones that did not propagate into the cover. In this contribution, we analyze the witnesses of paleo-fluid circulations associated with these crustal deformations, focusing particularly on Bourg d'Oisans and Mizoën basins (external Western Alps). On the basis of structural and microstructural observations coupled to geochemical analyses (cathodoluminescence, O and C stable isotopes, trace elements) of vein versus host-rock minerals, we show that in the cover, fluids mainly circulated over short distances (closed-system). However, trace element data also show that percolation of small amounts of basement-derived fluids occurred over several tens of meters in cover rocks right above basement shear zones. Indeed, the three successive vein sets recognized in the field display enrichments in basement-derived Ni, Co, and Cr, which indicate that fluid transfer from the basement was efficient since the beginning of basin inversion, therefore confirming the synchronous deformation of cover and basement. Fluid temperatures and pressures are estimated (microthermometry coupled to δ18O of vein minerals) to about 250–400 °C and 2–5 kbar for veins that most likely formed at or close to metamorphic peak conditions. These results coupled to literature data are finally integrated into a model of fluid circulation evolution through progressive deformation of the whole external Western Alps.In convergent settings (subduction or collision), large amounts of fluids are released in rocks by successive metamorphic dehydration reactions occurring during burial (e.g., ). The occurrence of fluids in rocks, in particular water, has crucial effects not only on the scale of mass transfer processes and fluid–rock interactions (e.g., ), but also on the deformation mechanisms and rock rheology (e.g., ; and references therein). Moreover, there is a strong link between mass transfer and deformation mechanisms (). Indeed, the scale of fluid circulation and mass transfer through rocks is mainly controlled by the size and connectivity of the deformation structures, and their evolution through time (). The permeability of high-pressure rocks being low and fluid pressure close to lithostatic (), most rocks may behave as almost closed-systems, experiencing only small-scale (mm–dm) diffusive mass transfer through the pervasive fluid produced locally by dehydration reactions (). In these rocks, fluid flow may be channelized in highly deformed zones (shear zones, faults), which form localized preferential pathways for large-scale (m–km) advective mass transfer; open-system fluid–rock interactions are restricted to mm–m scale halos in the surrounding rocks (However, it is still unclear how open systems develop in previously closed-system rocks and what are the transitional stages. Yet these stages are keys for understanding the fluid system evolution and constraining the (early) rheological evolution of the continental crust when it is subducted or underthrusted. It is therefore a major issue to characterize the fluid system(s) at that time, i.e., the fluid source, the circulation timescale and pathways, the intensity of fluid–rock interactions along pathways, as well as the evolution of the fluid system with progressive deformation, especially during the early stages.Such a study has been performed in the External Crystalline Massifs (ECM) of the external Western Alps. During the Alpine collision phase, this proximal part of the European passive margin was buried to mid-crustal depth below the internal Alpine units at Oligo-Miocene times (). The ECM experienced mainly thick-skinned deformation during collision: shortening was accommodated by hundred meter-wide shear zones in the basement, while the overlying sedimentary cover was disharmonically folded (). In the northern ECM, i.e., the Mont-Blanc and Aar–Gothard massifs (), the fluid system in the sedimentary cover nappes remained closed to external fluid infiltration, even in highly cleaved metasediments (e.g., ). However, the major cover thrusts or mylonites and associated veins record the infiltration of important amounts of basement-derived fluids (). Indeed, the major basement shear zones propagated as thrusts into the cover, resulting in local opening of the fluid system: ascendant fluids were then channelized within the thrust zones and flowed through both basement and cover (). Further North, in the Glarus thrust, a localized fluid flow occurred at the basal contact of the sedimentary nappes, as attested by a clear isotopic front due to northward metamorphic fluid flow (Was the fluid evolution similar in the southern ECM (i.e., North Oisans), which underwent less burial and shortening? In the north-eastern part of the northern Oisans massif (), the cover is locally detached from its basement, which is consequently not involved in crustal shortening (i.e., local thin-skinned tectonic style; ). There, cover rocks behaved as a closed-system during the whole deformation process; fluid circulations were restricted to the sedimentary unit scale (). On the contrary, the north-western part of the Oisans massif is characterized by thick-skinned deformation (). However, basement shear zones did not propagate into the cover. Each of these structures having accommodated an amount of shortening of only a few hundred meters they are most likely key features on which one can study the early fluid circulations and probably the transition from closed to open fluid system.The questions we address in this contribution are the following: what is the scale of the fluid system and its evolution with progressive deformation? As the basement shear bands did not propagate into the cover, how is the deep fluid circulation (if any) accommodated at the basement–cover interface? Can the fluid system give information about the relative timing of basement and cover shortening during the margin inversion? In order to answer these questions, we combine the structural and microstructural analysis of basement shear zones and the overlying metasedimentary cover in the northern Oisans massif with geochemical and microthermometric investigation of the successive vein filling material and host-rocks.The external zone of the Western Alps arc consists of fold-and-thrust belts (Vercors, Chartreuse, Bauges, Bornes, Aravis, Haut Griffre) and External Crystalline Massifs (ECM, Argentera, Oisans, Grandes Rousses, Belledonne, Mont Blanc, Aiguilles Rouges, Aar, Gothard; ). It corresponds to the proximal part of the European margin, thinned during Liassic to Dogger times, with the formation of tilted blocks () limited by normal faults oriented N–S to NE–SW in the ECM (During the collisional phase of the Alpine orogeny, the ECM were buried down to mid-crustal depth below the internal (Penninic) units. The ECM burial was deeper in the North (400 °C, 5 kbar in the Mont Blanc massif; ) than in the South (270–360 °C and 2–5 kbar in the Oisans massif; During collision, basement shortening was accommodated by West-verging reverse shear zones (e.g., ) or anastomosed steep shear zones (e.g., ). In the northern Oisans area, the cover was mainly disharmonically folded over basement West-verging shear zones, without significant décollement between the basement and its sedimentary cover (). The main difference between the northern and the southern ECM is that, in the North (Mont Blanc, Aar), basement shear zones propagated into the cover, while in the South (Oisans) shear zones were restricted to the basement and did not propagate into the cover. Moreover, basement shear zones and the associated significant crustal shortening are mainly observed in inverted pre-orogenic Liassic basins (). On the contrary, where the crust was not pre-structured by Liassic extensional basins (e.g., La Grave area in the north-eastern Oisans; ), there is less field evidence of significant crustal East–West shortening in the basement, and the cover is detached from the basement, i.e., the shortening style is locally thin-skinned. A striking point in all the basins is the absence of any significant reactivation of the inherited normal faults (The Bourg d'Oisans and Mizoën basins are pre-orogenic Liassic–Dogger N–S extensional basins bounded by East-dipping normal faults, the Col d'Ornon and the Mizoën faults, respectively (). Their normal offset values were of 3 and 2.5 km, respectively (). The base of the sedimentary cover is composed of a thin Triassic layer (few tens of m) of sandstones and dolomites with scarce gypsum lenses (). The Bourg d'Oisans basin underwent important subsidence during Liassic (Sinemurian) to Dogger (Bajocian) times, which resulted in the deposition of thick marls or marly limestones. The Mizoën basin was also filled with marls and shales but mainly during Domerian to Bajocian subsidence (In the northern ECM (Mont Blanc and Aar massifs), most studies so far concluded that Alpine fluids were channelized in the major basement and cover shear zones (). Indeed, at the proximity of large thrusts (i.e., within a few meters), oxygen isotopic values (δ18O) in shear zones and associated veins are locally much lower than in the surrounding rocks within the cover nappes (e.g., ), which is interpreted as the result of large-scale fluid flow from deeper (basement) parts of the shear zones. However, apart from the basal thrusts of the largest nappes, similar values of oxygen isotopic compositions (δ18O) were observed in the cover shear zones and the surrounding rocks within the nappes (), suggesting that these rocks underwent little to no advection of isotopically distinct fluid.In the southern ECM and especially in the Oisans area, few geochemical studies have been performed and none of them provided any evidence of large-scale fluid circulation. In particular, carried out a geochemical study of the metasedimentary rocks of La Grave (), where the cover is detached from its probably undeformed underlying basement (). Isotopic and trace element signatures of metamorphic vein filling material and their host-rocks show that the metasedimentary cover behaved as a chemically closed-system, locally-derived fluid circulations being restricted to the metasedimentary Jurassic units (i.e., few tens of meter scale; Microthermometric studies all over the Bourg d'Oisans basin carried out on fluid inclusions trapped within vein mineral phases also suggest the absence of large-scale fluid circulation in the Oisans Jurassic cover, although high salinities recorded in some fluid inclusions in the deeper Liassic layers were interpreted as due to a local percolation of fluids from the underlying Triassic gypsum (In both the Bourg d'Oisans and Mizoën basins, basement shortening was accommodated by West-verging 100–500 m wide shear zones () restricted to the basement (i.e., they did not propagate significantly into the cover) and that cut across the Variscan foliation (A), without any evidence for its reactivation. The two basins have slightly different styles of shortening. In the Bourg d'Oisans basin, there are five units () that are separated by four West-verging shear zones named Bourg d'Oisans, Huez, Col de Cluy, and Croix de Cassini shear zones (A). Above these basement shear zones, the sedimentary cover is disharmonically folded. The observed meter-to-kilometer scale fold axes are N–S and synchronous with the main cleavage. The Mizoën basin is divided into two parts: the eastern part is a basement horst named the Emparis Plateau, with a thin metasedimentary cover (A), whereas the western part is composed of a thicker disharmonically folded metasedimentary cover (A) underlain by the main West-verging basement shear zone (the Mizoën shear zone). Just West of the Emparis Plateau, the Mizoën metasedimentary cover is composed of two structural units separated by the Alp décollement (A). Note that the base of the sedimentary cover is composed of a thin Triassic layer (a few tens of meters wide) attached to the basement, i.e., not disharmonically folded.Several cleavages affecting the metasedimentary cover are documented in the field (). In the Bourg d'Oisans basin, one main cleavage (S2) is observed (), striking N–S and dipping eastward. This cleavage is associated with the main folds whose axial surface is East-dipping (). Close to the basement of the Bourg d'Oisans basin, a previous cleavage (S1) can also be documented (B), striking N–S and dipping westward. The chronological relationships between the two cleavage sets are described in : during the initiation of the basin inversion, the base of the sedimentary cover underwent a top-to-the-East shearing that locally formed the West-dipping S1 cleavage. This cleavage was then overprinted by the S2 main cleavage resulting from the top-to-the-West shearing of the entire basin (). In the Mizoën basin, the S1 cleavage is only recorded below the Alp décollement, while the S2 cleavage developed only above it (During the development of the two cleavages, metamorphic veins were continuously formed and deformed in the metasedimentary cover. Vein deformation occurred either by transposition into cleavage or by folding (i.e., ptygmatic vein; ). In the field, three successive vein sets could be distinguished owing to their crosscutting relationships with S1 and S2 cleavages (). V1 veins were deformed by both cleavages (C), V2 veins cut across S1 but were deformed by S2 cleavage (A), and V3 veins are undeformed and cut across both cleavages (B). V1 veins are interpreted as pre- to syn-S1 cleavage, V2 veins as pre- to syn-S2 cleavage and V3 veins as late- to post-S2 cleavage. Indeed, most V3 veins are perpendicular to S2 but undeformed, therefore compatible with the end of the ductile event forming the S2 cleavage. Therefore, most of the veins are related to the main ductile deformation event associated to the development of S1 and S2 cleavages (), except the latest V3 veins that can be related to the basin exhumation.Locally, where only one cleavage is expressed, only two vein generations can be distinguished, e.g., far from the basement in both basins, where S1 and S2 cannot be distinguished. In this case, V1 and V2 veins cannot be discriminated and are labeled as “early veins” (B–D). Moreover, under the Alp décollement in the Mizoën basin, where S1 only is expressed, V2 and V3 veins are undistinguishable and considered as “late veins”. The structural relationships between vein sets and cleavages are summarized in In order to investigate the scale of fluid circulation in the metasedimentary cover rocks in the Bourg d'Oisans basin and to identify potential basement-derived fluid infiltration, we collected samples in the Huez basement shear zone and in the cover right above, at different distances above the basement–cover interface (). Basement samples from the Huez shear zone itself were sampled on the road between La Garde and l'Alpe d'Huez, a few meters under the basement–cover contact (location 3 in A), along with Triassic samples. Cover samples were collected in two places: (1) a few tens of meters above the basement–cover contact on the road between La Garde and l'Alpe d'Huez (called “Huez samples” thereafter; location 1 in A) and (2) about 300 m structurally higher, on the road between Huez and Villard-Reculas (called “Villard-Reculas samples” in this study; location 2 in We also collected samples from the cover in the Mizoën basin, between the Mizoën village and the Emparis plateau, from both tectonic units separated by the Alp décollement, i.e., above (location 4 in A), in order to investigate the possible effect of this tectonic contact on potential fluid circulations. Unfortunately, the underlying basement shear zone could not be sampled, because (1) it does not crop out around the Romanche valley (A) and (2) further South, in the Veneon valley, the sheared basement is accessible, but according to , the lithology is quite different than in the Romanche valley.Most analyses were performed at ISTeP (Paris, France), except when specified in the following. Whole-rock analyses were performed at the SARM (Nancy, France) on six representative cover and basement samples from the Huez area (locations 1, 2, and 3 in A). The vein and host-rock mineralogy was determined by X-ray diffraction (D2 Phaser Bruker) and EDS analyses (SDD detector PGT Sahara) associated to scanning electron microscopy (SEM Zeiss Supra 55VP). Cover vein microstructures were characterized under SEM and optical microscope. Cathodoluminescence (Cathodyne OPEA) on cover veins and host-rocks was used to identify successive crystallization generations. To investigate local versus external fluid infiltration in metasedimentary cover rocks, the isotopic signature (δ13C, VPDB and δ18O, SMOW) of vein calcite was compared to that of their surrounding host-rocks. The δ18O of quartz (laser fluorination at IPGP, Paris, France) and calcite in textural equilibrium in the different vein sets were combined in order to constrain the fluid temperature during vein mineral crystallization. Trace element analyses were performed (ICP-MS, following the methodology of ) on the different vein sets and their host-rocks, and on samples from the underlying basement (for the Huez shear zone) in order to detect any basement-derived fluid infiltration into cover rocks. Besides, microthermometric measurements were performed on primary fluid inclusions in quartz and calcite from Huez and Mizoën (locations 1 and 5 in A, respectively) cover veins, providing fluid salinity and isochoric evolution paths. Finally, the combination of calculated isochors and quartz–calcite equilibrium temperature provided fluid pressure estimates during the vein formation. More details about the analytical procedures can be found in In the Bourg d'Oisans basin, the Huez Alpine shear zone (location 3 in A) presents a thick talc-rich zone of a few meters width. This talc-rich zone is composed of rocks with very mafic Ca-rich composition () exhibiting the following minerals: talc, Cr-rich chlorite, Mg–Ca-amphibole, plagioclase, Ni-rich pyrite, sphene, chromite, rutile and titanomagnetite (). Talc, chlorite and sphene are observed to have co-crystallized in shear zones, where they clearly replace amphibole (A) and plagioclase. Pyrite cores included in amphibole crystals are rimed by iron-oxide or hydroxide (B), which formation is associated and coeval with talc crystallization in cracks through the amphibole crystals. Zoned chromite crystals are partly dissolved and embedded in Cr-rich chlorite (C, D, E). These mineralogical reactions required both rock hydration (to form phyllosilicates) and progressive evolution towards more oxidizing conditions (i.e., pyrite rims).Therefore, two successive parageneses can be identified: (1) the assemblage amphibole + plagioclase + pyrite + chromite + rutile corresponds to the Variscan paragenesis also observed by in the unsheared basement and is in agreement with description by , and (2) the assemblage talc + chlorite + sphene + Fe-oxide/hydroxide + titanomagnetite is obviously related to basement shearing, accompanied by external fluid influx, during the Alpine collision. Indeed, this second paragenesis grew in Alpine shear zones, and the presence of chlorite is characteristic of the greenschist facies P–T conditions undergone by basement rocks during the Alpine collision.In the talc zone, numerous quartz veins and scarce calcite veins (e.g., sample Alp10-5 in the following results) are kinematically consistent with Alpine shearing, and thus considered of Alpine age.The Triassic layers, still attached to the basement rocks, are mainly composed of sandstone, dolomite and scarce lenses of gypsum (). Above the Huez basement shear zone, fractures in the Triassic layers are mainly filled with calcite, dolomite and quartz, with local enrichments in galena (PbS), pyrite (FeS2) and/or sphalerite (ZnS). These minerals can be locally concentrated, e.g., in the La Gardette () ancient gold mines: these sulfur minerals crystallized in a 500 meter long quartz vein affecting both the basement (granite and gneiss) and the lowermost Triassic dolomites. In La Gardette, similar sulfur minerals are also concentrated in a specific level of the Triassic dolomite, where they were interpreted as syn-sedimentary enrichments (In both the Huez and Mizoën basins, the metasedimentary cover is made of marls () containing calcite, quartz, phyllosilicates (mainly pyrophyllite, chlorite, phengite and rare cookeite), iron oxides, rutile, and pyrite (). Framboidal pyrite is observed and is interpreted as formed during early diagenesis (). The pyrites are often partially to completely oxidized (A). Detrital albite grains and rare dolomite (B) and apatite crystals were also observed. The phyllosilicate paragenesis is characteristic of the Alpine greenschist facies metamorphism. The analyzed samples () show a large compositional variability, which results in variable proportions of calcite (30–90 vol.%) versus quartz (0–20 vol.%), phyllosilicate fraction (5–40 vol.%) and organic matter (0.5–1.5 wt.%; modal composition calculations according to Finally, host-rocks show calcite dissolution zones (A), attesting material displacement by pressure-solution into the sedimentary cover.In both basins, veins are mainly filled with calcite and quartz, the quartz amount decreasing from V1 to V3, so that many V3 veins are devoid of quartz and only filled with calcite. Calcite and quartz display either fibrous crystals elongated perpendicular to the vein walls (“fibrous texture” in the classification of C) or euhedral morphologies (“blocky texture” in C, D, E), whatever the vein generation. Both textures can coexist in synchronous veins in the same thin section (e.g., late veins in C), and mixed textures are also observed.Quartz and calcite crystals show multiple evidences of textural equilibrium: they often show mutual micro-indentations (intergrowth textures), and some calcite crystals are embedded within quartz crystals, themselves surrounded by calcite crystals of similar composition (F). The textural equilibrium of calcite and quartz in veins indicates their co-precipitation.Scarce accessory micrometric phases such as dolomite, apatite, oxidized pyrite and chlorite can be recognized with SEM.Several samples from the three vein generations (from both the Bourg d'Oisans and Mizoën basins, locations 1–2 and 4–5 in A, respectively) were analyzed under cathodoluminescence. Two representative V1 and V2 veins are illustrated in G–H. In V1, V2 and their host-rock, calcite displays the same red color, with the same intensity (H). Quartz in V1 vein and other phyllosilicate phases in the host-rock appear in black and lighter red, respectively (H), as they do not incorporate the same amount of luminescent elements. A similar red color was observed in V3 veins and their host-rocks on other samples. These observations indicate that the nature and concentration of luminescent elements incorporated in calcite did not vary during the successive vein crystallization, and are similar to that of the host-rock calcite. In these samples, calcite appears red although pure calcite is usually orange under cathodoluminescence microscopy (): this may indicate a high concentration in red luminescent elements such as manganese or magnesium.The absence of variation in either color or intensity of calcite within the veins precludes any evidence of successive growing phase in any vein generation.δ18O and δ13C signatures of calcite from cover veins and their host-rocks from both basins (sample locations 1, 2, 4, and 5 in ) along with one calcite vein from the talc-rich zone of the Huez basement shear zone (sample location 3 in . In the Bourg d'Oisans basin, the isotopic ratios, whatever the vein generation, are very close to that of their host-rock (B–C). δ18O and δ13C composition ranges of most veins and host-rocks are relatively narrow: + 18 to + 23.5‰ and − 2 to + 1.5‰, respectively (A). These variations mainly reflect lithological variations: the most pelitic (i.e., phyllosilicate-rich) samples exhibit the lowest δ18O and δ13C signatures. In the Mizoën basin, the isotopic signatures of each vein are also very close to those of their host-rocks (B–C). Two groups can be identified: one group has isotopic composition within the range of Bourg d'Oisans rocks (δ18O = + 21.5 to + 23.5‰, δ13C = − 1 to + 1.5‰, A), while the second group has lower δ18O (+ 19.5 to + 21.5‰) and δ13C (− 2.5‰ to − 6‰, A). This bimodal distribution is correlated to the rock lithology: all samples with negative δ13C (pelitic layers on A) belong to the Aalenian layer, which is particularly rich in phyllosilicates and organic matter, while other compositions correspond to samples collected in marls.Cover vein and host-rock calcite presents very homogeneous isotopic signatures (B–C). On the contrary, the isotopic signature of the calcite vein from the Huez basement shear zone shows a strikingly different value (δ13C = − 9.2‰ and δ18O = + 13.3‰, A) compared to the cover vein calcite. In the Triassic layer, the isotopic composition of the vein (δ13C = − 5.3‰, δ18O = + 19.5‰) is very different from that of its host-rock (δ13C = − 1.5‰, δ18O = + 25.7‰, A), and intermediate between Triassic host-rock and basement vein composition (The δ18O signatures of quartz crystals in textural equilibrium with calcite in cover veins from both basins (sample locations 1, 2, 4, and 5 in A) are relatively homogeneous for most veins (δ18O = + 22.3 to + 26.3‰, ) whatever the sampling site or vein generation.Microthermometric measurements were performed on about 80 fluid inclusions in either quartz or calcite crystals from V1 or early (V1/V2) cover veins from both basins (locations 1 and 5 in ). In order to characterize the fluid phase present in cover veins during mineral crystallization, we focused on primary fluid inclusions, which are interpreted as being trapped during crystal growth (). Primary fluid inclusions are distributed either in trails along successive crystal growth surfaces (). In veins showing fibrous textures, we also analyzed the pseudo-secondary trails of fluid inclusions (For each fluid inclusion, the salinity of the fluid was calculated from the measured melting temperature (Tm) with the equation presented in . Measured Tm is between − 1 °C and − 10 °C (dispersion of ± 3 °C for each sample), which corresponds to 1.74wt.% eq. NaCl and 13.94wt.% eq. NaCl, respectively. Measured homogenization temperatures (Th) are between 125 °C and 198 °C (dispersion of ± 12 °C in a single sample). Then, isochoric P–T relationships were calculated for each sample from the mean measured Th and mean calculated salinity, using the equation of ) are almost parallel, with P–T relationships of 16.5 MPa·C− 1 to 21 MPa·C− 1, whatever the basin (Bourg d'Oisans or Mizoën).For both basement and cover rocks, trace element concentrations of 3d transition elements (Sc, V, Cr, Co, Ni, Cu and Zn) and HFSE (Zr, Hf, Nb, Ta, Mo, Sb) were normalized to mean crust composition (A, B) to allow comparison. However, basement rocks have Ca-rich basic compositions, and cover rocks are Triassic dolomites and Liassic marls, thus these rocks display accordingly specific patterns relative to the mean crust composition. As the aim of this study was to trace potential basement-derived fluid infiltration in cover rocks, trace element patterns normalized to mean crust composition are discussed hereinafter in terms of relative variations of shapes rather than in terms of absolute values. The trace element patterns of sheared and unsheared basement, basement and cover rock, and cover vein versus host-rock are also compared. Trace element values can be found in Trace elements were analyzed in samples collected both in the Huez basement shear zone (including the talc-rich area, D) and just outside the shear zone in less deformed rocks.Basement trace element concentrations are presented on ). For most elements, trace element patterns of samples from the sheared basement are similar to those of the unsheared basement with however lower absolute contents, except in Ni, Co and Cr (A). These lower concentrations in sheared basement samples reflect either trace element depletion in basement rocks during shearing or trace element dilution effects due to depletion or enrichment in some major elements.Concerning the sheared basement, trace element compositions in samples from the shear zone outside of the talc-rich zone follow the same trend than the talc-rich zone samples (A–D), except for the Cr, Co, and Ni concentrations that are clearly higher in the talc-rich zone: Cr, Co, and Ni patterns are even inversely correlated in both zones. This suggests that the shearing of basement rocks has mainly affected the compositions in Cr, Co and Ni, probably through fluid-assisted mobilization, while other 3d transition elements and HFSE (known as the least mobile) remained unaffected.In the talc-rich shear zone, the calcite vein from sample Alp10-5 has a similar pattern (except in Zn and Mo) than its host-rock, with bulk depletion in all trace elements.Trace element concentrations of cover rocks are presented normalized to the mean crust ones () whereas the vein trace element concentrations are presented normalized to their host-rocks (Composition patterns of Triassic dolomites are slightly enriched in Zn, Mo, and Sb compared to the mean crust composition (B). The associated veins are enriched in Zn ± Sb and V compared to their host-rocks (In Liassic layers above the Huez shear zone, the samples have very similar trace element patterns, whatever their distance to the contact (B): indeed, both Huez (a few tens of meters above the basement, location 1 in D) and Villard-Reculas (a few hundred meters from the basement, location 2 in D) samples exhibit similar concentrations in all elements (B), with higher concentrations in V, Zn, Mo and Sb than mean crust composition (B). The variability of trace element concentrations in marls may reflect their heterogeneous calcite content (30–90 vol.%, All the veins sampled just above the Huez basement shear zone (location 1 in D) present strong enrichments in Ni (up to 100 times) and/or Co (up to 10 times), Cr (up to 10 times), Zr and Hf (up to 100 times) compared to their respective host-rocks (C), for V1, V2 and V3 veins. On the contrary, veins sampled farther from the basement–cover interface (location 2 in D) display a bulk depletion in all trace elements compared to their host-rocks, and do not present any particular trace element enrichment (In samples from the metasedimentary cover of the Bourg d'Oisans and Mizoën basins, calcite shows a homogeneous isotopic composition in each lithological unit (A). However, the calcite isotopic signal is not that of a typical marine limestone, and plots between “normal marine limestones” and “metamarls-metapelites” isotopic values (. This means that isotopic reequilibration occurred among the different minerals (quartz, calcite, phyllosilicate fraction, organic matter) during burial. Following , we conclude that these cover rocks attained complete isotopic reequilibration under greenschist facies conditions.As a consequence, the isotopic signature of host-rocks seems to be mainly controlled by the lithology, i.e., it depends on the calcite–quartz–organic matter–phyllosilicate fraction relative amounts. This is the reason why the Aalenian samples from Mizoën (“pelitic layers” on A), which are much richer in phyllosilicates, present different isotopic signatures (i.e., lower δ18O and δ13C) than all other Liassic marls. These isotopic ratios are in agreement with those analyzed all along the ECM cover, in metapelites (Aalenian samples in . Note that rocks from the La Grave basin (East of our study area) being globally richer in pelitic fraction than our samples, their isotopic signatures are accordingly slightly lower than our Aalenian Mizoën samples (; reported as “ultra-Dauphinois metapelites” in However, it is noteworthy that although the samples from a specific lithological layer have highly variable calcite–phyllosilicate relative proportions (e.g., Bourg d'Oisans samples in ), their isotopic signatures are very close (), meaning that the rock isotopic composition was equilibrated at the scale of the lithological layer.In both the Bourg d'Oisans and Mizoën basins, calcite isotopic compositions are very similar in veins and associated host-rocks, whatever the vein generation, suggesting local fluid–rock equilibrium in cover rocks all along the successive vein formation. This vein–host-rock isotopic equilibrium precludes massive external fluid influx during deformation (e.g., ; Marquer and Burkhard, 1992). Indeed, basement-derived fluids may have very different isotopic signatures, as suggested by the calcite vein analyzed in the Huez basement shear zone (), and in agreement with the isotopic signature estimated for basement fluids by ). Therefore, the percolation of important amounts of basement-derived fluids into the overlying cover veins would have modified their isotopic signatures, as observed in the northern ECM basal thrusts of the sedimentary nappes (e.g., In our rocks, the absence of oxygen isotopic fractionation between host-rock and vein calcite suggests that vein formation occurred at P–T conditions high enough to allow the host-rock mineral isotopic reequilibration mentioned above (i.e., probably close to the peak P–T conditions). Therefore, as indicated by the scale of isotopic reequilibration mentioned above, fluid circulation in the cover rocks was restricted to the scale of the sedimentary formation, i.e., tens to a hundred meter scale. This is supported by observations under cathodoluminescence where calcite from veins and host-rocks exhibits very similar color, whatever the vein generation (G–H), suggesting no evidence of massive external fluid influx in those rocks. have drawn the same conclusion of a closed-system fluid circulation at the unit-scale for similar cover rocks from the La Grave basin.Both isotopic and cathodoluminescence homogeneity of calcite in veins and host-rocks thus suggest a local origin for the vein calcite. Dissolution patterns of calcite crystals in the host-rock close to vein walls (A) confirm that calcite was partly dissolved by pressure-solution processes and locally transferred to the successively opening fluid-filled veins where it crystallized. Quartz and calcite pressure-solution and recrystallization in adjacent veins are common deformation mechanisms observed during burial () and which have already been described in this area (). As there is no evidence for infiltration of an important amount of external fluid, diffusive mass transfer likely occurred through the pervasive fluid produced locally by the metamorphic dehydration reactions undergone by clay minerals (i.e., phyllosilicate fraction of the marls) during burial (). Making simplified assumptions on the clay mineralogy of the marls protolith (i.e., kaolinite, smectite, illite; see for further details), modal composition calculations suggest that about 0.1–2 vol.% of fluid may have been produced by metamorphic reactions during the Alpine burial.However, several authors state that the salinity increase in fluid inclusions from cover veins with the proximity to the Triassic dolomites () supports the idea of small-scale percolation of Triassic-derived NaCl-rich fluids in the overlying Liassic cover. This is consistent with our salinity data obtained from fluid inclusions in veins from Huez (i.e., just above the Triassic layer, location 1 in A), which correspond to the highest salinity values measured by these authors for the whole Bourg d'Oisans sedimentary cover. Moreover, the isochors calculated from our fluid inclusion microthermometric data () are in good agreement with those calculated from previous data and reported in On the contrary, the isotopic signature of calcite from Triassic layer veins is clearly different (i.e., lower δ18O and δ13C) from that of the host-rock calcite (A). This isotopic disequilibrium between Triassic vein and host-rock (B–C) suggests external fluid infiltration through the vein during calcite precipitation. In addition, the Triassic vein isotopic ratio is intermediate between Triassic host-rock and basement isotopic ratios (), suggesting that the thin Triassic layer underwent basement-derived fluid infiltrations.From isotopic data, it can be concluded that there was no massive influx of external fluids in Liassic layers. However, trace element analysis gives complementary insights on the source and scale of fluid circulation. Above the Huez shear zone, veins from Villard-Reculas, situated a few hundred meters above the basement (location 2 in D), have very similar composition patterns than their host-rocks (C), which is consistent with a local origin of both vein filling elements and fluid (D). On the contrary, in the Huez samples situated just above the sheared basement–cover interface (location 1 in D), veins present strong relative enrichments in Cr, Co, Ni, Zr, and Hf compared to their host-rock (C). Interestingly, the underlying mafic basement rocks (location 3 in D) present similar trace element patterns in both sheared and unsheared zones for all elements but Cr, Co, and Ni, suggesting that their concentration may have been modified during shearing: these elements are specifically enriched in the talc-rich zone of the sheared basement (A). Moreover, SEM analysis revealed that, in this talc-rich zone, Ni- and Cr-bearing minerals are Variscan pyrite cores and chromite crystals, respectively (). Both minerals were partially dissolved and replaced by Alpine hydrated phases, as described in detail below, suggesting a release of both Ni and Cr in the infiltrating fluid phase during the Alpine basement shearing.Therefore, there are clues that small amounts of basement-derived fluids percolated into the cover just above the Huez shear zone (D). However, these fluid amounts must have been very limited: the isotopic signature of this basement-derived fluid, which was obviously very different from that of cover rocks (i.e., calcite basement vein, A), affected only the Triassic layer (still attached to basement rocks), and was buffered by the isotopic signature of Liassic cover rocks. Moreover, the scale of infiltration of these basement-derived fluids was spatially limited to a few tens of meters in the cover (D). Indeed, in Villard-Reculas (location 2 in D), the trace element signal of basement-derived fluids is not detected anymore. The very limited flux of percolating fluids through the cover rocks is in agreement with the structural observation that veins are mostly isolated structures and do not constitute a connected network in the metasedimentary cover. It is interesting to note that the three successive vein generations of Huez samples present a similar basement-derived trace element signal. Therefore, there is no evidence for significant change in the scale of the fluid system during progressive deformation.Moreover, in the Triassic dolomites still attached to the sheared basement, sphalerite, pyrite, and galena were observed to have crystallized in calcite fractures. Accordingly, compared to their host-rocks, veins in these Triassic layers show enrichments in some trace elements (Zn ± Sb; B) that are locally concentrated in the underlying talc-rich Huez basement shear zone (A), which supports the hypothesis of local basement-derived fluid percolations into the cover, through the basement–cover interface, above basement shear zones. Interestingly, in La Gardette ancient gold mine (situated in the Bourg d'Oisans basin), similar basement-derived fluid percolation across the unconformity was observed, although this zone is not situated above a basement shear zone: a 500 meter long quartz dyke enriched in sulfur minerals is rooted in the Pelvoux granite (source of the sulfur minerals; ), affects the overlying gneiss and ends in the first 30 m of the Triassic dolomite (). However, the source of this local basement fluid circulation remains unclear (Therefore, this study shows that fluid circulations in the Oisans sedimentary cover were restricted to the lithological unit scale, i.e., the fluid system was almost closed. However, when shortening started, small amounts of basement-derived fluids percolated above the basement shear zones into the cover on a few tens of meter scale. These conclusions are in agreement with those of who showed that Alpine marine sediments were generally metamorphosed under closed-system conditions whatever the locality and the metamorphic grade, i.e., element mobility from the sediments is negligible except locally, in narrow shear zones where some specific elements (including Ni and Cr) can be strongly mobilized by the circulation of ascendant fluids.The mobilization of specific elements (Cr, Co, Ni) from basement rocks provides insights into the basement fluid properties during shearing. Samples from the talc-rich area of the Huez basement shear zone () show enrichments in Cr, Co, Ni (± Zn and Sb; A) relative to the mean crust, similarly to talc and chlorite schists observed in other settings (e.g., ). The retrogression of the amphibole + plagioclase Variscan paragenesis to Alpine phyllosilicates (chlorite + talc; A) required fluid influx during shearing. Moreover, the partial replacement of the Variscan pyrite crystals by Fe-oxide or hydroxide (i.e., rims around pyrite cores, B) during Alpine fluid infiltration (i.e., linked to talc formation in amphibole cracks) and the partial dissolution of Variscan chromite crystals in Alpine shear zones (C–D–E) suggest a progressive evolution towards more oxidizing conditions in basement rocks with their shearing. Indeed, the internal fluid in equilibrium with the Variscan paragenesis was likely reduced, as shown by the presence of both pyrite and chromite, whereas the external fluid responsible for Alpine phyllosilicate formation had to be more oxidizing.Interestingly, the pyrite cores are Ni-rich whereas Ni is absent from the Fe-oxide or hydroxide rims, which means that Ni may have been released into the fluid phase during pyrite Alpine oxidation. Similarly, Cr-rich chromite crystals are partially dissolved and surrounded by Cr-rich Alpine chlorite, suggesting Cr release during the Alpine shearing (C–D–E). Therefore, the oxidizing fluid circulating in the basement shear zone may have become enriched in SO42 − while dissolving sulfur minerals such as pyrite, which was obviously a good complexing anion for metallic elements such as Cr and Ni that were released coevally by pyrite and chromite partial dissolution. Similar mobilization of metallic low-mobility elements (e.g., Ni, Cr, Zn) by S-rich fluids was observed by in subduction zone settings. This SO42 − fluid enrichment is supported by the presence of small barytine crystals (identified by SEM) in both the calcite vein in the basement talc-rich zone (Alp10-5, A) and in a Cr–Ni-rich V1 vein in Huez cover rocks, as well as the precipitation of various sulfur minerals (such as sphalerite, galena, pyrite) in Triassic veins above the shear zone. Element mobility was thus clearly controlled by both basement rock mineralogy (enriched in transition metals) and fluid composition, i.e., pH, fO2 and ligands (e.g., ), as was observed for REE complexing and mobility in the Mont Blanc basement shear zones (Small amounts of this basement-derived fluid enriched in transition metals (Cr, Ni) percolated into the overlying metasedimentary cover. Its mixing with the CO2-rich internally-produced cover fluid, locally at equilibrium with the cover carbonates, may have induced Ni and Cr release in veins within the Lias sediments, and similar sulfur mineral precipitation when percolating through the Triassic layer; the metallic element precipitation may have been caused by sudden changes in fluid pH conditions and XCO2 (e.g., Here, we combine the analysis of quartz and calcite isotopic signatures in veins to microthermometric measurements of fluid inclusions from the same veins to estimate both the temperature and pressure of the cover vein fluids above basement shear zones. Indeed, quartz and calcite present evidences of textural equilibrium in most veins (F), which implies their co-precipitation under similar P–T conditions. Oxygen isotopic fractionation being essentially dependent on temperature, the temperature of quartz + calcite co-precipitation in veins was estimated from the independent analysis of their δ18O signature (equation in As most late veins only contain calcite and no quartz, all the veins analyzed in terms of P–T conditions are V1 or early veins (except Alp10-18, ). Above the Huez shear zone (locations 1 and 2 in A), the quartz–calcite equilibrium temperatures range from 144 to 421 °C (± 30 °C, ), but most samples plot in the 250–360 °C range (locations 1 and 2 in ). Most of these calculated vein temperatures are consistent with the previous peak temperature estimates for the Bourg d'Oisans and the Mizoën basin cover rocks (300–350 °C, ). For a few veins from Huez, combining quartz–calcite equilibrium temperatures to the isochors determined from microthermometric analysis of fluid inclusions in the same quartz or calcite crystals () provides estimates of the vein fluid pressures (). The fluid pressures range from 1.7 to 5.1 kbar (), which is quite similar to the lithostatic pressure range estimated for the metamorphic peak conditions in these rocks (2–5 kbar, estimated fluid pressures of 2 kbar from microthermometric data coupled to temperature data estimated by for basement vein fluid inclusions (Na/K ratios), which corresponds to the low pressure bound estimated here. However, recalculating the isochors from the microthermometric data of ), and therefore would undoubtedly tend to higher pressure estimates.Moreover, as stated before, the δ18O values for vein calcite are very close to that of the host-rock calcite (A), suggesting that most veins formed close to the peak temperature conditions. However, some V1 veins may have formed during the end of prograde path because (1) the veins are filled by local infiltration of fluid released by prograde metamorphic dehydration reactions into the host-rock and (2) their fluid temperature can be as low as 144 °C.For the Mizoën basin (locations 4 and 5 in ), the calculated vein temperatures range from 306 to 436 °C, but most temperatures are above 400 °C, which exceeds the estimated peak temperature in the Oisans cover rocks. As discussed in , these higher values could be the result of an incomplete quartz powder decarbonation for some samples, due to the particularly fine micro-indentation of quartz and calcite in veins.For the Mizoën basin (locations 4 and 5 in ), the calculated vein temperatures range from 306 to 436 °C, but most temperatures are above 400 °C, which exceeds the estimated peak temperature in the Oisans cover rocks. As discussed in Appendix A.1, these higher values could be the result of an incomplete quartz powder decarbonation for some samples, due to the particularly fine micro-indentation of quartz and calcite in veins.In the talc-rich area of basement shear zones, the abundant crystallization of phyllosilicate minerals (talc and chlorite) required the influx of important amounts of external fluids. This talc-rich zone is only a few meters wide inside the shear zone, suggesting that fluid circulation in the basement may have been channelized in the core of the shear zone. Therefore, basement shear zones served as preferential pathways for crustal fluid circulation, as observed in the Mont Blanc () basement rocks, as well as in other collisional or subduction-related settings (e.g., ). In turn, the crystallization of talc and chlorite may have enhanced deformation localization by progressive softening of these shear zones () similarly to the weakening effect of the feldspar phengitization (e.g., In northern ECM (e.g., Mont Blanc massif, ), the largest basement shear zones may have propagated as thrusts into the overlying sedimentary cover with ongoing deformation, and became pathways for important amounts of basement-derived fluids, causing meter-scale opening of the fluid system, as recorded by isotopic data in the Aar–Gothard massifs (). In contrast, in the Oisans massif, which underwent less shortening than northern ECM, the basement shear zones did not propagate into the sedimentary cover, preventing important circulation of basement-derived fluids in cover rocks, as shown by isotopic and cathodoluminescence data. Nevertheless, the trace element pattern of cover veins situated just above the basement–cover interface (Huez, location 1 in A) recorded the infiltration of small amounts of basement-derived fluids, whereas in eastern Oisans (La Grave), where the basement is not sheared and the cover decoupled from the basement above a décollement, both isotopic and trace element patterns showed that cover rocks were isolated from basement fluid infiltration (i.e., closed system, Even if only small amounts of basement-derived fluids percolated through the interface in western Oisans, they give key insights into the timing of shear zone development and basement shortening. Indeed, the percolation of basement-derived fluids into Huez cover rocks was recorded in the three successive vein generations (V1 to V3), which are all related to the E–W collisional shortening and subsequent basin inversion (). Moreover, the basement shear zones are also kinematically consistent with the East–West shortening. Therefore, the percolation of basement-derived fluids into cover veins, which occurred during the entire basin deformation history, unambiguously witnesses that the shortening of the Oisans massif basement began very early during the inversion of the pre-orogenic extensional basins, and lasted during the whole shortening phase. This is supported by geochronology data: the basement shear zone activation was dated to 30–34 Ma in the Pelvoux massif (), and mineralizations into basement veins give a date of 36–39 Ma (; La Gardette gold mine) for basement fluid circulations. The simultaneous deformation of basement and cover basins during collisional shortening, hence the absence of decoupling between cover and basement, supports an overall thick-skinned structural style.If most of our cover veins are related to the ductile deformation event (V1, V2 and most V3), it cannot be excluded that part of the V3 veins could be associated to late brittle exhumation phase. Late fluid circulation, from 17 Ma to less than 10 Ma, was evidenced from late vertical veins in basement shear zones from the northern Belledonne and the Aar massifs (), showing that fluid circulation lasted until the brittle exhumation of the ECM.Our results, combined with results of other fluid system studies in the ECM () allow proposing a general fluid flow model for the Alpine external zone. This fluid flow model is illustrated together with a schematic kinematic model of the ECM shortening in The overall tectonic framework is the Alpine burial and collisional shortening of the European crust (see simplified section in A). Probably during the end of the crust burial beneath the internal units, i.e., the end of prograde path (B) or at the metamorphic peak, first ductile deformations initiate (S1 cleavage and associated veins, see for more details). At this stage, for example in the present-day eastern part of the Oisans massif (Ultra-Dauphinois, B), the cover has been locally detached from its basement and progressively overthrusted the western inherited basins. In both the allochthonous and autochthonous units, the fluid system is overall closed () with isotopic reequilibration restricted to each lithologic unit. However, it is possible that large-scale fluid circulations occurred in the cover thrusts–décollement (thick blue arrow in . Meanwhile, in the western part, the inherited basins (e.g., Oisans massif, B) were progressively shortened and inverted. Shear zones developed within the basement while the overlying cover was disharmonically folded (e.g., Bourg d'Oisans basin, B). Basement-derived fluids were channelized into basement shear zones, and locally infiltrated the overlying cover over a few tens of meters (e.g., the Huez shear zone described in this study, B). In the rest of the cover, i.e., far from basement shear zones, the fluid system remained closed, restricted to the sedimentary unit scale, and involved mainly formational and metamorphic fluids. To sum up, in the Oisans area (Bourg d'Oisans basin, Huez shear zone, Ultra-Dauphinois nappes, B), collisional shortening remained moderate, about 11.5 km (16.1%, ) and thus we face an “immature” fluid system in the cover of the inherited basins.With increasing shortening, coupled to increasing P–T conditions, basement shear zones developed through the increasing connection of anastomosed shear bands () and were most likely the location of channelized basement-derived fluids (e.g., Aar, ) with important fluid–rock interactions, highlighted by REE fractionation (). In areas where the cover was detached from the basement (e.g., Glarus thrust, C), fluid flow was channelized in the main thrust (). The related presently one meter-thick calc-mylonite () was the locus of δ18O-depleted fluid circulation, as attested by isotopic fronts (e.g., ), where metamorphic fluids derived from deeper (southern) parts of the thrust mixed with formational fluids northward. The extreme localization of the deformation (1 meter-thick thrust zone) may have permitted a very localized and fast fluid flow that explains the isotopic anomalies. Similar fluid flow probably occurred in the Moine Thrust that is also a very thin and localized thrust accommodating tens of km of shortening (). To sum up, in and around the northern ECM (Mont Blanc, Glarus thrust above the Aar massif) where the amount of shortening was higher than in Oisans (10.6 km, 38.3% in the Mont Blanc massif and 12 km, 40% in the Aar massif vs. 11.5 km, 16.1% in the Oisans massif; ), the structural style is still a mixed thin- and thick-skinned style. Large-scale fluid flow may occur both in basement shear zones and in large sedimentary nappe thrusts, and may have initiated during the previous step (In the last step of our model, the structural style switched to a pure thick-skinned style (D), i.e., the shortening was almost exclusively accommodated in basement shear zones. The fluid flow was still active in basement shear zones (). The cover of these massifs, the Morcles nappe (), respectively, were disharmonically folded as recumbent anticlines with strongly sheared overhanging limbs. Although showed that basement-derived fluids circulated in the major cover thrusts of the Aar massif, the reverse limbs of the above-cited nappes do not systematically bear evidence of large-scale fluid flow (D). These reverse limbs are interpreted as large shear zones (). Thus, it appears that the shearing geometry distribution made them less prone to large-scale circulation of important amounts of basement-derived fluids, in contrast to localized structures such as few meter-thick thrust zones, which were efficient drains for localized, large-scale, fluids.In this contribution, we analyzed the fluid system(s) in the inner part of the Alpine External zone, the External Crystalline Massifs (ECM). In this domain, pre-orogenic Liassic extensional basins were buried at mid-crustal depth and inverted during the collisional phase of the Alpine orogeny. During the basin shortening, the sedimentary cover was disharmonically folded over localized basement shear zones. Fluid circulations are associated to this deformation. Isotopic analyses and cathodoluminescence observations show that, as for the ultra-Dauphinois cover nappes, fluid circulations were restricted to local scale in the metasedimentary cover. In the basement shear zones, fluids were channelized as for the other ECM. On the basis of trace element analyses, we highlight the small amount of basement-derived fluid percolation in the metasedimentary cover above such shear zones. Stable 18O isotopic signature coupled to microthermometric analyses of fluid inclusions from cover veins above the Huez basement shear zone provides estimates of fluid temperature and pressure: around 250–400 °C and 2–5 kbar for veins that thus formed close to peak P–T conditions. These veins formed contemporaneously with the onset of and during the basin inversion and basement collisional shortening.Our study documents the evolution of the fluid system in an area undergoing limited crustal collisional shortening. It therefore nicely complements earlier studies of fluid systems further North in the Western Alps (Mont Blanc and Aar massifs) where the crust suffered more shortening, and allow one to build a consistent conceptual model of fluid flow at the scale of the whole external Alpine collision zone.The following are the supplementary data related to this article.Detailed analytical techniques and procedures.Trace element analyses (3d transition elements and HFSE) for basement and cover host-rocks and veins.Supplementary data to this article can be found online at http://dx.doi.org/10.1016/j.lithos.2014.07.005Ignition time of nanopowders during milling: A novel simulationDetecting the sudden temperature increase of the milling vial, detecting the sudden total pressure increase inside the vial as well as XRD analysis from the synthesized phases are techniques that can be used to determine the ignition time in mechanically self-sustaining reactions (MSRs) induced by ball milling. In the present study a novel technique based on the Gene Expression Programming (GEP) algorithm is presented to estimate the ignition time in MSRs induced by high energy planetary mills, without any experimental testing. In other words, only by knowing some of the milling and reaction parameters comprised of ΔH/CP, ball to powder weight ratio (BPR), vial spinning rate, arithmetic mean of melting points of reactants, average diameter of balls and amount of used process control agent (PCA), one can predict the ignition time in the mentioned systems. Accordingly, most of the systems that are based on the MSR mode were gathered from the literature, and the data obtained from them are trained and tested by the GEP modeling algorithm. The results indicated a very good agreement between the experimental data and the predicted ones. The biogeography based optimization (BBO) was also utilized to optimize the milling parameters. Experiments were performed at the optimized parameters to proof the validity of the analysis. Given the broad range of the parameters used, it was found that our analysis and model are fully functional to accurately estimate the optimal conditions for planetary mills experiments which show the potential application of these calculations and analysis in materials science and engineering.Mechanical alloying or the high-energy milling of (elemental) powders is a very low cost technique to obtain amorphous or nanostructured materials Ball milling can also produce mechanically induced self-sustaining reactions (MSRs) in many highly exothermic powder mixtures It should be noted that tig can be detected with the sudden temperature increase of the milling vial In this paper, Gene Expression Programming (GEP) and biogeography based optimization (BBO) algorithms as powerful tools have been utilized for modeling and optimizing of milling process, respectively.BBO was developed by Dan Simon in 2008 in the form of a computational algorithm As a generalization of genetic algorithm (GA), Genetic programming (GP) was proposed by Koza The GEP has been utilized as a new tool several times in the field of mechanical alloying. Predicting and maximizing the hardness of metal matrix nanocomposites produced by mechanical alloying In the present work, the ignition time of nanopowders in mechanically induced self-sustaining reactions has been estimated in high energy planetary mills (Given the significant differences between the mills The collected data from the previous works . Ignition time of several mechanically induced self-sustaining reactions in high energy planetary mills has been considered as the main objective of this study for prediction by GEP model. The input parameters consist of ΔH/CP, ball to powder weight ratio (BPR), vial spinning rate, average melting point of reactants, average diameter of balls and amount of used PCA, with the ranges given in . The ignition time of nanopowders has been calculated with three different methods including detecting the sudden temperature increase of the milling vial, detecting the sudden total pressure increase inside the vial and the XRD of analysis the synthesized phases.The GEP encodes the individuals of the created computer programs as linear strings of fixed length (the genome or chromosomes) which are afterwards expressed as nonlinear entities of different sizes and shapes called as expression trees (ET). Accordingly, two languages comprised of the language of the genes and the language of ETs, are utilized in GEP. A significant advantage of GEP is that it enables us to infer exactly the phenotype given the sequence of a gene, and vice versa which is termed as Karva language , for example. In addition, its ET and the corresponding mathematical equation are shown in same figure., the expression tree of GEP approach model was constructed for ignition time (tig) values of mechanically induced self-sustaining reactions. In , d0, d1, d2, d3, d4 and d5 represent the values for input layers including ΔH/CP, average melting point of components (Tm), ball to powder weight ratio (B), amount of used PCA (P), vial spinning rate (V) and average diameter of balls (D), respectively. In the GEP model, the number of genes (SUB-ETs) was five, and the linking function was multiplication. In the training and testing of GEP model, ΔH/CP, Tm, B, P, V, D were considered as input data and tig as independent output data. Among 36 experimental sets, 28 sets were randomly chosen as a training set for the GEP modeling and the remaining 8 sets were used as testing for the generalization capacity of the proposed model.For the GEP-based formulations, the fitness function should be chosen at first. For this problem, the fitness, fi, of an individual program, i, is initially measured by Eq. where M is the range of selection, C (i,j) is the value returned by the individual chromosome i for fitness case j (out of Ct fitness cases) and Tj is the target value for fitness case j. The advantage of this kind of fitness functions is that the system can find the optimal solution for itself. The second important step involves choosing the set of terminals T and the set of functions f to create the chromosomes. In this problem, the terminal set obviously consists of the independent variable, i.e., tig = {ΔH/CP, |
Tm, |
B, |
P, |
V, |
D}. The choice of the appropriate function set is not so obvious, but a good guess can always be made in order to include all the necessary functions In the present study, after several trials, for the GEP-based formulation, the number of genes and length of head is determined as given in . For example, for the GEP approach model, the number of chromosomes 30 was observed to be the best of generation individuals predicting ignition time of nanopowders of mechanically induced self-sustaining reactions in planetary mills.Explicit formulation based on the GEP approach model for f was obtained by:The related formulations could be obtained by the procedure shown in Mathematical models of BBO are based on the migration of species from one island to another, and on how species arise and become extinct. In BBO, any habitat that is geographically isolated from other habitats is called Island. Well suited habitats for species are said to have high habitat suitability index (HSI) while habitats that are not well suited are said to have low HSI. In high energy ball milling, high habitat suitability index is equal to the minimum of ignition time. Each habitat consists of features that decide the HSI for the habitat. These features are considered as independent variables and are called suitability index variables (SIV), which map the value of the HSI of the habitat. High HSI habitats have large number of species, while low HSI habitats have small number of species In BBO, each individual has its own immigration rate and emigration rate. In high energy ball milling both of immigration rate and emigration rate are all of the solutions that lead to the speeding up of the ignition of nanopowders. The immigration rate and the emigration rate are functions of the number of species in the habitat (). In high energy ball milling they are functions of all of the milling parameters. They can be calculated as follows where I: is the maximum possible immigration rate. E is the maximum possible emigration rate and k is the number of species of the k th individual in the ordered population according to the fitness.In BBO the mutation is modeled as SIV mutation using species count probabilities to determine mutation rate. Very high HSI and very low HSI solutions are likely to be mutated to a different solution using the mutation rate.In high energy ball milling when an ignition time is provided as a solution (solution is selected for mutation) then we replace a randomly chosen SIV in the habitat with a new randomly generated SIV to provide the next ignition time.A planetary ball mill (Model PM 2400) with the stainless steel vials and balls and nominally at room temperature under argon atmosphere was used for milling of the powders. Phase identification of the milled samples was performed by X-ray diffraction (XRD) with CuKα radiation at 30 kV and 25 mA. Morphology of the milled powders was monitored using a Hitachi S4160 scanning electron microscopy (SEM) operating at 15 kV and a ZEISS EM 10C transmission electron microscopy (TEM) operated at 100 kV.In the present study, the modeling variables or the input parameters should be chosen so that the model can detect the mechanically induced self-sustaining reactions from gradually ones. On the other hand, those milling parameters affecting the kinetic of these reactions should be considered. Accordingly, it appears that the mentioned demands are met by the following input parameters: shows the relation between ΔH/Cp and tig of the reactions during the ball milling of metal/CuO powder mixtures according to the values presented in . As can be seen, contrary to what has been mentioned in the literature shows the ignition time of nanopowders (tig) changes versus the arithmetic mean Tm1+Tm22 (b) of the melting points of reactants for the reduction of Cu-oxides with metals based on the reactions presented in a and b, one can see that the graph of the tig versus arithmetic mean of melting points has a lower error range than the graph of tig versus ΔH/Cp as well as the graph of tig versus molar mean of melting points. Accordingly, another parameter used in the model can be arithmetic mean of the melting points of the reactants.In many powder mixtures with brittle raw materials there is no need to use process control agents (PCAs) during milling process. However, it was used in a few specific instances with interesting results. According to the literature It should be noted that the kinetic energy of the milling balls can lead to the deformation of the powder during the impact The critical milling time for the onset of combustion was reported as a function of ball size for the milling of CuC12 |
+ 2Na There have been very few studies on the effect of the ratio of the constituent reactants (off-stoichiometry) on the combustion process. Takacs Given the formula above, all systems that have been studied in this research, were in the stoichiometric conditions. In other words, our predictive model is only applicable in the stoichiometric mode.In this study, in order to investigate the capabilities of the GEP-based formulation, the formula given by some national building codes and the developed regression-based formulation, mean absolute percentage error (MAE), root-mean-squared error (RMSE) and R-square (R2) were used as the criteria between the experimental and predicted values which are in accordance with Eqs. where “t” is the experimental value, “o” is the predicted value, and “n” is the total number of data. The related equations to the GEP model obtained from tig=ArctanTm/B2/PTm+B×Tm−B2×5.86Tm3−V×9.91/BV+6.5−B−6.5×−6.5/ΔH/CP×cosV−6.7+ΔH/CP−8.6+coscosV3×cos2D+sin2D.All the results obtained from experimental studies and predicted by using the training and testing results of GEP model are given in . As can be seen, the values obtained from the GEP model in the training set are very close to those in the experimental results proving that the proposed model has dramatically well learned the non-linear relationship between the input parameters and the ignition time of nanopowders with high correlation and comparatively low error values (a). On the other hand, in the testing stage a comparatively low error values were obtained indicating that the GEP model is capable of generalizing between input and output variables with reasonably good predictions (b). All of these findings exhibit a successful performance of the model for predicting the ignition time of nanopowders in self-sustaining reactions induced mechanically in high energy planetary mills in training and testing stages. R2, MAPE and RMSE values are shown in for the training and testing data. The entire R2, MAPE and RMSE values show that the proposed GEP model is suitable and can predict ignition time of nanopowders values of MSR mode in planetary mills very close to the experimental values.In the optimization process, we are going to find the best conditions for minimizing the ignition time. For this purpose, we can use Eq. as the cost function in which tig is dependent upon the milling parameters so that by optimizing the milling parameters, tig will be minimized. are B, D, P and V and the intrinsic parameters are ΔH/Cp, Tm. Since for any combustion system it is required that ΔH/Cp |
> 2000 K and Tm |
< 2993 K (according to the values presented in ), the values of 2000 K and 2993 K respectively are considered as default for ΔH/Cp and Tm during optimization process. shows the convergence of the BBO to the optimal solution based on the GEP model (Eq. ). It is evident that, BBO has achieved the optimal values of B, D, P and V in the 79th iterations. The minimum cost function tig is found to be 0.7654 after 79 iterations. In , the results of optimization are presented for the object of minimum cost function tig.Other milling parameters including number of the balls, vial radius, vial height, ball diameter distribution coefficient, plate spinning rate, vial spinning rate and the distance between the center of the plate and the center of the vial, should also be optimized. Accordingly, it is important to find a mathematical model to correlate the mentioned milling parameters with the ignition time of nanostructured powders and then optimize the mentioned model. In this work the Burgio's model If we assume that the total energy transferred by the planetary mill per gram of reactant mixture and required for synthesis of nano-structured powders is a constant value, A, the Burgio's model defines this amount of energy by the following expression:Et/g=NbφbfbKambWv3Rv−db/2/Wp+WPWvRPRv−db/2tmch=AJ/gwhere Nb is the number of balls; φb is a parameter that accounts for the degree of filling of the vial; fb is the frequency with which the balls are launched against the opposite wall of the vial (s− 1); Ka is a constant that accounts for the elasticity of collisions, and a value of 1 represents perfectly inelastic collisions; mch is the mass of the powder charge; Wp and Wv are respectively the absolute angular velocity of the plate of the mill and of one vial; Rp and Rv are respectively the vectorial distances from the center of the mill to the center of the vial and from the center of the vial to its periphery (vial radius) and t is the synthesis time (here the ignition time) measured (s).where Hv, |
ρb are respectively the height of the vial and the density of balls. K is a proportionality constant and is approximately equal to unity t=12AρbKaK×mch×1−Nb1−db3NbπRv2HvεWp‐Wvdb3Wv3Rv−db/2/Wp+WPWvRPRv−db/2where t is the synthesis time (here the ignition time) measured (s)., B as the first term is related to density of balls, elastic and inelastic collisions. Accordingly, one can say that B is a constant. mch, as the second term of the equation, is the mass of the powder charge which is determined by ball to powder weight ratio (BPR). The last and most important term in Eq. , f, depending on all variables of the milling so that by minimizing the f the synthesis time will be minimized as well. shows the convergence of the BBO to the optimal solution based on the Burgio's model. As can be seen, BBO has achieved to the optimal values of [Nb, |
db, |
Rv, |
Hv, |
ε, |
Wp, |
Wv, |
Rp] in number 103 of iterations. The minimum cost function f is found to be 0.02572 after 143 iterations. In the results of optimization are presented for the object of minimum cost function f. An optimized planetary mill can be designed by the values presented in . As can be observed from this table, Wv |
= 1.18Wp which means that the vial spinning rate should be higher than that of the plate (in the opposite direction). It is important to note that in both optimization process (), the velocity of the vial and the ball diameter are the same (). On the other hand, ε |
= 0.211 which means that the ball size distribution is close to 22%. The number of ball categories, s, is calculated as follows:Taking into consideration the balls diameter and the above interpretation one can say that:db=16mmNb=15s=3⇒15×16mm=5×14mm︷1+5×16mm︷2+5×18mm︷3.So it can be claimed that the balls diameter used in the proposed design should be 14 mm, 16 mm and 18 mm for the maximum energy transfer to the raw materials.To check the accuracy of the GEP model and the BBO algorithm in the experimental conditions, several experimental environments were provided and then the results obtained from the GEP model and the BBO algorithm were compared with those in the literatures.In this research, ΔH/Cp, Tm, B, D, P and V as input parameters for GEP approach model were 4337, 2486, 10, 5, 0 and 360, respectively. By putting these values into Eq. , ignition time of nanopowders as output parameter, is equal to: 5.75 h. This means that our model is completely successful for predicting the ignition time of nanopowders in the mentioned system.By applying the optimized parameters (according to ), one can see that the synthesis time (ignition time) of nanopowders in this system has fallen to less than 3 h (). To compare the optimal and non-optimal milling parameters two systems were introduced. System 1 was established based on the parameters used by Jianlin Li et al. and system 2 was established based on the optimized parameters provided by BBO algorithm. The sampling was performed after 3 h of milling and the related SEM images were analyzed (a) the work hardening of the milled powders leads to the decrease in the particle size (compared to the initial size). At this stage, fracture is a dominant factor. However in system 2 (b), the released heat from the MSR mode leads to the increase in powder temperature, plastic deformation, and agglomeration of powders. In other words since the milling energy in system 2 (the optimized system) is higher than that in system 1, the ignition occurred faster in this system resulting in agglomeration of powder particles at 3 h of milling.In a research that has been conducted recently by Rabiezadeh et al. The result of our modeling in this case is really stunning. With replacement input parameters (ΔH/Cp, Tm, B, D, P and V as input parameters for GEP approach model were 5000, 1257, 20, not reported,, the value of 8.27 h is obtained for the ignition time of nanopowders which is consistent with the experimental results.Here an important advantage of the model into XRD analysis is revealed. In other words in the XRD analysis it is not possible to estimate the ignition time of nanopowders exactly, whereas in presented model one can predict the ignition time of nanopowders.It is expected that by applying the optimized parameters in TiB2–Al2O3 system, the ignition (synthesis) time is reduced. Accordingly a system was established in optimal conditions (according to Tables 6 and ), and the sampling was performed at 1 and 3 h of milling. is the XRD analysis of the powder mixture during this period of time. As can be seen, the ignition time in optimal conditions has fallen to less than 3 h of milling. This means that more milling energy was available in optimized system so that the ignition (synthesis) occurred almost 2 h earlier.Alumina-based nanocomposite powders with tungsten carbides particulates were synthesized by ball milling WO3, Al and graphite powders Our theory results based on the GEP approach suggest that placing the input parameters in Eq. (ΔH/Cp, Tm, B, D, P and V as input parameters for the GEP approach model were 7661, 1695, 10, not reported, 0 and 500, respectively) leads to the ignition time of nanopowders to be 4.5 h, which is consistent with experimental results reported by Zakeri et al Optimization results based on the BBO algorithm showed an increase in the energy transferred to the nanopowders and hence reduction of the ignition time. shows the change of vial temperature during milling of powder mixture in optimized and non-optimized systems (in non-optimized system the milling parameters were established based on Zakeri et al. A solid state reduction reaction of ZnO and Al powder mixtures induced by mechanical alloying has been investigated by J.M. Wu indicates that in this system there is a potential for the occurrence of a combustion process around 5 h of milling. However, XRD results are inconsistent with the predictions of the model. It is noteworthy that in the mentioned system the adiabatic temperature of 2530 K is much higher than the melting point of zinc (419° C) and even the boiling point of zinc (907 °C). Therefore, if the local temperature does exceed the adiabatic one during the combustion reaction, it is very likely that the zinc phase evaporates as a result of the heat release and, consequently, the adiabatic temperature decreases remarkably. This suggests that unforeseen circumstances may exist in any system that affects the accuracy of the model.In the present work, by the use of a modeling algorithm based on artificial intelligence, we were able to predict the ignition time of nanopowders in mechanically self-sustaining reactions induced by planetary ball mills. Modeling results showed high regression values for training and testing sets (respectively, 0.89 and 0.96) with low error rate for both of them (respectively, 1.61 and 1.55 for MAPE and 2.29 and 1.91 for RMSE). Case study results for TiB2–TiC, TiB2–Al2O3, and Al2O3–W2C systems also proved that the mentioned model is fully functional to accurately estimate the ignition time of nanopowders for ball milling experiments which shows the potential application of such calculation and analysis in materials science and engineering. Mismatch of the ignition time of nanopowders in Zn–Alumina system with the one in the proposed model implies that unforeseen circumstances may exist in any system that affect the accuracy of the model. The optimization results based on the BBO algorithm showed that a minimum of ignition time can be obtained in optimal conditions. Two GEP and Burgio models were used as cost function in BBO algorithm. Experimental results based on optimization showed that by utilizing the mentioned models as cost function we can obtain the ideal conditions for minimizing the synthesis time in high energy planetary mills.Segmental copolymers of condensation polyesters and polylactideL-Lactide (LA) polymerizations carried out in the presence of various types of polyesterodiols and polyesterocarbonatediols obtained from 1,4-butanediol, 1,3-propanediol and dimethyl esters of adipic, terephthalic and carbonic acids were studied. It was found, based on 1H NMR and MALDI ToF analysis of the chains microstructure that the reactions carried out in bulk at 190 °C in the presence of tin (II) 2-ethylhexanoate as a catalyst led to the formation of a mixture of polylactide (PLLA) and block copolymers of LA with the applied macrodiols with high yield. DSC studies of the obtained products indicate that the segments comprising of adipic and carbonic acid derivatives were fully miscible with PLLA causing the reduction of its glass transition temperature. The segments containing terephthalic acid monomeric units in its structure showed only slight miscibility with PLLA segments and the obtained products consist of two phases of glass transition temperatures close to those of PLLA and the macrodiol. These systems are characterized by better thermal stability and higher elasticity than PLLA.Poly(L-lactide) (PLLA), biodegradable and compostable aliphatic polyester is derived from renewable resources such as corn, potatoes, and sugar beets A number of methods have been proposed to improve the mechanical properties of PLLA, among which plasticization, addition of rigid fillers and blending with a variety of flexible polymers are the most commonly used approaches Taking into account that higher molecular weight of the plasticizer causes a reduction in its migration rate and thus improves the morphological stability of PLLA during storage, the combination of small molecule plasticizers with polymeric ones of complementary advantages was proposed. Ren et al. used a combination of glycerol triacetate with poly(butylene adipate) to plasticize PLLA Blending with flexible polymers is another most extensively used strategy to improve the ductility of PLLA. Typical polyolefins are immiscible with poly(L-lactide). Thus, they should be modified prior to blending to improve miscibility. Oyama presented a dramatic improvement in the mechanical properties of PLLA by its reactive blending with poly(ethylene-co-glycidyl methacrylate) (EGMA) containing 30 wt% of methyl acrylate and 3 wt% of glycidyl methacrylate Biodegradable blends of PLLA which exhibited improved impact strength are obtained with use of flexible aliphatic polyesters such as poly(ε-caprolactone) (PCL) Very recently, Dubois's group has found that random copolyesters of δ-valerolactone (VL) and ε-caprolactone (CL) added at 10 wt.% into a commercially available PLA matrix led to a significant improvement in toughness of PLA materials To improve miscibility, instead of homopolyesters, copolymers with L-lactide were used for blending with PLLA. Poly(butylene succinate-co-L-lactate) was reported to blend with PLLA. It was found that the blends showed higher elongation at break than their parent polymers Grijpma et al. showed also that the TMC block (Mn = 65 000) in the tri-block copolymers was also effective in toughening PLLA. With varying weight content of TMC rubber from 10.9 to 21.4 wt%, the elongation increased from 135 to 210% It was found that some biodegradable and flexible polyurethanes containing polylactide as well as other polyester type polyols can also be used to blend with PLLA to improve its flexibility Taking into account that aliphatic-aromatic copolyesters like poly(butylene adipate-co-terephthalate) (PBAT) are fully biodegradable and displayed high flexiblity and ductility, Jiang et al. used PBAT for blending PLLA. The presence of aliphatic-aromatic copolyester lead to toughening of the polylactide material. The elongation of PLLA containing 5 wt% of PBAT increased from 3.7 to 115% To improve the compatibility, a reactive compatibilizer was used by Zang et al. To achieve enhanced miscibility as well as improved mechanical properties of the blends with PLLA, various compatibilizers are additionally introduced. Wu et al. studied the influence of the presence of copolyester on mechanical properties of PLLA blend with poly(ε-caprolactone). It was shown that di- and triblock compatibilizer-copolymers (PLLA-b-PCL and PCL-b-PLLA-b-PCL) improve the interphase and mechanical properties of PCL/PLLA blends In this paper we present the results of our work on the synthesis of compatibilizers containing segments of PLLA and polyesters obtained in polycondensation reactions. The chosen process is based on LA ring opening polymerization (ROP) carried out in bulk in the presence of hydroxyl end-capped aliphatic or aliphatic-aromatic polyesters and tin(II) 2-ethylhexanoate. We assumed that incorporation of macrodiols may result mainly from– ROP of LA initiated by hydroxyl end-groups of polyesters (Eqns. (1a),(1b)),– Intermolecular chain transfer reaction between the growing PLLA chains and hydroxyl end-capped macromolecules (Eq. (1c)),However, one cannot exclude that these elementary reactions are accompanied by several others processes like homopolymerization of LA initiated by hydroxyl-impurities and transesterification reactions leading to the segmental exchange and formation of macrocyclic products. In order to gain a greater understanding of these processes we have studied the reaction product be means of 1H NMR, MALDI ToF and GPC techniques. We report here also some preliminary data concerning the phase structure, thermal stability and mechanical properties of the products obtained.L-Lactide (Aldrich) was recrystallized consequently from dry isopropanol, then toluene and vacuum dried before polymerization. Toluene and isopropanol were dried with sodium and distilled. Tin (II) 2-ethylhexanoate [Sn(EH)2], adipic acid (AA), dimethyl terephthalate (DMT), dimethyl adipate (DMA), 1,4-butanediol (1,4-BD), 1,3-propanediol (1,3-PD), dimethyl carbonate (DMC) and titanium tetrabutoxide, were purchased from Aldrich and used without further purification. Poly(1,4-butylene adipate) (PBA) of Mn 1000 and poly(1,4-butylene terephthalate) (PBT) of Mv 38000 were supplied by Aldrich. Dichloromethane and methanol were used without purification.All condensation copolyesters were obtained in two-step transesterification–polycondensation reaction between appropriate diesters and diols mixed in the molar ratio of 1:2, respectively. The substrates of the same type were used in equimolar amount with the exception of PPAT synthesis, where the molar ratio of DMA/DMT was 3. The process was catalyzed by 1 mol % of Sn(EH)2 (with regard to the total amount of diesters). The first step was carried out under nitrogen at 160 °C and methanol was distilled off as the reaction by-product. The transesterification step was completed after the methanol yield was close to 85%. In the second step the temperature was increased to 180 °C and the pressure was slowly reduced to 10−3 mbar. The polycondensation reaction has been carried out for 10 h while diols were distilled off. Then the reaction mixtures were cooled to room temperature and dissolved in CH2Cl2. The crude polymers were precipitated and washed with cold methanol. The final products were dried under vacuum at 40 °C for 48 h. This procedure was successfully applied for the synthesis of the following polymers: poly(1,3-propylene adipate) (PPA), poly(1,4-butylene adipate) (PBA), PPAC poly(1,3-propylene adipate-co-carbonate), poly(1,3-propylene-co-1,4-butylene adipate) (PPBA), poly(1,3-propylene-co-1,4-butylene adipate-co-carbonate) (PPBAC), poly(1,3-propylene adipate-co-terephthalate) (PPAT), poly(1,4-butylene adipate-co-terephthalate) (PBAT), poly(1,3-propylene-co-1,4-butylene adipate-co-terephtha-late) (PPBAT) and poly(1,3-propylene-co-1,4-butylene adipate-co-terephthalate-co-carbonate) (PPBATC). On the other hand, PBAT (, no. 9) was obtained by polycondensation reaction of PBA and PBT. In this case, PBA was synthesized during polycondensation reaction of AA with 1,4-BD, catalyzed by tetrabutoxy titanium (0.1 mol % of AA). The mixture of PBA and PBT was heated up to 240 °C (at the rate of 7 °C/min) with pressure reduced to 0.031 Torr and then the polycondensation reaction was carried out for 90 min.“Pure” PLLA was synthesized as a reference by ROP of LA (29.5 g) with 0.04 wt. % of Sn(EH)2 as catalyst in a 250 mL reactor. The reaction was carried out at 190 °C for 3 h under nitrogen atmosphere. The final product was dissolved in methylene chloride, precipitated in cold methanol and dried in a vacuum oven at 40 °C for 48 h.The solubility parameters δ = (E/V)0.5 of PLLA and condensation polymers were calculated in order to distinguish between homogeneous and heterogeneous systems. The energy of cohesion E can be substituted with F2/V, then δ = F/V. The group attractive constant (F) and molar volume (V) are additive quantities and, therefore, can be calculated for each m.u. from respective values estimated for substructures.1H NMR measurement was performed on Varian Mercury 400 MHz spectrometer using CDCl3 as solvent. The molecular weight and molecular weight distribution were determined by GPC on a Viscotek system comprising GPCmax and TDA 305 triple detection unit (RI, IV, LS) equipped with one guard and two DVB Jordi gel columns (102–107, linear, mix bed) in CH2Cl2 as eluent at 35 °C at a flow rate of 1.0 mL/min. Triple detection was used for determination of absolute molecular weight and DI of condensation copolyesters, whereas RI detector and PS calibration were applied for block copolymers characterization. The sample of block copolymer was fractionated using a LabAlliance GPC system equipped with one DVB Jordi gel column in CHCl3 as eluent at 25 °C at a flow rate of 0.75 mL/min. MALDI–ToF mass spectrometry was performed on AXIMA Performance instrument. Dithranol was used as MALDI matrix. The DSC measurement were performed using a DSC Q200 V24.2 Build 107 apparatus. The first heating run from −100 °C to 200 °C was performed at a heating rate of 5–10 °C/min in order to study crystallinity, then cooling at the rate of 20 °C/min was applied. The second heating run was measured at the rate of 20 °C/min to determine glass transition temperatures. TGA measurements were conducted using a TA Instruments SDT Q600 instrument, at heating rate of 5 °C/min under air atmosphere. The tensile properties were examined with an Instron mechanical tester, model 5566, at tensile speed of 4 mm/min at room temperature. The specimens were cut into strips with approximate dimensions 80/20/0.3 mm (l/w/d) using films prepared by hot pressing at 170 °C in a hydraulic press. Five specimens of each material were used.The oligo- and polyesterodiols which were used as precursors of elastic segments (ES) were prepared from dimethyl esters and diols in one-batch two-step polycondensation in the presence of tin (II) 2-ethylhexanoate as a catalyst. The 1,3-propanediol, 1,4-butanediol and methyl esters of adipic, terephthalic and carbonic acids were employed as starting materials. The composition of all of these multicomponent aliphatic (a) polymers was determined by the means of 1H NMR (). The same technique was applied for estimation of Mn by the end-group analysis method; however, these results were verified by absolute GPC measurement (Mn and DI) using the triple detection system (RI, IV, LS). The differences between these two techniques usually did not exceed 30% of the chromatography data. The value of Mn,GPC varies in the range of 1000–16500. The distribution of molecular weight is moderate broad (DI 1.3–2.5) except the commercial sample of PBA (The materials obtained can be divided into two general categories depending on the composition. The first one includes aliphatic polyesters based on adipic and optionally carbonic acids derivatives which exhibit very low Tg in the range of −70 to −50 °C. The other group are aliphatic-aromatic polyesters containing 30–50 mol % of terephthalate units which reveal slightly higher Tg in the range of −35 to −20 °C. Most of the products were semi-crystalline excluding aliphatic polyesterocarbonate - PPAC, which existed in the form of a viscous liquid. It should be noticed that solubility parameters (δ) calculated for all the obtained materials (19.6–21.1 MPa0.5) are very close to that of PLLA (20.7 MPa0.5).In order to obtain segmental copolymers, ring opening polymerization (ROP) of LA was carried out in the presence of ES-diols and 0.01 wt. % of tin 2-ethylhexanoate as catalyst. The synthesis was performed in bulk at 190 °C (which resemble the conditions applied in industrial technologies) for 3 h. The 1H NMR studies of the reaction products reveal that under these conditions signals of terminal units −Ca, signal a) present in ES molecules a decay and in the case of aliphatic ES a new kind of signals coming from −C(CH3)OH end-groups derived from PLLA chains could be clearly observed (b, signal j). Therefore, one can expect that ES molecules are totally incorporated into the polylactide chains. It should be noticed that the other signals characteristic for m.u. of ES did not change, which suggests that these segments were not involved in transesterification processes (Figs. b). To support this hypothesis, we have studied the LA polymerization in the presence of PBA at 220 °C for 8 h and we found the presence of an additional signal that can be attributed to the diad LAc-A (c, signal o) which should not be observed in the expected block copolymer. One can also observe a significant increase in the intensity of the signal characteristic for the link between PLLA and ES blocks (B-LAc, signal d'). Taking these into consideration we can treat the signals characteristic for ES m.u. as internal standards in the analysis of the product obtained at 190 °C. the LA conversion and weight fraction of PLLA m.u. in the products obtained determined on the basis of 1H NMR spectra are shown. In most cases the PLLA weight fraction constituted as much as 85–95%, whereas LA conversion varied from 60 to 90%. However, on the basis of 1H NMR study it is impossible to determine the concentration of LA m.u. in block copolymers and homopolymers that can be also formed in these systems. In the case of systems containing aliphatic ES one can only roughly estimate the molar fraction of LA homopolymer chains xPLLA by analyzing the integrals of the terminal ∼CH(CH3)E–OH signal (b, j) and that of ∼CH2OC(O)CH(CH3)∼ in linking unit (d'). Due to the relatively low intensity of both signals the precision of this analysis is not high (in the case of aliphatic-aromatic ES, the signal of j protons overlaps with that of methylene protons in ester groups d), nevertheless it shows clearly that we are dealing actually with a mixture of homo- and copolymers of LA. This observation was also supported by MALDI ToF analysis of low molecular weight product based on PBA (. presents the spectrum of sodium-cationized ions. The m/z range of the highest signal intensity is from 1500 to 3500, however, the overall distribution in the sample covers the m/z range from 1000 to 5000. We analyzed three regions in the m/z ranges of (a) 1000–2000 (c). The following peak assignments (notifications) for various populations were used: x/y - for linear macromolecules and x/y M - for macrocyclics, where x is the degree of polymerization of ES and y is the number of lactic acid units in the molecules (In the first range (a) no separated signal of the unreacted PBA macroinitiator was found (x/0), which confirms the very high incorporation efficiency. On the other hand, that area is rich of the products of LA polymerization initiated with water or LA hydrolysis product (0/y), with the odd and even values of y (lactic acid units), which suggests segmental exchange processes occurring in the system. One can also observe the set of peaks that can be assigned to the macrocyclization products (x/y M; x = 1–2, y = 5–20). Finally, distributions of signals derived from the block copolymers (x/y; x = 1–3) were present. The middle range (b) of the spectrum contained all the same populations as described previously, but with higher x and y values. The most intensive population was found for LA homopolymer (0/y; y = 27–40) comprising a larger even y series than the odd ones. The series of linear block copolymer (x/y; x = 1–3, y = 18–37) was present at medium strength. The population of macrocycles (x/y M) was low and decreasing for higher m/z values. In the third analyzed range (c) of the spectrum the major populations were linear block copolymers (x/y; x = 1–8, y = 32–54) which are expected to be the main products in the batch. According to 1H NMR analysis they should constitute about 99% of the chains population in this sample, however, only small portion of them give rise to the signals that can be observed in MALDI ToF spectra.. shows the GPC traces of reaction products containing short PBA segments (Mn ∼ 1000) (, 1b). They reveal bimodal distribution of molecular weight. Mp values (determined according to PS calibration) equal to 30,400 and 56,700. The sample was additionally fractionated into 4 portions using a GPC column. On the principles of GPC theory and our measurements, Mn was decreasing from fraction a to d as follows: 54800, 46600, 29100, 23400. 1H NMR analysis of these fractions showed that the concentration of butylene adipate (BA) units comparing to lactic acid units decreases in the same order: 4.7, 2.7, 0.33 and 0.30 mol %, respectively. These results are consistent with MALDI ToF analysis and confirm that in these systems we are dealing indeed with two populations of chains, which correspond to LA homopolymer and block copolymer, respectively (Mp,PLLA-PBA-PLLA ≅ 2 ⋅ Mp,PLLA). On the other hand, GPC traces of the sample which contained less homopolylactide chains (, no. 2b, xPLLA = 13%) and the other one obtained in the presence of aliphatic-aromatic ES, were monomodal but broad and “tailed” (Generally, DI for the products obtained in the studied systems may differ significantly (1.2–3.6, see ) depending on the degree of polymerization and the content of homopolymer chains. It should be noticed, however, that the weight average molecular weight (Mw) of the products may be as high as 104–105 g/mol, which allows to assume them as useful for practical applications.Taking into account the calculated values of δ one could expect that all ES used in this study should be miscible with PLLA. However, DSC studies showed that only the systems based on aliphatic copolyesters reveal one Tg, which is close to the data calculated on the basis of the Fox's equation (). Therefore, this type of ES can be regarded as an internal plasticizer. The ES containing terephthalic acid units exhibits only partial miscibility with PLLA, which results in the formation of PLLA and ES rich phases of Tg slightly lower and higher, with respect to the values determined for pure segments.Preliminary observations show that materials containing aliphatic ES have thermal stability similar to that of PLLA, whereas the introduction of aliphatic-aromatic ES into the PLLA chains leads to the improvement of thermal stability (). The mechanical studies show that this latter material is characterized by higher elasticity indicated by lower Young's modulus and higher elongation at break than those for PLLA (From studies carried out in this work it appears that various types of polyesters or poly(ester carbonates) terminated with hydroxyl groups can be incorporated with high yield into the PLLA structure under conditions similar to those as in classical bulk LA polymerization. On the basis of 1H NMR and MALDI ToF studies it can be assumed that these copolymers are of a triblock structure. When carrying out the reaction at 190 °C in the presence of tin(II) 2-ethoxyhexanate, practically complete conversion of macrodiols is achieved after about 3 h, and the LA conversion was close to 90%. Under these conditions, however, part of LA undergoes homopolymerization, resulting from the reaction, in which small amounts of water or the monomer hydrolysis product are the co-initiator. On the basis of the polylactide chains molar content it can be estimated that in the studied by us systems based on aliphatic ES, the water content (or that of the LA hydrolysis product) was from 20 to 600 ppm. An analysis of the chains structure by means of MALDI ToF shows that at 190 °C transesterification processes proceed in PLLA segments, whereas macrodiol segments practically do not participate in transesterification processes. However, introductory observations show that such processes are of essential importance for the structure of products when the polymerization is carried out at 220 °C.Segments built of polyesters comprising in their structure butylene adipate m.u., propylene adipate m.u., butylene carbonate m.u. and propylene carbonate m.u. are completely miscible with PLLA and act as polymeric plasticizers.Segments containing 30–50 mol % of terephthalic acid m.u. show a limited miscibility with PLLA. Products of LA polymerization in the presence of this type of segments form two-phase systems containing an elastic phase of Tg in the −10 to −20 °C range and a stiff phase of Tg in the 45–60 °C range. Initial observation shows that these heterogeneous systems are characterized by a slightly higher thermal stability than that of PLLA obtained in analogous conditions. The present studies aim to the utilization of thus modified PLLA for the formation of blends with commercially available biodegradable polyesters and estimation whether the in situ block copolymers are efficient compatibilizers.Evaluation of the ductile–brittle transition temperature in the NESC-I material using small punch testingSmall punch (SP) fracture testing with subsequent SEM fractographic analysis was applied to an A 508 Class 3 reactor pressure vessel (RPV) steel, used originally in the NESC-I spinning cylinder experiment, in order to determine the SP ductile–brittle transition temperature of both the base material and the sub-clad heat affected zone. In addition to the evaluation of the SP transition temperature, TSP, corresponding to the mean value of upper and lower shelf SP fracture energies, three alternative procedures were assessed. The SP transition temperatures were then compared to original NESC-I data obtained using conventional testing methods, which yielded values of the empirical correlation factor α consistent with those reported in literature for RPV steels of similar composition. The evaluation of the fracture mode transition temperature TFM from the SP test was identified as a viable procedure, in particular for the case when lower shelf energy data cannot be obtained due to technical limitations associated with very low testing temperatures.Reliable structural integrity assessment of nuclear power plant components plays a key role in ensuring safe plant operation, in particular at present, when a large number of currently operating nuclear power stations are approaching the end of their design lives. NESC-I was the first in a series of projects launched by the Network for Evaluating Structural Components (NESC), aimed at the validation of the entire structural integrity assessment process in an international framework with 34 participating organizations worldwide In the NESC-I project a wide range of mechanical testing techniques was employed to provide input data for the structural mechanics assessment procedures, including tensile tests, Charpy V-notch impact testing, Pellini drop-weight tests, and fracture toughness testing using compact tension (CT) and single edge notched bend (SENB) specimens. In recent years, Small Punch (SP) testing — a miniaturized mechanical testing technique first introduced by Manahan et al. The present work has two principal objectives: first to assess the methodology for the evaluation of the ductile–brittle transition temperature (DBTT) from the SP fracture test, and second to revisit the NESC-I project in terms of SP fracture testing in order to establish a correlation between the DBTT values from SP tests and those obtained with conventional testing techniques for an RPV steel. In the following section, the experimental material/equipment used in the present study is described, and four different methods for the evaluation of the ductile–brittle transition temperature from SP test data are introduced. Results of the SP fracture tests and corresponding SEM fractography are presented in Section and correlated with the original transition temperature values available from the NESC-I project in Section also offers a more detailed discussion of the four methods for SP transition temperature determination assessed in the present study.The material for the small punch tests was taken from a segment of the original NESC-I cylinder arc used in the pre-test material property characterization phase of the project. It was fabricated from A 508 Class 3 RPV steel with internal 316 stainless steel cladding. The chemical composition of the base material is given in . The steel was subjected to complex non-standard heat treatment in order to simulate radiation embrittlement. The objective was to obtain the strength, fracture toughness and DBTT typical for an in-service aged RPV, rather than the specific microstructure associated with radiation embrittlement. Details concerning the fabrication of the NESC-I cylinder and the heat treatment procedure can be found in Ref. To fabricate the SP specimens, cylinders with an 8 mm diameter were extracted from a segment of the NESC-I cylinder wall, both from the base material and from the sub-clad heat affected zone. The positions for the extraction of the material for SP tests were deliberately chosen so as to coincide (in terms of the distance from the cladding) with the positions of the Charpy and CT specimens used in the NESC-I experiment to obtain the DBTT and fracture toughness properties – see schematic in . During microstructural examination carried out within the framework of the NESC-I project, the width of the HAZ was found to be 7.5–8.8 mm The small punch fracture tests were carried out in accordance with ). The cooling was carried out by means of liquid nitrogen vapor in a temperature chamber. For the lowest testing temperature, a modification allowing for direct submersion of the testing assembly in liquid nitrogen was employed. The temperature was measured using a type T thermocouple with a precision of ±2 °C. The tests were performed with a constant rate of cross-head displacement equal to 0.005 mm/s. The puncher displacement was measured using a Class 2 LVDT. Following the SP tests, fractured specimens corresponding to 6 testing temperatures between RT and −196 °C for base material as well as the sub-clad HAZ, i.e. 12 specimens in total, were subjected to fractographic examination using a scanning electron microscope (SEM) equipped with a field emission gun. Both the macroscopic fracture appearance of the SP discs and fracture surface features were examined.For each test, the SP fracture energy ESP was calculated by integrating the area under the measured load-displacement curve up to fracture. The small punch transition temperature TSP was obtained from the temperature dependence of ESP as the temperature corresponding to the mean value of upper and lower shelf SP fracture energies, according to the Code of Practice First, the SP transition temperature was determined based on a specific energy level TSP(E) rather than the mean energy between the upper and lower shelf, as proposed by Kim et al. Based on the results of the SEM fractography, a small punch fracture appearance transition temperature TSP, FATT was estimated as the temperature with a 50% shear fracture area on the fracture surface.The SP ductile–brittle transition was also characterized by means of the so called fracture mode transition temperature TFM, evaluated from the distinctive change of shape in the SP load-displacement curves, which signals the onset of brittle behavior. This procedure was suggested more than 15 years ago in EPRI technical reports shows typical small punch load-deflection curves obtained for the base material (a) and the sub-clad HAZ (b) at temperatures from RT down to −196 °C. To retain clarity of the plots, only selected curves, representative of the material behavior at the given temperature, are presented. The appearance of the curves for both base material and HAZ follows the same general trend: With decreasing temperature, the maximum load values increase, up to a point when the fracture mode starts to change – the individual curves start to exhibit abrupt load drops. From this point on, the maximum load begins to decrease with decreasing temperature, along with the displacement at fracture (uf) values. A gradual increase in the apparent strain hardening rates with decreasing temperature can be observed as well. Careful inspection of all measured load–displacement curves allowed for the evaluation of the fracture mode transition temperature TFM as −110 °C for the base material and −130 °C for the HAZ.The dependence of SP fracture energies on temperature used for the determination of the SP transition temperature TSP and TSP (1.2 J) is shown in . For temperatures where multiple tests were carried out, only one point corresponding to the average value of the obtained SP fracture energies is shown. It should be noted that some of the load–displacement curves at lower temperatures exhibited multiple load drops – see examples in . The part of each curve following the first load drop was not taken into account for the calculation of the SP fracture energy and was not included in . In the temperature dependence of fracture energies, a 4-parameter sigmoidal fit was used for the base material data. In the case of the HAZ, such a fit would not be meaningful due to the lack of lower shelf energy data, given by the technical limitations of the cooling. It was not possible to carry out tests at temperatures below −196 °C. For this reason, the TSP transition temperature was evaluated only for the base material. The SP transition temperature at a specific energy TSP (1.2 J) could be determined for both base material and HAZ. A comparison of the obtained small punch transition temperature values is presented in show typical SEM micrographs obtained for fractured specimens of the base material, with the macroscopic fracture appearance of the SP discs in (a) and the corresponding fracture surfaces in (b). A clear difference can be seen between the macroscopic appearance of the specimens fractured at room temperature (upper shelf), with a circumferential crack and a large number of deformation bands visible on the specimen surface – (a), and at −196 °C (near lower shelf), where the specimens exhibited a star-shaped crack pattern – (a). As regards the fracture surface features, a gradual transition was observed over the range of temperatures from RT down to −196 °C: from purely ductile fracture manifested by the presence of dimples at room temperature ( (b)) to purely brittle fracture with large cleavage facets and no evidence of ductile crack initiation at −196 °C ( (b)). At temperatures corresponding to the transition region, ductile crack initiation followed by brittle fracture was observed. The total area of fracture surface exhibiting ductile features was found to gradually decrease with decreasing temperature. The small punch fracture appearance transition temperature for the base material was estimated as TSP, FATT = −120 °C, with the shear fracture area ≈50%, documented on (b). At this temperature, a large area with fully ductile features was identified, while the surrounding areas exhibited predominantly cleavage facets, frequently with localized zones of dimples at the facets' edges. Analogous observations made it possible to estimate the TSP, FATT for the sub-clad HAZ as −140 °C. shows the macroscopic fracture appearance and a typical fracture surface of the HAZ specimen strained at −130 °C, i.e. the value of the fracture mode transition temperature TFM, together with the corresponding SP load–displacement curve. The curve exhibits both a gradual change of slope, passing through a maximum in load, and a subsequent abrupt load drop. The appearance of the fractured specimen ( (b)) resembles the specimens fractured in a fully ductile manner, with a circumferential crack and visible deformation bands on the surface (compare with (a)). A brittle continuation of the ductile circumferential crack can be identified at the bottom of the micrograph. An inspection of the fracture surface in the specimen shown in (b) revealed a prevalence of ductile tearing. Areas with small cleavage facets were present but relatively scarce. A typical fracture surface appearance can be seen in The appearance of the small punch load–deflection curves over the temperature range studied () is typical for a material with a ductile–brittle transition. The material strengthens (due to increasing strain hardening rate) with decreasing temperature, but this effect starts to be overcompensated by the gradual loss of ductility in the transition regime. The multiple load drops () occurring at the lower testing temperatures, where the material is brittle, have been reported in the authors' recent work While the SP fracture energies presented in and used for the evaluation of the transition temperatures were calculated using the first load drop criterion, the authors alternatively applied the remaining two criteria to the base material data in order to assess the effect on the obtained value of TSP. It was found that while applying the CLD criterion led to a slightly higher lower shelf energy value as compared to the procedure using the first load drop criterion, the TSP itself did not change. Statistics from a larger number of lower shelf tests would be needed to confirm this observation. By contrast applying the CoP fracture criterion yielded clearly unrealistic lower shelf energies, so TSP could not be evaluated reliably.However, even using the first load drop fracture criterion to determine uf and consequently ESP, a robust value of the small punch transition temperature TSP could only be obtained for the base material, as the absence of lower shelf data for the sub-clad HAZ would lead to considerable uncertainty in TSP. It is not uncommon for the SP transition temperature to be determined based on one half of the upper shelf SP energy in the (frequent) situation when the lower shelf energy is not known, but this procedure can lead to errors and should be avoided if possible. Instead, alternative procedures can be used, which do not rely on the availability of lower shelf data. One possibility is to evaluate the transition temperature at a specific energy. This method was applied in the present study for both base material and HAZ data, but it should be kept in mind that the lack of lower shelf data points can still influence the value of TSP(E) to a certain extent through the error related to the interpolated slope of the energy–temperature dependence in the transition region. A second drawback of the method is the lack of a standard energy value to be used for the TSP(E) determination (analogous to the 41 J energy level in a Charpy impact test). The energy level of 1.2 J for a specimen with 0.5 mm thickness used in the present investigation was chosen so as to coincide with the value proposed by Kim et al. for DBTT evaluation in radiation embrittled RPV steels using the SP test Determining the fracture appearance transition temperature is another option of how to avoid problems associated with the lack of lower shelf data. The results of the present work have clearly demonstrated that this procedure can be applied also to the SP test (see ). However, it should be emphasized that due to the geometry of the SP specimens and the crack patterns, the evaluation of the percentage of shear fracture in the total fracture surface area is not as straightforward as in the case of Charpy specimens (see standard procedure in e.g. Ref. The last method employed in the present investigation is the determination of the fracture mode transition temperature TFM, which combines two principal advantages: no need for testing at lower shelf temperatures and at the same time great simplicity, both in terms of the experimental requirements and the data evaluation. At present, a rigorous definition of the procedure for TFM evaluation still does not exist. It was originally proposed by the authors of the concept to determine TFM as the median temperature between successive test temperatures such that half of the SP load–displacement curves exhibit an abrupt load drop (indicating brittle crack propagation), and the other half do not In the transition region, some of the SP load–displacement curves exhibit both a gradual change of slope, passing through a maximum in load, and subsequently an abrupt load drop – the former being characteristic for ductile and the latter for brittle crack propagation. However, SEM fractography carried out on a specimen corresponding to such a curve revealed a strong prevalence of ductile fracture with only a limited amount brittle crack propagation (compare (a), (b) and (c)). The position of the brittle crack propagating outwards from the ductile circumferential crack, documented on (b), even suggests that the abrupt load drop recorded on the load-displacement curve may possibly be associated with the “opening” of the specimen following mostly ductile fracture and further propagation of the crack. Additional tests would need to be carried out to confirm this hypothesis, nevertheless even the present results already demonstrate that ductile fracture can strongly prevail in the specimen despite the presence of an abrupt load drop on the load–displacement curve. For this reason, it is proposed to evaluate TFM as the lowest test temperature, at which the SP load–displacement curves smoothly pass through a maximum in load and subsequently exhibit a sudden load drop. In the case that no such behavior is observed, the original definition should be applied (median temperature such that one half of the curves exhibit a sudden load drop). The authors believe that this modification of the procedure for the evaluation of the fracture mode transition temperature should reduce the overestimation of its value in particular when only a limited amount of test data is available, with appreciable differences in successive testing temperatures.The viability of SP testing as a method to determine the ductile–brittle transition temperature strongly depends on the correlation of the transition temperature values with those obtained using conventional testing techniques. The SP transition temperatures are known to be systematically lower than the transition temperatures obtained with Charpy impact testing. This is mainly caused by a combination of the size and strain rate effects. Empirically, it has been shown for a wide range of steels (see e.g. Refs. where α is an empirical correlation factor. However, this general relationship leads to a scatter band, which is too broad to allow practical application In the NESC-I project, Charpy impact testing was carried out for the base material, according to DIN 50115 shows a direct comparison of the transition temperature values from the NESC-I project . For the base material, both procedures based on fracture energy and the procedure based on fracture appearance yield consistent values of α in the vicinity of 0.4. This result is in good accord with the values reported by Kim et al. for RPV steels of similar composition (SA508 Class 3, SA508 Class 2, and SA533B Class 1) When correlating the SP transition temperatures with the T0 reference temperature, which was the only parameter available from the NESC-I project also for the HAZ, it can be seen from that for the fracture mode transition temperatures TFM, nearly identical values of α were obtained for both BM and HAZ (0.48 and 0.49 respectively). This confirms the approach based on the determination of TFM as an applicable procedure, in addition to its simplicity and the advantage of avoiding tests at lower shelf temperatures.In a practical application of the correlation procedure, only the small punch testing is carried out and the factor α (already known for the given material class) is used to predict the conventional transition temperature. It should be noted that for the presented results, an uncertainty of ±0.01 in the value of α still translates into uncertainties of approximately ± 5 °C to ± 10 °C in the transition temperature (depending on the chosen method and specific transition temperature value). However, this still represents a major improvement compared to the scatter band of ±56 °C for SP transition temperature correlations in steels reported by Norris The present study confirms the promise of SP fracture testing as a method for the determination of the ductile–brittle transition temperature, provided that a reliable value of the empirical correlation factor α is available for the class of materials in question.Small punch fracture tests were carried out on embrittled A 508 Class 3 reactor pressure vessel steel, originating from the NESC-I spinning cylinder project. The SP testing technique was applied to both the base material and material extracted from the sub-clad heat affected zone. Fractured specimens were subjected to SEM fractographic analysis. Ductile–brittle transition temperatures were evaluated from the SP test data and compared to the original results from the NESC-I project. The main conclusions of the study can be summarized as follows:It was demonstrated that four different methods can be employed to evaluate the ductile–brittle transition temperature using the SP test. In addition to the two parameters based on fracture energies TSP and TSP (1.2 J), the SP transition temperature can be evaluated from fracture appearance (TSP, FATT) and from the change in characteristic appearance of the SP curves (TFM). The TFM method appears as very promising due to its simplicity and the benefit of avoiding tests at very low temperatures.For the base material, a correlation between the SP transition temperature values and the original data from the NESC-I experiment obtained by means of Charpy impact testing was established using the empirical relationship:with consistent values of the correlation factor α in the vicinity of 0.4 for both procedures based on fracture energy as well as the procedure based on fracture appearance. The established values of α are in accord with those reported in literature for RPV steels of similar composition, including ones subjected to radiation embrittlement, thus confirming the applicability of the above empirical relationship to this class of materials.The transition temperatures determined by SP testing were also correlated with the T0 reference temperature evaluated in the NESC-I project for both base material and HAZ using the Master Curve Approach. For the fracture mode transition temperature TFM, nearly identical values of α were obtained with both base material and the HAZ (0.48 and 0.49 respectively). A larger number of tests and smaller intervals between successive test temperatures in the transition region are expected to further increase the robustness of the correlation.Processing and mechanical properties of a molybdenum silicide with the composition Mo–12Si–8.5B (at.%)Alloys with the nominal composition Mo–12Si–8.5B (at.%) were prepared by arc-melting or powder-metallurgical processing. Cast and annealed alloys consisted of approximately 38 vol.% α-Mo in a brittle matrix of 32 vol.% Mo3Si and 30 vol.% Mo5SiB2. Their flexure strengths were approximately 500 MPa at room temperature, and 400–500 MPa at 1200°C in air. The fracture toughness values determined from the three-point fracture of chevron-notched specimens were about 10 MPa m1/2 at room temperature and 20 MPa m1/2 at 1200°C in air. The relatively high room temperature toughness is consistent with the deformation of the α-Mo particles observed on fracture surfaces. Three-point flexure tests at 1200°C in air and a tensile test at 1520°C in nitrogen indicated a small amount of high temperature plasticity. Extrusion experiments to modify the microstructure of cast alloys were unsuccessful due to extensive cracking. However, using powder-metallurgical (PM) techniques, microstructures consisting of Mo3Si and Mo5SiB2 particles in a continuous α-Mo matrix were fabricated. The room temperature fracture toughnesss of the PM materials was on the order of 15 MPa m1/2.Molybdenum silicides such as MoSi2 offer excellent oxidation resistance, but their fracture toughness tends to be low. The room temperature fracture toughness of MoSi2 is only on the order of 3 MPa m1/2. The range of stoichiometry of the Mo5Si3–B phase at 1800°C was also investigated in considerable detail by Huebsch et al. One way to toughen silicides is by the incorporation of ductile Mo. For example, brittle Mo3Si may be toughened by Mo inclusions. Its oxidation resistance can be improved by boron additions. The resulting class of Mo–Si–B alloys was developed by Berczik The purpose of the present work is to investigate the processing and microstructure of Mo–12Si–8.5B, which is an example of a ductile phase toughened molybdenum silicide. In Mo-Si-B alloys, the fracture toughness is expected to increase as the Mo-content is increased. On the other hand, to obtain some degree of oxidation resistance, relatively high Si and B concentrations (e.g. Mo–14Si–10B Alloys with the composition Mo–12Si–8.5B (at.%) were prepared by arc-melting in a partial pressure of argon (70 kPa) from high purity elements. The elemental Mo, Si, and B were 99.95, 99.99 and 99.5 wt.% pure, respectively. After melting several times on a water-cooled copper hearth, the alloys were dropped into a cylindrical water-cooled copper mold with a diameter of 25 mm and a length of 50 mm. The ingots were subsequently homogenized in vacuum (10−4 Pa) for 24 h at 1600°C. Some ingots destined for extrusion were dropped into MgO molds in order to reduce their cooling rates.Several Mo–Si–B ingots were inserted into molybdenum extrusion cans with an outer diameter of 50 mm, which were evacuated and sealed by electron beam welding. The sealed cans were extruded through zirconia-coated H13 steel dies with an inner diameter of 25 mm at temperatures ranging from 1450 to 1600°C.For the powder-metallurgical processing, arc-cast ingots of Mo3Si and Mo5SiB2 were crushed into −100 mesh (< 150 μm diameter) powders. These powders were then mixed with appropriate amounts of 2–8 μm Mo powders and sealed in an evacuated Nb can, which was subsequently hot isostatically pressed (HIPed) for 1 h at 1650°C and 200 MPa.Microstructural analysis was carried out by optical metallography and scanning electron microscopy (SEM) of polished specimens etched with Murakami's etch (an aqueous solution of potassium ferricyanide and sodium hydroxide). Phases were identified by powder X-ray diffraction and by measuring the Mo to Si ratio by energy-dispersive spectroscopy (EDS) in the SEM. Phase volume fractions were determined by full pattern fitting (Rietveld refinement) of the powder X-ray diffraction patterns A powder-metallurgical specimen was also examined by transmission electron microscopy (TEM). Specimens were prepared by mechanical dimpling to approximately 30 μm thickness followed by ion milling. Final thinning was carried out on a cryogenic stage using 8 kV argon ions until perforation occurred.Flexure specimens with a cross section of 3×4 mm and a length > 20 mm were electro-discharge machined and ground, and subsequently tested in a screw-driven testing machine in a three-point bend fixture with a 20 mm span at different temperatures in air. It should be pointed out that, generally, three-point fracture strengths tend to be higher than four-point fracture strengths. The cross-head speed in all tests was 10 μm/s. Fracture toughness values were determined by three-point bend tests using chevron-notched specimens. The advantage of using such specimens is the relatively small amount of material required. The specimens were 3 mm wide and 4 mm high with a notch tip depth of 2 mm resulting in a notch tip angle of approximately 75°. During testing, a crack initiated at the tip of the chevron-notch and propagated gradually through the specimen until fracture occurred. Fracture toughness values were determined based on an energy criterion by integrating the load-displacement curves. The fracture toughness was determined as , where W is the energy absorbed during the fracture of the area A swept out by the crack. Assuming the material to be linear elastic, the fracture energy (energy criterium) can also be expressed in terms of the stress intensity (stress criterium) by is the plane strain Young's modulus and ν is Poisson's ratio. Representative values for A and W were 2.8 mm2 and 1 mJ, respectively.High temperature tensile tests were carried out in flowing nitrogen using frictionally loaded miniature specimens A Mo–12Si–8.5B alloy was cast and annealed for 1 day at 1600°C in vacuum. , which includes data for the powder-metallurgical alloy to be discussed later, shows that the cast alloy had a low carbon and oxygen content. X-ray powder diffraction verified the presence of α-Mo, Mo3Si, and Mo5SiB2. shows that the α-Mo formed isolated particles in the Mo3Si/Mo5SiB2 matrix. shows the phase volume fractions calculated from the nominal composition Mo–12Si–8.5B by assuming the 3 phases to be pure Mo, and stoichiometric Mo3Si and T2. The volume fractions of the 3 phases were also determined by Rietveld analysis ). According to the Rietveld analyis, the α-Mo contained approximately 1.2 at.% Si and 1.2 at.% B. The Si concentration agreed well with the EDS concentration determined by using Mo3Si or T2 as standards, namely, 1.3 at.%. However, since the overall composition derived from the Rietveld analysis was Mo–12.4Si–9.4B, the Rietveld analysis appears to underestimate the α-Mo volume fraction. A 5–10% error in the Rietveld analysis is possible because of uncertainties in the peak profile function.The elastic constants were determined ultrasonically (E=327 GPa and ν=0.29). The fracture toughness values calculated with these elastic constants are shown in . A comparison with more precise work by Choe and Ritchie ) shows evidence for debonding and limited ductility of the α-Mo particles in the alloy. The 1200°C fracture toughness is considerably higher than that at room temperature or 500°C. This increase may be due to the high ductility of α-Mo at 1200°C, as well as limited ductility of Mo3Si and T2.Three-point flexure strengths were determined at three different temperatures in air (). The significantly higher strength at 500°C, as compared to room temperature, may be due to the increased ductility of the α-Mo phase at 500°C and the associated reduction in notch sensitivity of the specimen. The alloy retained a reasonably high strength at 1200°C. At 1200°C and loads near the fracture load, the load-displacement curves showed a slight deviation from linear elastic behavior, suggesting a small degree of plasticity (). In tensile tests in flowing nitrogen at 1200°C, on the other hand, no evidence for plasticity was observed and fracture took place in a brittle manner. The premature fracture may have been caused by bending moments during the tensile testing. However, with a combination of increased temperature (1520°C) and reduced loading rate (0.04 μm/s or 7×10−6 s−1, respectively), a small amount of plasticity was observed. The loading curve in indicates a plasticity-related deviation at about 195 MPa and brittle fracture at about 315 MPa. Assuming that the plastic elongation occurred only in the gage section, the plastic strain was only about 0.4% in spite of the very high testing temperature. SEM examination showed significant ductility of the α-Mo particles. The α-Mo was also very ductile at 1200°C and a strain rate of 1 μm/s as illustrated in . The relatively brittle behavior of Mo–12Si–8.5B even at very high temperature is therefore attributed to the brittleness of the intermetallic phases Mo3Si and T2.Extrusion of Mo–Si–B alloys was attempted to refine the microstructure and reduce or eliminate micro- or macro-cracks formed during casting. Two Mo–12Si–8.5B ingots cooled in Cu molds were canned in Mo. Extrusion at 1450°C was aborted because the load capacity of the press was reached once the Mo near the nose of the can had extruded. A second extrusion attempt with a glass lubricant at 1550°C proceeded to completion. However, the alloy was severely cracked. Slowly cooled ingots (MgO mold) with the composition Mo–14Si–10B were also extruded at 1600°C, but the extruded product was again severely cracked. Research carried out at Universal Energy Systems (UES) is in qualitative agreement with our results A powder-metallurgical (PM) specimen with the nominal composition Mo–12Si–8.5B was prepared in the following manner: arc-cast ingots of Mo3Si and T2 were pulverized into -100 mesh (< 150 μm) powders. These powders were mixed with 2–8 μm Mo powder and sealed in an evacuated Nb can, which was subsequently hot isostatically pressed (HIPed) for 1 h at 1650°C and 200 MPa. The HIPed material had a low carbon content and, as expected, a relatively high oxygen content (). A back-scattered SEM micrograph (SEM) of a polished and etched specimen is shown in . The large intermetallic particles were often cracked. Whereas the corresponding cast material contained α-Mo particles in a brittle matrix, the PM material consisted of intermetallic particles embedded in an α-Mo matrix. The dark spots in the α-Mo contained Si and O (B could not be analyzed by EDS) and are probably borosilicate glass. The brightest phase with the highest mass contrast is α-Mo. T2 can be distinguished from Mo3Si by the etching contrast at the periphery of T2.Transmission electron microscopy showed that, locally, the microstructure was free of SiO2. In particular, no evidence was found for silica films at the interphase boundaries, which would reduce the high-temperature creep strength. The SiO2 appeared to occur in the microstructure in the form of isolated particles or clusters. The three crystalline phases were identified by a combination of X-ray spectroscopy and selected area diffraction. TEM images of the α-Mo phase showed fine black speckles (a few nm in size) which were attributed to radiation damage from the ion milling. None of the intermetallic phases showed signs of radiation damage. The few dislocations found were in the α-Mo or T2 phases and were typically associated with small inclusions or arranged into low angle boundaries. shows the room temperature flexure strength and fracture toughness of PM Mo–12Si–8.5B. Due to the high density of cracks it is not surprising that its strength is lower than that of cast and annealed material with the same nominal composition (compare ). Its fracture toughness, on the other hand, is distinctly higher. There are three reasons for this. First, the microstructure consists of a continuous α-Mo matrix instead of individual α-Mo particles. This type of structure is similar to that of cemented tungsten carbide. In particular, a propagating crack cannot avoid the toughening α-Mo. The ridge indicated by the arrow in shows some evidence for plastic deformation of the α-Mo on a room temperature fracture surface (although not as clearly as for the α-Mo on the fracture surface in ). The PM processing did therefore not unduly embrittle the Mo. Second, the microstructure in the PM material is much coarser than that in the cast and annealed alloys, and it is well known that coarse microstructures result in higher ductile-phase toughening than fine microstructures Cast and annealed molybdenum silicides with the composition Mo–12Si–8.5B consist of α-Mo particles in a matrix of Mo3Si and T2. They exhibit room temperature fracture toughness values on the order of 10 MPa m1/2. These relatively high values are consistent with the observed debonding and deformation of the α-Mo particles observed on fracture surfaces. Even at temperatures as high as 1200°C substantial flexure strengths on the order of 400–500 MPa were observed. Hot extrusion of cast Mo–12Si–8.5B was unsuccessful since the extruded material was cracked severely. Microstructures containing Mo3Si and T2 particles in a α-Mo matrix were obtained by powder-metallurgical processing. Because of the coarse microstructure, and because of the inability of the cracks to avoid ductile α-Mo particles, these microstructures were associated with significantly improved room temperature fracture toughness values. Such PM materials offer potentially more ductility than the corresponding cast and annealed materials.Exploration of the high order theory for functionally graded beams based on Legendre's polynomial expansionA high order theory for functionally graded (FG) beams based on expansion of the two dimensional (2-D) equations of elasticity for functionally graded materials (FGMs) into Legendre's polynomials series has been developed. The 2-D equations of elasticity have been expanded into Legendre's polynomials series in terms of a thickness coordinate. In the same way functions that describe functionally graded relations also has been expanded. Thereby all equations of elasticity including Hook's law have been transformed to corresponding equations for coefficients of Legendre's polynomials expansion. Then system of differential equations in term of displacements and boundary conditions for the coefficients of Legendre's polynomials expansion coefficients has been obtained. Cases of the first and second approximations have been considered in more detail. For the obtained boundary-value problems solution, a finite element method (FEM) has been used and numerical calculations have been done with MATHEMATICA, MATLAB and COMSOL Multiphysics software. Numerical results are presented and discussed.A FGMs with material properties varying continuously, possess remarkable advantages over classical laminated composites in maintaining the integrity of the structure subjected to the action of thermo-mechanical loads, due to the absence of distinct interfaces. The FGMs are heterogeneous materials in which the elastic properties change from one surface to the other, gradually and continuously to achieve a required function []. They have been presented as an alternative to laminated composite materials that show a mismatch in properties at material interfaces. This material discontinuity in laminated composite materials leads to large interlaminar stresses and the possibility of initiation and propagation of cracks []. This problem is reduced in FGM because of the gradual change in mechanical properties as a function of position through the composite laminate.The FG thin-walled structures, such as beams, rods, plates and shells, have numerous applications in sciences and engineering, especially in micro- and nanotechnology []. At the micro- and nano-scale surface effects may have a significant effect on the physical properties and behavior of material and structures []. The study of the stress-strain state of FG beams is an important aspect in the successful applications of them as structural elements []. Stability and vibration analysis is very important to ensure the safe operation of thin-walled structures. For FG beams, plates and shells such analysis has been presented, for example in Refs. []. Various theoretical models of FG beams, plates and shells have been developed in the last decades []. Most of the proposed models of FG beams are based on the Euler–Bernoulli, Timoshenko [] hypotheses or have used more complicated high order theories such as the third-order shear deformation plate theory []. Some researchers explore 2-D or 3-D equations of elasticity to analytically solve some particular problems for FG beams []. These problems can be used as benchmarks for comparison and testing of numerical solutions obtained using new theories. Mathematically rigorous and take into account mechanical properties that are important for the engineering applications approach to the creation of high order hierarchical models of beams is based on the expansion of the 2-D equations of elasticity in Legendre polynomials series in terms of thickness. Such an approach has been used for the development of various theories of isotropic [] plates and shells. The method of Legendre polynomials series expansion has been used widely in our previous publications for developing the theory of thermoelasticity of plates and shells with considering close mechanical and thermal contact []. More specifically, the problem of heat conducting and unilateral contact of plates and shell through the heat-conducting layer with considering a change of layer thickness in the process of the shell deformation has been formulated in Refs. []. The developed approach has been applied to the laminated composite materials with the possibility of delamination and thermoelastic contact in the temperature field in Refs. [], the pencil-thin nuclear fuel rods modeling in Ref. [] and modeling of MEMS and NEMS in Refs. [In this study we are developing a high order theory for FG beams based on the expansion of the 2-D equations of elasticity for FGMs into Fourier series in terms of Legendre polynomials. More specifically, we expanded functions that describe functionally graded relations into Fourier series in terms of Legendre polynomials and find Hook's law that related Fourier coefficients for the expansions of stress and strain. Then using techniques developed in our previous publications we find a system of differential equations and boundary conditions for Fourier coefficients. Cases of the first and second approximations have been considered in more detail. Calculations have been done using our computer codes prepared in MATLAB and also using implemented the in the commercial software MATHEMATICA 10 and COMSOL Multiphysics FEM. Verification of the presented models was carried out by comparison with experimental data and theoretical result obtained using 2-D elasticity models. Analysis of the stress-strain state and examples of numerical calculations using FEM are presented.We consider a linear elastic beam in 3-D Euclidian space domain V=Ω×[−h,h] with a smooth boundary ∂V. Here 2h is a beam's thickness, Ω=[−l,l] is the middle surface of the beam and 2l length of the beam. Boundary of the beam ∂V can be presented in the form∂V=S∪Ω+∪Ω−, where Ω+ and Ω− are the outer and inner sides and S is a sheer side. We introduce here the coordinate system x1,x2 that related to the middle surface of the beam Ω as shown in Stress-strain state of the beam as 2-D elastic body is defined by stress σij and εij strain tensors with components {σ11,σ22,σ12}and {ε11,ε22,ε12} respectively, and displacements ui, traction pi, and body forces bi vectors with components {u1,u2}, {p1,p2} and {b1,b2} respectively. These quantities are not independent; they are related by equations of linear elasticity. In the introduced system of coordinates the equations of equilibrium have the form∂σ11∂x1+∂σ12∂x2+b1=0∂σ21∂x1+∂σ22∂x2+b2=0ε11=∂u1∂x1,ε22=∂u2∂x2,ε12=12(∂u1∂x2+∂u2∂x1)Material of the beam is inhomogeneous with elastic modulus of the formThen Hooke's law can be written in the formHere E(x)is Young's modulus which depend on coordinates, and ν is a Poisson ratio, δij are the Kronecker's symbols.These equations will be used for elaboration of the 1-D equations for FG beam.We expand the physical parameters, which describe the stress-strain state of the bean as 2-D elastic body into the Legendre polynomials series along the coordinate x2 and represent them in the formσij(x)=∑k=0∞σijk(x1)Pk(ω),σijk(x1)=2k+12h∫−hhσij(x1,x2)Pk(ω)dx2,εij(x)=∑k=0∞εijk(x1)Pk(ω),εijk(x1)=2k+12h∫−hhεij(x1,x2)Pk(ω)dx2,ui(x)=∑k=0∞uik(x1)Pk(ω),uik(x1)=2k+12h∫−hhui(x1,x2)Pk(ω)dx2,pi(x)=∑k=0∞pik(x1)Pk(ω),pik(x1)=2k+12h∫−hhpi(x1,x2)Pk(ω)dx2,bi(x)=∑k=0∞bik(x1)Pk(ω),bik(x1)=2k+12h∫−hhbi(x1,x2)Pk(ω)dx2.Substituting these expansions in equations we obtain corresponding relations for Legendre polynomials series coefficients. For details one can refer to the monographs [] and mentioned above our publications, for example [∂σ11k∂x1−σ12k¯+f1k=0,∂σ21k∂x1−σ22k¯+f2k=0.fik(x1)=bik(x1)+2k+1h(σi2+(x1)+(−1)kσi2−(x1)),σi2k¯(x1)=2k+1h(σi2k−1(x1)+σi2k−3(x1)+…)ε11k=∂u1k∂x1,ε12k=∂u2k∂x1+u1k¯,ε22k=u2k¯,uik¯(x1)=2k+1h(uik+1(x1)+uik+3(x1)+…).In order to transform Hooke's law in 1-D form following [], we expand Young's E(x) into Legendre polynomials seriesE(x)=∑r=1∞Er(x1)Pr(x2),Ek(x1)=2k+12h∫−hhE(x1,x2)Pk(ω)dx2..Substituting this expansion and expansions for the stress and strain tensors into Hooke's law (6) we obtain the 1-D Hooke's law for the Legendre polynomials series coefficients in the formσijn(x1)=cijkl0∑r=1∞∑m∞∈nrmEr(x1)εklm(x1)Substituting Cauchy relations (10) and Hooke's law (12) into equations of equilibrium (8) we obtain 1-D differential equations of equilibrium in the form of displacements. This system of equations contains infinite number of equations which can be written in the form∂∂x1(C˜(x1)⋅∂u(x1)∂x1)+B˜(x1)⋅∂u(x1)∂x1+A˜(x1)⋅u(x1)=f(x1)Here infinite dimensional matrixes have the form C˜=E⋅C, B˜=E⋅B and A˜=E⋅A,C=|Cij000⋯0Cij11⋯⋮⋮⋱|,B=|Bij00Bij01⋯Bij10Bij11⋯⋮⋮⋱|,A=|Aij00Aij01⋯Aij10Aij11⋯⋮⋮⋱|u=|uj0uj1⋮|,f=|fj0fj1⋮|Matrixes C, B and A correspond to the case of homogeneous elastic beam, matrix E characterized inhomogeneous properties of the beam.E=|Eij00Eij01⋯Eij10Eij11⋯⋮⋮⋱|,Eijnm=|Enm00Enm| has to be considered together with corresponding boundary conditions, which have to be formulated in the term of the Legendre's polynomial expansion. Dirichet's and Neumann's boundary conditions have the formuik(x1)=uik,0(x1)andpik(x1)=pik,0(x1)∀x1∈∂Vrespectively. Here uik,0(x1)andpik,0(x1) are coefficients of the expansion Legender's polynomial of the displacement and traction vectors components.We obtain infinite set of 1-D differential equations for coefficients of the Legandre's polynomial series expansion (15). In order to simplify the problem we have to construct approximate theory and keep only finite set of members in the expansions (7). Order of the approximation depends on assumption regarding thickness distribution of the stress-strain parameters of the beam. We consider here the case of relatively small thickness in comparison with other dimensions of the beam. Therefore we can keep only two and three members in polynomial expansion (7). In this case we will get first and second order approximation equations for FG beams.In this case only the first two terms of the Legendre polynomials series expansions (7) have to be taken into account. Then the parameters, which describe the stress-strain state of the beam, have the formσij(x)=σij0(x1)P0(ω)+σij1(x1)P1(ω)εij(x)=εij0(x1)P0(ω)+εij1(x1)P1(ω)pi(x)=pi0(x1)P0(ω)+pi1(x1)P1(ω)bi(x)=bi0(x1)P0(ω)+bi1(x1)P1(ω)Equations of equilibrium (8) now have more simple form∂σ110∂x1+f10=0,∂σ111∂x1−3σ120h+f11=0,∂σ210∂x1+f20=0,∂σ211∂x1−3σ220h+f21=0fi0(x1)=bi0(x1)+1h(σi2+(x1)+σi2−(x1)),fi1(x1)=bi1(x1)+3h(σi2+(x1)−σi2−(x1)),σi20¯(x1)=0,σi21¯(x1)=3hσi20(x1).Cauchy relations (10) also have simple formε110=∂u10∂x1,ε120=∂u20∂x1+1hu11,ε220=1hu21,ε111=∂u11∂x1,ε121=∂u21∂x1,ε221=0. for the coefficients ∈nrm Hooke's law for coefficients of the Legendre polynomials series expansion has the formσij0=cijkl0(E0εkl0+E1εkl1),σij1=cijkl0(E1εkl0+E0εkl1)Now system of the equations for displacements has the same form as (14), it contains only four equations. Expressions for corresponding matrixes and vector are presented in the Appendix.Thus we have obtained system of differential equations which together with boundary conditions (17) can be used for the stress-strain calculation for the H1BT approximation.In the second order approximation first three terms of the Legendre polynomials series have to be taken into account. In this case the parameters, which describe the stress-strain state of the beam, can be presented in the formσij(x)=σij0(x1)P0(ω)+σij1(x1)P1(ω)+σij2(x1)P2(ω),εij(x)=εij0(x1)P0(ω)+εij1(x1)P1(ω)+εij2(x1)P2(ω),ui(x)=ui0(x1)P0(ω)+ui1(x1)P1(ω)+ui2(x1)P2(ω),pi(x)=pi0(x1)P0(ω)+pi1(x1)P1(ω)+pi2(x1)P2(ω),bi(x)=bi0(x1)P0(ω)+bi1(x1)P1(ω)+bi2(x1)P2(ω).Equations of equilibrium (8) in this case have the form∂σ110∂x1+f10=0,∂σ111∂x1−3hσ120+f11=0,∂σ112∂x1−5hσ121+f11=0,∂σ210∂x1+f20=0,∂σ211∂x1−3hσ220+f21=0,∂σ212∂x1−5hσ221+f21=0fi0(x1)=bi0(x1)+1h(σi2+(x1)+σi2−(x1)),fi1(x1)=bi1(x1)+3h(σi2+(x1)−σi2−(x1)),fi2(x1)=bi2(x1)+5h(σi2+(x1)+σi2−(x1)),σi20¯=0,σi21¯=3hσi20,σi22¯=5hσi21.ε110=∂u10∂x1,ε120=∂u20∂x1+1hu11,ε220=1hu21.ε111=∂u11∂x1,ε121=∂u21∂x1+3hu12,ε221=3hu22ε112=∂u12∂x1,ε122=∂u22∂x1,ε222=0. for the coefficients ∈nrm Hooke's law for coefficients of the Legendre polynomials series expansion has the formσij0=cijkl0(E0εkl0+E1εkl1+E2εkl2),σij1=cijkl0(E1εkl0+(E0+E2)εkl1+E1εkl2),σij2=cijkl0(E2εkl0+E1εkl1+(E0+E2)εkl2).Now system of equations for displacements has the same form as (14), it contains only four equations and corresponding matrixes and vector are presented in the Appendix.Thus we have obtained system of differential equations which together with boundary conditions (17) can be used for the stress-strain calculation for the H2BT approximation.FGMs are advanced materials, whose mechanical properties can be changed continuously in the most suitable way for specific applications, therefore they are very useful for applications in engineering science. In the simplest FGMs, two different material ingredients change gradually from one to the other. Discontinuous changes such as a stepwise gradation of the material ingredients can also be considered as FGM. The most familiar FGM is compositionally graded from a refractory ceramic to a metal. Typically, FGMs are made from a mixture of ceramic and metal or from a combination of different materials. The ceramic in an FGM offers thermal barrier effects and protects the metal from corrosion and oxidation, and the FGM is toughened and strengthened by the metallic composition. FGMs are now developed for general use as structural elements in extremely high temperature environments and other applications.Material properties of an FGM are the functions of volume fractions and they are managed by a volume fraction []. For FG elastic beam the volume fraction is assumed asWhen the beam is considered to consist of two materials with Young's modulus E1 and E2respectively, the effective Young's modulus E(x3) given by the following power-law expressionFrom these relationships, the following facts are noted: at the lower surface, the FGM properties are those of the constituent material 2; at the upper surface, they are those of the material 1. Thus, FGM properties change continuously from the material 2 at the lower surface to the material 1 at the upper surface.Substituting function (29) into equation we obtain expressions for the Legendre polynomials coefficients for the effective Young's modulusE1=E1sinh(βh)βh,E2=3E1(βhcosh(βh)−sinh(βh))β2h2,E3=5((3+h2β2)sinh(βh)−3βhcosh(βh))β3h3.In this study we consider the constituents of the FGM beam that include metal and ceramics with Young's modulus E1=75GPaandE2=150GPa, and with Poison ratio ν=0.25respectively.In order to validate developed theory we compare results obtained using the H1BT and H2BT FG beam theories with experimental data presented in Ref. [] and results obtained using 2-D theory of elasticity.] are presented experimental data of the deflection of the simply supported FG beam loaded with concentrated load at the center of beam. Two samples of the five-layer FG specimens made with Al/SiCand Ni/Al2O3 have been considered. The length and width of each specimen are 2l=125mm and b=15mm, respectively. Geometrical parameters of the FG specimens are the following: the thicknesses of the Al/SiCand Ni/Al2O3 five-layer specimens are 10 mm and 2.5 mm respectively. The material properties of the basic constituents Al, SiC, Ni and Al2O3 of the FGM systems, are listed in The results of comparison of the experimental results presented in Ref. [] and theoretical ones obtained using the H1BT and H2BT are plotted in for Al/SiCand Ni/Al2O3 systems respectively on the left and right plots. Following [] load (vertical axis) is presented in (Kgf) and deflection (horizontal axis) is presented in (mm) respectively. It is observed that the results obtained using the H1BT and H2BT are in good agreement with the experimental results for all cases.Boundary-value problems for the H1BT and H2BT FG beam theories are defined by a system of differential equation with corresponding matrices presented in the Appendix and by the boundary conditions which depend on the method of the ends of the beam fixation. To solve these boundary-value problems we will use here the 1-D FEM author's implemented in MATLAB codes and compare the obtained results with MATHEMATICA 10, and COMSOL Multiphysics FEM software. For simplicity, dimensionless coordinates, displacements and stress have been introduced in the formWe consider a beam under uniform load applied to the upper side. The calculations have been done for Young's moduli equal to E1=1Pa and E1/E2=2, Poison's ratio ν=0.25respectively and for widness b=1, hl=0.1.Our study shows that quadratic and cubic polynomial approximations have much better convergence than linear ones. The calculated results are presented in , where the calculated vertical dimensionless displacements for the cases mentioned above have been presented. it is evident that good convergence is achieved even with 6 elements for both simply supported and clamped-clamped beams. All of the presented results have been compared with calculations obtained with MATHEMATICA 10, and COMSOL Multiphysics FEM software using H1BT and H2BT. The same results have been obtained in all cases considered here.In order to validate the proposed models we compare the solutions for simply supported and clamped-clamped beams obtained using the H1BT and H2BT FG beam theories with the solution obtained using 2-D equations of elasticity, we call it 2-D beam theory (2DBT). In the case of the H1BT and H2BT FG the differential equation with matrix coefficients presented in the Appendix and boundary conditions which correspond simply supported and clamped-clamped beams respectively have been used. In the case of 2DBT standard FEM software and the exponential-law expression (29) for effective Young's modulus has been used. For details of FEM implementation using MATHEMATICA 10 and COMSOL Multiphysics FEM software one can refer to corresponding tutorials.The results of calculations are presented in where the vertical displacements calculated for the cases above mentioned have been compared. One can see a very good agreement of the results obtained by using proposed theories and those obtained using the 2-D theory of elasticity. show the Legendre polynomials coefficients for the normalized displacements distribution versus normalized length for the simply supported and clamped-clamped FG beam for the H1BT and H2BT theories respectively. These coefficients are FEM solutions of the systems of differential equation for the first and second approximation respectively.Displacements have been calculated using Legendre polynomials coefficients for the displacements and representations (18) and (23) for the H1BT and H2BT respectively. For stress calculation the same representations have been used, but Legendre polynomials coefficients for stress have been calculated first using Cauchy relations (21) and (26) and Hook's law (22) and (27) for the H1BT and H2BT respectively. show normalized vertical displacements and normal stresses distribution versus normalized length and thickness for the H2BT for the simply supported and clamped-clamped FG beams respectively. For H1BT the qualitative nature of the patterns of the displacements and stress distribution is the same as for the H2BT. Therefore they have not been presented here. Obtained results show that the displacements qualitatively and quantitatively distributed in the same way, but stresses are slightly different quantitatively. More accurate results give us second approximation theory. present of the normalized vertical displacements u2 distribution at the bottom surface of beams Ω− with coordinate ξ2=−1 calculated with MATHEMATICA using FEM for the H1BT and H2BT respectively versus the Young's moduli ratio E2/E1 for simply supported and clamped-clamped FG beam respectively. As it was expected displacements and stresses decrease as the Young's moduli ratio increase.The high order theory for FG beams, which is based on the expansion of the 2-D equations of elasticity for FMs into Legendre's polynomials series has been developed. Starting from 2-D equations of elasticity for FGMs, the stress and strain tensors, vectors of displacements, traction and body forces and also function that describe functionally graded relations for Young's modulus have been expanded into Legendre polynomials series in terms of the shell thickness coordinate. Then all equations of elasticity including Hook's law have been transformed to the corresponding equations for the Legendre's polynomials series expansion coefficients. The system of differential equations in terms of displacements and boundary conditions for the coefficients of expansion has been obtained. The cases of the H1BT and H2BT approximations have been considered in more details. All necessary equations and their coefficients have been written explicitly and corresponding boundary-value problems have been formulated. For the numerical solution of the formulated problems the FEM has been used. Calculations have been done using our computer codes in MATLAB and using the ones implemented in commercial software MATHEMATICA 10 and COMSOL Multiphysics FEM. For validation of the proposed models and obtained equations comparison with experimental data and the results obtained using 2-D equations of elasticity has been done for exponential function for graduation law. The influence of different parameters on the stress-strain state of the FG beams has been studied.Matrices and vectors from in (15) and (16) for the H1BT have the formE=|E00E1/300E00E1/3E10E000E10E0|,A=|0000000000−3μh20000−(λ+2μ)3h2|,B=|000λh00μh00−3μh00−3μh000|C=|λ+2μ0000μ0000λ+2μ0000μ|,u=|u10u30u11u31|,f=|f10f30f11f31|Matrices and vectors from in (15) and (16) for the H2BT have the formE=|E00E1/30E2/500E00E1/30E2/5E10E0+2E2/502E1/500E10E0+2E2/502E1/5E202E1/30E0+2E2/700E202E1/30E0+2E2/7|,A=|00000000000000−3μh2000000−(λ+2μ)3h2000000−15μh2000−000−(λ+2μ)15h2|,B=|000λh0000μh0000−3μh0003λh−3λh0003μh0000−5μh0000−5λh000|,C=|λ+2μ000000μ000000λ+2μ000000μ000000λ+2μ000000μ|,u=|u10u30u11u31u12u32|,f=|f10f30f11f31f12f32|Anisotropy of magnetic susceptibility (AMS)Pull-apart emplacement of the Margeride granitic complex (French Massif Central). Implications for the late evolution of the Variscan orogenA microstructural, magnetic fabric and gravity study is performed on the Carboniferous Margeride granitic complex that crops out in the central part of the Variscan French Massif Central. This complex consists of three facies, namely, a main porphyritic monzogranite, a two-mica granite, and late leucogranite dykes and stocks. In spite of local variations, the magnetic lineation mainly trends NW–SE with a shallow plunge throughout the complex. The magnetic foliation pattern is more complex with various strikes and generally a moderate dip. This fabric pattern complies with the late-orogenic NW–SE regional extensional deformation already recognised in this part of the Variscan Belt. New gravity measurements complete available data and are used to build up a gravity map and a cross-section with which four gravity minima are identified. Three gravity minima are interpreted as extensional fractures consistent with a NW–SE maximum stretching. The last gravity low corresponds to an extensional jog between the two westernmost fractures. The structural and gravity features of the complex are used to propose a feeding and emplacement model controlled by the regional late-orogenic extensional tectonics. The Margeride complex is interpreted as a kilometre-scale laccolith-like pluton emplaced in a transtensional setting controlled by a NW–SE opening direction. Such a model strengthens the relationships between pluton emplacement and late-orogenic collapse of the Variscan Belt.Anisotropy of magnetic susceptibility (AMS)The internal fabric of granite plutons is now routinely used to determine the magmatic processes occurring in crystallising magma chamber and also, in some cases, to determine if regional deformation prevailed during the emplacement and crystallisation of the magma (). Such studies are made easier by the use of the anisotropy of magnetic susceptibility (AMS) method, which allows a fast and accurate measurement of the magnetic fabrics of plutons (). Granite bodies can be very useful as kinematic markers, since they may record structural features during a short time interval allowing an accurate reconstitution of the tectonic evolution of orogenic belts (). However, since fabrics are often acquired quite late during the crystallisation process of granite, the internal fabric is less helpful to characterise the emplacement mechanisms of plutons (). In addition, the knowledge of the 3D shape of a pluton can bring important information on the emplacement process, for example by locating and characterising the likely feeder zones (). Several studies illustrate the usefulness of combining gravity and structural data when investigating the emplacement mode and structural evolution of a granite pluton (During Middle and Late Carboniferous times, numerous plutons emplaced in the Variscan Belt of the French Massif Central attest to a significant crustal melting event resulting from the Early to Middle Carboniferous nappe stacking (). This granitic magmatism occurred mainly during the last stages of the compressional event or during the late-orogenic extensional collapse (). Both events begin earlier in the northern part of the massif than in the southern part (). The southward shifting with time of the two deformation events is still not accurately constrained. Moreover, these two events are partly synchronous; the extensional event begins in the northern part of the massif, whereas the southern part is still in compression. The change from the compressional tectonic event to the extensional tectonic regime must be investigated in more detail in order to understand the evolution of the Variscan Belt.This study, based on the combined results of several methods, such as field and microstructural observations, low-field AMS measurements and gravity data modelling, was carried out on one of the largest pluton of the French Massif Central, namely the Margeride granitic complex, located in the central part of the French Massif Central (). This complex consists of several granitic bodies with distinct facies emplaced at the end of the Variscan Orogeny when the two tectonic events, extensional and compressional, occur in the northern and southern parts of the massif, respectively. The choice of the Margeride complex as a study area is supported by: (1) the central location of this granitic complex in the Massif Central between the northern and southern parts of the massif, (2) the large size of the granite body, and (3) the occurrence of two facies that emplaced in an interval of about 10 Ma. The location and the period of emplacement of these facies allow us to investigate the change of tectonic deformation in the Massif Central. The aim of this study is to better constrain the internal structure and the emplacement processes of this granitic complex bringing new insights not only on the late-orogenic tectonic evolution of the French Massif Central, but also on the mechanisms controlling the emplacement of such large plutons. Our feeding and emplacement model is different than that previously proposed for this pluton () in the sense that emphasis is placed here on an extensional tectonic setting.The Variscan Belt of Western Europe has long been recognised as a collisional orogen resulting from the convergence between Gondwana and Laurussia supercontinents, to the south and north, respectively (). In the French Massif Central, the collision is characterised by the succession of several tectonic events between Late Silurian (ca. 410 Ma) and Early Permian (ca. 300 Ma) (e.g. ). From Late Silurian to Middle Devonian, the northward continental subduction of the Gondwanian margin beneath the Armorica microplate is followed by the exhumation of high- and ultra-high pressure rocks (). A nappe stacking coeval with the exhumation of high-pressure rocks is evidenced by NE–SW-trending lineations with top-to-the-SW shearing. During Late Devonian, the occurrence of calc-alkaline magmatic rocks in the NE part of the French Massif Central is interpreted as the mark of a new subduction event and a rifting period (). The true collision between Laurussia and Gondwana occurred in the Late Devonian–Early Carboniferous. In the northern part of the Massif Central, compression ended in the Early Carboniferous (ca. 360–350 Ma), whereas it lasted up to the Middle Carboniferous (ca. 330 Ma) in the southern part. This Middle Carboniferous compressional event is characterised by south-directed ductile shearing (). The entire Massif Central experienced its first extensional event due to late-orogenic collapse during the Middle to Late Carboniferous (ca. 325–310 Ma). Extension is diachronous, beginning earlier in the north than in the south of the Massif Central. This extension was orogen-parallel with a NW–SE maximum stretching direction (). A second extensional event, coeval with the formation of coal-bearing half grabens, occurred during the Late Carboniferous–Early Permian. The general direction of this extension is N–S in the Massif Central (The southern central part of the French Massif Central, in the Margeride area, shows several lithological and tectonic units (). The Palaeozoic metamorphic rocks are unconformably covered by several Mesozoic to Quaternary volcanic and sedimentary formations. The Margeride area comprises a huge granitic pluton, the Margeride massif, surrounded by several metamorphic units separated by thrust contacts marked by mylonites and top-to-the-south shearing (). To the NE of the Margeride pluton, the uprise of the Late Carboniferous Velay granite–migmatitic dome tilted the previous structures to the west (The Margeride massif is the largest post-Visean granitic complex in the French Massif Central. The massif, elongated in an E–W direction, covers an area of 3200 km2 and is cut into two parts by the Tertiary Aubrac volcanic plateau (). The westernmost part is called the Entraygues appendix. One of the peculiar features of this pluton is the numerous NW–SE-trending joints and faults. These structures are also found on the pluton borders because most of the faulted-contacts have this NW–SE trend. The Margeride granitic complex consists mainly of a porphyritic monzogranite facies, called Margeride granite sensu stricto (a) and to the east, the Chambon-le-Château granite, which compared with the monzogranitic facies shows grain size reduction and biotite enrichment (). Due to the petrological and structural similarities between the Margeride granite sensu stricto and Chambon-le-Château granite, these two types will be described as a single one. The main facies consists of K-feldspar megacrysts enclosed in a quartz, K-feldspar, plagioclase (andesine–oligoclase), biotite and locally muscovite, groundmass. K-feldspar megacrysts generally do not show any particular preferred orientation; however, locally, especially along the pluton margins, the megacrysts can be oriented ( defined three sub-facies in the Margeride porphyritic granite according to the amount of biotite, namely the ‘dark’, ‘medium’ and ‘light’ sub-facies. The ‘dark’ sub-facies is observed only in the eastern part of the Margeride pluton. Moreover, the porphyritic facies encloses some mafic enclaves with variable compositions (c). It is also cross-cut by numerous leucogranite bodies and dykes that trend generally NW–SE or NE–SW (d). The texture of leucogranites range from aplitic to pegmatitic and their composition is quartz, K-feldspar, plagioclase (oligoclase–albite), muscovite and minor biotite. Tourmaline (schorlite) grains are locally observed in some leucogranite dykes. Due to N–S-trending large-scale upright folding related to the emplacement of the Velay dome (), the massif floor is well observed along the eastern margin. There, the Saint-Christophe-d'Allier two-mica granite (SCA in ) exhibits a well-marked subsolidus fabric with a NW–SE mineral and stretching lineation. This two-mica granite is composed of quartz, K-feldspar, plagioclase, biotite and muscovite; those two latter being roughly in equal contents.The Margeride complex was emplaced at the end of the Variscan orogeny as indicated by the geochronological ages. Previous data on the Margeride porphyritic monzogranite yield older ages, namely 323±12 Ma (Rb/Sr on whole rock; ). Moreover, new 40Ar/39Ar results on biotite indicate younger cooling ages at 310±3 and 306±3.1 Ma (). The Chambon-le-Château granite yields an age of 311±6 Ma with U/Pb on monazite (). Although the formation of the Margeride granitic complex as several stages cannot be totally excluded, the most recent and reliable data suggest that the emplacement of the porphyritic granite occurred around 310 Ma. Only one date of 305±14 Ma (U/Pb on monazite) exists for the Saint-Christophe-d'Allier two-mica granite (). Regardless of the method, the geochronological ages for the cross-cutting leucogranites are more tightly grouped and range from 307 to 298 Ma (). In agreement with our structural observations, the leucogranites are younger than the porphyritic granite and were emplaced in a host granite already solidified. Therefore, the emplacement of the Margeride complex took place in Late Carboniferous times and post-dates nappe emplacement.Previous analyses of the preferred orientation of K-feldspar megacrysts show two imbricate fabrics at the pluton scale in the Margeride porphyritic granite. One trends N60–100° and is named ‘longitudinal’, whereas the other trends N140–160° and is ‘oblique’ relative to the general elongation of the pluton (). The ‘oblique’ fabric is the only one developed in the eastern part of the pluton, whereas the ‘longitudinal’ fabric is present more in the central and western parts. The two directions of K-feldspar megacrysts preferred orientation were interpreted to reflect emplacement and crystallisation processes of the magma (, the ‘medium’ and ‘light’ sub-facies were emplaced through a feeder zone located in the eastern part of the pluton. Then the magma expanded along an E–W direction. The ‘longitudinal’ fabric is interpreted as reflecting this E–W-trending magma flow. The later emplacement of the ‘dark’ sub-facies through the same eastern feeder zone would change the initial orientation of the primary fabric into an ‘oblique’ direction in the eastern area of the pluton (). In a more detailed study carried out in the eastern border of the Margeride pluton, the statistical analysis of K-feldspar megacrysts preferred orientation is reinterpreted in a new structural model () according to which the mylonitic contacts observed between the Margeride, Chambon-le-Château and Saint-Christophe-d'Allier plutons indicate a top-to-the-southeast shearing. Both preferred orientations would be coeval and would correspond to the same strain field. The ‘longitudinal’ orientation, developed in the centre of the pluton, is associated with coaxial flow, whereas near the floor and roof contacts of the granites, the ‘oblique’ orientation develops with a non-coaxial strain related to the regional southeastward shearing (). Regardless of the emplacement model and related kinematics, the previous studies consider the Margeride granite as a laccolith-like pluton (Furthermore, some local variations of the preferred orientation of K-feldspar megacrysts are observed, mainly near the northeastern border. It has been suggested that, in this area, the N–S-trending large-scale upright folding related to the emplacement of the Velay dome has reoriented the primary planar and linear structures of the Margeride complex (Conversely to the porphyritic granite, very few previous structural works dealt with the leucogranite bodies in the Margeride complex. However, it is worth noting that the leucogranite dykes exhibit a peculiar orientation related to the NW–SE-trending regional extensional strain field. Leucogranite dykes trend generally NW–SE or NE–SW. The dominant NE–SW-trending dykes, perpendicular to the maximum extensional direction, might be interpreted as steep-dipping faults open like ‘tension gashes’. Conversely, the NW–SE-trending dykes emplaced along NW–SE faults.In the porphyritic facies, three kinds of microstructures are observed (Magmatic microstructures. Many samples exhibit microstructures that are typical of magmatic flow (a and b). However, some signs of a very weak solid-state overprint are sometimes present. Millimetre-sized quartz grains show weak undulose extinction and are almost free of sub-grain boundaries. A few quartz grains underwent a slight dynamic recrystallisation, as indicated by the occurrence of rare, well-individualised sub-grains. Generally, biotite grains are characterised by the lack of any evidence of solid-state deformation, such as undulose extinction or bending (a). Ductile deformation is completely lacking in feldspars. Compositional zonings (b) and syneusis of plagioclase are locally observed and K-feldspars show perthitic texture. A few late fractures indicate that a low temperature brittle deformation was recorded in these samples.Moderate solid-state deformation microstructures. Quartz grains systematically exhibit undulose extinction, often with a chess-board texture, and evidence exists for dynamic recrystallisation, such as lobate grain boundaries (c). Large quartz grains are rarely observed since they are often replaced by numerous small recrystallised grains with irregular boundaries. Biotite is more frequently ‘kinked’ than in the first type of microstructures and some grains are elongated (c). K-feldspar displays features of intracrystalline deformation, such as undulose extinction. Some myrmekites are also observed. All these criteria are typical of a solid-state deformation (e.g. Significant solid-state deformation microstructures. All the primary quartz grains, without exception, are recrystallised and replaced by aggregates of newly formed grains, elongated along the same direction (e) and may sometimes be compared with ‘mica-fish’. Locally, biotite and quartz grains are organised in ribbons forming a gneissic-like microstructure (f). Feldspar often displays intense undulose extinction.At the pluton scale, the distribution of these different types of microstructures shows a well-defined spatial organisation (). The magmatic microstructures are observed mostly in the central and western part of the Margeride granite. On the contrary, in the eastern border, most samples show a solid-state deformation and this is the only part of the massif where significant solid-state deformation microstructures occur. In the leucogranites, microstructures range from magmatic to moderate solid-state deformation, but no geographic distribution can be determined at the scale of the massif. Conversely, in the Saint-Christophe-d'Allier two-mica granite, moderate to intense solid-state deformation microstructures are observed throughout this pluton.The identification of the phases that contribute to the magnetic signal and its anisotropy is important in every AMS survey, because the occurrence of some minerals may lead to anomalous magnetic fabrics that cannot be easily related to kinematics (). These anomalous magnetic fabrics can result from the crystallisation of new minerals after fabric formation, from the alteration of primary crystals, or due to anomalous intrinsic anisotropies of some minerals, such as tourmaline or single-domain magnetite, whose magnetic fabric orientations do not correspond to crystallographic or shape preferred orientations. Furthermore, as mentioned by , some phases, so-called ferromagnetic sensu lato, such as magnetite or hematite, even in small amounts, can strongly influence the orientation and shape of the ellipsoid of the magnetic susceptibility. In order to identify the main ‘carriers’ of the magnetic fabrics, several methods have been used.The magnetic mineralogy survey was essentially carried out on the porphyritic facies. Observations have been realised with a petrographic microscope and also with a scanning electronic microscope (SEM). Biotite is the main Fe-bearing silicate phase observed, but, locally, Fe-bearing muscovite is also present. Biotite is sometimes altered to chlorite. Because some opaque minerals are also observed in many samples, their determination has been made with SEM observations in a few samples. The main accessory phases observed were pyrite and hematite. This latter phase is generally associated with biotite crystals. The primary or secondary character of hematite has not been established. SEM observations also show the presence of rutile and tourmaline inclusions in some biotites.Thermomagnetic experiments show different types of magnetic susceptibility vs. temperature curves. In half of the samples, heating curves display a progressive decrease of magnetic susceptibility with increasing temperatures showing that only paramagnetic phases, such as biotite or muscovite, are present (a). However, the other samples reveal a decrease of susceptibility in general around 570 °C, indicating the occurrence of magnetite. Nevertheless, this decrease follows, in general, an increase of susceptibility that begins around 500 °C (a). This may suggest that magnetite could be formed during the heating by a mineralogical transformation, perhaps from pyrite, but in some samples, magnetite is undoubtedly primary. Other sharp decreases, sometimes observed for 200 and 630 °C, suggest that pyrrhotite and hematite are locally present. Furthermore, cooling curves indicate that mineralogical transformations at high temperature can sometimes produce some hematite and magnetite crystals (a). On the contrary, the relation between the applied magnetic field and the induced magnetic moment of a sample (hysteresis loops) shows similar behaviour for most samples. Magnetic moment is linear for both increasing and decreasing magnetic fields. This complies with a predominance of paramagnetic Fe-bearing silicate phases (b). This last point is also evidenced by the low values of the magnetic susceptibility, Km ( in the supplementary data in the online version of this article). For the porphyritic granite, Km ranges from 33.3 to 461.3 μSI with an average of 123.9 μSI. Such values are typical of granite whose magnetic signal mainly comes from a paramagnetic source.Based on these data, we can conclude that in the porphyritic granite, the magnetic fabric is most likely controlled by a combined contribution of biotite±muscovite±hematite±pyrite±traces of magnetite. In spite of local alteration and probable formation of minor new phases, the AMS fabric corresponds mostly to the rock primary fabric. In the porphyritic facies, anomalous fabrics are rare, and, furthermore, due to its high content relative to other magnetic carriers, biotite is the most important contributor to the AMS.The magnetic mineralogy for the Saint-Christophe-d'Allier two-mica granite is roughly the same as the porphyritic granite. However, since this granite contains more Fe-bearing muscovite, the contribution of this latter phase is more significant. On the contrary, the magnetic mineralogy of leucogranites is more complex since the biotite content may strongly vary from one sample to another. Fe-bearing muscovite is always the most abundant Fe-bearing silicate phase. As for the porphyritic facies, traces of hematite and magnetite are encountered. Locally, occurrence of tourmaline leads to the development of an anomalous fabric since tourmaline has an inverse magnetic fabric. The long axis of tourmaline crystals corresponds to the minimum axis of magnetic susceptibility ellipsoid. In tourmaline-rich samples, AMS is a mix between a normal fabric due to muscovite, hematite, etc. and an inverse fabric due to tourmaline. Thus, the fabric of such a sample will be more or less ‘anomalous’ according to the relative abundance of the different phases.The magnetic fabrics were mapped throughout the Margeride pluton and for all the facies. However, this study concerns more particularly the porphyritic facies since this is the most represented one in the pluton. Furthermore, the eastern part of the massif was more explored due to better outcrop conditions. For the porphyritic facies, AMS was measured in 213 sampling sites by following the classical sampling and measurement methods described in . The data processing was carried out according to the statistics described in In the porphyritic facies, the magnetic lineation pattern is quite well organised (). Magnetic lineations generally trend N120–160° with a shallow plunge (<30°) (a). This trend is particularly clear in the SE part of the pluton, but is also present in the central and western parts. In these latter areas, the lineation pattern is more scattered both in trend and plunge. For example, in the western part, two lineation trends are observed, namely NW–SE and E–W. The NE area, around the Saint-Christophe-d'Allier two-mica granite, also presents a complex structure. Indeed, the AMS lineation forms an arcuate pattern with a change from N–S in the north to E–W in the south. In the Saint-Christophe-d'Allier two-mica granite, the lineation pattern is subparallel with that of the porphyritic granite host. Lineations trend NW–SE in the southern part of the Saint-Christophe-d'Allier granite, whereas they trend E–W in the central part (). The plunges are always shallow ranging from 0 to 25° (The analysis of magnetic lineation results from leucogranites shows a great scattering. However, NW–SE and NE–SW are the two dominant directions (a). Only to the southeast of the Saint-Christophe-d'Allier granite, lineations of leucogranites present a constant E–W trend. However, at the pluton scale, no preferred orientation can be clearly established for the leucogranite facies. This can be the result of several processes. In some leucogranite samples, the potential carriers of the AMS signal (muscovite, traces of ferromagnetic minerals) are so rare that the AMS measurement may either be influenced by diamagnetic phases such as quartz and feldspars or may not be representative of a petrofabric from a statistical point of view. Furthermore, as stated above, in samples where tourmaline is present, an anomalous AMS fabric develops.The main feature of the magnetic foliation, in the porphyritic facies, is its conspicuous subhorizontal dip (In the Saint-Christophe-d'Allier two-mica granite, only foliations located in the central part of the pluton show a clear pattern (). There, foliations strike E–W with a moderate to strong southward dip. In the leucogranites, the magnetic foliations do not have a clear attitude. For example, dips are highly variable (To summarise, the main structural feature of the Margeride pluton inferred from our AMS analysis is clearly the conspicuous NW–SE trend and subhorizontal attitude of the magnetic lineation (a). Such a pattern complies with those already described in other plutons of the Cévennes area (The Margeride granitic complex has already been the topic of several gravity studies (). From the 1:1,000,000 gravity map of France, observed that the Margeride pluton corresponds to a large negative gravity anomaly and proposed a laccolithic-like shape for it. A more detailed analysis carried out by with numerous measurements in the western and central parts of the pluton led this author to assume an emplacement of the prophyritic granite through three NE–SW-trending feeding zones, deduced from the first vertical derivative of the anomaly and located in western, central and eastern parts of the pluton. have also modelled some 2D cross-sections along the southern border of the pluton, oriented in the N–S-direction only. The eastern border was less explored and the deep structural relationships between the porphyritic granite and the Saint-Christophe-d'Allier two-mica granite are still unknown. Furthermore, none of the previous studies have built E–W-trending models in order to visualise the shape of the pluton along its largest dimension, cross-cutting the possible feeder zones proposed by . Therefore, 520 new gravity stations have been surveyed along the eastern border of the pluton to complete the existing gravity data (a). The resulting gravity coverage allowed us to improve the gravity anomaly all over the pluton and to build up gravity models. Joint interpretation of the gravity map and cross-sections gives insights into the 3D shape of the pluton, which is a prerequisite to propose an emplacement model.Previous and new gravity data have been tied to the Carte Gravimétrique de la France 1965 (CGF65) gravity reference network and reduced to the Hayford 1930 ellipsoid. Usual free air and plateau corrections have been made. A Bouguer reduction density of 2.60 g/cm3, close to the granite density, was used. The terrain corrections were computed out to 167 km in order to obtain the complete Bouguer anomaly. Therefore, a good integration of the new dataset within the French gravity database is achieved. In order to highlight the granite related anomalies, the long wavelengths of the complete Bouguer anomaly were removed from the signal. This was achieved using a low-pass filter of Gaussian-type with a cutoff wavelength of 250 km over the entire Massif Central. The resulting residual Bouguer anomaly represents the effect of density heterogeneities located below the topography, down to a few kilometres (a). To the first order, negative anomalies can be correlated to the granitic pluton. To the south, the pluton–host rock boundary generally coincides with a strong gradient that suggests a steep contact between the two units (). On the contrary, to the north, despite the very weak measurement coverage on this border, the negative anomaly extends over the metamorphic units, suggesting that the pluton plunges gently below the host rock (). The negative anomaly is in continuity with the other negative anomalies observed to the west and to the SE, upon the Veinazès and the Mont-Lozère plutons, respectively (). However, more detailed gravity studies and structural constraints in the host rocks preclude any link at depth between those plutons (). The eastern border displays the highest gravity values that suggest a thinning of the granite eastward. More negative anomalies on the rest of the pluton correspond to thicker parts or lower density facies. The most negative values are observed in the west of the pluton, but the central part of the pluton and the Aubrac area (dense basalts) exhibit very negative anomalies as well. This latter anomaly is surprising: in our modelling, we assumed a large amount of granite beneath the Aubrac volcanics in order to balance the high density of basalts. However, this solution is certainly not the only way to explain the anomaly and would certainly require more detailed study.To build 2D models, some constraints, such as pluton structure and densities of lithological units, are needed. Geological information, such as geological contacts and structural data, are the main constraints that can be used in the Margeride area. This information is deduced from geological maps and field observations. For example, in host rock, several lithological units have been distinguished: micaschists, paragneisses and orthogneisses. Densities of these units were taken from previous studies (). General density values, particularly for the basalts, were extracted from the literature. Previous and new density measurements are in good agreement and are also concordant with those known for other similar plutons around the study area (). The density values used in our model are given in the lower part of b. Furthermore, one E–W-trending seismic profile crossing the central part of the pluton yields a maximum thickness ranging from 7.5 to 8.85 km in this area (In order to image the deep structure of the Margeride pluton, the residual gravity anomaly has been modelled along a WSW–ENE-trending profile across the whole pluton (see location on a). Although not strictly perpendicular to the major gravity anomalies, this profile has been chosen in order to provide an overall image of the pluton at depth. Other more detailed gravity models located on the eastern border of the pluton can be found in . As seen below, because the nature (and therefore the density) of the host rocks vary around the Margeride batholith, we thought that performing directly the 3D inversion of the gravity anomaly in terms of an interface between the granite and its basement would not have been correct. Rather, we preferred a 2D direct modelling approach, which would let us better explain the assumptions of the model.The western host rock consists of micaschists, whereas the eastern one is dominated by orthogneisses (b). As indicated by the geological maps, some paragneissic lenses are located between the pluton and the host rock orthogneisses, but also into the granitic pluton. In the central part of our model, where the real petrological nature of the host rock is unknown, micaschist is the most likely lithology and consequently a density of 2.72 g/cm3 was assumed for a so-called ‘undifferentiated host rock’ metamorphic unit. The profile cross-cuts several thin units, the Aubrac basaltic plateau and the Grandrieu leucogranite massif. Their influence is almost negligible due to their minor thickness. The thickness of these units was determined using the geological maps and field observations. For the Aubrac basalts, a maximum thickness of 200 m, consistent with the geological map (The main feature of the model is the variation in thickness of the porphyritic granite (b). Four zones are characterised by a thickness of 5–7 km of granite and are located (i) on the western border, (ii) beneath the Aubrac basalts, (iii) in the central part, and, finally, (iv) to the east of the pluton. Between these areas, the granite is thinner, ranging between several hundred metres and 3 km. The progressive thinning observed on the eastern border complies with the decrease of the gravity anomaly.Three of the thickened zones, namely in the Entraygues appendix, the central and the eastern parts, correspond to the NE–SW-trending minimum gravity axes identified by a), the negative gravity anomalies corresponding to these thickened zones trend also NE–SW. Conversely, the fourth thickened zone, beneath the Aubrac basalts, complies with a negative anomaly trending NW–SE.From the geological contacts, the gravity map and modelling, it is possible to establish the likely overall 3D shape of the pluton in order to build a reliable emplacement model. The Margeride complex presents some peculiar features on the geological map. Indeed, the eastern part is much larger (∼55 km) than the Entraygues appendix (∼15 km), resulting in a rough triangular shape of the massif. Furthermore, the southern margin exhibits a peculiar shape with two important offsets (). The western one separates the Entraygues appendix from the central part, presently it corresponds to the Aubrac plateau. The second one, to the north of Mende, separates the central and eastern parts of the pluton. It is worth noting that these two offsets are in the same sense.The WSW–ENE-trending gravity profile allows us to identify four deepening zones of the pluton (). Three of them, trending NE–SW, are located in the centre of each of the three parts of the massif, that is the Entraygues appendix, the central and the eastern parts. These negative anomalies correspond to the three minimum gravity axes identified by . The fourth negative gravity anomaly, trending NW–SE, corresponds to the Aubrac plateau and also to an offset zone. Such negative anomalies and deepening zones in granites are often interpreted as feeder zones ( have already proposed that the three NE–SW-trending negative anomalies could be three pluton feeder zones. In agreement with this interpretation, we consider that the Margeride pluton emplaced through four feeder zones, three trending NE–SW and the fourth NW–SE. From these structural and geophysical features, a 3D diagram of the Margeride pluton can be schematically sketched out (The microstructures observed in the porphyritic facies are typical of magmatic-state and high temperature solid-state deformation (). This suggests that the magnetic fabric of the prophyritic facies developed during, or just after, the complete crystallisation of the magma. However, the lack of significant subsolidus deformation at the microstructural scale does not preclude that late tectonic deformation occurred. To interpret the magnetic fabrics, it is necessary to compare the internal fabrics of the plutons and the coeval host rock structures (During the emplacement and crystallisation of the Margeride complex, in Namurian–Westphalian times, the French Massif Central experienced a general NW–SE extension that is also inferred from the internal structures of several plutons () and the stretching lineation in the metamorphic aureole. This maximum stretching direction is concordant with the general trend of the magnetic lineation of the porphyritic facies of the Margeride complex, suggesting that the internal structure of the pluton recorded some increments of the regional strain (). The NW–SE-trending lineation is observed in areas where microstructures indicate a solid-state flow (e.g. the southeastern part of the pluton), but also in some areas where magmatic flow is dominant (e.g. the central part of the pluton). This suggests a continuum of deformation during the emplacement, the crystallisation and the cooling of the porphyritic granite. Conversely, magnetic lineations (except those near the northeastern border), which are not concordant with the regional deformation, are observed only in areas of magmatic flow (e.g. the western part of the pluton), suggesting that they are likely related to internal dynamics of the magma chamber. In the northeastern border of the pluton, the arcuate shape of the lineation is likely a consequence of late deformations associated with the emplacement of the Velay dome (). Although it is quite difficult to interpret the complex pattern of the magnetic foliation, the general low dip of the foliation throughout the pluton is, however, in agreement with an extensional deformation that is characterised by a vertical maximum shortening axis (Z-axis) of the finite strain ellipsoid.The magnetic fabrics of leucogranites are scattered and different from one leucogranitic body to another. However, the general NW–SE direction is also represented suggesting that leucogranites probably record some increments of the regional deformation. As shown in , the structural and kinematic pattern of the leucogranite dykes is consistent with the extensional setting proposed for the porphyritic monzogranite. Since the leucogranites are younger than the porphyritic granite, the occurrence of NW–SE-trending lineations in both lithologies would indicate that the regional extension occurred during the emplacement and crystallisation of every facies of the Margeride complex.Based on the gravity and AMS results presented above, an emplacement model can be proposed. Although a full 3D modelling has not been performed, the combination of an E–W cross-section and the gravity map gives an overall visualisation of the shape of the pluton. From east to west, it is a 1–3-km-thick laccolith-like pluton, with four 5–7-km-thick axes that are interpreted as probable feeder zones. The geometric features and the organisation of these feeder zones are integrated in the regional tectonic framework. Furthermore, the microstructures indicate that the internal fabric was acquired during, or just after, the crystallisation of the magma. Moreover, as already discussed, this fabric is mainly related to the extensional regional deformation. Consequently, the regional stretching direction is the same during the emplacement, the crystallisation and the cooling of the Margeride complex. Finally, geochronological data allow us to rebuild the history of the pluton and the successive episodes of emplacement and cooling experienced by the different facies (A four-stage feeding and emplacement model is proposed (Stages 1–2. Around 310 Ma, the magma chamber of the Margeride complex was formed by three main feeder zones trending NE–SW to NNE–SSW, which correspond to three minimum gravity axes (a). These zones can be interpreted as ‘tension gashes’ or extensional fractures perpendicular to the NW–SE maximum stretching direction. These feeder zones are linked by sinistral transfer faults trending NW–SE, which are deduced from the peculiar shape on the geological map of the Margeride pluton as described above (). The fourth minimum gravity anomaly, located below the current Aubrac plateau, is also interpreted as a feeder zone (b). In this case, the NW–SE-trending Aubrac feeder zone worked as a sinistral transtensional jog between the two other opening feeder zones of the central part and Entraygues appendix (c). The magma coming from the different feeder zones gathered in a single large magma chamber. On the eastern border of the granite complex, the Saint-Christophe-d'Allier two-mica granite emplaced below the porphyritic granite, but still controlled by the same tectonic setting. In agreement with the microstructural observations, it is likely that at this stage, during the magma crystallisation, the internal fabric of the porphyritic granite recorded some increments of the NW–SE-trending regional extensional deformation.Stage 4. Around 300–305 Ma, after the complete crystallisation and cooling of the porphyritic granite, leucogranite stocks and dykes cross-cut this one (d). The emplacement mode of the leucogranites is not really constrained by our study. However, leucogranite dykes generally trend NW–SE and NE–SW and the stocks are often limited by contacts having the same, NW–SE and NE–SW, directions suggesting that the extensional regional tectonic and the previous structure of the porphyritic granite may have influenced the emplacement of leucogranites.The emplacement of the Velay dome around 290 Ma is responsible for late deformations such as the westward tilting of the eastern border of the pluton. Presently, the tilting is consistent with the westward dip of the magnetic fabrics, the eastward thinning of the porphyritic granite and the exposure of the lower level of the pluton (This emplacement model can explain the E–W difference in shape of the Margeride complex. The eastern and central parts likely experienced a greater NW–SE stretch during the emplacement period resulting in a larger size of these areas relative to the narrower western part. The magma supply was also more important in those larger parts. It is worth noting that throughout the late Variscan times, the eastern part of the complex shows a more important magmatic activity than the western one. Indeed, after the emplacement of the Saint-Christophe-d'Allier two-mica granite, most of the leucogranites are emplaced in this eastern part of the complex. Then, later, around 290 Ma, the Velay dome was emplaced more eastwards. This suggests a repetition of magmatic events in this area during at least 25 Ma.Our feeding and emplacement model of the Margeride complex controlled by the late-Variscan extensional regional strain is not in agreement with previously proposed models (). The NW–SE-trending magnetic lineation is consistent with the N140–160°-trending ‘oblique’ fabric of the K-feldspar megacrysts, but unlike this one, the NW–SE-trending magnetic lineation is well developed throughout the pluton. Furthermore, new feeder zones are suggested by our gravity results. Indeed, the interpretation of our data is made easier by a better knowledge of the tectonic setting of this part of the French Massif Central during Namurian–Westphalian times and by analogy with models emphasising the compatibility of multiple feeder zones with an extensional setting (e.g. Our model is also not compatible with the emplacement model proposed for the Veinazès and Marcolès monzogranitic plutons () located to the NW of the Entraygues appendix (Our model takes place in a larger scale setting taking into account the Late Carboniferous emplacement of plutons and the regional tectonics in the SE part of the French Massif Central. NW–SE-trending AMS and mineral lineations are also observed in the two other plutons of the Cévennes area namely, the Mont-Lozère—Borne and Aigoual—Saint-Guiral—Liron plutons (). The NW–SE-trending AMS lineation in the southeastern part of the Margeride complex is parallel to the lineation measured in the northwestern area of the Mont-Lozère–Borne complex located 10 km more southward. However, in the two plutons of the Cévennes area, the magnetic lineation turns E–W, parallel to the E–W-trending late-Variscan stretching and mineral lineation observed in the host rock surrounding the two plutons. It seems that a rotation of the extensional direction, from NW–SE to E–W, occurs south of the Margeride complex up to the southeastern termination of the Massif Central.). This interpretation is consistent with results obtained on contemporaneous and more southern plutons in the Cévennes area (). This suggests that the late-orogenic extension already acted in the central and the southern parts of the French Massif Central around 315 Ma.Anisotropy of magnetic susceptibility data. n: number of specimens; Km: Bulk magnetic susceptibility in μSI; K1: Magnetic lineation; K3: Pole of the magnetic foliation; Dec, Inc: declination, inclination, respectively, in degrees. Data in normal type, bold and italic correspond to the porphyritic granite, the Saint-Christophe-d'Allier two-mica granite and the leucogranites, respectively.Supplementary data associated with this article can be found, in the online version, at Multiphysics modeling of thorium-based fuel performance with a two-layer SiC cladding in a light water reactorThe performance of thorium-based mixed oxide fuel ((Th,U)O2 fuel and (Th,Pu)O2 fuel) with a two-layer SiC cladding in a light water reactor have been investigated by finite element multiphysics modeling method. The material properties of specific thorium-based Th0.923U0.077O2 fuel, Th0.923Pu0.077O2 fuel and SiC cladding were firstly reviewed and implemented into the multiphysics model. Then the performance of Th0.923U0.077O2 fuel, Th0.923Pu0.077O2 fuel and UO2 fuel separately combined with Zircaloy cladding and two-layer SiC cladding have been investigated and compared under PWR normal operation conditions. Finally, the thorium-based fuel with Zircaloy cladding was found to decrease the fuel centerline temperature, especially for the Th0.923U0.077O2 fuel, but with a much earlier gap closure time. And the two-layer SiC cladding was found to effectively mitigate the pellet-cladding mechanical interaction (PCMI) but also greatly increase the fuel centerline temperature. So the combination of the thorium-based fuel with two-layer SiC cladding is expected to improve the reactor safety by keeping a moderate fuel centerline temperature and meanwhile greatly delaying the PCMI time.Recently, thorium-based mixed oxide fuel applied in pressurized water reactor (PWR) has received continuous attention, leading to several fuel performance investigations with fuel applied in light water reactors (). Thorium-based mixed oxide fuels show several potential advantages over uranium fuels: higher thermal fission factor, η, (as Thorium breeds U-233, which is better than U-235 and Pu-239 from a neutronic standpoint), lower capture-to-fission ratio, low actinide production, plutonium reduction, and higher chemical stability (). Moreover, thorium is three times more abundant than uranium in the earth’s crust. There are many thorium utilization scenarios in the nuclear fuel cycle including application in many nuclear reactor types (such as light water reactor (LWR), supercritical water reactor and high-temperature gas-cooled reactor). It has already been proved that the (Th,Pu)O2 fuel can be applied in PWRs without any significant modification in the nuclear reactor design by , and the (Th,Pu)O2 fuel was found to effectively decrease the fuel centerline temperature by , which the feasibility of the use of thorium based fuel in a PWR was further examined from a neutronics standpoint by . Previously, Long investigated the thermal–mechanical and chemical behavior of thorium-uranium and thorium-plutonium mixed oxide fuels under reactor in-service and hypothetical accident conditions in LWRs by FRAPCON code (), which concludes that the performance of thorium-uranium mixed oxide fuel is generally equal to or better than that of UO2 fuel from the point view of safety, and the safety margin of thorium-plutonium mixed oxide fuel is equal to or greater than that of UO2 fuel. Recently, the fuel performance of thorium-based (Th,Pu)O2 fuel with monolithic SiC cladding was well studied by based on the FRAPCON 3.4-MIT code. Then Björk investigated thermal–mechanical behavior of thorium-based nuclear fuel under test irradiation conditions, and the preliminary simulation results are compared with experimental irradiation data (), which good agreement has been found for the fuel temperature profile. More recently, Bell collected the best physical-based data and implemented the (Th,U)O2 and (Th,Pu)O2 fuel models into MPM-FAST code and the corresponding fuel performance under normal operation conditions of a SCWR was investigated, which consistent behaviors were found when compared with the experimental irradiation data (Currently, Silicon carbide (SiC) is being investigated as an accident tolerant candidate cladding to replace Zircaloy in the operating LWRs, as SiC offers several advantages over Zircaloy: higher melting temperature, high strength, chemical stability, lower thermal neutron capture cross section and four orders of magnitude lower corrosion rates, which makes it economically appealing and potentially allows for higher achievable fuel burnups (). However, SiC, as a ceramic, is a brittle material, resulting a lower toughness and potential for dramatic failure, which necessitates the use of a SiC/SiC composite material to provide strong tensile strength and mitigate the catastrophic crack propagation. There are many accident tolerant cladding designs composed of CVD-SiC and SiC/SiC composite materials for PWR fuel cladding (). Even though the SiC/SiC composite provides improved mechanical strength, it is typically fabricated with a degree amount of porosity and thus cannot prevent the release of radioactive gases or the ingress of the water. Additionally, the SiC/SiC composite is expected to react with cooling water. So the SiC/SiC composite alone is also not considered as a viable cladding solution. Then based on the analysis work done by , the developed model is a combination of monolithic CVD SiC and SiC/SiC composite in a multi-layer structure form, namely the two-layer and three-layer SiC cladding. The proposed duplex cladding with an inner fiber-reinforced SiC/SiC composite layer and an outer monolithic CVD SiC layer was found to greatly reduce steady-state cladding failure probability compared with the conventional three-layer cladding design with an inner monolithic CVD SiC, SiC/SiC composite interlayer and outer thin environmental barrier overcoat, which has been analyzed by . The outer monolithic CVD SiC layer of the two-layer SiC cladding is able to provide a dual function of providing hermeticity and corrosion protection, making it a promising candidate for the application of Accident Tolerant Fuels (ATFs). Moreover, the stress distribution and two types of layer thickness fraction (750/250 μm and 600/400 μm) of the two-layer cladding with inner SiC/SiC composite and outer monolithic CVD SiC have been analyzed and optimized by , respectively. Then another design of an equal layer thickness of 400 μm (800 μm in total) was adopted and analyzed by . More recently, the fuel rod behavior of the two-layer SiC cladding with UO2 fuel under PWR normal operating conditions has been well studied by , and effects of the SiC layer thickness on the fuel performance have also been analyzed.Currently, few studies have been found on investigating the fuel performance of thorium-based fuel with two-layer SiC cladding, inspired by the work done by and also the analysis of pellet-cladding mechanical interaction on U3Si2 fuel with a multi-layer SiC cladding (), in this work we focus on assessing the thorium-based fuel performance under normal LWR operation conditions with innovative two-layer SiC cladding and together with Zircaloy cladding for comparison. Namely, the (Th,U)O2 fuel and (Th,Pu)O2 fuel were selected together with two-layer SiC cladding with an inner SiC/SiC ceramic matrix composite (CMC) layer and an outer monolithic CVD SiC layer and Zircaloy cladding.The material properties of the UO2 fuel and Zircaloy cladding were implemented the same as those in our previous work (). In this part, the thermal–mechanical properties of the (Th,U)O2 fuel and (Th,Pu)O2 fuel, monolithic SiC and SiC/SiC composite are presented in the following sections.The same method of simulating the thermal conductivity of (Th,U)O2 fuel from was adopted in this work, which suggests that the multiplicative factors proposed by could be able to evaluate the irradiation effects on the thermal conductivity of (Th,U)O2 fuel. The thermal conductivity of (Th,U)O2 fuel in units of W/m/K is calculated as:where the multiplicative factors from Lucuta are given in the following Eqs. , and k0,Belle is the (Th,U)O2 fuel thermal conductivity correlation from κ1p=1+0.019∙βp3.0-0.019∙βp1.01.0+exp-(T-1200)100κ1d=1.09βp3.265+0.0643∙Tβp∙arctan1.01.09βp3.265+0.0643∙TβpB1=1.597×10-4+6.736×10-4MUO2-2.155×10-3MUO22where βp is the fuel burnup in atom percentage, MUO2 is the mole fraction of UO2 in the (Th,U)O2 fuel within the range of 0–30%, and is valid under 2200 K.The same modeling method is adopted to the simulation of the thermal conductivity of (Th,Pu)O2 fuel under irradiation given in the following equation.where k0,Cozzo is the thermal conductivity of un-irradiated (Th,Pu)O2 fuel with 5% porosity presented by where wtPuO2 is the PuO2 weight percentage in the (Th,Pu)O2 fuel matrix. The thermal conductivity of un-irradiated Th0.923U0.077O2 fuel and Th0.923Pu0.077O2 fuel are depicted in , and the thermal conductivity of UO2, PuO2 and ThO2 fuel are also plotted for comparisons (The heat capacity of (Th,U)O2 fuel is calculated based on Neumann-Kopp rule summarized in IAEA technical document (where CpThO2 is the ThO2 fuel heat capacity from CpThO2=(55.962+0.05126T-3.6802×10-5T2+9.2245×10-9T3-5.74031×105T-2)And CpUO2 is the UO2 fuel heat capacity from MATPRO-11 (). Similarly, the heat capacity of (Th,Pu)O2 fuel is calculated based on the Vegard’s law (where CpPuO2 is the PuO2 fuel heat capacity, which is also from MATPRO-11. The heat capacity of Th0.923U0.077O2 fuel and Th0.923Pu0.077O2 fuel are depicted in with heat capacity of UO2, PuO2 and ThO2 fuel for comparison.The density of thorium-based fuel is calculated by the expression of volumetric fraction multiplied by single component density for ThO2, PuO2 and UO2:Thermal expansion coefficient and Young’s modulus, Poisson’s ratioThe thermal strain of (Th,U)O2 fuel is given in the following equations derived by -0.179-0.087wtUO2+5.097+4.705wtUO2×10-4T+3.732-4.002wtUO2×10-7T2+-7.594+11.98wtUO2×10-11T3-0.179-0.149wtUO2+5.097+6.693wtUO2×10-4T+3.732-4.002wtUO2×10-7T2+-7.594+19.784wtUO2×10-11T3where the first equation is applied in the temperature range of 273K≤T<923K, the second equation in the temperature range of 923K≤T<2000K, and wtUO2 is the UO2 weight percentage in the (Th,U)O2 fuel.The thermal strain of (Th,Pu)O2 fuel is given in the following Eq. εthm,Th,PuO2=-0.179-0.049wtPuO2+5.079+225wtPuO2×10-4T+3.732-2.257wtPuO2×10-7T2+(-7.594+12.454wtPuO2)×10-11T3The Young’s modulus of (Th,U)O2 fuel is calculated based on vegard’s law (where VThO2 and VUO2 are the volume fraction of ThO2 and UO2, respectively, and the ThO2 Young’s modulus is adopted from EThO2=2.491×1011(1-2.21×P)×1.023-1.405×10-4exp181Twhere P is the volume fraction of porosity in the range from 0.06 to 0.4.The Young’s modulus of UO2 fuel recommended by EUO2=(2.334×1011)×(1-(1.091×10-4)×T×exp(-1.34Xdev))where Xdev is oxygen to metal ratio deviation and T is temperature in K.The Young’s modulus of (Th,Pu) O2-x fuel is given in equation , which is also adopted in FRAPCON 3.4 code.E(Th,Pu)O2=EThO2(1+0.0284wtPuO2)e-1.75Xdevwhere wtPuO2 is the weight percentage of PuO2 in the (Th,Pu) O2-x fuel.Due to the fact that no Poisson’s ratio data has been found for the (Th,Pu)O2 fuel and (Th,U)O2 fuel (, the Poisson’s ratio of ThO2 was used both for the Poisson’s ratio of the (Th,Pu)O2 fuel and (Th,U)O2 fuel.The UO2 fuel fission gas diffusion coefficient is calculated by the weighted summation of diffusion coefficients contributed by three types of mechanisms given in the following equations.where Dthrm is the diffusion coefficient determined by thermally activated processes, Dirr is the diffusion coefficient determined by irradiation induced vacancies, and Dathrm is the diffusion coefficient determined by athermal effects, and all the diffusion coefficients are in units of m2/s and the detailed expression of these three diffusion coefficients can be found in (, the net diffusion coefficient of UO2 fuel is given in the following equation:where b' is the intragranular resolution rate in unit of s−1,ga is the trapping rate in s−1, and D0,UO2 is the single fission gas atom diffusion coefficient of a fully dense UO2 crystal. The detailed expression of D0,UO2 was adopted from the work of . The fission gas diffusion coefficients of thorium-based fuels were calculated by changing the weighting of the three components of D0,UO2. For (Th,U)O2 fuel, the fission gas diffusion coefficient is scaled by a factor of 0.1 applied to the effective diffusion coefficient of UO2 to account for the low fission gas release at the same temperature level, which is also the same as previous models proposed by Besides, according to the post-irradiation annealing tests on thoria-urania fuel by , it was found that the diffusion of 133Xe in poly-crystalline (Th,U)O2 was approximately an order of magnitude less than that of UO2. These findings provided the rationale for selecting the 0.1 scaling factor for the (Th,U)O2 fission gas diffusion model used. According to , the fission gas diffusion coefficient of (Th,Pu)O2 fuel was calculated by fitting the fission gas release based on the irradiation history from the BDL-422 experiment, which is given as:D0,(Th,Pu)O2=1.415Dthrm+0.1604Dirr+0.1604Dathrmwhere Dthrm is the diffusion coefficient due to thermally activated processes, Dirr is the diffusion coefficient due to irradiation induced vacancies, and Dathrm is the diffusion coefficient due to athermal effects. The net diffusion coefficient of (Th,U)O2 and (Th,Pu)O2 fuel are calculated with the same method as given in equation The thermal mechanical properties of monolithic CVD SiC and SiC/SiC composite are reviewed and presented in the following section.The irradiated thermal conductivity of SiC cladding is calculated based on the model which the inverse of the thermal conductivity is calculated by the summation of the thermal resistivity of both unirradiated and irradiated SiC (), and it is given as following equation:where k is the total thermal conductivity of irradiated SiC, R0 is the resistivity of unirradiated SiC, and Rirr is the resistivity of irradiated SiC. R0 is calculated as the inverse of the unirradiated thermal conductivity. The unirradiated thermal conductivity of monolithic SiC is given by kT=-3.70×10-8T3+1.54×10-4T2-0.214T+153.1 also introduces the resistivity of monolithic SiC under irradiation:where S is the volumetric swelling strain. The unirradiated thermal conductivity of SiC/SiC composite is also presented by kT=-1.71×10-11T4+7.35×10-8T3-1.10×10-4T2+0.061T+7.97Similar to that of monolithic SiC, the resistivity of SiC/SiC composite under irradiation is calculated as:And the irradiation swelling model of both monolithic SiC and SiC/SiC composite are assumed to be the same and it is given by based on recent swelling evaluation which accounts for both dimensional change and irradiation temperature:where S is the SiC irradiation swelling rate strain,γ is displacement damage of SiC in unit of dpa, Ss is the saturation swelling given in the following temperature dependent polynomials:Ss=5.8366×10-2-1.0089×10-4T-6.9368×10-8T2-1.8152×10-11T3And γc is the characteristic dose for the SiC swelling saturation given as:γc=-0.57533+3.3342×10-3T-5.3970×10-6T2+2.9754×10-9T3As the monolithic CVD SiC and SiC/SiC composite were expected to exhibit similar thermal expansion behaviors, according to , the coefficient of thermal expansion (CTE) of these two components is given as:αT=10-6(-0.7765+1.435×10-2T-1.2209×10-5T2+3.8289×10-9T3)where α is the CTE in units of 1/K with a reference temperature of 293 K, T is the temperature in K, and the correlation is valid from 293 K to 1273 K.Based on experimental data, the specific heat capacity of CVD SiC developed by Cp=925.65+0.3772T-7.9259×10-5T2+3.1946×107T-2where Cp is in J/kg-K and T in K. And this formula is assumed to be valid from 200 K to 2400 K. The heat capacity of SiC/SiC composite is assumed to be identical to that of CVD SiC and based on , irradiation has been observed to have no effect on the heat capacity of SiC.The Young’s modulus of non-irradiated CVD SiC is recommended by where ECVD,non-irr is the Young’s modulus in GPa, VP is the fractional porosity, which is assumed to be 0.02, and T is the temperature in K.For SiC/SiC composite, the non-irradiated Young’s modulus is adopted as 230 GPa based on the measurement by The Young’s modulus of both CVD SiC and SiC/SiC composite have been found to be degraded with irradiation by several studies, the following relationship has been adopted in this study to account for the irradiation effect (where E0 is the as fabricated Young’s modulus and ΔVV is the swelling strain in %vol.Based on the fact that no significant impact of either temperature or irradiation on the Poisson’s ratio of both CVD SiC and SiC/SiC composite, which is assumed to be a constant 0.21 and 0.13 for CVD SiC and SiC/SiC composite, respectively. to be 3.21 g/cm3, and the density of SiC/SiC composite with a fiber volume faction of 31% and a porosity of 11% is given as 2.74 g/cm3 by The steady-state creep rate is given by ε̇=2×103σ191×1032.3∙exp-1740008.314T+2.7×10-35σ∅And based on the fact that the corrosion rate of SiC is several orders of magnitude lower than that of Zircaloy, the SiC cladding is assumed to have no corrosion during operation (In this work, the modeling geometry adopted a 2D axisymmetric plane with a Zircaloy-4 cladding or SiC cladding (see (a)), a typical ten pellets geometry was considered and a single pellet was considered to represent all the ten pellets with a periodic boundary condition in the axial direction, as shown in (b), with a mapped mesh, which is computational economically and was found to be accurate based on our simulation results (). This geometry is applied for UO2 and thorium-based mixed oxide fuel, with the model specifications summarized in , which is based on the application discussions by and the stress analysis and probabilistic assessment of multi-layer SiC-based cladding done by , then a two-layer SiC cladding with 750 μm inner SiC/SiC composite and 250 μm outer monolithic CVD SiC was adopted in our simulation case, and an additional two-layer SiC cladding with 400 μm inner SiC/SiC composite and 400 μm outer monolithic CVD SiC was also considered for fuel performance comparison.The UO2 fuel, Th0.923U0.077O2 fuel and Th0.923Pu0.077O2 fuel separately with Zircaloy and two-layer SiC cladding are modeled in this work by properly modifying the CAMPUS code developed in our previous work (), from which the CAMPUS code was developed and introduced in detail. The multiphysics models considered in this work include: heat generation and conduction, species diffusion, thermomechanics (thermal expansion, elastic strain, densification, and fission product swelling strain), grain growth, fission gas diffusion and release, gap heat transfer, pellet-cladding mechanical contact, gap/plenum pressure with plenum volume, thermal and irradiation creep of fuel and cladding, cladding corrosion and fuel burnup calculation. Most of the model’s governing equations (such as heat transfer model, solid mechanics model, fission gas diffusion model) were kept unchanged from CAMPUS code but with the customized coefficients updated, such as material properties, fission gas diffusion coefficients, boundary conditions. Part of the models such as fuel and cladding swelling, cladding thermal and irradiation creep and together with the cladding geometry were modified based on the different combination of the fuel and cladding.All the considered models were solved by using the non-linear backward-difference formulation (BDF) as implemented in COMSOL platform for calculating the time-derivatives. A direct solver called Multifrontal Massively Parallel sparse direct Solver (MUMPS) has been applied to solve a system of linear equations generated from the combinations of the weak-form equation definitions and the finite-element mesh. All the dependent variables defined in COMSOL are scaled before solved to improve the numerical stability. The scaling factors are typically taken as the approximate expected magnitude of each variable in order to improve the numerical accuracy when solving a large set of equations.In this part, the simulated performance of UO2 fuel, Th0.923U0.077O2 fuel, and Th0.923Pu0.077O2 fuel separately combined with Zircaloy and two-layer SiC cladding are analyzed for a 2D axisymmetric LWR fuel rodlet, as depicted in (b), the geometry is consist of an individual fuel pellet (Th0.923U0.077O2 fuel or Th0.923Pu0.077O2 fuel or UO2 fuel), and Zircaloy or two-layer SiC cladding with an inner SiC/SiC composite layer and an outer monolithic CVD SiC layer. An initial 80 μm pellet-cladding gap together with an upper plenum was adopted and the plenum to fuel length ratio was set to be 0.045. A uniform convective boundary condition at the cladding outer surface is applied to simulate the heat transfer from the cladding to the outer flowing coolant. A typical PWR normal operation conditions were adopted as summarized in ). Then the performance of UO2 fuel, Th0.923U0.077O2 fuel and Th0.923Pu0.077O2 fuel separately with Zircaloy or two-layer SiC cladding were further analyzed and compared, provided that the two layers of the SiC cladding are assumed to be perfectly contacted with no fabrication gap. shows temperature evolutions of the fuel and cladding for UO2-Zircaloy, Th0.923Pu0.077O2-Zircaloy and Th0.923U0.077O2-Zircaloy, respectively. The Th0.923U0.077O2 fuel was simulated to get the lowest fuel centerline temperature with an average temperature being about 100 K lower than that of UO2 fuel, while the Th0.923Pu0.077O2 fuel was found to get an equal fuel centerline temperature compared with that of UO2 fuel before fuel burnup of 300 MWh/kgU and a much lower fuel centerline temperature than that of the UO2 fuel between fuel burnup of 300 MWh/kgU and 900 MWh/kgU, and then a little bit lower fuel centerline temperature than that of UO2 fuel after fuel burnup of 900 MWh/kgU, which is a combined results mainly influenced by the fuel thermal conductivity, thermal expansion and fission gas released. Lastly, no big temperature difference at the outer surface of the fuel and inner surface of the cladding were found for the Th0.923Pu0.077O2 and Th0.923U0.077O2 fuels, while the outer surface temperature of UO2 fuel was found to be the lowest at the early stage of the fuel burnup and the highest after 300 MWh/kgU. In summary, compared with the traditional UO2 fuel, the reactor fuel core temperature could be greatly decreased by using the Th0.923U0.077O2 fuel and could be decreased a little bit by using the Th0.923Pu0.077O2 fuel.Then the gap size evolutions were calculated, as shown in , a latest gap closure was observed for UO2 fuel case, which is mainly caused by the different thermal expansion and swelling behavior of the fuel. And an increase of gap size can be found among all the fuels as the fuel densification process happened during fuel thermal expansion. Then the gap size was shown to decrease again and a much earlier gap closure was observed for Th0.923Pu0.077O2 fuel (about 100 MWh/kgU fuel burnup earlier than UO2 fuel), then followed by the Th0.923U0.077O2 fuel. The gap size for Th0.923U0.077O2 fuel is found to be the largest before the fuel burnup of about 800 MWh/kgU. While the use of both Th0.923Pu0.077O2 fuel and Th0.923U0.077O2 fuel were found to get an earlier gap closure time compared with that of UO2 fuel, which would induce an earlier PCMI and may not be good for nuclear reactor safety improvement.The fission gas release and plenum (gas) pressure profiles were also calculated, as depicted in , The starting time of the fission gas release for the Th0.923U0.077O2 fuel and Th0.923Pu0.077O2 fuel was found to be both greatly delayed, and the fission gas release fraction of the Th0.923U0.077O2 fuel was calculated to be the smallest, which further induces the lowest gas pressure. While the UO2 fuel was calculated to have the largest fission gas release fraction and highest plenum pressure, this is because the Th0.923U0.077O2 fuel gets the lowest centerline temperature and the UO2 fuel gets the highest, and also the fact that the fission gas diffusion coefficient of Th0.923U0.077O2 fuel is only ten percentages of that of the UO2 fuel. The fission gas release and plenum pressure of Th0.923Pu0.077O2 fuel were calculated to be between that of the Th0.923U0.077O2 fuel and UO2 fuel, which is also much smaller and lower than those of UO2 fuel, respectively. In all, both the Th0.923U0.077O2 fuel and Th0.923Pu0.077O2 fuel were found to greatly decrease the fission gas release and plenum pressure.The fuel performance of thorium-based fuel with innovative two-layer SiC cladding were further assessed and discussed in the following. The same operating conditions summarized in were adopted in this case, and the fuel performance with a 1000 μm thick two-layer SiC cladding was investigated firstly, as shown in , the temperature profile for the fuels with two-layer SiC cladding was found to be much higher compared with that of fuels with Zicaloy cladding. Namely, it is about 200 K higher for all the three types of fuel with two-layer SiC cladding at the beginning of fuel burnup. And the fuel centerline temperature of all the fuels is found to increase dramatically with the increase of fuel burnup, which is caused by the thicker SiC cladding compared with Zircaloy cladding and also the SiC thermal conductivity degradation effect and SiC swelling effect under fuel irradiation. Additionally, the fuel centerline and fuel outer surface temperature of Th0.923U0.077O2 fuel with two-layer SiC cladding were found to be increased gradually and were found to be the lowest, with fuel centerline temperature found to be about 100 K lower than that of UO2 fuel at the beginning of the fuel burnup and 400 K lower than that of UO2 fuel at the fuel burnup around 600 MWh/kg-U, which is mainly ascribed to the higher thermal conductivity of the Th0.923U0.077O2 fuel and its lower fission gas diffusion coefficient compared with that of the UO2 fuel. The fuel centerline and fuel outer surface temperature for Th0.923Pu0.077O2 fuel and UO2 fuel was found to evolve in a similar way but with a bigger temperature difference from the fuel burnup of 200 MWh/kgU to 700 MWh/kgU. Meanwhile, for the Th0.923U0.077O2 fuel case, about a 200 K temperature difference between the fuel outer surface and cladding inner surface was observed in , and a much larger temperature difference between the fuel outer surface and cladding inner surface was found for both UO2 fuel case and Th0.923Pu0.077O2 fuel case (after fuel burnup of 200 MWh/kgU), which is caused by the different thermal expansion, thermal conductivity and fission gas diffusion coefficient properties of the UO2 fuel, Th0.923Pu0.077O2 fuel and Th0.923U0.077O2 fuel.This temperature difference can be further demonstrated by the gap size evolution shown in , all the gap size evolutions were found to have a sudden decrease and short sharp increase, which is ascribed to the fuel thermal expansion and fuel densification, respectively. The Th0.923Pu0.077O2 fuel was found to have the smallest gap size before 300 MWh/kgU and the Th0.923U0.077O2 fuel was found to have the largest gap size (greatest delayed PCMI) during almost all the considered irradiation time and then followed by Th0.923Pu0.077O2 fuel and UO2 fuel, respectively, with only 5 micros difference between these two types of fuel after 300 MWh/kg-U, which is mainly influenced by the combined factors of fuel thermal expansion, thermal conductivity, fission gas release, fuel densification, fission products induced swelling, fuel thermal and irradiation creep, cladding thermal and irradiation creep and so on.The gap size can further influence the reactor plenum pressure. As shown in , the fission gas release and plenum pressure were checked and compared with those depicted in , a giant increase of fission gas release and plenum pressure was found in both UO2 fuel and Th0.923Pu0.077O2 fuel, and the Th0.923U0.077O2 fuel was found to have the lowest fission gas release and plenum pressure, which is the same conclusion to those of Zircaloy cladding cases. Apparently, the main factor for the lower fission gas release and plenum pressure is ascribed to the higher thermal conductivity of the Th0.923U0.077O2 fuel compared with that of UO2 fuel. So it is suggested that the future advanced fuel equipped with two-layer SiC cladding should have very high thermal conductivity, such as U3Si2 fuel and UN fuel.In order to assess and compare the fuel performance of PCMI for the Zircaloy and two-layer SiC cladding with different fuels, the radial displacement of fuel and cladding was further analyzed. shows the radial displacement of the UO2 fuel and thorium-based fuel with Zircaloy cladding, firstly, the radial displacement of fuel outer surface was found to have a quick increase due to fuel thermal expansion, and then have a short decrease due to fuel densification, and finally keep on expand during irradiation. A continuous and dramatic inward displacement was found to happen at the cladding inner and outer surface, which is mainly caused by the Zircaloy cladding creep, and an inflexion point was observed at cladding inner and outer surface, indicating the gap has been closed and the PCMI gets started. The Th0.923Pu0.077O2 fuel and Th0.923U0.077O2 fuel were found to have an earlier PCMI time than that of UO2 fuel. Then the cladding was found to further expand outward together with the fuel and the radial displacement of the Zircaloy inner surface and outer surface was found to deform in the same pace, as we can see the displacement difference of the Zircaloy inner surface and outer surface was found to be a constant value. However, the radial displacement of the UO2 fuel and thorium-based fuel with two-layer SiC cladding was found to behave quite different from that of Zircaloy cladding. As depicted in , the radial displacement of fuel outer surface was found to increase dramatically and was found to be much larger than that of Zircaloy cladding, especially for the UO2 fuel case with a maximum radial displacement increase about 50 μm at the end of fuel burnup, and the Th0.923Pu0.077O2 fuel and Th0.923U0.077O2 fuel were found to have a maximum radial displacement increase about 35 μm and 30 μm, respectively. And the radial displacement of two-layer SiC cladding was found to increase at the beginning of fuel burnup and then keep constant during irradiation due to the thermal expansion and swelling behavior of the two-layered SiC cladding, but the radial displacement of two-layer SiC cladding with UO2 fuel was found to be much larger than that of Th0.923Pu0.077O2 fuel and Th0.923U0.077O2 fuel. Due to the cladding thermal expansion and swelling under a higher fuel temperature condition, for the models using the two-layer SiC cladding, the UO2 fuel was found to expand a lot compared with the Th0.923Pu0.077O2 fuel and Th0.923U0.077O2 fuel. So basically, the use of two-layer SiC cladding was found to effectively delay the PCMI time but at the expense of a much higher fuel centerline temperature.The hoop stress in the different layers of the SiC cladding are shown in , together with the SiC/SiC composite proportionality limit stress (refers to the point at which matrix cracking and fiber sliding begins) and monolithic SiC fracture stress for comparisons (, tensile stresses were observed in the SiC/SiC composite side of the two-layer SiC cladding and compressive stresses were found in the monolithic SiC side of the two-layer SiC cladding, which is a combined results of thermal strain and swelling strain in the SiC cladding, the gap gas pressure and the coolant pressure applied in the SiC cladding. The transition from compressive to tensile stress in the monolithic SiC cladding was found to gradually occur when the gap is going to be closed (, the hoop stress in the SiC/SiC composite cladding side was found to gradually decrease from -a-c, which the hoop stress of UO2 fuel case was found to be larger than the SiC/SiC composite proportionality limit stress around fuel burnup of 850 MWh/kgU, while the hoop stress of Th0.923Pu0.077O2 fuel case and Th0.923U0.077O2 fuel case in the SiC/SiC composite cladding was found to be gradually decreased and was found to be lower than the SiC/SiC composite proportionality limit stress before the fuel burnup of 1200 MWh/kgU, indicating a lower cladding failure probability when compared with the UO2 fuel case. As discussed in , the PCMI of Th0.923U0.077O2 fuel with two-layer SiC was found to be greatly delayed, which allows for the fuel to operate longer with lower cladding hoop stresses (with lower PCMI failure probability), and then followed by the Th0.923Pu0.077O2 fuel and UO2 fuel. So the reactor safety would be expected to be improved by using the thorium-based mixed oxide fuel with two-layer SiC cladding.Lastly, the fuel performance with 800 μm two-layer SiC cladding was further investigated and compared with that of 1000 μm two-layer SiC cladding. As we can see from , at all the considered fuel burnup, the fuel performances of fuel centerline temperature, gap size, fission gas release and plenum pressure from the fuels with 800 μm two-layer SiC cladding were found to have no big changes compared with those from the fuels with 1000 μm two-layer SiC cladding. Specifically, as shown in (a), the fuel centerline temperature of fuels with 800 μm two-layer SiC cladding was found to be a little bit lower than that of fuels with 1000 μm two-layer SiC cladding while the gap size of fuels with 800 μm two-layer SiC cladding is found to a little bit larger than that of fuels with 1000 μm two-layer SiC cladding, indicating a delayed gap close process. Additionally, the fission gas release and plenum pressure of fuels with 800 μm two-layer SiC cladding were found to be a bit lower than those of fuels with 1000 μm two-layer SiC cladding. So the fuel performance of fuels with 800 μm two-layer SiC cladding may be somewhat better than that of fuels with 1000 μm two-layer SiC cladding. However, the thinner SiC layer puts forward higher requirements of SiC fabrication, and it is still challenging to fabricate longer and thinner SiC cladding.In summary, the performance of thorium-based fuel with two-layer SiC cladding and Zircaloy cladding have been investigated in a light water reactor under normal operation conditions, together with those of UO2 fuel with two-layer SiC cladding and Zircaloy cladding for comparison. The thorium-based fuel includes Th0.923Pu0.077O2 fuel and Th0.923U0.077O2 fuel and two types of SiC cladding thickness were chosen for investigation, as it is reported to be the most expedient scheme for making use of the world’s thorium resources and also the fact that the monolithic CVD SiC and SiC/SiC composite have been considered as promising cladding materials for PWR. The conclusions are summarized below based on the investigation of the fuel performance of the studied cases:Compared with the UO2-Zircaloy system, the Th0.923U0.077O2-Zircaloy system was found to greatly decrease the fuel centerline temperature (about 100 K), but no big fuel centerline temperature decrease was found for the Th0.923Pu0.077O2 fuel with Zircaloy cladding, which facilitate a reduced fission gas release and plenum pressure together with an earlier gap closure time. However, all the three types of the fuels combined with two-layer SiC cladding were found to have a much higher fuel centerline temperature (about 50 K-200 K) than the fuels with Zircaloy cladding, especially at a higher fuel burnup, resulting in a much larger fission gas release and much higher plenum pressure.2 The use of two-layer SiC cladding was found to greatly delay the gap closure time, indicating a delayed PCMI could possibly improve the reactor safety.The fuel performance of fuels with the two-layer SiC cladding was found to be not very sensitive to the thickness of the two-layer SiC cladding, which a little bit better fuel performance could be achieved by using a thinner SiC cladding. So the thickness of each layer of the SiC cladding was found to be not detrimental to the great improvement of the fuel performance.The presented results showed the thorium-based fuel with Zircaloy cladding could reduce the fuel centerline temperature but with an earlier PCMI time, and the two-layer SiC cladding was found to greatly delay the gap closure time. Then the fuel performance of thorium-based fuel with two-layer SiC cladding could be expected to be improved by reducing the fuel centerline temperature and meanwhile delaying the PCMI time. Further core neutronics analysis should be performed to assess the fuel economics and thorium-based fuel and SiC cladding material properties under irradiation should also be investigated further.Ultrasonic transducers play important role in nondestructive testing systems, ultrasonic machining technology, ultrasonic welding instruments, and ultrasonic imaging and diagnostic systems. Recently, ultrasound enhanced transdermal drug delivery technology (sonophoresis) has received increasing attention. Transdermal drug delivery (TDD) offers several advantages over traditional delivery methods including injection and oral delivery. When compared to oral delivery, TDD avoids gastrointestinal drug metabolism, reduces first-pass effects and provides sustained release of drugs for up to 7 days The precise mechanism by which the acoustic waves help to enhance permeability through the skin is not fully understood. It is hypothesized that the acoustic waves cause microcavitation in the drug medium and the skin itself, and this action helps the drug molecules to diffuse into and through the skin. Furthermore, the ordered lipid bilayers of the skin maybe temporarily disrupted by the acoustic waves induced cavitation thus permitting molecules to pass Development of sonophoresis devices of varying types of transducers have been patented, namely the horn-type and the disk-type (a)). The height of the body is decided by the sonophoresis experimental setup. A shoulder screw (MMSB 8-20, SANSHO, Singapore) is attached to the threaded hole of ultrasound transducer through an elliptical hole in an acrylic plate of thickness 13.3 mm. The acrylic plate is placed on the top of the donor compartment of the vertical Franz diffusion cell, so the sonophoresis ultrasound device is submerged in the donor compartment and is placed about 1.0 mm away from the skin as shown in Ultrasound frequency at a range of 20–100 kHz has been shown to enhance transdermal transport of a variety of drug molecules. The enhancement is determined by various parameters, including intensity, duty cycle and application time. The most important parameters are ultrasound frequency and acoustic intensity applied on the skin surface (a), the following sizes must be determined: the thickness tp, inner and outer diameter (di and dp, respectively) of ring-shaped PZT, and the thickness tm and diameter dm of the vibrating zone of the metal vibration plate as shown in The finite element method was used for the vibration mode analysis of a flexible ultrasound transducer and the key parameters are determined. The liquid interface forms the boundary condition at bottom surface of the device (see (b)). Thus the vibration of the structure in contact with liquid is transferred to the liquid motion and results in a discernible increase in the kinetic energy of the total system. It is generally known that the natural frequencies of structure that are in contact with liquid decrease significantly compared to the natural frequencies in air Because the proposed flexible ultrasound transducer exhibits axis-symmetry about its central axis, it was modeled as a two-dimensional axis-symmetric body. The two-dimensional model offers the simplicity in the configuration of the nodes over the three-model. Moreover, a standard sized ring-shaped PZT (i.d.=6.0 mm, o.d.=12.0 mm, and thickness=1.2 mm) was selected and only the parameters of three different types of materials, which are stainless steel, brass and aluminum, were changed to determine the structure of the transducer.(a) shows the detailed two-dimensional axis-symmetric simulation model that consists of the metal vibration plate (indicated by number 2), ring-shaped PZT (indicated by number 1), stainless steel body (indicated by number 3) and fluid elements (indicated by number 7). For the metal vibration plate and stainless steel body, it was assumed that their properties were linear and isotropic. The piezoelectric material properties were taken to be linear and anisotropic. The values of the required material properties necessary for this analysis are listed in . The boundary condition was such that any translational motion of the model was prohibited. In addition, absorption coefficient of sound at the interface of liquid and skin (indicated by number 5), which was obtained empirically (b) shows the first vibration mode of the structure at the frequency of 20.14 kHz. When the metal vibration plate deformed, the elements at the fluid–structure interface (indicated by number 4) changed their size following the deformation of the metal vibration plate. The fluid pressure also changed due to the vibration of the plate. So the solutions output associated with the element are average pressure, pressure gradient, fluid velocity and sound pressure level ) of proposed device based on the first resonance frequency of about 20 kHz. By using the harmonic analysis, details of first resonance frequency of three different types of materials are calculated as listed in The piezoelectric ring-shaped material (C-203, Fuji Ceramic) has a thickness 1.2 mm with inner diameter and outer diameter 6.0 and 12.0 mm, respectively. The piezoelectric ring is poled in the thickness direction. The silver electrode on the piezoelectric ring is ground with an abrasive paper to remove the oxide layer and cleaned with acetone. Stainless steel sheet is selected as a vibration plate based on the simulation results as shown in . The stainless steel sheet is cut by using a wire-cut machine and ground with an abrasive paper. The piezoelectric ring is then bonded to the stainless steel vibration plate. The bonding material is electrically conductive silver epoxy (Acoustic Technologies). The whole assembly of flexible transducer is cured in the oven for 75 min at 130 °C and then it is kept at room temperature for 24 h. The stainless steel vibration plate with ring-shaped PZT as shown in (a) is bonded to the stainless steel body using the same conductive epoxy and method. The entire structure of the sonophoresis device after fabrication is shown in (b). The diameter of the device is 29.6 mm and weighs only 71.5 g, which is much less than that of ultrasonic probe or converter from a commercial sonicator which is about 1 kg.. A cap screw (CB 6-15, SANSHO) is attached to the threaded hole of ultrasound transducer through a hole on the wall of acrylic water tank (measuring ) with 1 l degassed deionized water, where the ultrasonic device was mounted (see (c)). In the first method, the input impedance was measured with Solartron impedance/gain-phase analyzer (SI 1260, Solartron Analytical). The driving frequency versus input impedance diagram (. The frequency difference is probably due to the presence of conductive epoxy layer between PZT material and metal vibration plate which was not considered in the simulation model. The geometrical error during fabrication and assembly of the device may also be responsible for the difference.The proposed dominant mechanism for the sonophoresis, although not completely understood, has been suggested to be the result of cavitation which refers to the formation and the subsequent dynamic life of bubbles in liquids In order to observe the bubble phenomena, the proposed transducer device was driven by a pulsed signal with 20% duty cycle generated by an arbitrary function generator (AFG320, Sony Tektronix) and amplified by a power amplifier (Model EPA-102, Piezo Systems). The function generator operated at 17.466 kHz with 4, 6 and 8 (amplitude) output and amplified 20 times by the amplifier. A SONY CCD (charge coupled device) camera was used to record the activity of the bubbles and the results are shown in . It is shown that the acoustic bubble vibrated on the surface of the metal vibration plate and the bubble quantities increased with the increase of voltages. Although the acoustic intensity generated by the vibration metal plate is beyond the cavitation threshold, no large-sized bubbles were observed at the vicinity of the vibration plate. Most of the acoustic intensity exists at the interface of the metal vibration plate and water. Because of the mismatch of the acoustic impedance between stainless steel and water, there is inadequate acoustic intensity radiating into the water and to allow sub-micron or micro-sized bubbles to grow in a manner visible by naked eyes.However, for ultrasound enhanced transdermal drug delivery, in general, the ultrasound transducer is placed about 1.0 mm above the skin surface. The acoustic intensity transmits through the metal–liquid interface, radiates into the liquid and applies on the skin surface. So the value of the acoustic intensity applied on the skin surface is much attractive. In order to measure this kind of radiated acoustic intensity, a calibrated miniature omni-directional reference hydrophone (Model TC4013, RESON OFFSHORE Ltd.) was used in the same acrylic water tank containing degassed, distilled water. A computer controlled linear stage (M462 series, Newport, Irvine) was used for automated scanning. The scanning step was set to 1.0 mm and the scanning area was . Using same driving condition, the spatial peak-temporal-peak intensity (Isptp) was determined at a plane 1.0 mm from the metal vibration plate based on the standard guidance plots the results of maximum spatial peak-temporal-peak intensity with different driven voltages. When the acoustic transmission coefficient is decided, the maximum spatial peak-temporal-peak intensity increased with the increment of driven voltages. The maximum spatial peak-temporal-peak intensity under three different driven voltages (80, 120 and 160 V) was approximately 41.13, 69.66 and 105.09 mW/cm2, respectively. Thus, the proposed ultrasonic device produced intensity is comparable to those achieved using a commercial sonicators. Further experiments will be conducted to test the performance of this proposed ultrasonic device for transdermal drug delivery applications.All these values and features showed that the proposed sonophoresis device is feasible for use in practical application. Further experiments will be conducted to test the performance of this proposed sonophoresis device for transdermal drug delivery applications.H.Y. Zhang received BS degree in Mechanical Engineering from Beijing University of Technology in 1993. He has worked for over 8 years as mechanical engineer in several companies. He currently is PhD student in School of Mechanical and Production Engineering, Nanyang Technological University, Singapore. His research interests are in piezoelectric transducers and actuators, ultrasonic biomedical devices.S.H. Yeo graduated from Queen Mary College, London in 1981 with a BSc degree from the Mechanical Engineering Department. He then received his ME and PhD degree from National University of Singapore. Dr. Yeo has been with the Nanyang Technological University since 1986 and his research interest is in precision manufacturing and biomedical engineering application based on ultrasonic and magnetic principles.Diameter dependence of the apparent tensile modulus of hemp fibres: A morphological, structural or ultrastructural effect?The aim of this paper is to investigate the origin of the diameter-dependence of Young’s modulus in hemp fibres. In view of the considerable experimental difficulties encountered when determining the 3D morphology of elementary fibres, the influence of the fibre morphology and size on the E-modulus is studied using a mathematical model. An approach based on the 3D elastic theory is used to construct a model of the fibre structure, and to predict its mechanical properties. We clearly show that the modulus is dependent on the size of the lumen and on the outer fibre diameter. This structural effect, induced by the cylindrical geometry, the multi-layered organisation, and the orientation of the cellulose microfibrils only partly explains the large, experimentally determined dispersion of apparent E-modulus, as a function of fibre diameter. Ultrastructural parameters, such as cellulose crystallinity and microfibril angles, are identified to be the main factors involved in this dependence.Ligno-cellulosic fibres have an intricate structure and organisation. Stem fibres such as hemp are made up from single cells (unitarian or elementary fibres). The elementary fibres are glued together by a pectin interface, to form technical fibre bundles. These bundles are separated from one another through partial decomposition of the cell wall, induced by bacteria or mechanical processes. The elementary fibres have a typical cell plant structure, with a lumen and a particularly thick wall. They have a rounded polygonal outer shape, which is irregular and non-uniform along length of the fibre, and also varies from one fibre to another. Typically, the fibres have a diameter lying between 10 μm and 50 μm, and a length of approximately 8–14 mm ). Each layer is comprised of a mixture of three main types of polymer, i.e. cellulose, hemicellulose and lignin. The cellulose unit cells are organised in a crystalline network, which forms microfibrils with a lateral thickness of about 2–5 nm and a length of 30 nm . The cellulose microfibrils are spirally wound, at an angle with respect to the fibre axis. This angle varies between layers (Natural fibres are characterised by a broad scattering of their mechanical properties, which is generally attributed to: the methods used for single fibre tensile tests and the computation of their mechanical properties As a consequence of the experimental difficulties, in this paper, we propose an analysis of the influence of lumen size on the macroscopic mechanical properties of fibres, through the use of a comprehensive model. Most of the models available in the literature, developed over the past decades for wood or plant cells, are based on the classical laminated theory (CLT). Plate models, with an antisymmetric laminated structure, are somewhat different to the realistic structure of a natural fibre. Only more sophisticated tools, based on the thick laminated composite tube model, are able to take the hollow structure of the stem fibres into account, and more precisely to formulate the 3D stress–strain relationships in the laminate, under loading. Gassan et al. Hemp fibres (Cannabis sativa L.) were procured from the LCDA Company in France. They were delivered in a jumbled state. Bundles of fibres were washed in water for 72 h at 30 °C, as recommended by Bourmaud et al. The isolated single fibres were firstly examined using polarised light microscopy (Nikon Eclipse LV 150), to determine their outer diameters. The average diameter of each fibre was computed by taking ten measurements along its length. The outer diameter was also measured in the rupture area, following the tensile test.A Dynamic Mechanical Analyser (Bose Electroforce 3230) was used to perform the tensile tests. Thirty elementary fibres were tested at a constant crosshead displacement rate of 2 μm s−1. The clamping length was 8 mm. According to the ASTM standard, the paper frame supporting each elementary fibre was clamped onto the testing machine, and the paper frame was cut before the beginning of each test. The tests were carried out at a controlled temperature of 23.5 ± 1.5 °C and a relative humidity of 50.5 ± 1%. The relative humidity was controlled in the DMA chamber using a HumiSys humidity generator from Instruquest Inc. Tensile properties: the apparent Young’s modulus, ultimate strength, and failure strain were determined for each fibre. The effective cross-sectional area was calculated from the average diameter in the failure area for the ultimate strength and from the average diameter of the fibre for the apparent Young’s modulus, assuming the fibre to be perfectly cylindrical. The apparent Young’s modulus was computed from the final linear section of the stress–strain curve, and the strain at failure was determined from the crosshead displacement at rupture.Following the tensile tests, the fibres were coated on their outer surface with a thin layer of gold, in a Jeol-JFC1100 Sputter Coater, and were then analysed with a Jeol 840 Scanning Electron Microscope (SEM). The samples were observed using the second electron mode, and the images were digitally recorded. Some untested fibres were also observed.Our proposed model is based on macroscopic considerations, and uses the continuum mechanics formulation. Knowledge and understanding of the fibre structure and ultrastructure is of great importance to derive a suitable micro-mechanical model. There are still some uncertainties regarding the structural arrangement of the fibre constituents within the wall. So, some relatively far-reaching hypotheses are formulated, and the elementary fibres are somewhat idealised. The elementary hemp fibre is considered to be a multi-layered tube, made from a stack of five perfectly cylindrical, concentric layers: the primary wall, the first sub-layer S1 (which is itself split into two layers S1a and S1b to take into account the variation in the sign of rotation of the cellulose microfibrils) and the last two sub-layers S2 and S3, whose geometric properties are provided in . Each layer is modelled as if it were a long fibre reinforced composite material. A three-phase model is used (). The microfibrils are considered as a crystallite comprised of an aggregate of approximately 6 × 6 unit cells in transverse section Various significant hypotheses must be made, in order to use laminate theory to model the plant fibres. Geometric considerations prevented us from using plate theory to model the fibres, especially since their curvature is small with respect to the fibre thickness. A second aspect to be taken into account is the coupling between the radial and hoop components of a cylindrical structure: in practice, the deformation energy due to radial effects can be neglected. The third aspect to be considered is the thickness of the different layers, since the thickness of layer S2 is high compared to that of the other layers. In order to use classical laminate theory, each layer must be sufficiently thin with respect to the external radius of the fibre. A 3D model was thus developed to model the fibre. No major assumptions were made for the integration of the balance and compatibility equations. In addition to the assumption of small displacements and displacement continuity between each layer, only the shear components on the tube were considered to be negligible. The assumption of displacement continuity between each layer is only acceptable at the beginning of the deformation. Actually, the interfaces between layers are likely to be relatively weak, since micro-observation near rupture plans of the fibres confirm that the failure occur along these interfaces. Assuming that the fibre is restricted from rotation when tensile tested, the cylinder assembly was constrained not to twist.In addition to the global coordinate system related to the fibre (X→,Y→,Z→), a local coordinate system (e→ϑ,e→z,e→r) was defined at each point of the fibre (). The working plane is defined as (e→ϑ,e→z), and the MFA in each layer is measured with respect to the fibre axis (z→≡e¯z), and defined by βj. A material coordinate system n→,t→,e→r is defined by a rotational angle of π2-βj, in order to express the isotropic transverse behaviour of the composite, in a material coordinate system related to the orientation of the microfibrils. Four strain components were used: hoop strain εϑϑ, longitudinal strain εZZ, radial strain εrr and shear strain γzϑ=2εzϑ. The two other shear strain components were neglected.To obtain the layer elastic properties (longitudinal and transverse moduli EL, |
ET, Poisson’s ratio νLT, and shear modulus GLT), mixture laws (ML) were used with the crystalline cellulose and amorphous cellulose (microfibril: ML1), with the hemicelluloses and lignins (matrix: ML2) and with the microfibrils and matrix . The layer stiffness properties in local coordinates are summarised in Finally, the behaviour law governing each ply in the material coordinate system (with the plane stress state), where n→ is the microfibril direction, is:S̲̲=1EL-νTLET-νTLET000-νLTEL1ET-νTTET000-νLTEL-νTTET1ET0000001GLT0000001GLT0000001GTT{σ̲}=σnnσttσrrσntσnrσrtand{ε̲}=εnnεttεrrγntγnrγrt. The apparent elastic properties are global quantities, and are calculated using the homogenised stresses and strains, according to the loading path. As an example, in a tensile test the apparent longitudinal modulus Ezz∗ is defined as the ratio of the homogenised longitudinal stress to the longitudinal strain.The longitudinal strain is obtained as a constant in the proposed 3D model.A thorough review of the literature relating to the composition, organisation, dimensions, and mechanical properties of the cell wall components is given in . The values for the input parameters of our model were determined using this review.A reliability approach based on the second-moment-FORM (First Order Reliability Method) was used to determine the parameters sensitivity. This method is well-described in Carbillet et al. provide a summary of the diameters and mechanical properties of 30 single hemp fibres, determined from tensile tests. The table compares the average value and standard deviation of the experimental values with the data found in the literature. The average diameter of the fibres is close to 27.6 μm, which appears to be consistent with the typical dimensions found in the literature (). The average E-modulus is approximately 24.7 GPa, for a UTS of 640 MPa and a strain to failure of approximately 2%. These values are in agreement with the data found in the literature (). A wide dispersion of the mechanical properties, as a function of fibre diameter (), is also observed. The apparent Young’s modulus varies between 8.7 and 47.6 GPa, the UTS between 230 and 3100 MPa and the strain to failure between 0.9 and 4%. These values are in agreement with the data collected by Thygesen et al. shows that the fibres’ tensile properties depend significantly on their diameter. The effective cross-sectional area of the fibres was determined using their mean diameter, whilst assuming their shape to be perfectly cylindrical (ignoring the presence of the lumen). In practice, in most cases SEM observations of the fractured end surface, as proposed by Hu et al. provides an example of significant modifications to the geometry of the fibre section, in which the lumen almost disappears in the rupture area. Using optical microscopy, the lumen can be revealed looking at the fibre against the polarised light (). Unfortunately, the contrast between the cell wall and the lumen is sometimes inadequate and the cell wall transparency not enough to ensure a quantitative analysis. The strength of these hemp fibres appears to comply with Griffith’s theory, i.e. the UTS decreases when the fibre diameter increases. Microscopic observations () made under polarised light reveal the well-known dislocations of hemp fibres For an external fibre diameter of 27.6 μm (representing the average experimental value), our model leads to a computed value for the elastic modulus of approximately 42.2 GPa when the rotation of the fibre is free and 64.6 GPa when the rotation is prohibited. Although, the rotation of the fibre is prohibited by the testing device during tensile tests, the experimentally determined value of the apparent E-modulus is more closely related to the numerical value determined using a free rotation. This could partly be explained by the relatively strong hypothesis of cellulose microfibril continuity formulated when constructing the model. Indeed, in addition to the debate on the arrangement of cell wall components, there is a parallel discussion of exactly how alternate regions of amorphous and crystalline cellulose in the cross-section and along the microfibrils Apart from this question of extension–twist coupling, the numerically determined value of E-modulus is over-estimated in comparison to the average experimentally determined value. Uncertainties in the input parameters and model assumptions could provide a partial explanation for this difference. In fact, the input parameters used in our model (), such as the layer thicknesses, wall composition, or MFA values, were taken from the literature. These data were determined by different authors, using various species of annual plants, or wood, and a range of experimental techniques. Although the order of magnitude of these values is clearly in agreement with those relevant to the properties of hemp fibres, there are certainly some discrepancies. The accurate identification of the cell wall components and organisation of natural fibres requires a sophisticated experimental set-up, and is still an openly studied field of research. As aforementioned, many questions and uncertainties also remain concerning the structure of the cell wall, in particular the structure of cellulose and other polymers, and their interactions. The assumptions made in our model concerning the elementary fibre are somewhat idealised, and these approximations can lead to significant inconsistencies. The model nevertheless ensures that the solution is comparatively free of geometrical or mechanical assumptions. Only the shear components of strain and stress are affected by some assumptions.Furthermore, the experimental value of apparent Young’s modulus is clearly under-estimated, due to the fact that the presence of the lumen is not taken into account. As it is extremely difficult to determine the lumen’s cross-sectional area experimentally, most authors, including ourselves, consider the full cross-section of the fibre without taking the lumen into account. In the following, we propose a calculation based on the hollow cylinder model to estimate the resulting error in fibre stiffness, i.e. for the case of the fibre being considered to be a full cylinder. plots the resulting error as a function of the lumen diameter. This calculation shows that the assumption of a cylinder without a lumen can lead to a significant under-estimation of the longitudinal fibre stiffness: between 15% and 25%, for a surface area ratio of the lumen between 10% and 20%. When viewed under the light of this result, the longitudinal modulus computed using our model could be considered to give a consistent estimation of the average properties of hemp fibres. shows the E-modulus calculated using the 3D model and the Classical Laminate Theory (CLT), for a lumen diameter varying from 2 to 44 μm, and a constant fibre diameter of 50 μm. A significant decrease in E-modulus of approximately 11%, is found when the diameter of the lumen increases. The same calculations were made for a wide range of fibre and lumen diameters, according to range of typical values reported in the literature (), i.e. 10–50 μm for the fibre diameter, and approximately 10–20% for the lumen ratio. The E-modulus is computed to vary by approximately 10%, for this range of geometrical properties. This behaviour is attributed to a structural effect resulting from the fibre’s cylindrical geometry, the multi-layered organisation of the fibre itself, and the orientation of the cellulose microfibrils. Only a model based on a thick-walled, multilayered cylindrical organisation is able to express this type of structural behaviour ( plots the variation in E-modulus computed using the two extreme cases of rotation, as a function of fibre diameter, for four different laws describing the evolution of the lumen surface area: (I). Ri=3.5μm. The lumen area is constant, whatever the fibre’s diameter. The lumen diameter lies in the range 4–10 μm withΩthe surface area ratio of the lumen:Ω=Ri2Re2For these four cases, the corresponding changes in cell wall thickness are reflected by variations in the geometry of layer S2 only. For fibre diameters between 10 and 50 μm and for a restricted rotation, the apparent E-modulus increases with fibre diameter, by approximately 19–47.5%, according to the law used to express variations in lumen geometry. The cell wall thickening acts as a stiffening factor. In case of a free fibre rotation, the E-modulus increases with the fibre diameter, by approximately 24%, if the lumen surface area proportion decreases (case I and IV) but it slightly decreases when the lumen surface area proportion is constant or increases (case II and III). For the considered microstructural parameters and biochemical composition, the E-modulus is mainly influenced by: (i) the thickness of the S2 layer, (ii) the proportion of the S2 layer to the total cell wall thickness, (iii) the mean diameter of the S2 sub-layer and (iv) the boundaries conditions (the ability of the fibre to rotate).The structural effect cannot explain the large experimentally observed scattering of E-modulus vs. fibre diameter. Only an increase in surface area proportion of the lumen, as a function of fibre diameter, could justify a decrease in E-modulus. This assumption is, however, not currently supported by morphological studies of hemp fibres The sensitivity analysis on model parameters clearly shows the dominant influence of the ultrastructural parameters on the longitudinal elastic modulus in comparison to morphological parameters (). The parameters ranges used for the sensitivity analysis are detailed in . They were determined according to range of typical values reported in the literature (). The sensitivity index basically defines the ratio of the change in the output, the E-modulus of the fibre, to the change in input parameter from its minimum value to its maximum value. The inputs which contribute most to output variability are: the fraction of crystalline cellulose, the longitudinal elastic modulus of crystalline cellulose, the angle of cellulose microfibrils and the shear modulus of amorphous cellulose. The high influence of this last parameter was particularly unexpected and points out the need to strengthen the knowledge of the mechanical properties of amorphous cellulose. The shear modulus of amorphous cellulose appears to be a dominant parameter especially when the rotation of the fibre is not restricted. This analysis allows for unimportant parameters such as some of the morphological and geometrical parameters (lumen and fibre diameter) to be ignored and provides a direction for future research in order to reduce ultrastructural parameter uncertainties and increase model accuracy. The sensitivity ranking of some of the most dominant parameters also points out the need to strengthen the description of the distribution law.Concerning the predominance of the ultrastructural parameters, it must be kept in mind that the natural fibres play an essential role in the original plant, before becoming a potential engineering material. It is well known that vegetable organisms control the shape and size of cells during growth, adjusting the mechanical performances of tissues, inducing bending, and also influencing the orientation of the cellulose microfibrils illustrates the significant impact of these ultra-structural parameters on the E-modulus, based on the calculations provided by our 3D model. As an example, considering that the rotation of the fibre is free, a change in E-modulus from 10 to 70 = 3.5 μm GPa is computed, for an MFA of respectively 50° and 0°, from 15 to 60 = 3.5 μm GPa for a fraction of crystalline cellulose of respectively 10% and 100%, and from 28 to 42 = 3.5 μm GPa for a mass fraction of cellulose of respectively 10% and 100%.From experimental measurements, a high correlation is observed between the UTS and the apparent Young’s moduli (r2 |
= 0.81) for such hemp fibres (). This correlation was revealed many years ago by Mc Laughlin and Tait shows that the fibres with the smallest diameters have the highest Young’s modulus and UTS. This result appears to support our hypothesis of a diameter-dependence of the ultrastructural parameters, such as MFA or cellulose content and crystallinity, of these fibres.This hypothesis could be confirmed experimentally, only if a test campaign were carried out on elementary fibres derived from the same plant. In order to enrich this discussion and establish a representative model, experimental data concerning the morphology, microstructure and mechanical properties of such fibres will be imperative. Such an approach will require the use of sophisticated measurement techniques and devices, and represents a new field of research.The results presented in this paper show that a 3D hollow model with exact solution is able to predict the tensile modulus of hemp fibres, in relatively good agreement with experimental measurements. The importance of the boundary conditions, especially the ability of the fibre to rotate, on the elastic modulus is pointed out.This paper also highlights the likely origins of the diameter-dependence of the E-modulus of elementary hemp fibres. Numerical results collected using the 3D model show that morphological, dimensional and structural parameters evidently affect the Young’s modulus of these fibres. The assumption of a cylinder without a lumen can lead to a significant under-estimation of the longitudinal fibre stiffness: between 15% and 25%, for a surface area ratio of the lumen between 10% and 20%. The Young’s modulus of elementary fibres is also particularly influenced by the lumen surface area and the fibre diameter. For a constant fibre diameter, an increase in lumen size produces a decrease in the value of Young’s modulus. This structural effect is a consequence of the cylindrical geometry, the multi-layered organisation of the fibre itself, and the orientation of the cellulose microfibrils. However, only an increase in surface area proportion of the lumen, as a function of fibre diameter, could justify the decrease in E-modulus experimentally observed. This assumption is not supported by morphological studies of hemp fibres and the diameter dependence of the E-modulus cannot be attributed to the fact that the presence of the lumen is not taken into account. So, this diameter-dependence can only be partially explained by morphological and structural parameters. Other parameters, such as ultrastructural parameters, are certainly involved. A parameter sensitivity analysis clearly shows the main role of MFA and cellulose crystallinity. From an experimental point of view, a significant correlation is observed between the UTS, Young’s modulus and fibre diameter, and tends to support the possibility of a diameter-dependence of the cellulose’s microfibril angle and crystallinity.The parameter sensitivity analysis also identifies several parameters which are significant and require additional research for strengthening the knowledge base and thereby reducing output uncertainty, such as the longitudinal elastic modulus of crystalline cellulose and the shear modulus of amorphous cellulose.To obtain the mechanical properties of the cellulose microfibrils (longitudinal and transverse moduli ELf,ETf, Poisson ratio νLTf, and shear modulus GLTf), a classical mixture law was implemented, using the crystalline cellulose (CC) and amorphous cellulose (AC) properties.ELf=VCCELCC+VACEAC;1ETf=VCC1ETCC+VAC1EAC;1GLTf=VCC1GLTCC+VAC1GAC;νLTf=VCCνLTCC+VACνACwith Vi the volume fraction of the constituent i.In like manner, the mechanical properties of the matrix (longitudinal and transverse moduli ELmat,ETmat, Poisson ratio νmat, and shear modulus GLTmat) were calculated using an other mixture law with the lignin (LG) and hemicelluloses (HC) properties. The mechanical properties of the other constituents such as pectins, waxes and extractibles are ignored.ELmat=VHCELHC+VLGELG;1ETmat=VHC1ETHC+VLG1ELG;1Gmat=VHC1GLTHC+VLG1GLG;νmat=VHCνLTHC+VLGνLGFinally, to obtain the mechanical properties of each cell wall layer, (longitudinal and transverse moduli EL, |
ET, Poisson’s ratio νLT, and shear modulus GLT), a new mixture law was used with the cellulose microfibril and matrix properties, respectively.EL=VfELf+VmatELmat;1ET=Vf1ETf+Vmat1ETmat;1GLT=Vf1GLTf+Vmat1Gmat;νLT=VfνLTf+VmatνmatThe porosity of each layer was assumed to be approximately 3%. As a simple mixture law usually tends to minimise the transverse modulus, an improved formulation was proposed, to provide a better representation of the transverse properties. This is based on a calculation initially made by Jacquet et al. Taking into account the problem’s independence with respect to ϑ, and considering the strain–displacement and compatibility relationships, as well as the balance equations along the circumferential and axial axes, a possible solution is:where A∗ and P∗ are constants and q(r) is a function to be determined using the balance equation in the radial direction. γrϑ and γrz are neglected.In order to obtain the function q(r), the stress needs to be computed in the local coordinate system. The stiffness matrix [K] is obtained by rotating the stress tensors of the material (along the radial axis) into the local coordinate system, such that [K]=[Tα][S]-1[T-α′] and finally:σϑϑσzzσrrσzϑσrϑσrz=K11K12K13K1400K21K22K23K2400K31K32K33K3400K41K42K43K44000000K55K560000K65K66εϑϑεzzεrr2εzϑ2εrϑ2εrzAs a result, the [K] matrix is different from one layer to the next, depending on the orientation of the fibril in each layer.By integrating the third balance equation, the following solution is found:q(r)=C∗rα1+D∗rα2+(2K34-K14)(4K33-K11)A∗r2-P∗r(K32-K12)(K33-K11)whereα1,2=±K11K33ur=rεϑϑ=C∗rα1+D∗rα2+2K34-K14(4K33-K11)A∗r2-P∗r(K32-K12)(K33-K11)There are six integration constants per layer C∗,D∗,A∗,P∗,b∗,g∗.Once the strains are known, the stresses can be obtained for each layer. As there are many layers, the displacements and stresses are needed for each layer, in order to express the displacement continuity between each layer. The five layers must be calculated in the same way, according to the appropriate [K] matrix:urj=rεϑϑj=Cj∗rα1j+Dj∗rα2j+(2K34j-K14j)(4K33j-K11j)Aj∗r2-Pj∗r(K32j-K12j)(K33j-K11j)The 30 integration constants are determined according to the following boundary and continuity conditions. describe the free lateral internal and external boundaries (internal radius = |
Ri, external radius = |
Re).The following two equations describe the possible loading on the fibres. The torsional torque is taken to be nil when the rotation is free at both ends of the fibre, but could be used to prevent the fibre from rotating if the ends are clamped.[e4]∫02π∑j=15∫rj-rj+rσzϑjrdrdϑ=Cforz=±h2The symmetry conditions at the middle of the fibre lead to:Only 20 integration constants remain unknown.The continuity conditions for the displacements are u→(rj+)=u→(r(j+1)-) whatever the axial location.(rj+ and r(j+1)- are the positions of M at the transition between two layers j and (j |
+ 1)).Now, only 12 integration constants need to be foundur(r=rj)j=ur(r=rj)j+1→[e5]-[e6]-[e7]-[e8]The continuity conditions for the stresses are T→er(rj+)=T→er(r(j+1)-)σrr(r=rj)j=σrr(r=rj)j+1→[e9]-[e10]-[e11]-[e12]By solving the 12 × 12 equation system ([e1] to [e12]), the stresses, strains and displacements are known throughout the structure.Failure criteria: Old wines in new bottles?Aim of the present work is to communicate a part of the experience gathered by the first author after 40 years of involvement in Fracture Mechanics and Failure of Materials in general, through close collaboration with his colleagues. From this point of view, a few prerequisites for the, say, correct formation of a failure criterion are considered. For example, are of equal validity deductive and inductive approaches? Furthermore, some common characteristics of existing failure criteria are discussed and their effect on the quality and the general applicability of failure criteria is presented. For example, how and why does geometry affect failure predictions? Do cracks or other singularities require a special treatment? Is the characterization of materials through usual constitutive equations adequate? In a more practical level, the necessity of introducing as more as possible stress/strain components in the formation of a failure criterion is emphasized, driving directly to strain energy density (SED) considerations. The deterministic requirement of “cause-effect” demands available (i.e. elastic) SED, excluding plastic work, and, consequently, plastic strains from the formation of any criterion. Considering the only two mechanisms of storing SED in materials (volume-lengths and shape-angles changes of the elementary volume), we arrive into a dilemma regarding the “behavior” of the SED parts been spent for volume or shape changes. In case they are collaborative, their sum (i.e. the total elastic SED) is adequate to describe failure. Contrarily, in case of competitive behavior, each SED component has its-own importance and must be traced separately. Finally, existing groups of criteria are commented and some conclusions are presented.In the Euclidian–Newtonian world – where we and other materials live – source of knowledge and wisdom is the direct observation of our environment. We put an order/explanation to phenomena by either (a) introducing an axiomatic hypothesis governing a class of them and concluding, in a deductive manner (top down), their observed behavior or (b) comparing many similar cases in order to locate behavioral similarities allowing for the extraction of a rule in an inductive way (bottom up). Both methods are almost equivalent but in case of induction it is not clear if the prerequisite of causality is satisfied. All observed parameters may not play role in the phenomenon, being in fact side-effects. In other words, a series of experiments may result in a dilemma concerning the “space” of description of the experimental data, namely which of the parameters are causally connected, the remaining being simply “present”. Classical example of a “wrong” space is fatigue where a purely linear elastic parameter (stress intensity factor KI) is used to describe a purely non-linear elastic/plastic strains phenomenon. The outcome, after almost two centuries of hard work, is disappointing. A Keplerian approach is, still, missing to put an order to the Ptolemaic chaos of the world of fatigue. This is due to the fact that causality is not satisfied (KI is irrelevant to, at least, elastoplastic phenomenon of fatigue and after a few cycles is meaningless) and Paris law for fatigue is in fact an interpolation procedure. Why is causality important? Because the Euclidian–Newtonian world is deterministic and in practical terms, no one can guarantee that an even huge number of experiments is wide enough to cover extreme cases belonging apparently to the same group. Hence the absence of a rational cause drives to a kind of “conceptual extrapolation”, often accompanying inductive approaches.On the other hand, deductive approaches fail to predict exactly experimental results, showing however a more or less acceptable agreement with them. In any case, they are free from the dangerous and unpredictable conceptual extrapolation and, with a proper safety factor, can cover a really wide range (if not all) of similar phenomena. In other words, deductive approaches serve as the thin main line passing through experimental data. This way, axiomatic-deductive approaches can be considered as “Laws”, although inductive ones may be called “Results”.The space of description of a class of similar phenomena may be wrong as in case of fatigue but, also, it may be “incomplete”. The dimensions of such spaces are fewer than the number of independent physical parameters being involved. Constitutive equations are a typical example. In the friendliest case of an isotropic material two mechanical properties are necessary to describe its behavior, implying the execution of two independent experiments. Instead, one experiment is executed (usually uniaxial tension to obtain σ1 |
= |
f(e1)) assuming identical results for all three axes. The area under this curve represents the total SED. The second independent experiment (usually torsion) is required to evaluate, for example, volume expansion or shear modulus. Following the rationale of von Mises, four “equivalent” quantities (equivalent stress/strain σeq, eeq and hydrostatic pressure/volume expansion p, Θ are introduced, which in terms of principal stresses/strains are:σeq=12[(σ1-σ2)2+(σ2-σ3)2+(σ3-σ1)2]1/2,p=(σ1+σ2+σ3)/3eeq=12(1+v)[(e1-e2)2+(e2-e3)2+(e3-e1)2]1/2,Θ=e1+e2+e3They form two constitutive equations, σeq |
= |
f(eeq) and p |
= |
f(Θ). The first of them, being the classical von Mises equation, describes the behavior of a material under shear stresses per se and the second one the respective behavior under normal stresses per se. This way, an interpretation of the response of materials in physical terms is obtained and the space of description is complete.The problem of selection of the correct space of description of a class of experiments becomes more difficult when guidance from the appropriate constitutive equations is absent. One has to make a choice of a pair of mechanical quantities (stresses, strains, SED, plastic work, etc.) between about 170 different combinations of these quantities! Most of them are a priori rejected in terms of taste and experience of the researcher. Yet, enough room remains for the introduction of many failure criteria. So, it is rather conservative to conclude that even a single set of objective experimental data can be the launching point of appearance of some dozens of criteria to interpret them. At eighties and nineties, whole issues of reputable journals, like Int. J. of Fracture, Engng. Fracture Mechanics and many others, were solely devoted to “new” fracture criteria Let us now make some short comments on existing classes of failure criteria.We start with an example of inductive treatment of a familiar problem in Strength of Materials, namely sheet metalforming. There a Forming Limit Diagram (FLD) is asked to describe limiting strains. This FLD is obtained by either in-plane stretching or out-of-plane (punch) stretching of the sheet. Pairs (eA, |
eB) of such strains seem to follow two empirical rules, namely eA |
≈ 1 + |
eB or eA |
≈ 1 − 0.5eB (see for example However, a problem of incomplete space appears. A, through-thickness third strain e3 exists as obtained from equivoluminal changes of plastic strains. Then, in general we have:e1=1+e2e1+e2+e3=0⇒e3=-1-2e2ande1=1-0.5e2e1+e2+e3=0⇒e3=-1-0.5e2Symmetry considerations (eA |
= |
ei, |
eB |
= |
ej, |
i, |
j |
= 1, 2, 3, |
i |
≠ |
j), valid for isotropic materials, drive to six straight lines forming a hexagon symmetric to the diagonals of the space (eA, |
eB) as shown in The hexagon is, simply, the old Tresca failure surface satisfying Eq. in each quadrant. Obviously, an ellipse (e.g. the von Mises one) could be obtained by replacing Eq. by second order expressions. The failure mechanism is a typical shear one (as it was, obviously, expected), regardless “localized” or “diffuse” necking, etc.An overspill of failure criteria of the general form ∫0eeq,maxf(eeq)deeq=c was introduced in order to interpret the same results (see again Atkins We distinct two main classes of deductive (axiomatic) criteria. The first of them launches from a mathematical assumption of the type: “The failure surface must be a closed, smooth, continuous surface in the stress or strain space, enveloping zero”. Typical examples of this class are Hill The second class starts with a physical assumption of the type: “Materials fail because their mechanisms of reaction to external stimuli collapse”. Such an assumption requires the a priori knowledge of the set of specific mechanisms of reaction and their inter-relationships. For that the as complete as possible description of the, so called, constitutive behavior of materials is needed. It was, already, mentioned that the complete space of constitutive behavior must be described by two constitutive equations and not only one, as almost exclusively is assumed. Contrarily to the mathematical class of criteria, the complete history of loading/stretching paths is required now, an often boring but elucidating task.Both classes of deductive failure criteria require a selection of the space of description. The mathematical class is, usually, based on the three-dimensional stress or strain fields, a not very practical choice to perform graphical measurements. The physical class of criteria has the advantage of availability of four quantities obtained from the two constitutive equations (Eq. ) which represent the full stress/strain fields. In addition, curves σeq |
= |
f(eeq) and p |
= |
f(Θ) inscribe the areas of SED for distortion and dilatation of the material, respectively.At this point some provisional conclusions are worth:It is desirable for failure criteria to be based on all stresses and strains. Otherwise, what should be the criterion for the exception of any of them?Causality demands the presence of an available cause for failure, namely elastic SED.Plastic work is elastic SED already transformed to “work”. So, plastic work is a result not a cause of failure. It is failure itself.Cracks and other geometrical singularities must be solely understood as modulators of the stress/strain fields and nothing more. It is not clear to us the distinction between criteria suitable for either “Fracture Mechanics” or “Strength of Materials”.It is obvious that we prefer SED for the development of a failure criterion with two precautions:To distinguish between elastic and total SED, in order to satisfy causality.To check whether or not the two parts of elastic SED (SEDD for distortion and SEDV for dilatation) are collaborative or competitive.A rule helping towards the selection between collaboration and competition is the observation of the failure behavior of materials. Namely, they change shape and/or volume depending on stress or strain triaxiality, i.e. the ratio of normal to shear stresses or strains. Eventually, they fail by either brittle fracture or plastic collapse. Mixed failures are a sequence of distortion or dilatation dominance. A classic example of mixed failure is that of a cylindrical ductile specimen under tension, where failure starts as brittle (cleavage) in the core of the specimen and integrates as plastic collapse in the remaining ring of the cross-section, forming the well-known cup-cone failure. It seems that the mutual behavior of SEDD and SEDV is rather competitive than collaborative.As “size effect” is defined the dependence of strength of materials on the size and/or shape of specimens. Size dependence is not permitted, by definition, in Mechanics of Continua. Contrarily, shape effect might be addressed to the difference in stress/strain states between surface and interior of the specimen e.g. plane stress in the surface and plane strain inside. If so, specimens having self-similar dimensional changes could not show size effect, since the ratio volume/surface is constant. This is not true. The only alternative is to assume that mechanical properties of the material are different between surface and interior.This alternative is well-supported theoretically (e.g. Eshelby Consider a specimen of an isotropic, homogeneous, linear elastic material with modulus of elasticity E0 and cross-section area A0 submitted to uniaxial tensile load P0. The stress and strain are:Assuming, now, that a specimen from this material consists of a “core” with cross-section area AC and a “surface” of finite thickness t and area AS at the periphery of the specimen, compatibility of strains requires that strain ɛS in the surface is equal to ɛC in the core, say ɛm. Consequently, modulus of elasticity changes from EC in the core to ES in the surface. For this new specimen the externally applied load P is separated into PC in the core and PS in the surface. Thus:where Em is the apparent mean or equivalent value of elasticity modulus. From Eq. εm=εC=εS=σm(AC+AS)/(ECAC+ESAS)σC=ECεm=ECσm(AC+AS)/(ECAC+ESAS)σS=ESεm=ESσm(AC+AS)/(ECAC+ESAS)In what follows, we replace symbol SED by T for simplicity. Then, at each loading step, total T equals to:Assuming that T takes a critical value for various core and surface areas AC, |
AS, we obtain the relation between critical failure stress σm and specimen geometry.) of core radius r and surface thickness t, with:Em=ECr2+ES(2r+t)t(r+t)2σm=ECr2+ES(2r+t)t(r+t)2TConsidering as starting point a homogeneous specimen with dimensions r0 |
= 1 m, |
t |
= 0 m, satisfying Eq. and critical total SED T |
= 1 MPa, from Eq. we obtain σ0=2E0. Then, from the second of Eq. a relation σm/σ0 |
= |
f(r/r0, |
t/r0, |
ES/EC) is obtained as it is plotted in Curves for various combinations of elastic moduli plotted here are sigmoid, tending to EC for r |
→ + ∞ and to ES for r |
→ 0 independently of t. It implies that they converge to the behavior of the core material for increasing r, although for decreasing r they converge to the behavior of the surface material, a quite reasonable behavior.In case the present approach is valid, then it is free of scale (not size) effect, in the sense that scalar (reduced) values of the quantities used here do not affect the results, as far as these quantities can be defined in Mechanics of Continua. The present results depend on the moduli of elasticity and surface thickness t (not to be confused with the independence from t at zero or infinite core radius r). It is worth to note that the product ESt is measured in Force/Length, as Eshelby force It is out of the scope of the present work to compare theoretical predictions with experimental data. But, it seems that high differences between moduli of elasticity correspond to materials with weak homogeneity, the opposite being true for reasonably homogeneous ones.It is concluded that the present analysis gives an answer to the size effect phenomenon in pure terms of Mechanics of Continua by assuming that specimens are not as we believe they are. They, always, have a finite thickness skin created by environmental and/or human activities (e.g. specimen machining). In terms of Mechanics of Continua any specimen (not material in general) is a special type of bi-material composite structure. In this respect and to our opinion, size effect does not exist. It is a typical example of “incomplete space” of description, as we have, already, mentioned.The title of this section, implicitly, accepts the validity of the previous one. Consequently, we adopt the approach described in Section 3. We deal with a specimen described by Eqs. , but now we separate the total strain energy density T into its components TV for dilatation and TD for distortion. This separation is necessary, because the type of failure of core and surface is not a priori known. Then, we introduce four strain energy density quantities as:TVC=1-2ν6ECσC2,TDC=1+ν3ECσC2TVS=1-2ν6ESσS2,TDS=1+ν3ESσS2for core and surface,respectively.Poisson ratio, ν, has the same value in both core and surface to satisfy the compatibility of strains (first of Eq. ). The variation of the reduced to T values of TVC, TVS vs AS/AC is shown in . Qualitatively identical results are obtained for the second pair TDC, TDS of distortion.It is observed that as the areas ratio increases, core values decrease and surface values increase, an expected result, since the dominance of the core decreases as surface area increases. However, an important observation concerning the cross points satisfying TVC |
= |
TVS and TDC |
= |
TDS, is worth mentioning. These points (black dots in ) represent geometries and material properties where, in energy terms, the specimen behaves as made of a typical isotropic material with identical mechanical properties, in spite of the different moduli of elasticity in core and surface. This happens independently of surface thickness. The locus of “isotropic” points of is plotted in the space elastic moduli ratio ES/EC vs areas ratio AS/AC (Out of all the points of this figure, Point A with coordinates (1, 1), represents a material-geometry configuration with the same value of elastic moduli (EC |
= |
ES |
= |
Em) and equal surface-areas (AS |
= |
AC) i.e. a real isotropic material without skin and “quotes”.Finally, the dependence of dilatational, TVS and distortional TDS SED is plotted in vs areas ratio AS/AC. It is interesting that both quantities show a maximum value exactly at the point of real isotropy of . This implies that there exist an optimum material-geometry configuration, for SED to be stored. This configuration is the simplest possible, i.e. the full isotropy. Maxima appear at AS/AC |
= 1 with reduced values:For the time being, the physical rationale behind the maxima in is not clear. Perhaps, it is true that Nature follows the rule “the simpler the better”. In any case, it allows us to have a glance at a more practical level. Really, measurements in a uniaxial tension experiment in an assumingly isotropic material give a value EExp of elastic modulus, which, in fact, is the mean one Em given by Eq. . In addition, the physical diameter dExp of the specimen is known at every loading step. Thus, we have from Eq. A third equation is obtained by the condition of maximization of TVS or TDS in , i.e.: ∂(TVS)∂(AS/AC)=0, which after a boring algebra results to a lengthy equation of the form:Unfortunately but obviously, maximization conditions are linearly dependent meaning that we have three equations with four unknowns r, |
t, |
EC, |
ES. The missing information may be obtained by determining ES through, for example, micro-hardness experiments. However, the present approach can serve as a launching point for the study of failure of materials in more realistic terms.In the present work we tried to refresh and emphasize some points widely known but, sometimes, mis- or un-considered. It is out of our imagination that a clearly observed and measured phenomenon, like size effect, is not explainable in terms of Mechanics of Continua. If so, this part of Natural Sciences has a Gödel-like “hole” as pure Mathematics have. There this hole is allowed, since pure Mathematics deal with abstract ideas which, sometimes, are far beyond any type of “experimental” verification. Contrarily, Mechanics of Continua deals with everyday reality. If so, Gödel-like holes are not allowed in Mechanics of Continua and a kind of explanation for size effect must exist. The explanation presented here is, perhaps, completely wrong but, a better explanation based on the familiar terms we use, will appear sooner or later. Already existing respectable and rigorous microscopic theories transport an old wine (size effect) to a new bottle (microstructure) leaving un-answered the question of whether or not, for example, Young’s modulus has the same meaning in macro- and micro-scale. To our opinion, the apparent weakness of Mechanics of Continua lies on the wrong question we ask when we search for the mechanical behavior of a material. We do not refer to unavoidable simplifications like linearity but to the confusion between a material and a specimen from this material. Both have a skin as any natural construction has. But in the large block of material the skin is a small proportion of the total volume. That is why size effect disappears at very large specimens. Contrarily, in small laboratory specimens skin volume is comparable with total volume. Consequently, the approximately homogeneous material becomes a bi-material small specimen. This observation implies that constitutive equations deserve much higher importance than we, usually, pay to them. Most researchers use a single constitutive equation, some others (including us) insist on a couple (like Eq. ) and, now, we are afraid that two couples are needed for core and skin. In this sense, failure criteria reduce to a trivial observation of the limiting points of constitutive equations. Finally, size effect is strongly connected to volume and surface energy, being the two sides of the same coin. As a post scriptum, minimal list of references is included, citing, only, those been really used.Numerical modeling of non-planar hydraulic fracture propagation in brittle and ductile rocks using XFEM with cohesive zone methodWith the increasingly wide use of hydraulic fracturing in the petroleum industry, it is essential to accurately predict the behavior of fracture propagations based on the understanding of fundamental mechanisms governing the process. For unconventional resources exploration and development, hydraulic fracture pattern, geometry and associated dimensions are critical in determining well stimulation efficiency. In shale formations, non-planar, complex hydraulic fractures are often observed, due to the activation of pre-existing natural fractures. The propagating of turning non-planar fractures due to re-fracturing treatment and unfavorable perforation conditions have also been reported. Current numerical simulation of hydraulic fracturing generally assumes planar crack geometry and weak coupling behaviors, which severely limits the applicability of these methods in predicting fracture propagation under complex subsurface conditions. In addition, the prevailing approach for hydraulic fracture modeling also relies on Linear Elastic Fracture Mechanics (LEFM), which uses stress intensity factor at the fracture tip as fracture propagation criteria. Even though LEFM can predict hard rock hydraulic fracturing processes reasonably, but often fails to give accurate predictions of fracture geometry and propagation pressure in ductile rocks, such as poorly consolidated/unconsolidated sands and clay-rich ductile shales, even in the form of simple planar geometry. In this study, a fully coupled non-planar hydraulic fracture propagation model in permeable medium based on the Extended Finite Element Method (XFEM), Cohesive Zone Method (CZM) and Mohr-Coulomb theory of plasticity is developed for the first time, which is able to model fracture initiation and propagation in both brittle and ductile formations. To illustrate the capabilities of the presented model, example simulations are presented on both near wellbore and far field scale. The results indicate that the in-elastic deformations induced by propagating hydraulic fracture have significant impact on propagation pressure and fracture geometry, and the prediction of fracture propagation behaviors can be extremely erroneous if ductile formations are simply treated as soft rocks with lower Young's modulus. The method discussed in this article represents a useful step towards the prediction of non-planar, complex hydraulic fractures and can provide us a better guidance of completion design and optimizing hydraulic fracture treatment that will better drain reservoir volume in formations with complex stress conditions and heterogeneous properties.Hydraulic fracturing has been widely used as a common practice to enhance the recovery of hydrocarbons from low permeability reservoirs and prevent sand production in high permeability reservoirs (). During the last decade, the shale revolution has transformed United States into the world's fastest-growing hydrocarbon producer and will continue impacting global energy market substantially in the foreseeable future. Hydraulic fracturing, which creates extensive and conductive fracture path that extended from wellbore to the deep formation, combined with horizontal drilling, allow formerly uneconomical shale layers to be commercially viable. Understanding hydraulic fracture initiation and propagation from wellbores is essential for performing efficient hydraulic fracture stimulation design and treatment.As a reservoir stimulation tool to enhance production, the problem of hydraulic fracturing is in essence one of predicting the shape of the fracture as a function of time, given the fluid pressure at the wellbore or the flow rate into the fracture. Even today, modeling fluid-driven fracture propagation is still a challenging problem, due to its moving boundaries and strong coupling effects among different mechanisms. The mathematical formulation of the problem is represented by a set of nonlinear integro-differential equations, with moving boundaries where the governing equations degenerate and become singular. The complexity of the problem often restricts researchers to consider only simple fracture geometries, such as the KGD model (). However, under certain circumstances, the hydraulic fracture may evolve in a complex non-planar fashion due to various reasons, such as heterogeneous formation properties, intersection with natural fractures, initiated from an unfavorable orientation and stress interference with other hydraulic fractures. This in turn can significantly alter the relationship between the injection history and the crack geometry that predicted by planar fracture models, so it is crucial to model non-planar hydraulic fracture propagation in order to understand its impact on the completion process and/or to ensure that undesirable situations do not arise.In recent years, the extended finite element method (XFEM) has emerged as a powerful numerical procedure for the analysis of fracture problems. This method was developed to ease difficulties in solving problems with localized features that are not efficiently resolved by mesh refinement and help alleviate shortcomings of the finite element method and has been used to model the propagation of various discontinuities, such as cracks and material interfaces. A key advantage of XFEM is that in such problems the finite element mesh does not need to be updated to track the crack path, and discontinuous enrichment functions are used to approximate the displacement discontinuity across the crack surface, which significantly reduced the computational costs and projection errors associated with conventional finite element methods that restricting the discontinuities to mesh edges (). Since the introduction of this method, the XFEM has been used to model complex, non-planar hydraulic fracture propagations by many authors (However, an assumption commonly made in all these proposed XFEM models are that the loss of fluid into the rock is evaluated using a one-dimensional diffusion equation and the mechanics of fracture opening and fluid loss are considered independent and their interactions are ignored, so the pressure diffusion and porous behavior of the rock deformation are not fully coupled. In addition, all these models are under the assumption of linear elastic fracture mechanics (LEFM), which uses stress intensity factor at the fracture tip as fracture propagation criteria. Although hydraulic fracturing simulators based on LEFM can give reasonable predictions for hard rock formation, they fail to predict fracture net pressure and geometry with enough certainty in ductile formations, even in the case of simple, planar fracture geometry. Numerous study and surveys have indicated that the net pressure observed in the field of ductile formations are much higher than that predicted by LEFM and this disparity is even larger in poorly consolidated rocks. These observations have triggered a series of dedicated studies which looked into the importance of the plastic deformations in hydraulic fracturing (Besides the commonly known poorly consolidated/unconsolidated sands, the fact that shale rocks can also exhibit in-elastic behavior has been largely ignored in both industry and academia. Laboratory studies have shown that deformation of weakly cemented sands and shales can even occur through elastic–visco-plastic constitutive behavior, and the deformations of these types of rocks cannot be predicted by linear elasticity (). Not all shales are as brittle as the Barnett shale; depending on mineral compositions (such as clay, carbonates, quartz, etc.), many shales can exhibit a pronounced ductile behavior, such as the Eagle Ford and the Haynesville Shale. The ductile nature of these shale plays poses a challenge in hydraulic fracturing design and execution. Field studies indicate that it is more difficult to initiate a hydraulic fracture in ductile Haynesville shales (), and the production from hybrid fracturing treatments (creating highly conductive primary fractures) wells outperforms the water-frac (creating large fracture surface area) wells in the Eagle Ford shale (); this demonstrates that creating a high-conductivity fracture may be more critical to long-term production than creating large surface area in ductile shales. Such trends can be explained by the fact that in clay-rich ductile shale, the level of complexity of the induced hydraulic fracture geometry is relatively low, dominant “bi-wing” fractures are more likely to occur and unpropped fractures will most likely not be conductive after initial production. Thus, being able to predict fracture geometry and fracturing pressure with enough confidence is not only important in poorly consolidated/unconsolidated sands, but also crucial in ductile shales, because of the long term production hinges on the portion of the primary fracture that can be propped with enough size and concentration of proppants. It should also be mentioned that not only ductile rock itself can lead to difficulties in initiating hydraulic fracture, field study () also indicates that perforations-associated damaged rock can behave like a plastic rubber buffering between the reservoir and fracture entry, which can lead to significant increase of formation breakdown pressure.In order to model hydraulic fracture growth in quasi-brittle and ductile materials, cohesive zone method (CZM) has been adopted by many authors to model fracture initiation and propagation. The conception of cohesive zone was first introduced by to investigate fracture propagation in perfectly brittle materials. In order to investigate fracture damage behavior in ductile materials with small scale of plasticity, a fracture process zone was proposed by , that adopt a critical opening condition as a fracture criterion. The physical meaningless tip singularity predicted by LEFM can be resolved with the idea of cohesive zone, which is a region ahead of the crack tip that is characterized by micro-cracking along the crack path, and the main fracture is formed by interconnection of these micro cracks due to damage evolution. Based on this conception, proposed a 3D finite element model for horizontal well hydraulic fracture; their work shows the simulation results from cohesive zone model matches well with field data. developed a nonlinear fluid-solid coupled hydraulic fracture cohesive zone model, which is capable of modeling fracturing propagation in multiple formation layers. proposed an analytical solution for hydraulic fracture with Barenblatt's cohesive tip zone, the results demonstrate that the derived solutions from cohesive zone model can fit the pressure log much more accurately than LEFM for the case of fracturing soft rock. developed a cohesive zone model to predict fracture propagation in brittle and ductile rocks and the effective fracture toughness method was proposed to consider the fracture process zone effect on the ductile rock fracturing. The results show that in ductile formations, the cohesive zone model gives predictions that are more conservative on fracture length as compared with pseudo 3D and PKN models. developed a poro-elasto-plastic hydraulic fracturing model for both brittle and ductile formations, which combines the cohesive zone method and Mohr–Coulomb flow theory of plasticity. Unlike previous studies, their model not only considers microscopic in-elastic behavior at the fracture tip inside the cohesive zone, but also captures plastic deformations in the bulk formation. Their work indicates that plastic deformations during fracturing execution can lead to higher initiation, propagation pressure and shorter, wider fracture geometry. It also found that effects of formation plasticity on fracture propagation are mostly controlled by in-situ stress, rock cohesion strength and pore pressure, and the effects of shear failures in the bulk formation cannot be fully represented by imposing an artificial increased toughness at the fracture tip (i.e., increase critical fracture energy in the cohesive zone). However, despite the successful implementation of CZM in hydraulic fracturing modeling, all these CZM models proposed in literature require a pre-define path for fracture propagation, which severely limits the applicability of these methods in predicting complex fracture geometry.The aim of this study is to develop a numerical hydraulic fracture model that can overcome these challenges discussed above. The structure of this article is as follows. First, the mathematical framework for the fully coupled, non-planar hydraulic fracture model developed in this study is presented, in which the concept of XFEM combined with CZM was introduced. Then, simulation cases are established and the simulation results are analyzed to demonstrate the capabilities of the presented model and gain some fundamental insights of the importance of in-elastic rock behaviors. Finally, conclusion remarks and discussions are presented.Following fracture initiation, further fluid injection results in fracture propagation. the geometry of the created fracture can be approximated by taking into account the mechanical properties of the rock, the properties of the fracturing fluid, the conditions with which the fluid is injected (rate, pressure), and the stresses and stress distribution in the porous medium. In describing fracture propagation, which is a particularly complex phenomenon, two sets of laws are required: (i) fundamental principles such as the laws of conservation of momentum, mass and energy, and (ii) criteria for propagation that include interactions of rock, fluid and energy distribution.A wide variety of fluids have been used for fracturing including water, aqueous solutions of polymers with or without crosslinkers, gelled oils, viscoelastic surfactant solutions, foams, and emulsions. Many hydraulic fracturing fluids exhibit power law rheological behavior and temperature-related properties. In order to avoid additional complexity added by fluid behavior, incompressible and Newtonian fluid is assumed in this study. shows a sketch of fluid-driven hydraulic fracture with vary aperture.For a flow between parallel plates, local tangential flow rate qf can be determined by the pressure gradient to the fracture width for a Newtonian fluid of viscosity μ (where w is the crack aperture, pf is the fluid pressure inside the fracture and μ is the average fluid velocity over the cross-section of fracture. Pressure drop along the fracture can be determined by Eq. with local flow rate and local fracture width. The conservation of the fluid mass inside the fracture can be described by the Reynolds (lubrication) equation:where ql is the local fluid loss in rock formation per unit fracture surface area. The local flow rate qf can be determined by taking fluid leak-off into consideration. Pressure Dependent Leak-off Model can be used to describe the normal flow from fracture into surrounding formations:where pm is pore pressure in the adjacent formation and cl is the pressure dependent leak-off coefficient, which can be interpreted as the permeability of a finite layer of filtrate cake on the cracked surfaces. For high permeability formations, such as poorly consolidated/unconsolidated sands, cl can have a large value, and for low permeability formations, such as tight sandstone or shales, the value of cl can be very small. Alternatively, Carter's Leak-off Model can be used to describe the leak-off phenomena (where vL is leak-off velocity, CL is the Carter's leak-off coefficient, t is the time elapsed since the start of the leak-off process. VL is the cumulative leak-off volume that passes though the surface AL during the time period from time zero to time t. The integration constant Sp is called spurt loss, which represents a sudden fluid loss into the formation when new fracture surface area are created and a stable leak-off status has not been established yet within the newly create fracture surface area. The two coefficients CL and Sp, can be determined from laboratory tests or, preferably, from the evaluation of Fracture Calibration Tests (FCT), which is also often referred to as MiniFrac, Mini-Fall-off (MFO) and Diagnostic Fracture Injection Test (DFIT) in literature. presented vast theoretical modeling and field example analyses on this specific subject. For high permeability formations, spurt loss can contribute to the majority of the fluid leak-off volume even with the addition of leak-off control additives. For low permeability formations, spurt loss is negligible unless the fissure system is considered, if present. Setting AL to be the unit surface area, and discretizing Eq. with respect to time, the difference of cumulative leak-off volume VL between each time intervals can be calculated. When the discretized time intervals are small enough, the changes of cumulative leakoff volume VL divided by time interval are approximate to the leak-off rate per unit surface area, ql, within that specific time interval.In essence, the leak-off phenomenon described by the Pressure Dependent Leak-off Model using Eq. and the Carter's Leak-off Model through Eqs. represents the fundamental constitutive equations for pressure driven leak-off. While Eqs. are empirical relationships that represent the average leak-off behavior over the entire fracture surface during FCT, where the effects of variations in lithology among multiple layers and formation heterogeneity are included. For practical applications, Eqs. are preferred to directly model leak-off rate and calculate injection fluid efficiency in hydraulic fracture design, by honoring the representative subsurface data from FCT, because the field scale fracture leak-off behavior cannot be quantified by any other means. For the purpose of numerical simulation, Eq. can also be used to model leak-off behavior and the value of the pressure dependent leak-off coefficient, cl, can be estimated by history matching against the FCT data, if the resistance of fracture surface to the penetration of leak-off fluid does not change with time. This can be the case in low permeability, homogenous formations where leak-off control additives are normally absent. have successfully implemented Carter's Leak-off Model in their study. In this article, the Pressure Dependent Leak-off Model that described by Eq. is used in the following simulations. Inside the reservoir formation, Darcy's Law is used to describe fluid diffusion in the porous mediawhere qm is the fluid flux velocity vector in the porous media, k is formation permeability vector.The basic theory of poroelasticity in which the fully coupled linear elastic rock deformation and pore pressure equations was initially introduced by the pioneering work of . Since then, many researchers have contributed to its further development. In fluid filled porous media, the total stresses σi,j are related to the effective stresses σi,j`throughThe effective stresses govern the deformation and failure of the rock, the poroelastic constant α is a rock property that is independent of the fluid properties. demonstrated that variations in the value of poroelastic constant α has negligible influence on fracture geometry. In this study, the poroelastic constant α is assumed to be 1 and the equilibrium equation in the form of virtual work principle for the volume under its current configuration at time t can be written as (where σ' are effective stress and virtual rate of deformation respectively. t and f are the surface traction per unit area and body force per unit volume, I is unit matrix. This equation is discretized using a Lagrangian formulation with displacements as the nodal variables. The porous medium is thus modeled by attaching the finite element mesh to the solid phase that allows liquid to flow through. A continuity equation required for the fluid, equating the rate of increase in fluid volume stored at a point to the rate of volume of fluid flowing into the point within the time incrementwhere ρf and ϕ are the density of the fluid and porosity of the porous media respectively. n is the outward normal to the surface S. The continuity equation is integrated in time using the backward Euler approximation and discretized with finite elements using pore pressure as the variable.The extended finite element method was first introduced by to help alleviate shortcomings of the finite element method in modeling the propagation of various discontinuities. By introducing the concept of partition of unity () and extend local constructions to the whole space under the conventional finite element method context, the presence of discontinuities can be presented by special enriched functions in conjunction with additional degrees of freedom. Thereby, it enables the accurate approximation of fields that involve jumps, kinks, singularities, and other non-smooth features within elements. In the context of fracture analysis, the enrichment functions generally consist two parts: asymptotic functions that capture the singularity around the crack tip and a discontinuous function that depicts the displacement jump across the fracture surfaces. The displacement vector u can be approximated with the partition of unity enrichment (where NI(x) are the usual nodal shape functions, which have a value of 1 at the node whose number it bears and zero at all other nodes. uI is the usual nodal displacement vector associated with the continuous part of the finite element solution, it is applicable to all the nodes in the model. aI is the nodal enriched degree of freedom vector and H(x) is the associated discontinuous jump function across the crack surfaces. The product of aI and H(x) is valid for nodes whose shape function support is cut by the fracture interior. bIα is also the nodal enriched degree of freedom vector, but only exist at the fracture tip, and Fα(x) is the associated elastic asymptotic crack-tip functions. The product of bIα and Fα(x) is only used in the situation for the nodes whose shape function support is cut by the fracture tip. illustrates the discontinuous jump function across the crack surfaces.The discontinuous jump function has the following form:where x is a sample (Gauss) point, x*is the point on the crack closest to x, and n is the unit outward normal to the crack at x*. In , r and θ denote local polar coordinates system with its origin at the crack tip and θ=0 is tangent to the crack at the tip. The asymptotic crack tip function, Fα(x), can be determined by Fα(x)=[rsinθ2,rcosθ2,rsinθsinθ2,rsinθcosθ2]The above functions can reproduce the asymptotic mode I and mode II displacement fields in LEFM, which represent the near-tip singular behavior in strains and stresses. The use of asymptotic crack-tip functions is not restricted to crack modeling in an isotropic elastic material. The same approach can be used to represent a crack along a bimaterial interface, impinged on the bimaterial interface, or in an elastic–plastic power law hardening material. However, in each of these three cases different forms of asymptotic crack-tip functions are required depending on the crack location and the extent of the in-elastic material deformation. The different forms for the asymptotic crack-tip functions are discussed by , respectively. However, accurately modeling the crack-tip singularity requires constantly keeping track of where the crack propagates and is cumbersome because the degree of crack singularity depends on the location of the crack. However, this problem can be avoided if the moving cracks are modeled with the cohesive zone method, which will be discussed in later sections.A key development that facilitates treatment of cracks in an XFEM analysis is the description of crack geometry, because the mesh is not required to conform to the crack geometry. The level set method (), which is a powerful numerical technique for tracking interfaces and shapes, fits naturally with the XFEM and makes it possible to model arbitrary crack growth without remeshing. To fully characterize a crack, two orthogonal signed distance functions are needed. The first, ∅, describes the crack surface, while the second, ψ is used to construct an orthogonal surface so that the intersection of the two surfaces gives the crack front, as illustrated in . Crack growth is modeled by appropriately updating the ∅ and ψ functions at each iteration. In the context of XFEM, no explicit representation of the boundaries or interfaces is needed for the crack geometry because they are entirely described by the nodal data.There is a number of fracture propagation criteria have been proposed by previous studies, which is usually described by either a stress condition or an energy condition. The propagation criterion introduced within the context of LEFM assumes that the process zone, a region near the fracture tip where behavior of the material in not elastic (e.g. region of plastic deformation, microcracking, etc.), is small compared to the fracture size, and a fracture can propagate only if the stress intensity factor exceeds the material toughness. However, fracture propagation in ductile formations can induce a significant plastic deformation around the fracture due to share damages, which put the adequacy of such assumption into questioning. The cohesive zone model that takes the process zone into consideration is able to capture non-linear fracture mechanics behavior based on energy condition, and a fracture will propagate when the energy release rate in the process zone reaches the critical fracture energy (The constitutive behavior of the cohesive zone is defined by the traction–separation relation, which includes the stage of initial loading, initial damage, and the damage evolution that leading to the final failure at the bounded interface. The behavior of the interface prior to initiation of damage can be described as linear elastic but with a penalty stiffness that degrades under tensile and shear loading if the corresponding stress reaches a critical value, but the stiffness is not affected by pure compression. Laboratory experiments can be used to derive these relations by investigating post-peak behavior with principle/shear stress and axial/shear strain data.In this study, bilinear cohesive law proposed by . Where tn,ts,tt refer to the nomal, the first, and the second shear stress components; and tn0,ts0,tt0 represent the tensile strength of the rock material when the deformation is purely perpendicular to the interface and the shear strength of rock material in the first and the second shear direction; δn0,δs0,δt0 corespond to the displacement of initial damage in the normal, the first, and the second shear stress direction and δnf,δsf,δtf are the displacement of complete failure in these three directions. It assumes that the material exhibit linear elastic behavior before the traction reaches the tensile strength/shear strength or the separation of cohesive surfaces exceeds the displacement of damage initiation. Beyond that, the traction reduces linearly to zero up to the displacement of complete failure, and any unloading takes place irreversibly.Based on loading conditions, a crack can generate according three different failure modes, which are often named as Mode I (tension failure), Mode II (shear failure under sliding) and Mode III (shear failure under tearing). For mode-I plane strain fracture, the critical fracture energy GIc equals the area under the traction–separation curve, which can be related to the rock fracture toughness KIC by (where E is Young's modulus of formation and ν is Poisson's ratio. Typically, conventional hydraulic fracture models based on LEFM employ Mode-I based fracture criteria that only consider tensile failure mechanism in fracture propagation. However, shear failure can play an important role in ductile formations under certain loading conditions. In this study, the combined effects of different modes will be used to define the damage initial and the propagation criteria in the following discussions.Damage is assumed to initiate when one of the stress components reaches the value of maximum strength of rock material in that direction, which can be represented by a quadratic lawThe symbol ⟨□⟩ used in the above equation represents the Macaulay bracket with the usual interpretation. The Macaulay brackets are used to signify that a pure compressive deformation or stress state does not initiate damage. The stress components of the traction–separation model are affected by the damage according towhere t are stress components, t̅ are the stress components predicted by the elastic traction–separation behavior for the current strains without damage. D is a scalar damage variable, which has an initial value of 0 and monotonically increases to 1 as damage developing, represents the overall damage that comes from the combined effects of different traction–separation modes in the rock material. For linear softening as shown in , the evolution of the damage variable, D, reduces to (where δm is effective displacement, defined asThe mode mix of the deformation fields in the cohesive zone quantify the relative proportions of normal and shear deformation. The the Benzeggagh–Kenane fracture criterion () can be implemented to determine the mixed-mode damage evolution during fracture propagation. This criterion is suitable for situation when the critical fracture energy of rock material along the first and the second shear directions are similar. The combined energy dissipated due to failure Gc, is defined aswhere Gshear=GIIc+GIIIc, Gtotal=Gshear+GIc. And GIc,GIIc,GIIIc are the work done by the tractions and their conjugate relative displacements in the normal, first, and second shear directions. For an isotropic formation, the traction-separation responses in different modes are assumed to be the same in this study, where GIIc=GIc, so the cohesive response is insensitive to parameter η. The fracture will propagate when the energy release rate reaches the value of Gc. And the newly introduced crack is always orthogonal to the maximum local tensile stress direction when the fracture propagation criterion is satisfied.Unlike these methods in previous studies (), which require that the cohesive zone surfaces align with element boundaries and the cracks propagate along a set of predefined paths, the combination of XFEM and cohesive zone method can model fracture initiation and propagation in any arbitrary path, by taking the advantages of both XFEM and cohesive zone methods. Under such combination, the near tip asymptotic singularity is replaced by cohesive zone and only the displacement jump across an entire fractured element is considered. Therefore, the crack has to propagate across an entire element at a time to avoid the need to model the stress singularity.Phantom nodes, which are superposed on the original real nodes, are introduced to represent the discontinuity of the cracked elements (. When the element is intact, each phantom node is completely constrained to its corresponding real node. When the element is cut through by a crack, the cracked element splits into two parts. Each part is formed by a combination of some real and phantom nodes depending on the orientation of the crack. Each phantom node and its corresponding real node are no longer tied together and can move apart.The magnitude of the separation is governed by the cohesive law until the cohesive strength of the cracked element is zero, after which the phantom and the real nodes move independently. To have a set of full interpolation bases, the part of the cracked element that belongs in the real domain, Ω0, is extended to the phantom domain,Ωp. Then the displacement in the real domain can be interpolated by using the degrees of freedom for the nodes in the phantom domain. The jump in the displacement field is realized by simply integrating only over the area from the side of the real nodes up to the crack; i.e.,Ω0+ and Ω0− (). This method provides an effective and attractive engineering approach and has been used for simulation of the initiation and growth of multiple cracks in solids by . It has been proven to exhibit almost no mesh dependence if the mesh is sufficiently refined.In this part, two simulation sections are presented and discussed to demonstrate the capability of XFEM based CZM to model non-planar hydraulic fracture propagation in both brittle and ductile rocks. The first section focuses on fracture initiation and propagation in the near wellbore region, where the direction of perforation is not perpendicular to the minimum principle stress. The second section investigates fracture propagation in rock formations with different properties on a field scale, to demonstrate the importance of plastic failure on fracture propagation behavior.Because of well stability problems, isolating well from unwanted zones and several other operational considerations, usually wellbores are cased and then perforated; therefore, majority of hydraulic fracturing treatments are performed through perforations. Hydraulic fractures initiate and propagate along a preferred fracture plane (PFP), which is the path of least resistance. In most cases, stress is largest in the vertical direction, so the PFP is vertical and is perpendicular to the minimum horizontal stress. Because perforations control the initial onset of fracture, if the perforations are misaligned with the direction of PFP, the fracture will reorient itself to propagate in the direction of least resistance. This can result in a tortuous fracture path and smaller fracture width, which can lead to early screen out from proppant bridging and, eventually, not optimal stimulation treatments. shows the re-oriented fractures due to unfavorable perforations.In order to demonstrate the presented model′s capability of capturing non-planar hydraulic fracture propagation, a near wellbore region with fractures initiated from an unfavorable perforation angles is simulated, as shown in . The fracture path at the perforation points are defined as initially open to allow entry of the fluid at the perforation tunnels, so that the initial flow and fracture growth are possible. A dynamic pressure that equals to the fluid pressure at the perforation entry is imposed on the inner surface of wellbore, which is implicitly calculated by the model during analysis. All the outer boundaries have zero displacement along the direction that perpendicular to its surface, and constant pore pressure condition is imposed on the outer boundaries. The whole simulated regime is saturated with reservoir fluid and incompressible Newtonian fluid is injected at a constant rate at the perforation entries.Besides modeling fracture propagation in brittle formation that undergoes purely elastic deformations, fracture propagation in ductile formation will be also included in this study. Because the plastic failure does not restrict itself to the very vicinity of propagating fracture tip, the plastic behaviors in the whole stimulated reservoir rocks should be considered as well (). The in-elastic rock material behavior in the formation follows the Mohr–Coulomb flow theory of plasticity and the associative behavior with constant dilatation angle is considered in the following simulations. These assumptions are justified by the presence of high confining stresses prior to crack propagation and to a decrease in the initial in-situ mean pressure near the crack tip during propagation. For the values of cohesion strength, ductile shales and sandstones can have ranges between 3 MPa and 8 MPa, while the cohesion strength of poorly consolidated sands and shales with planes of weaknesses is normally below 3 MPa ( shows all the input parameters for the simulation cases presented in this section.The solution procedure is schematically illustrated in . The coupled system of non-linear equations is solved numerically by Newton–Raphson method. All the variables in the system are updated at the end of each time increment and input as initial values at the start of the next increment. The program for numerical calculations was developed using FORTRAN and finite element package ABAQUS. Because the fracture has to propagate across an entire element at a time during the simulation, mesh size has to be sufficient small enough (compared to the investigated domain) to minimize this effect. In this section, the mesh size monotonically decreases from the outer boundary (0.05 m) to the wellbore (0.004 m). shows the fracture propagation path and the induced shear stress distribution, it can be noted that the fracture first initiated along the direction of perforations, and then it gradually changes its propagation direction to align itself with the direction of PFP until it hits the simulation boundary. It can be also observed that the in-situ shear stress is intensified in front of fracture tip, where a shear zone (appears as a two wing pattern) is developed due to local stress disturbance by propagating fracture. It should be mentioned that the displace field presented in all the following simulation results (except for ) are enlarged by a scale factor of 50, so we can have a better visual observation of the fracture geometry and formation deformations. shows the formation pore pressure distribution at two different stages. The left figure depicts a moment before hydraulic fracture initiates, when pressure inside the wellbore and perforation channel gradually builds up with the constant injection rate of fracturing fluid. Then the fluid begins diffuse into the surrounding formation, which leads to higher pore pressure around the wellbore. The right figure depicts a moment when fracture is propagating. It can be observed that a high pressure zone is developed at the fracture tip and a low pressure zone is developed in in front of the fracture tip. The higher pressure zone can be explained by the fact that when a complete new fracture surface is generated within a cell at the fracture tip, there is a sudden fluid leak off into the adjacent formation, which results from a large pressure difference between the fluid inside the fracture and the surrounding formation pore fluid. The low pressure zone is developed because of the newly generated fracture volumes by the microcracks (partially damaged cells) in front of the fracture tip, where it cannot be filled by the fracturing fluid before complete failure happens. shows the leak-off rate and cumulative leak-off volume through the surfaces of fractured cells. In order to have a better visual inspection of the simulation results in the split elements, the displacement field presented in this figure is kept original. From the left figure, it can be noticed that the leak-off rate is higher in the newly fractured cells and lower when it is closer to the wellbore (except for those cells that affected by the simulation boundary conditions). This is mainly because the pressure difference between the fluid inside the fracture and the surrounding pores is higher within the newly fractured region. However, as time goes on, leak-off rate will gradually decline because of the increase of surrounding pore pressure due to fluid diffusion. From the right figure, it can be observed that the total cumulative leak-off volume is higher within the fractured cells that closer to the wellbore, this is due to the fact that these cells are fractured earlier, so the exposure time the fracture surface to the fracture fluid is longer and more fracturing fluid has been leaked off though these surfaces. Even though, the Pressure Dependent Leak-off Model is implemented in the simulation, the time dependent leak-off behavior actually resembles the general trend that depicted by the Carter's Leak-off Model, as reflected through shows the plastic strain and plastic deformation area during hydraulic fracture propagation when formation in-elastic behavior is considered in ductile rocks. It can be observed that the closer to the fracture surface, the more server the plastic damage, and plastic deformation area becomes larger when the fracture aligned itself with the direction of PFP. This is because horizontal stress contrast is largest along the direction of PFP and generally, shear failure is more likely to occur when horizontal stress contrast is large (). The reason lies behind the plastic deformation is the intensified shear stress in front the fracture tip, as depicted by . When the stress conditions within these area reaches beyond the failure envelope that set by the Mohr–Coulomb failure criteria, irreversible plastic deformation occurs. summaries the evolution of pressure and fracture width at the wellbore during simulation. It can be observed that the pressure is almost the same in both brittle and ductile formations at initial stage, but the pressure in the ductile formation gradually surpass the pressure in the brittle formation. This is because the plastic deformation region is small initially, so its impact on fracture breakdown pressure is limited, however, as fracture propagates deeper into the formation, more area deforms plastically and in turn, the fracture geometry and wellbore pressure are impacted, and this is also the reason for the wider fracture geometry (initial fracture width at wellbore is not zero due to the open perforation tunnel), that is observed in the ductile formation. It can also be noticed that the simulation stops around 16 s in the brittle formation, but it runs 1 more second in the plastic formation, before the fracture tip hits the simulation boundary. This can be explained by the fact that with wider fracture geometry in the ductile formation, the fracture length is shorter, with the same fluid injection volume in both formations. For more details on how the formation plastic properties and fluid/rock interactions can impact fracture geometry and fracture propagation pressure beyond the near wellbore region, when fracture propagating along the PFP direction, readers can refer to . It should be mentioned that the oscillation of the simulation curves correspond to halts and sequels in the fracture propagation process step by step for the series of time increments. Even though smaller mesh size and simulation time interval can smooth the curves, it can also exert much burden on computational efforts. In this study, the discretization of time and space is a compromise between solution accuracy, numerical convergence and computation time, and the simulation results presented here are valid to draw general conclusions.) shows that it is more difficult to initiate and propagate hydraulic fracture in ductile formations due to additional energy absorbed by plastic deformations and strong fluid-rock interactions. However, there is a common misconception that the cause of these difficulties in executing hydraulic fracturing in ductile formations is because the rock is “soft” (low Young's modulus). In order to differentiate the impact of Young's modulus and plastic failures on hydraulic fracture propagation, three cases of hydraulic fracture propagation in multiple layers are presented in this section. shows the base case simulation model with layers of different Young's modulus. Each layer is 20 m in width and 60 m in length. Because the scale of the simulation domain is much larger than the near wellbore region, only the perforations are explicitly modeled. All the input parameters are the same as presented in the previous section (provided by ) unless otherwise specified, and the whole simulation domain is uniformly discretized (mesh size is 1 m). Symmetric displacement and constant pore pressure conditions are imposed along the outer boundaries. shows the fracture geometry and pore pressure after 1000 seconds of injection. The fracture initiated from the perforation inside the middle formation layer (E=15 GPa), then propagates into the formation along the direction of PFP. However, when the fracture tip hit the lower formation layer (E=20 GPa) boundary, it is arrested and only propagates into the upper formation layer (E=10 GPa) where the Yong's modulus is lower. The results indicate that compare to hard rock with high Young's modulus, hydraulic fracture indeed prefers to propagate into soft rock with lower Young's modulus, where less pressure and energy is needed to propagate the fracture.Next, The Young's modulus in the lower formation layer is set to be the same as the upper formation layer (E=10 GPa), but plastic failure is allowed to occur in the lower formation layer, as shown in shows the simulation results of fracture geometry and pore pressure after 1000 s of injection. As expected, fracture propagates into the upper formation layer (E=10 GPa) and arrested at the boundary of the lower formation layer (E=10 GPa with plastic deformations). It clearly demonstrates that with same Young's modulus, hydraulic fracture has difficulties in propagating into formations that can undergo plastic deformations, where higher pressure and energy is needed.Realizing that both Young's modulus and plastic deformations can significantly impact hydraulic fracture propagation behavior, the effects of Young's modulus and plastic deformations are compared in this final simulation case, as shown in . The upper formation layer has the highest Young′s modulus (E=20 GPa) and the lower formation layer has lowest Young's modulus (E=10 GPa) but can undergo plastic deformations. Both the upper and lower formation layers can be served as a potential barrier to contain hydraulic fracture growth into their perspective formations. show the simulation results of fracture geometry and pore pressure after 1000 and 2400 s of injection. demonstrates that fracture arrested at the lower formation layer boundary (E=10 GPa with plastic deformations) and only propagates into the upper formation layer (E=20 GPa) where no plastic deformation occurs, and after 1000 s of injection, the fracture has reached the upper simulation boundary. Compared to , the upper formation layer has higher Young's modulus, which leads to longer and thinner fracture geometry. The results also reveal a very interesting phenomenon: even though, hydraulic fracture prefers to propagate into soft rocks with lower Young's modulus, as presented in previous simulation results, but when the formation exhibits in-elastic deformations, it completely changes the fracture propagation behavior. shows that fracture finally propagates into the lower ductile formation when it arrested at the upper simulation domain. It can be also noted that fracture is wider after fracture enters the lower formation layer. shows the plastic strain and plastic deformation area in the lower formation layer that induced by the propagating fracture. It can be noticed that even a small amount plastic deformation (with equivalent plastic strain on the order of 10−3) can make a significant difference in fracture propagation behavior. shows the hydraulic fracture width and pressure at the wellbore (perforation tunnels) in the middle formation layer during simulation. Three fracture propagation stages can be observed, as marked on the figure. In the first stage, fracture initiated from the perforation and propagates inside the middle formation layer (E=15 GPa). The pressure declines after the breakdown, which reflects a typical pressure evolution trend in PKN fracture geometry. In the second stage, fracture continues propagating into the upper formation layer (E=20 GPa) until it hits the upper simulation boundary, the pressure stabilizes in this period due to the increase of the overall stiffness along the penetrated formation. In the third stage, when the fracture arrested at the upper simulation boundary, it has to propagate downward into the lower formation layer (E=10 GPa with plastic deformations), and the pressure starts increase gradually. At the boundary of different stages, pressure discontinuity can be observed, which resembles the breakdown behavior when fracture breaks into new formation layers. The corresponding changes in fracture width can be also noticed in the figure, which reveals that fracture width at the wellbore increases rapidly after the fracture enters the ductile formation.In general, it is true that ductile rocks often exhibit soft behavior that associated with lower Young's modulus. However, the cause of these well-documented challenges in initiating and propagating hydraulic fracture in ductile formations is due to the plastic deformations induced by propagating fracture, rather than the soft nature of the rock. On the contrary to the common misconception, hydraulic fracture actually prefers to propagate into soft rocks if it behaves linear elastically. So if we simply treat ductile formations just as soft rocks with lower Young′s modulus, then our assessment and prediction about the hydraulic fracture propagation pressure and geometry can be extremely erroneous, because the in-elastic deformations in these ductile formations may completely change fracture propagation behavior and the final fracture geometry.In this study, a fully coupled, non-planar hydraulic fracture numerical model was developed and the concept of XFEM based CZM, that combined with Mohr-Coulomb theory of plasticity, was introduced together for the first time, to model hydraulic fracturing process in brittle and ductile rocks. The physical process involves fully coupling of the fluid leak-off from fracture surface and diffusion into the porous media, the rock deformation and the fracture propagation. XFEM is implemented to determine the arbitrary solution dependent fracture path and CZM is used to model fracture initiation and damage evolution to avoid stress singularity. Mohr–Coulomb flow theory of plasticity is used to model plastic deformations in the formations. Example simulations indicate that the presented model is able to capture the initiation and development of non-planar hydraulic fracturing propagation in both brittle and ductile formations. The model developed in this study represents a useful step towards the prediction of non-planar, complex hydraulic fracture evolution and can provide us a better guidance of completion design and optimizing hydraulic fracture treatment that will better drain reservoir volume in formations with complex stress conditions and heterogeneous properties.Even though the in-elastic behavior of poorly consolidated/unconsolidated sands and its impact on hydraulic fracturing has already draw much attention, the plasticity of ductile shales has been largely overlooked in hydraulic fracturing design and execution, despite many laboratory and field studies have shown that shale rocks can exhibit plastic or even time dependent elastic–visco-plastic behavior. For heterogeneous formations, the rock properties can transit between brittle and ductile along the section of a horizontal well and the formation themselves can as well be laminated with both brittle and ductile rock layers. A better understanding of the properties of ductile rocks and their impact on hydraulic fracture propagation and well performance is crucial to optimize hydraulic fracturing design in these formations.This study reveals that the plastic failures induced by propagating hydraulic fracture have significant impact on propagation pressure and fracture geometry, and the prediction of fracture propagation behavior can be extremely erroneous if ductile formations are simply treated as soft rocks. Field observations have demonstrated that higher pressure is needed to initiate and propagate hydraulic fracture in clay-rich ductile shales, such as Eagle Ford and the Haynesville Shale, even though these shale formations are generally soft with low Young's modulus. Because hydraulic fracture prefers to propagate in formations with lower Young's modulus if these formations behaves elastically, then the explanation for these abnormal high treatment pressure encountered in these fields indicates that plastic deformations are actually happening during the execution of a stimulation job, and these effects induced by the in-elastic rock behaviors cannot be neglected. The content of this paper is not intended to replace or diminish the extraordinary current research effort focusing on “unconventional fracture modeling” of complex fracture networks, but rather provide some fundamental insights on the almost-always neglected importance of the plastic deformations. Because even small scale of in-elastic rock deformations (with equivalent plastic strain on the order of 10−3) may completely change the expected fracture propagation behavior and propagation path in the scenarios of hydraulic fracture interactions (stress shadow effects), fracture height containment and interactions with natural fractures.Spinodal decomposition in (CaxBa1−x)yFe4Sb12The thermoelectric and structural properties of a double filled skutterudite solid solution (CaxBa1−x)yFe4Sb12 were investigated. Using X-ray powder and X-ray micro analyses (EPMA) an immiscibility gap was established with a critical point at x |
≈ 0.45 and a critical temperature that depends on the filling level (TC |
= 590 ± 5 °C at y |
= 0.8 and TC |
= 610 ± 5 °C at y |
= 0.9). The thermoelectric properties were measured for samples prepared in four different states: (i) a single phase solid solution (CaxBa1−x)yFe4Sb12; (ii) a two phase microcrystalline mixture of CayFe4Sb12 and BayFe4Sb12; (iii) a single phase structure obtained after annealing of the latter sample at 600 °C for 200 h; (iv) a spinodally demixed sample after annealing at 400 °C for 672 h. The thermoelectric properties of the phase mixture (ii) are compatible with data reported for the microcrystalline end members (CayFe4Sb12 and BayFe4Sb12), whilst the single phase (i and iii) and spinodally decomposed (iv) samples show increased thermopower and decreased thermal conductivity, similarly to those observed for nano-structured CayFe4Sb12 and BayFe4Sb12.Skutterudites crystallize as structures of LaFe4P12 type, a filled variant of the CoAs3 prototype (space group Im3¯). In the general chemical formula EyT4X12 T is a transition element of group VIII in the 8c position (¼, ¼, ¼), X is a pnicogen atom in sites 24e (P, As, Sb) and electropositive E atoms fill the icosahedral void 2a (0, 0, 0) Reducing the lattice thermal conductivity is an important issue in achieving high ZT values. Studies report that alkaline earth or/and rare earth filler elements, particularly in the form of double or multifillers, are efficient in reducing the lattice thermal conductivity λphAnother route to produce materials with an ultrafine microstructure involves so-called spinodal decomposition, as shown for the thermoelectric systems Pb1−xSnxTe–PbS The current investigation tries for the first time to exploit binodal/spinodal decomposition in thermoelectric double filled skutterudites (CaxBa1−x)yFe4Sb12. The influence of heat treatment on the physical properties of (CaxBa1−x)yFe4Sb12 was investigated and compared with that for the pure end members CayFe4Sb12 and BayFe4Sb12.Samples with the general formula CayFe4Sb12, BayFe4Sb12 and solid solution Ca0.46Ba0.46Fe4Sb12 (sample B) were prepared from a master Fe4Sb12 alloy via a melting reaction technique from stoichiometric amounts of high purity elements (iron wire with a minimal purity of 99.5 mass% and bulk antimony with a minimal purity of 99.95 mass% from Alfa Aesar, Germany) by mixing, sealing in evacuated quartz tubes, melting at 960 °C and air quenching. Then stoichiometric amounts of Ca/Ba (both with a purity of 99.99 mass% from Alfa Aesar, Germany) were added, the samples were sealed under vacuum in quartz tubes, heated for 4 days at 600 °C, melted at 960 °C and air quenched. The bulk materials obtained were ground to fine powders in an argon filled glovebox (O2 and H2O below 5 ppm) in a tungsten carbide/Co mortar and filtered through a microstrainer to obtain particles of <20 μm. Alternatively double filled (CaxBa1−x)yFe4Sb12 samples were prepared by mechanical mixing of Ca0.91Fe4Sb12 and Ba0.83Fe4Sb12 (samples A1 and A2) and of Ca0.84Fe4Sb12 and Ba0.93Fe4Sb12 (sample A3). The mixtures were loaded in a 10 mm diameter graphite die inside the glovebox and hot pressed (HP) under Ar employing a uniaxial FCT hot press system (HP W 200/250-2200-200-KS) at 600 °C and 56 MPa for 2 h. The cylindrical specimens obtained (1 cm diameter × 8 mm high) were cut into several pieces, sealed in evacuated quartz tubes and subjected to heat treatment as described in . The bulk of sample B was treated in the same way as described for samples A1, A2 and A3.X-ray powder diffraction data were collected with Cu Kα1 radiation and a Huber Guinier image plate recording system using pure Ge (99.9999%) as an internal standard (aGe |
= 0.5657906 nm). X-ray spectra were used to calculate the lattice parameters, using the programs STRUKTUR For transmission electron microscopy (TEM) investigations disc-shaped specimens were prepared by spark erosion followed by mechanical grinding. The discs were subsequently mechanically dimpled on one side to a final thickness of about 30 μm and thinned to perforation using a Leica RES101 ion mill. Ion milling was carried out from the top and bottom simultaneously at an inclination angle of 15°. Both guns were operating at a voltage of 3 kV and a current of 1 mA. During the milling procedure (about 12 h) the sample was rotated at 1 rpm. The TEM study was carried out using a Philips CM200 microscope operating at an acceleration voltage of 200 kV.Thermodynamic boundary curves for the spinodal (∂2G/∂x2 |
= 0) and the binodal (∂2G/∂x2 |
> 0) regions were calculated assuming a simple regular solution for the Gibbs free energy G, exploiting Eqs. for the binodal curve, with x2 |
= 1 − |
x1, the mole fractions, R is the gas constant R |
= 8.3144621 J K−1 |
mol−1, T is the temperature in K, TC is the temperature in K of the critical point for decomposition, and α is the regular solution parameter characterising the interaction of Ca and Ba atoms in solution The spinodal is at T |
= |
TC, and instability sets in atwhere η is the linear expansion per unit composition change, E is the Young’s modulus, ν is the Poisson ratio, k is the Boltzmann constant (k |
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