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> 1 MeV respectively. For these fluence levels, the yield strength is calculated using the mathematical fitting of each of the data series.In comparison to the above mentioned alternative, this normalization process reduces the uncertainties because the data points used for comparison are derived from the whole data sets rather than interpolation between two adjacent individual data points.The same procedure was applied to the eight materials and the results are summarized in where for identical fluence levels between low and high flux, the yield strength of the surveillance material is compared to the high flux irradiated yield strength. A 1:1 line is added for clarity. The experimental uncertainty bounds of ±25 MPa are also added as they must be considered for a full assessment. A first glance shows that there is no evidence of a significant flux effect, although there is a trend to have a slightly lower hardening in high flux irradiation than in low flux irradiation for some materials. A detailed assessment will be performed in Section On thermodynamic bases, Odette and Lucas The post-irradiation annealing treatment was performed in an appropriate furnace located in the hot cells. The specimens were introduced in the furnace once the temperature has reached 350 °C and the 5 h countdown was triggered only once the specimens temperature had reached a stable 350 °C. Then the specimens were cooled to ambient with a rate of 10 °C/min. The temperature profiles shown in indicates an average temperature of about 351 °C ± 5 °C. The three temperature–time profiles correspond to three batches of specimens because of the limited number of specimens that can be accommodated in the specimen holder.As already mentioned, the tensile specimens were retrieved from the RADAMO irradiation program spare specimens. The neutron fluence levels range between about 2.5 × 1019 and 1.5 × 1020 |
n/cm2. All tests were performed at room temperature at a strain rate of about 1 × 10−4 |
s−1. where the yield strength of the annealed material is compared to the as-irradiated value. Clearly, there is no evidence of any effect of the post-irradiation heat treatment, confirming the previous results This was not unexpected, because for a long time, the hardening model The objective of the work presented herein is to investigate the flux effect on irradiation hardening. It is important, therefore, to examine any other difference that might play a role.First, neutron flux is not the only variable. Indeed, neutron spectrum is also different when comparing MTR to PWR. Typical neutron flux distribution in a PWR (Doel-IV) and an MTR (BR2), is shown in . As it can be seen, the actual flux dependence varies with neutron energy as well. In practice, the fast neutron flux, typically above 1 MeV, is taken as the damage parameter. At this stage, the issue of which appropriate neutron damage parameter should be used is out of the scope of the present paper and we will consider flux effect without taking spectrum into account. In addition, the damage parameter is based on the fast neutron (neutron energy > 1 MeV) only. Except in very special conditions that are usually not observed in operational PWRs, we believe, without bringing evidence, that these two assumptions, no spectrum effect and fluence above 1 MeV, are reasonable.Second, it is important to question the suitability of using the small tensile specimen and the eventual uncertainties it might bring. The use of small specimens was motivated by two main reasons. The first one is related to the possibility to fabricate tensile specimens from broken surveillance specimens, for example 4 small size tensile specimens can be taken out of a half broken Charpy specimen. The second is related to the suitability of irradiation of a large number of specimens in a limited space. Indeed, space limitations and flux gradient are usually encountered in MTR irradiation programs and therefore, small test specimens are usually preferred.In this work, most of the irradiated surveillance specimens were manufactured from the broken Charpy and tensile specimens (specimen heads). The irradiation rig is designed such as three specimens are irradiated per material and condition. The small tensile geometry was qualified on both unirradiated and irradiated materials but, as it will be seen later, it is not unusual to find few outlier behavior probably due to excessive deformation during fabrication in hot cells. An example of the qualification exercise on unirradiated steel is shown in . Large typical surveillance specimens with a cross-section diameter of 6.35 mm are used as reference and two sub-sized tensile geometries with a diameter of 3.6 mm and 2.4 mm, respectively, were used for comparison. In addition, few samples with a diameter of 2.4 mm were also fabricated from the undeformed heads of the surveillance specimens. For illustration, the large surveillance specimen (6.35 mm) and the small size (2.4 mm) are shown in , the overall agreement between the various specimens is very satisfactory.Contrary to unirradiated materials where there is a full control of the fabrication process, the fabrication of small size tensile specimen from broken Charpy and large tensile specimens in hot cell environment is not that obvious. This is inherent to the hot cell manipulations which reduce the accuracy in comparison to workshop fabrication using unirradiated materials. As a result, some specimens exhibit occasionally excessive deformation during fabrication. Because of lack of direct contact between the operator and the specimen, it is difficult to unambiguously detect such occasional outliers. This is the reason of duplicating the tensile tests in order to reduce the probability of erroneous measurements but increasing therefore the scatter. With time and experience, more attention was given to the fabrication procedure and consequently the scatter was somewhat decreased. In , we gathered a number of experimental data over the last decade where tests were performed on both large and small size tensile specimens. As it can be seen, most of the data are located around the 1:1 line and associated ±25 MPa uncertainty bounds despite the large scatter.The ±25 MPa uncertainty is also supported by other tensile tests that are gathered in where the 2σ-deviation of 108 data sets were plotted in a frequency diagram. As it can be seen, the average 2σ-deviation is ±25 MPa (2/3 of the database) but sometimes it can significantly increase. These deviations should be kept in mind when comparing average values.This justifies the use of this small size geometry which leads to satisfactory results as long as multiple specimens are tested. In order to reduce the uncertainties, single data points are not considered individually, but the trend derived from multiple specimens is used.Third, it should be emphasized that the two sets of data, MTR versus PWR, that are compared were actually submitted to two very different thermal and neutron histories. The irradiation temperature is usually more stable in an MTR than in an operational PWR. In the BR2 reactor, the irradiation duration is 21 days (single cycle) with a quasi-constant power of about 60 MW. The temperature can easily be controlled within this period of time with deviations typically not exceeding ±5 °C. On the other hand, for the surveillance materials, even though the power level is rather constant in PWR units used in baseload operation, the temperature varies slightly between beginning and end of cycle and the surveillance materials are exposed to the successive startup and shutdown of the reactor. The temperature in the surveillance capsules is not continuously recorded, only temperature monitors giving indication on the maximum temperature on whether the temperature remained below or exceeded a specific temperature. In addition the neutron environment is also changed with the successive reconfigurations of the fuel assemblies. It is therefore obvious that the thermal and neutron histories to which the MTR samples and surveillance samples are submitted might be different and will contribute to the observed scatter. So, beside the inherent experimental scatter associated with the material variability and specimen fabrication, one should consider an additional contribution associated with the different thermal and neutron histories of MTR (high flux) in comparison to PWR (low flux). Having this in mind, it is interesting to assess the statistical significance of the results of A simple statistical significance test, such as the t-test, cannot be applied to our experimental data shown in because of the different variables into play. A more appropriate method is given by the general linear model (GLM) which allows to treat a multi-variable problem. As a matter of fact, most of the commonly used simple statistical tests such as the t-test, the analysis of variance (ANOVA) … can be derived from the GLM.A computer tool called STATISTICA is used where the multi-variable database in introduced. Three variables were used: material, fluence and flux. Material and flux are categorical variables while fluence is a continuous variable. No averaging or normalization procedure is performed on the raw data such as no additional treatment of the database was done leaving the data in their original state. The STATISTICA-10 The statistical program allows to investigate the effect of a specific individual variable but also the combination or interaction of the involved variables. Concerning the sole flux effect, as it can be seen, the probability of no flux effect is equal to 0.85 (85%), well above the 5%-significance level confirming therefore the null hypothesis or, in other words, there is no statistically-significant effect of flux on the yield strength. On the other hand, it is found that the two other variables, material and fluence, lead to a very low probability in comparison to the 5%-probability significance level, rejecting the null hypothesis. In other words, the yield strength depends on both material and fluence. Finally, in order to examine eventual interactions between the various effects, the probability levels given in clearly indicate significant interactions (p |
< 0.05) between the variables, i.e., a complex interaction of material, flux and fluence effect on the yield strength. Note that similar conclusions are obtained if the tensile strength rather than the yield strength is considered.Microstructure investigations are scheduled on two of these materials, but the results are not available yet. We already know from the works reported by a number of authors In summary, there is a consistency between the absence of flux effect and the absence of unstable matrix damage. It is believed that two mechanisms are dominating the hardening and embrittlement of RPV steels, precipitation damage (Cu-rich precipitates, P-rich precipitates for large P-concentrations) and matrix damage involving Ni. The precipitation component is fast and reaches its saturation after about 1 × 1019 |
n/cm2. The precipitation kinetic depends on neutron flux by delaying peak hardening occurrence at high flux. On the other hand, matrix damage has a slower kinetic but its saturation is much higher which value and rate depend on the Ni-content and irradiation temperature. The matrix damage kinetic does not depend on neutron flux. However, it is recommended to generate additional experimental data including both mechanical properties and microstructural examination to unambiguously assess flux effect qualitatively but also quantitatively as the latter has a direct impact on the use of MTR data to investigate irradiation effects of structural materials. In a recent paper, Kempf et al. The experimental program on eight RPV materials irradiated at 290 °C to a large fluence range presented in this paper demonstrated that the effect of neutron flux on the tensile properties is not statistically significant. The differences that were observed can be attributed to material variability and are within the experimental uncertainties. The importance of incorporating the experimental uncertainties in the flux effect assessment was clearly pointed out as a result of a large testing program.The post-irradiation annealing treatment at 350 °C/5 h applied to fourteen specimens of nine different materials failed to induce any measurable recovery of the tensile properties. The so-called unstable matrix damage that was suggested in literature to occur at high flux was not found in these materials, supporting the previous results related to flux effect.Based on published microstructural observations showing different size distributions of low and high flux specimens on the one hand and a one-to-one relationship between the irradiation-induced hardening and the square root of the volume fraction of the induced-defects on the other hand, it is suggested that thermal ageing effects are probably contributing in modifying the defect size and density distribution but leaving their volume fraction unaffected. Indeed, it is likely that under low flux irradiation, exposition to long time at high temperature promotes defect coarsening at the expenses of their number density, the product between average size and density remaining essentially similar. Finally, it is recommended to further pursue flux effect investigations both qualitatively and quantitatively to eventually support data transferability form MTR to LWR conditions.Stabilization of metallic supercooled liquid and bulk amorphous alloysBulk metallic materials have ordinarily been produced by melting and solidification processes for the last several thousand years. However, metallic liquid is unstable at temperatures below the melting temperature and solidifies immediately into crystalline phases. Consequently, all bulk engineering alloys are composed of a crystalline structure. Recently, this common concept was exploded by the findings of the stabilization phenomenon of the supercooled liquid for a number of alloys in the Mg-, lanthanide-, Zr-, Ti-, Fe-, Co-, Pd–Cu- and Ni-based systems. The alloys with the stabilized supercooled liquid state have three features in their alloy components, i.e. multicomponent systems, significant atomic size ratios above 12%, and negative heats of mixing. The stabilization mechanism has also been investigated from experimental data of structure analyses and fundamental physical properties. The stabilization has enabled the production of bulk amorphous alloys in the thickness range of 1–100 mm by using various casting processes. Bulk amorphous Zr-based alloys exhibit high mechanical strength, high fracture toughness and good corrosion resistance and have been used for sporting goods materials. The stabilization also leads to the appearance of a large supercooled liquid region before crystallization and enables high-strain rate superplasticity through Newtonian flow. The new Fe- and Co-based amorphous alloys exhibit a large supercooled liquid region and good soft magnetic properties which are characterized by low coercive force and high permeability. Furthermore, homogeneous dispersion of nanoscale particles into Zr-based bulk amorphous alloys was found to cause an improvement of tensile strength without detriment to good ductility. The discovery of the stabilization phenomenon, followed by the clarification of the stabilization criteria of the supercooled liquid, will promise the future definite development of bulk amorphous alloys as new basic science and engineering materials.Since the first synthesis of an amorphous phase in the Au–Si system by a rapid solidification technique in 1960 shows the relationship between the critical cooling rate (Rc), maximum sample thickness (tmax) and reduced glass transition temperature (Tg/Tm) for amorphous alloys reported to date shows the relationship between Rc, tmax and the temperature interval of a supercooled liquid defined by the difference between glass transition temperature (Tg) and crystallization temperature (Tx), (Tx−Tg) summarizes typical bulk amorphous alloy systems reported to date and the calendar years when details of each alloy system were published. Bulk amorphous alloys can be divided into nonferrous and ferrous alloy systems. The nonferrous alloy systems are Mg–Ln–M (Ln=lanthanide metal, M=Ni, Cu or Zn) . The first group (i) consists of ETM (or Ln), Al and LTM as exemplified for Zr–Al–Ni and Ln–Al–Ni systems. The second group (ii) is composed of LTM, ETM and metalloid as indicated by Fe–Zr–B and Co–Nb–B systems. The third group (iii) is LTM (Fe)–(Al,Ga)–metalloid systems and the fourth group (iv) is indicated by Mg–Ln–LTM and ETM(Zr, Ti)–Be–LTM systems. However, the Pd–Cu–Ni–P and Pd–Ni–P systems (v) are composed only of two kinds of group element (LTM and metalloid), which are different from the combination of the three types of group elements for the alloys belonging to the four previous groups (i)–(iv). Consequently, we must consider two different mechanisms for the achievement of the high stability of the supercooled liquid for the alloys of the first four groups and of the fifth group of alloys.Firstly, the reason for the stabilization of the supercooled liquid for the alloys belonging to the groups (i)–(iv) is discussed. All the alloy systems in these groups are based on the following three empirical rules summarizes the densities of some bulk amorphous alloys in as-cast and fully crystallized states. The difference in the densities between the as-cast amorphous and fully crystallized states is in the range of 0.30–0.54% summarizes the coordination numbers and atomic distances calculated from (a) the ordinary radial distribution function (RDF), and from the environmental RDFs for (b) Zr and (c) Ni in the Zr60Al15Ni25 amorphous alloy , in a new type of supercooled liquid with a higher degree of dense randomly packed atomic configurations, new local atomic configurations and long-range homogeneous atomic configurations, we have high solid/liquid interfacial energy which is favorable for the suppression of nucleation of a crystalline phase. The new type of liquid can have the difficulty of atomic rearrangement, leading to a decrease of atomic diffusivity and an increase of viscosity. Therefore, the new liquid can have high Tg. The supercooled liquid also has the necessity for atomic rearrangements on a long-range scale for crystallization, which causes the suppression of growth of a crystalline phase. In any event, the multicomponent alloys with the three empirical rules always have very deep eutectic valleys with low melting temperatures, leading to the appearance of high Tg/Tm and large ΔTx. Therefore, a high thermal stability of supercooled liquid for the multicomponent amorphous alloys which satisfies the three empirical rules is observed., the Pd–Cu–Ni–P and Pd–Ni–P amorphous alloys do not satisfy the three empirical rules because the heats of mixing are nearly zero for Pd–Cu and Pd–Ni pairs shows the ordinary radial distribution function and environmental radial distribution functions around Ni and Cu atoms for the Pd–Cu–Ni–P amorphous alloy . The coexistence of the two large clustered units seems to play an important role in the stabilization of the supercooled liquid for the Pd-based alloy, because of the strong bonding nature of metal–metalloid atomic pairs in the clustered units, high stability of metal–metalloid clustered units and difficulty of rearrangement among the clustered units. Furthermore, the GFA of the Pd–Cu–Ni–P alloy is much higher than that for the Pd–Ni–P alloy and the difference is due to the coexistence of the two large clustered units of the trigonal prism and tetragonal dodecahedron rather than the existence of only the trigonal prism.By choosing the above-described multicomponent alloy systems, we can produce bulk amorphous alloys by using two kinds of production techniques of solidification and consolidation summarizes tmax by use of the solidification techniques and approximates Rc in typical alloy systems. The value of tmax is about 10 mm for the Ln- shows the shape and outer surface appearance of typical bulk amorphous alloys 17 mm in diameter and 600 or 120 mm in length for the Zr60Al10Ni10Cu20 amorphous alloy and 75 mm in diameter and 80 mm in height for the Pd40Cu30Ni10P20 amorphous alloy. These bulk amorphous alloys have smooth outer surfaces and good metallic luster. By using these bulk amorphous alloys, Rc is determined accurately through the construction of continuous cooling transformation (CCT) curve. shows the CCT curves of the Pd40Cu30Ni10P20 alloy in the non-fluxed and B2O3 fluxed states In addition to the importance of basic science, it is important in applications as engineering materials to clarify the mechanical properties of bulk amorphous alloys. shows the relationship between the tensile fracture strength (σf) and Young's modulus (E) for the cast bulk amorphous Zr–Ti–Al–Ni–Cu alloys in sheet and cylinder forms with thicknesses (or diameters) of 1–5 mm shows the relationship between σf or Hv and E for bulk amorphous Mg–Cu–Y, La–Al–Ni, La–Al–Co–Ni–Cu, Zr–Ti–Al–Ni–Cu, Pd–Cu–Ni–P and Ti–Zr–Ni–Cu–Sn alloys together with the data of conventional crystalline alloys. The bulk amorphous alloys exhibit higher σf, higher Hv and lower E than those of any kinds of crystalline alloys. The feature of the mechanical properties for the bulk amorphous alloys is significantly different from that for the crystalline alloys. We further examined the compositional dependence of Charpy impact fracture energy for the cast bulk amorphous Zr70−x−yTixAlyCu20Ni10 sheets with a thickness of 2.5 mm summarizes the bending and rotating beam fatigue strength as a function of number up to failure for the bulk amorphous Zr65Al10Ni10Cu15, a vein fracture pattern is observed and no appreciable embrittled pattern is seen, indicating that these bulk amorphous alloys have high plastic deformability even near the pre-existent crack tip. These data were obtained for the sheet specimen with a thickness of 3.0 mm. It is not certain whether or not the K values correspond to the plain strain fracture toughness (KIc). However, it is noticed that the K values are much higher than those (24–36 MPa√m) When bulk amorphous alloys for their good static and dynamic mechanical properties are used as structural materials, it is essential for the bulk amorphous alloys to have good corrosion resistance in various kinds of corrosive solutions. There have been no data published on the corrosion resistance of Zr-based bulk amorphous alloys in any kinds of corrosive solution. We examined the corrosion resistance of melt-spun amorphous alloys in Zr–TM–Al–Ni–Cu (TM=Ti,Cr,Nb,Ta) systems in HCl and NaCl solutions and noticed that the Nb- and Ta-containing amorphous alloys exhibit good corrosion resistance in their solutions at room temperature shows the corrosion rates of the Zr60−xTMxAl10Ni10Cu20 (TM=Ti,Cr,Nb,Ta) amorphous alloys subjected to immersion for 16 and 64 h in 6M HCl solution at 295 K. The corrosion rates of the Zr–Al–Ni–Cu and Zr–Cr–Al–Ni–Cu amorphous alloys are too high to measure after immersion for 64 h, even though the corrosion rates of their amorphous alloys in 1 M HCl show very low values below 0.01 mm/year. Even in severe corrosive solution, the Nb- and Ta-containing alloys exhibit rather low corrosion rates below 0.1 mm/year after immersion for 64 h. also shows that the effect of the additional elements on the corrosion resistance is largest for Nb, followed by Ta, Ti and then Cr.It is also important to clarify the corrosion resistance of the Zr-based amorphous alloys in NaCl solution. shows the current density as a function of potential in 3 mol% NaCl solution at 298 K for Zr60Al10Ni10Cu20, Zr55Ti2.5Al12.5Ni10Cu20 and Zr55Nb5Al10Ni10Cu20 amorphous alloys, together with the data of pure crystalline Zr metal. The solution was exposed in air. The corrosion resistance is largest for the Nb-containing alloy, followed by the Ti-containing alloy and then the Zr–Al–Ni–Cu alloy. The corrosion resistance of the Nb-containing alloy is also superior to that for pure Zr metal, indicating the remarkable effectiveness of Nb addition on the improvement of corrosion resistance even in the NaCl solution. Considering the previous data As described above, the bulk amorphous alloys have a wide supercooled liquid region of more than 60 K before crystallization. shows the change in viscosity with reduced temperature (Tr=T/Tm) for the Pd40Cu30Ni10P20 amorphous alloy shows the relation between flow stress and strain rate for the Zr65Al10Ni10Cu15. The viscosity in the supercooled liquid region is independent of strain rate in a wide strain-rate range up to about 0.1 s−1, indicating the achievement of Newtonian flow in the wide strain-rate range. With further increasing strain rate, the viscosity is dependent on strain rate and decreases almost linearly, indicating that the deformation behavior of the supercooled liquid changes from Newtonian flow to non-Newtonian flow. Furthermore, the deformation mode of the amorphous solid occurs through the non-Newtonian flow and hence it is concluded that the Newtonian flow can occur only in the supercooled liquid region. There is no appreciable difference in the features of non-Newtonian flow mode in the supercooled liquid and amorphous solid. The independence of viscosity with strain rate is important for practical use of viscous flow working in the supercooled liquid region of bulk amorphous alloys. With the aim of clarifying in more detail the Newtonian flow behavior in the supercooled liquid, we examined the relation between true stress (σ) and in the supercooled liquid for the Zr55Al10Ni5Cu30 amorphous alloy. A good linear relation is recognized in the relation between σ and and the linear relation is expressed by We also tried to determine a deformation condition in which a maximum elongation by viscous flow is obtained. shows the relation between elongation at different testing temperatures and strain rates for the Zr65Al10Ni10Cu15 and La55Al20Ni25 amorphous alloys, together with data for ductility of the deformed samples Based on the three empirical rules for achievement of high GFA, a new bulk amorphous alloy with ferromagnetism at room temperature has been developed. As described in , soft ferromagnetic bulk amorphous alloys have been synthesized in multicomponent Fe–(Al,Ga)–(P,C,B,Si) shows the I–H hysteresis curves of the melt-spun Fe80P12B4Si4, Fe76Al4P12B4Si4 and Fe74Al4Ga2P12B4Si4 amorphous alloys in an optimally annealed state summarizes the magnetic properties of saturation magnetization (Is), residual magnetization (Ir), squareness ratio (Ir/Is), coercive force (Hc), effective permeability (μe) at 1 kHz and saturated magnetostriction (λs) of the three kinds of amorphous alloy ribbons, together with the data of Tg, Tm and Tx. The Is, Hc and μe values, which are important for a soft magnetic material, are about 1.1 T, 5–10 A/m and 12 000 to 19 000, respectively, indicating that the present Fe-based amorphous ribbons possess good soft magnetic properties. In particular, Hc and μe are superior to those In addition to Fe–(Al,Ga)–(P,B,Si) systems, Fe–(Al,Ga)–(P,C,B) shows both Is and Hc as a function of Si content for the four alloy series Fe72−xAl5Ga2P11C6B4Six, Fe72Al5Ga2P11−xC6B4Six, Fe72Al5−xGa2P11C6B4Six and Fe72Al5Ga2P11C6−xB4Six. One can see distinct Si dependence of Is and Hc values. The Is values increase from 1.04 to 1.13 T with increasing Si content for the latter three alloys where the metalloids are replaced by Si, whereas the replacement of Fe by Si decreases linearly Is values from 1.04 to 0.665 T. On the other hand, Hc values tend to increase in the range 1.93–18.2 A/m with increasing Si content for the four alloy series. From the compositional dependence of Tx, Tc, Is and Hc, it is concluded that the Fe72Al5Ga2P10C6B4Si1 and Fe72Al5Ga2P11C5B4Si1 amorphous alloys have a good combination of higher thermal stability of supercooled liquid and better soft magnetic properties with higher Tc.The Is values of melt-spun Fe63−xCoxNi7Zr10B20 (x=3, 7, 14 and 17 at.%) amorphous alloys We measured hysteresis I–H curves of Fe56Co7Ni7Zr10−xNbxB20 amorphous alloys annealed for 300 s at 800 K just below Tgshows the changes in μe at 1 kHz and λs as a function of Nb or Ta content for the Fe56Co7Ni7Zr10−xMxB20 (M=Nb or Ta) amorphous alloys in as-spun and annealed (300 s, 800 K) states shows Is, Hc and λs values as a function of Fe content for the Co70−xFexZr10B20 and Co72−xFexZr8B20 (x=0–21 at.%) amorphous alloys shows μe as a function of frequency for the Co69Fe3Zr8B20 and Co56Fe16Zr8B20 amorphous alloys, together with the data The Fe-based amorphous alloys with the large supercooled liquid region of over 60 K before crystallization are expected to have a high GFA which enables us to produce bulk amorphous alloys with diameters above 1 mm by the copper-mold casting process. The cast Fe72Al5Ga2P11C6B4The hysteresis I–H curve of the cast Fe72Al5Ga2P11C6B4 amorphous cylinder with a diameter of 1 mm was examined in the as-cast and annealed (723 K, 600 s) states shows the hysteresis I–H curves of the cast Fe72Al5Ga2P10C6B4Si1 amorphous cylinder with a diameter of 2 mm, together with the data of the cast Fe73Al5Ga2P11C5B4 amorphous cylinder with a diameter of 1 mm. The Is, Hc and Ir/Is values for the Si-containing cylinder are 1.14 T, 0.5 A/m and 0.38, respectively. In comparison with those shows the outer morphology of the bulk Fe61Co7Zr10Mo5W2B15 cylinders with diameters of 3 and 5 mm ) of around 29.6 nm−1 and no crystalline peak is observed even for the 5 mm sample. Besides, the optical micrographs of the cross-section of the two samples also revealed a featureless contrast in an etched state using hydrofluoric acid. These results indicate that the bulk cylinders are composed of an amorphous phase in the diameter range up to 5 mm. Considering that the bulk cylinder of 7 mm in diameter consists of an amorphous phase in an outer surface region with a thickness of about 2 mm and of amorphous and crystalline phases in the inner region, the tmax for the Fe61Co7Zr10Mo5W2B15 alloy is determined to be about 6 mm. It is noticed that the tmax is three times larger than the largest value (2 mm for Fe72Al5Ga2P10C6B4Si1) shows the DSC curves of the bulk amorphous Fe61Co7Ni7Zr8Nb2B15, Fe56Co7Ni7Zr8Ta2B20, Fe60Co8Zr8Nb2Mo5W2B15 and Fe60Co10Zr8Mo5W2B15 cylinders with diameters of 1–3 mm Tg/Tm was also evaluated. The Tm value was measured at 1420 K for Fe56Co7Ni7Zr10B20 and 1416 K for Fe61Co7Zr10Mo5W2B15 and Tg/Tm was evaluated to be 0.60 for the former alloy and 0.63 for the latter alloy. Considering that Tg/Tm is 0.54 for Fe80P12B4Si4summarizes tmax, Tg, ΔTx, Tg/Tm, Hv compressive fracture strength (σc,f), Is, Hc, μe at 1 kHz and λs for the new amorphous Fe–(Co,Ni)–(Zr,Nb,Ta)–B The single-stage crystallization mode typical for bulk amorphous alloys also implies that the amorphous phase containing homogeneously nanocrystalline particles is not formed. It has previously been pointed out that the mixed structure consisting of nanoscale crystalline particles embedded in an amorphous phase is formed in the satisfaction of the four following criteria existence of homogeneous nucleation sites in an amorphous phase;suppression of growth reaction caused by segregation of a solute element with low atomic diffusivity at the nanocrystal/amorphous interface; andhigh thermal stability of the remaining amorphous phase by the enrichment of solute elements from the primary crystalline phase. |
With the aim of changing the single-stage crystallization mode to the nanocrystallization mode in the maintenance of the supercooled liquid region, we examined the effect of additional M (M=Ag, Ti, Nb, Pd, Au or Pt) elements on the formation of the nanocrystalline structure for the Zr–Al–Ni–Cu amorphous alloys shows the DSC curves of the Zr–Al–Ni–Cu amorphous alloys containing Ag, Ti or Pd, together with the data of the Zr–Al–Ni–Cu alloy. The addition of all the elements causes the change from the single-stage crystallization mode to the two-stage crystallization mode in the maintenance of the supercooled liquid region before crystallization. The structure of the alloys annealed for different periods at the temperatures just below the onset temperature of the first exothermic reaction (Tx1) was examined by TEM. As shown in , the Ag- and Pd-containing alloys consist of nanoscale crystalline precipitates embedded in the amorphous matrix, indicating the effectiveness of the additional elements on the formation of the nanostructured amorphous phase. The precipitates appear to have a spherical morphology and their particle size is measured to be 10–20 nm for the Ag-containing alloy and 5–10 nm for the Pd-containing alloy. Furthermore, the redistribution behavior of alloy components by the precipitation of the primary crystalline phase was examined for the Ag-, Nb- and Pd-containing alloys by the nanobeam EDS technique. The EDS profiles indicate a common feature that the Al element, which plays a dominant role in the high stability of the supercooled liquid, is enriched into the remaining amorphous phase accompanying the significant segregation of Al in the amorphous phase just near the nanocrystal/amorphous interface. This is consistent with the previous interpretation that the redistribution of Al around Zr is essential for the progress of crystallization for the Zr–Al–Ni–Cu amorphous alloys.The precipitation behavior of the nanoscale Zr2Cu phase was also examined by some kinetic analyses shows the change in the first exothermic peak due to the precipitation of the Zr2(Cu,Pd) phase during isothermal annealing at different temperatures for the Zr60Al10Cu20Pd10 amorphous alloy. With increasing annealing temperature, the peak position shifts to a shorter time side and the peak intensity increases monotonously. The transformation ratio (y) corresponding to the crystalline fraction (Vf), from the amorphous to Zr2Cu phase as a function of isothermal annealing time (ta) excluding the incubation time, increases along the sigmoidal curve. The y values between 0.1 and 0.9 were used for the subsequent analysis in the framework of the Johnson–Mehl–Avrami equation It has been reported that the crystallized structure of the Zr65Al7.5Cu27.5 alloy caused by the polymorphic reaction consists of Zr2Cu phase with large grain sizes of 400–500 nm, while the addition of Pd or Au decreases drastically the grain size of the Zr2(Cu,Pd) phase to less than 10 nm . The addition of Pd or Au with much larger negative ΔHmix against Zr causes the change of the crystallization mode from the single stage to the two stages as well as the homogeneous generation of Zr–Pd-rich clusters, which can act as a nucleation site of the Zr2(Cu,Pd) phase. The subsequent growth of the Zr–Pd-rich cluster is also difficult because of the enrichment of Al in the remaining amorphous phase near the interface between amorphous and Zr2(Cu,Pd) phases resulting from the elimination of Al from the Zr2(Cu,Pd) phase. The difficulty of the growth reaction seems to result in the high activation energy for the precipitation of the Zr2(Cu,Pd) phase. Furthermore, the enrichment of Al also induces an increase of the thermal stability of the remaining amorphous phase against crystallization. The increase of the thermal stability also plays an important role in the maintenance of the nanoscale size of the primary crystalline phase.shows the σf, E and Hv as a function of volume fraction (Vf) of the primary crystalline phase for the melt-spun Pd-containing alloy . However, the ductile/brittle transition occurs at small Vf of 30–40% and the linear increase in Vf stops in the vicinity of the transitional Vf value. The maximum increasing amount in Vf is approximately 30% and the highest Vf is 1980 MPa for the Pd-containing alloy. The Vf of the mixed-phase alloy has been interpreted by the simple mixture rule between the amorphous matrix and the nanoscale compound. Tensile fracture occurs along the maximum shear plane, which is declined by about 45° to the direction of tensile load and the fracture surface consists mainly of a vein pattern, as exemplified by the Pd-containing alloy in . There is no distinct difference in the feature of tensile fracture behavior between the bulk nanostructured amorphous alloys and amorphous single-phase alloys. The similar fracture mode indicates that the intergranular amorphous phase keeps good ductile nature and the fracture occurs preferentially along the intergranular amorphous region, as illustrated in . The shear deformation, which is limited to the narrow width of 20–30 nm, can be effectively suppressed by the nanocrystalline (compound) particles with higher Hv and/or higher yield strength, leading to the increase in Vf. In this deformation and fracture mechanism, the remaining amorphous alloy must keep good ductility, which is comparable to the as-cast amorphous phase. The maintenance of the good ductility seems to result from the residual existence of the supercooled liquid region before crystallization for the remaining amorphous phase. The present annealing was made by heating in the supercooled liquid region, followed by water-quenching, which is capable of introducing a large amount of free volume. The decrease in the critical Vf of the ductile/brittle transition for the bulk alloy is presumably due to the difficulty in the re-entrance of free volumes owing to much lower cooling rates for the bulk alloys than for the ribbon alloys.More recently, the good ductility of the bulk nanostructured amorphous alloys has been confirmed by the appearance of slipping-off-type fracture mode for the bulk sample subjected to a compressive test at room temperature shows the compressive stress–elongation curves of the Zr60Al10Cu20Pd10 and Zr55Ni15Cu30Al10 cylindrical samples in as-cast amorphous single-phase and nanostructured amorphous-phase states. Although the amorphous single-phase alloy exhibits a high yield strength of 1770 MPa and a small plastic elongation of about 0.4%, the elongation increases remarkably to about 2.5% for the nanostructured amorphous alloy. The absence of data on the stress–elongation curve in the larger elongation range above 2.5% does not imply fracture of the specimen and results from damage to the strain gauge used in the measurement. shows the outer appearance of the nanostructured amorphous alloy subjected to the plastic elongation of 2.5%. A distinct slip step is seen along the maximum shear plane, which is declined by about 50° to the direction of the uniaxial compressive load. We have also confirmed that the vein pattern in the fracture surface of the nanostructured amorphous alloy is in a more developed and distinct state as compared with the corresponding amorphous single-phase alloy. The more distinct vein pattern is presumably because the shear sliding stress is localized to the intergranular amorphous-phase field and the localization causes a more distinct sliding deformation through the softening phenomenon caused by the increase in temperature.It was shown in the previous section that the most important point in obtaining high Vf for the mixed-phase alloys is attributed to the good ductility of the remaining amorphous phase. The above-described nanostructured amorphous alloys were obtained by annealing-induced partial crystallization in the supercooled liquid region, followed by water-quenching. In addition to the annealing treatment, as another route to producing the similar nanostructured amorphous structure, one can observe a control method of cooling rate from the liquid during casting into a bulk amorphous alloy. The control method was expected to result in the homogeneous dispersion of much smaller precipitates in an amorphous matrix containing a much larger amount of free volume shows small-angle scattering X-ray diffraction patterns of amorphous single phase and mixed phases in the as-cast state for the Zr60Al10Ni10Cu19Nb1 alloy. The mixed structure was produced by controlling the casting temperature. Some broad and low-intensity diffraction peaks are seen at the positions corresponding to Zr2Cu. From the high-resolution TEM image taken from the mixed-structure alloy, the compound phase has been identified to have a grain size of about 4 nm and an interparticle spacing of about 6 nm. The crystalline particles are much smaller than those for the nanostructured amorphous alloys obtained by annealing and hence the structure obtained by the controlling process is named as a clustered amorphous alloy.shows the flexural stress–deflection curves of the clustered amorphous Zr–Cu–Al–Ni–Nb alloy; together with the data of the corresponding amorphous single and nanostructured alloys. It is noticed that the bending flexural strength is 4300hsp sp=0.25>MPa for the clustered amorphous alloy, being much higher as compared with 2000 MPa for the amorphous single-phase alloy and 3300 MPa for the nanostructured amorphous alloy. The Vf of the precipitates in the clustered amorphous alloy is estimated to be 30%, which is considerably smaller than that for the nanostructured alloy, though the particle size is much smaller. The remarkable increase in the flexural strength cannot be explained by the mixture rule. The reason for the increase in flexural strength is presumably due to the generation of residual compressive stress in the amorphous region just near the clustered particles resulting from the high cooling rate from the melt owing to the difference in the coefficients of thermal expansion between the two phases summarizes fields of application in which the bulk amorphous alloys have expected uses. As particularly important application fields, one can list machinery/structural materials, magnetic materials, acoustic materials, somatologic materials, optical machinery materials, sporting goods materials and electrode materials. Finally, it is pleasing to introduce a successful example of a real application of bulk amorphous alloys as sporting goods materials. As exemplified in , the Zr–Al–Ni–Cu and Zr–Ti–Al–Ni–Cu bulk amorphous alloys have already been used as face materials in golf clubs A thermodynamically consistent, nonlinear viscoelastic approach for modeling glassy polymersA thermodynamically consistent nonlinear viscoelastic constitutive theory is derived to capture the wide range of behavior observed in glassy polymers, including such phenomena as yield, stress/volume/enthalpy relaxation, nonlinear stress–strain behavior in complex loading histories, and physical aging. The Helmholtz free energy for an isotropic, thermorheologically simple, viscoelastic material is constructed, and quantities such as the stress and entropy are determined from the Helmholtz potential using Rational Mechanics. The constitutive theory employs a generalized strain measure and a material clock, where the rate of relaxation is controlled by the internal energy that is likewise determined consistently from the viscoelastic Helmholtz potential. This is perhaps the simplest model consistent with the basic requirements of continuum physics, where the rate of relaxation depends upon the thermodynamic state of the polymer. The predictions of the model are compared with extensive experimental data in the following companion paper.Polymers in the glassy state exhibit complex, nonlinear, time-dependent relaxation of volume, enthalpy, and stress. A constitutive equation should be able to describe all of this relaxation behavior for arbitrary temperature and deformation histories, yet existing theories (see Refs. Two frameworks have been proposed for describing glassy polymers, plasticity and nonlinear viscoelasticity. While these two approaches may appear similar under certain situations, they have significant differences. Amorphous polymers are linear viscoelastic for infinitesimal strains, and no distinct change in the linear viscoelastic response is observed as rubbers are gradually cooled into the glass except that the relaxation times grow longer A number of nonlinear viscoelastic constitutive equations that have been developed for glassy polymers use the concept of a ‘material clock’ Existing clock models are based on two key assumptions. First, the instantaneous rate of relaxation is controlled by the current state of the material, and second, the material is thermorheologically simple (i.e., the shape of the relaxation spectrum does not change with temperature, specific volume or whatever feature of the current state controls the rate of relaxation). Thermorheological simplicity seems to be a reasonable assumption, although there are data indicating that polymers in the glass transition region may exhibit slight deviations Let us first examine several significant assumptions imbedded in the constitutive model of Lustig et al. In this paper, we identify and develop three critical advances that are necessary to obtain a theory capable of quantitative predictions: (i) the use of the Hencky strain measure to ensure reasonable volume changes during deformation, (ii) a new potential energy clock based on the ideas of Adam and Gibbs A brief review of the Rational Mechanics framework is presented in this section. A more complete discussion as applied to nonlinear viscoelasticity is available elsewhere Rational Mechanics starts from the basic conservation laws of continuum physics: the conservation of mass, linear momentum, angular momentum, and energy. The conservation of linear and angular momentum requires a constitutive equation for the stress, and the conservation of energy requires constitutive equations for the heat flux and the internal energy. Once these material-dependent constitutive relations are specified (e.g., a Newtonian fluid for the stress, Fourier's law for the heat flux and an equation-of-state for the internal energy) along with sufficient initial and boundary conditions, one can solve for the unknown temperature and velocity fields. This is the approach used in traditional fluid mechanics and heat transfer. However, there is one additional relationship, the second law of thermodynamics, which is an inequality specifying that the entropy can never decrease. The conservation laws, constitutive relationships, and appropriate initial/boundary conditions provide a balanced set of equations and unknowns making a solution possible; thus, the entropy inequality over-specifies the system. Rational Mechanics resolves this over-specification by restricting the constitutive equations to ensure the conservation laws and the entropy inequality are satisfied for any and all possible temperature and deformation histories. This is consistent with the requirement that a constitutive description be applicable for all situations that a material may encounter. Rational Mechanics does not provide the exact form of the constitutive equations, but only specifies relationships between the constitutive equations.The next step in Rational Mechanics is the choice of the independent variables. For an elastic material, we could assume that the Cauchy stress and the specific internal energy U (i.e., the constitutive variables) depend upon the deformation gradient the temperature T, and the temperature gradient ∇T. The temperature gradient is needed for the heat flux so that Fourier's law can be recovered. For this assumed set of independent variables, it can be mathematically proven (see is the second Piola–Kirchhoff stress, ρref is density in the reference state, η is the specific total entropy, Ψ=U−Tη is the specific Helmholtz free energy, and The Green–Lagrange strain measure is just one of the innumerable finite strain tensors that reduces to the well-known infinitesimal strain tensor in the limit of small deformations. The Cauchy stress is defined in terms of the material's current configuration whereas the second Piola–Kirchhoff stress is based on the original undeformed state. Two main results are derived from Rational Mechanics: (i) the stress and entropy are fully specified by appropriate derivatives of the free energy, and (ii) the free energy is a function of the strain and temperature but is independent of the temperature gradient. Thus, the stress and entropy cannot be independently specified, but are determined from the free energy. Eqs. are the restrictions that Rational Mechanics places on the constitutive equations to deal with the over-specification of the system discussed in the previous paragraph. When the deformation is isotropic (i.e., the material volume changes without distortion) Eqs. reduce to the more familiar results of thermostatics; specifically,where the isotropic component of the strain (stress) is the volume v (pressure p) and A is the well-known thermostatic Helmholtz free energy.For a viscoelastic material with fading memory, both the temperature and deformation histories are now important. Consequently, the stress, entropy, and heat flux do not depend solely on the current material state; instead, they are assumed to be functionals (i.e., a function whose argument itself is a function) of the temperature history T(s) and the deformation history and a function of the instantaneous value of the temperature gradient ∇T(t), where t indicates the current time and s≤t indicates times prior to t. The independent variable set define a rheologically simple material (note, this is not the same as a thermorheologically simple material). This implies that the stress, entropy, and heat flux only depend upon the deformation history at that point not upon a spatial gradient of the deformation history, and the deformation history of neighboring points is unimportant. Coleman Just like the elastic case, the free energy functional is independent of the temperature gradient, and the stress and entropy constitutive relationships are given by expressions analogous to the elastic descriptions given in Eqs. Thus, once the free energy functional has been specified, the constitutive equations for the stress and the entropy are defined by differentiation. Moreover, since all thermodynamic quantities can be determined from the stress, entropy, and Helmholtz free energy, we are now assured that the thermodynamic quantities are self-consistent, because they all arise from the same free energy functional. Eq. is the starting point for a thermodynamically consistent, nonlinear viscoelastic constitutive development.At this point, no restrictions have been made on the type of material that is being considered. For an isotropic material, an alternate, but equivalent, form of the free energy functional is given by is the left Cauchy–Green tensor at the current time and is a deformation measure at the current time. is a relative deformation measure that tracks the deformation backward from the current time using the current state of the material as the reference configuration. Through Eq. the free energy functional is expressed as the instantaneous (i.e., elastic) contribution associated with in conjunction with a time dependent (i.e., viscoelastic) contribution relative to this elastic response. Eq. is appropriate for both isotropic solids and isotropic fluids; however, fluids have an additional restriction that does not allow the material to support a shear stress when it is completely relaxed. After implementing this restriction Thus, only the density at the current time affects the free energy for a fluid. Finally, the free energy functionals defined in Eqs. also would be admissible as the argument in the functional definition. Thus, the free energies can be written more generally as respectively, and in the limit of small deformations must limit to the infinitesimal deformation measure. are the key results of Rational Mechanics. They state that if one assumes the response of a nonlinear viscoelastic material depends only upon the history of the temperature, the history of the deformation gradient (a very reasonable postulate for single phase polymers), then all constitutive quantities can be determined from a single Helmholtz free energy functional. This is an enormous simplification, since it says that we do not have to determine different response functionals for the various thermodynamic quantities (e.g., stress, enthalpy, or entropy). Moreover, it explicitly defines an intricate relationship among all the viscoelastic relaxation phenomena, which is one of the primary objectives in developing a consistent, fundamental constitutive model for glassy polymers. The problem now resides in specification of an explicit form for the free energy functional.The constitutive model is completely defined by the free energy functional; however, Rational Mechanics does not provide any information about the form of that functional. If one had a complete molecular theory of the glass, it might provide guidance on the functional form, but such a theory is not available. Consequently, all we know is that the functional depends upon the deformation history and thermal history as defined by Eq. . One way to represent the functional is to expand it about the equilibrium state in a Frechet series, which is the functional equivalent of a Taylor series. The first term in the Frechet series is the time-independent equilibrium contribution to Ψ, the second terms will be single integral terms over the temperature and deformation history, the next set of terms will be double integral terms over all products of the temperature and deformation histories, and so on. Each of these integral contributions in the Frechet series will contain a material function that must be determined from the experimental data. The determination of these material functions from experimental data can be overwhelming Based on the partial success of the clock models for describing nonlinear viscoelastic behavior, an expansion that incorporates ‘material time’ should be more effective than an expansion based on normal laboratory time. The material time where a is the generalization of the well-known shift factor (the ‘WLF’ ) used in time–temperature superposition that now can depend upon all independent variables of the system. If a generalized deformation measure is used, the shift factor can be represented byThe shift function, a(t), acts to compress or expand the apparent timescale of the material as compared to the actual time interval measured in the reference state. For example, if the material is at equilibrium in the undeformed state (i.e., a=aT) at a temperature greater than Tref, then aT is less than one; consequently, is greater than t, and the material time is greater than the laboratory time. Alternatively, if the temperature is less than Tref, then aT is greater than one, and is less than t. Since relaxation occurs via material time, the material will relax more when is greater than t, and relax less when is less than t. Thus, material time incorporates the classic time–temperature superposition analysis of polymers rather than C(t−s) and T(t−s). As discussed in , there are a host of deformation measures that could be used instead of thus, we now postulate a more general form of the free energy functional is the history in material time of the generalized deformation measure. Similar extensions to material time for the other free energy functionals given in Eq. are also possible. The Frechet expansion (through double integral terms) of the free energy functional given by Eq. where the pair of vertical double dots indicate a double dot product that contracts two second order tensors to form a scalar. The single integral terms are not present, since viscoelasticity always causes an increase in the free energy; that is, the system always decays to a minimum energy state. This is analogous to the requirement in thermostatics that the free energy is stable with respect to temperature and specific volume perturbations about any equilibrium state. ΨA is a fourth order tensor function, ΨB is a second order tensor function, and ΨC is a scalar function. Not all the components of ΨA, ΨB, and ΨC are independent. Specifically, ΨA must be insensitive to an exchange between and, since the free energy is a scalar, it must be independent of any coordinate rotation. Using these restrictions, Lustig et al. Expanding Ψ(t) through double integral contributions and neglecting higher order terms in we can write the free energy expansion asΨ∞ is the equilibrium contribution to the free energy and only depends upon the current value of the three invariants of and the current value of the temperature. The relaxation functions depend upon two backward looking material times and on the current value of the three invariants of In order to simplify determination of the material functions we will assume that the relaxation functions are separable; specifically,where similar expressions are used for the three other relaxation functions. The normalized relaxation behavior is contained in the fI(r,z) functions that decay from one at r=z=0 to zero when either argument goes to infinity. The ΨI prefactors depend only upon the current value of the invariants of the deformation tensor and the current temperature. Using the assumed separability between the time history and current state, the truncated expansion of the free energy is given byThis equation is an approximate representation of the free energy function for a solid expressed in terms of the generalized strain measure, As discussed earlier, a free energy expansion in terms of is equivalent. Thus, an alternative to Eq. We will show later that the two views expressed in Eqs. can appear quite similar. If we consider a fluid, the free energy functional can only depend on the first invariant of the relative deformation tensor. Since one of the relative invariants at t=0 will be related to the density, the free energy expansion for the fluid isIn contrast to the truncation of higher-order strain terms that was required to obtain the simplified forms of Eqs. for the representation of the free energy functional for a solid, only four terms are present for a fluid is the full expression for the Helmholtz free energy of a viscoelastic fluid through double integral terms, assuming that the relaxation functions are separable.To assign physical meaning to the prefactors and relaxation terms in the free energy approximation of Eq. , we first must define the equations for the entropy and stress using the relationships found in Eqs. . Examining the small strain limit behavior and comparing these results on a term-by-term basis to the known linear viscoelastic equivalents gives physical meaning to the prefactors. This allows us to rewrite the free energy in the following waywhere Λ and μ are the first and second Lame' coefficients, K=Λ+(2/3)μ is the bulk modulus, Cv is the constant volume heat capacity, and α is the volumetric coefficient of thermal expansion. The second Lame' coefficient is the shear modulus, G. The product Kα is the thermal stress, which is the pressure that is generated when a material held at constant volume is heated. The notation ΔZ represents the difference between the glassy Zg and rubbery Zr values of a particular quantity, Z, at the reference state. The analysis that led to this simple physical interpretation of the prefactors assumed that the prefactors were constant. If the prefactors are allowed to be functions of temperature or the invariants of more complex expressions for the stress, entropy and enthalpy result as will be shown later. Nevertheless, these prefactors are still physically connected to the linear properties of the material.If the prefactors remain constant, then the equations for the second Piola–Kirchhoff stress, and the specific entropy, η, through single integral terms are determined from Eqs. and η∞ are the stress and entropy that would occur if the system were equilibrated at the current temperature and deformation. Note that the price paid for expanding the free energy functional in terms of the generalized strain directly is the need to evaluate the fourth order tensor, for calculation of experimental forces. For infinitesimal deformations, is a fourth order identity tensor, and is just the second Piola–Kirchhoff stress, which for infinitesimal deformations is also equivalent to the more familiar Cauchy stress.The relaxation spectra can be identified by examining Eq. in the infinitesimal deformation limit. Since fi(t,0)=fi(0,t)=fi(t) for all spectra (i=1,2,3,4), f2 is just the normalized linear viscoelastic shear relaxation modulus, and f1 is a combination of the normalized linear viscoelastic shear and bulk relaxation moduli. The normalized thermal stress relaxation function, f3, and constant volume heat capacity relaxation function, f4, are less familiar but nevertheless represent well-defined linear viscoelastic functions. Although the double integrals appear menacing, we show in that these can be represented as the product of two single integrals whose coefficients are defined directly from the underlying linear viscoelastic response as well. In summary, all the prefactors and relaxation functions that are in the Helmholtz free energy as well as all constitutive relationships that are derived from the free energy can in principle be determined from linear viscoelastic data. Thus, the proposed theory is truly predictive for the nonlinear viscoelastic behavior with no nonlinear fitting parameters. The fact that the proposed constitutive equations do not employ nonlinear fitting parameters distinguishes this work from other constitutive models for polymer glasses.In an analogous fashion, the underlying equilibrium stress and entropy ( and η∞) are obtained from a Taylor series expansion of the equilibrium free energy using Eqs. . For a solid, the equilibrium free energy depends on the current strain and temperature taking the form:where the subscripts on Ψ indicate derivatives with respect to temperature and/or the first invariant of the stain. This expansion is complete through second order terms in temperature differences and strain (note that the second invariant IIx is quadratic in strain). However, in Eq. we have only explicitly shown those higher order terms that we found to be important for the materials that we investigated in the companion paper The equilibrated free energy of the liquid can similarly be expanded by a Taylor series in temperature and density only. It may seem inconsistent that the equilibrium free energy is expanded to fourth order terms whereas the Frechet expansion was extended only to second order. However, there is a fundamental difference between these two expansions. The equilibrium expansion approximates the equilibrated free energy surface in temperature and strain space about an arbitrary reference state. The actual temperature and strain differences from the reference state can, in fact, be quite large, and the expansion must capture this. On the other hand, the decaying expansion approximates the change in free energy away from the equilibrated state that would exist at the current temperature and strain. We have just assumed that the instantaneous free energy does not deviate significantly from its equilibrated value, which is the assumption implicit in including only the first nonequilibrium terms in the Frechet expansion of the free energy.In summary, in this section we have developed a particular expression for the Helmholtz free energy. Specifically, the material was assumed to relax on a material timescale governed by the shift functional, a(t), and the deformation was described by a generalized deformation tensor. A free energy functional that incorporates these two ideas was developed. The free energy functional was expanded in terms of a Frechet series that is guaranteed to converge if a sufficient number of terms are included in the expansion; however, evaluation of the higher order terms is prohibitive. The key idea of the constitutive development reported in this communication is that with an enlightened choice of both the generalized deformation measure and the shift functional, the leading nonequilibrium term in the Frechet expansion of the free energy can adequately describe the full range of relaxation behavior of glassy polymers.The constitutive development described in In this section, we will show the need for a generalized strain measure and identify the Hencky strain as the appropriate choice. First, one cannot use the infinitesimal strain measure (whose 1,1-component, ε11, would be defined as ΔL1/L1o=λ1−1, where λ1 is the extension ratio along the 1-axis) at large strains, because it incorrectly produces stresses for rigid body rotations. However, this deficiency is easily corrected. For example, Seth where n is an arbitrary exponent (not necessarily an integer). For n={2, 1, 0, −1, −2} one recovers the Green, Cauchy, Hencky, Swainger, and Almansi strain measures, respectively, One rationale for determining the strain measure is to select the n that provides the best fit of the experimental data ) is an extremely strong function of the specific volume. For the truncated free energy expansions discussed in , isochoric behavior is not observed in materials well above Tg, when the ratio of the shear modulus to the bulk modulus tends to zero, except when a tensorial form of the Hencky strain measure is employed for the generalized strain measure. Thus, in the proposed constitutive equation for glassy polymers we must employ the Hencky strain measure. will provide a complete discussion developing the need for the Hencky strain measure, and will discuss an approximation to the Hencky measure that may be useful for uncross-linked polymer glasses that are formally fluids. If the reader is not interested in the more detailed development of these two ideas, one can move directly to Motivation for the first necessary improvement to the Rational Mechanics framework, the need for the Hencky strain measure, is now presented. Note that in Eq. , the glassy first Lame' coefficient, Λ, is the prefactor of the rate of change in the first strain invariant, dIX/dt. The first Lame' coefficient is dominated by the glassy bulk modulus which typically multiplies the volumetric strain rate, dΔv/dt, in linear elastic and viscoelastic constitutive equations To see this clearly, examine the simplest constitutive equation for an elastic solid, which is a special isothermal case of Eq. For a uniaxial deformation in the ‘1’ direction, S2=S3=0 and the lateral stretches, λ2 and λ3, are equal sowhere ν=Λ/2(Λ+μ)=(3K−2G)/(6K+4G) is Poisson's ratio, G is the shear modulus and K is the bulk modulus. The volumetric strain is found to beFor infinitesimal strain ε12→0, and the limiting behavior is isochoric for incompressible materials (i.e., when K⪢G or equivalently when ν=1/2). However, for very large strains, dilatation is predicted even when ν=1/2. This is unphysical. As an example, the volumetric strain for a material with a very large first Lame' coefficient, Λ, (approximately the bulk modulus) is predicted to decrease as the tensile extension increases. For a slightly compressible material such as a glassy epoxy (i.e., ν≈0.43), the volume increases for tensile strains less than roughly 10%, but then decreases as the extension increases further. It is important to understand that this behavior comes from a second-order truncation of the free energy, where the first invariant serves as an approximation to the volumetric strain. Such volumetric response is not only inconsistent with experiment, but would wreak havoc in a free volume formulation of a viscoelastic material clock. The source of the problem is that the invariant IE is not the volumetric strain; specifically, when Λ tends to infinity IE is forced to zero, since the stresses are finite. However, when IE is zero, then Δv as defined in Eq. cannot be zero; thus, it is impossible to obtain isochoric behavior for large deformations, using the finite elastic constitutive equation given by Eq. To remedy this deficiency higher order terms could be added to Eq. . In general, the higher order expansion is given bywhere the higher order terms collectively conspire to yield ΛΔv. No difficulties of the type associated with Eq. now arise. However, the severe difficulty with this approach lies in defining the prefactors required to parameterize the stress, entropy, and energy consistently. That is, while it is trivial to write Eq. where the stress is defined in terms of strain-dependent Lame' coefficients, determining the free energy that yields this constitutive equation is more complicated. And without the free energy, the consistent thermodynamic formulation is lost.An alternative solution for elastic systems is available by pursuing a hyperelastic approach rather than the Taylor series approach of Eqs. . The constitutive equation for a hyperelastic material isIf the isothermal free energy (i.e., the work function) is assumed to be of the form is well behaved in volume as strain increases, since the first Lame' coefficient, Λ, now multiplies the true volumetric strain.The difficulties in the volumetric response for elastic analysis described above also will occur for viscoelastic solids. If the free energy is expanded in a functional Taylor series (i.e., a Frechet expansion) to second order in the isothermal viscoelastic analogue of Eq. The volumetric inconsistencies inherent in Eq. unfortunately parallel the elastic development, since Λ still multiples the time integral of IE rather than Δv. Expanding to higher order Frechet terms in the stress just like the elastic example of Eq. , again does not clearly indicate the associated free energy needed for thermodynamic consistency and adds multiple integrals that are difficult to evaluate.The viscoelastic extension of hyperelasticity (i.e., Eq. where the isothermal free energy is given byThe deformation history is defined in terms of the relative left Cauchy–Green tensor If a viscoelastic free energy is assumed to have a form that is equivalent to the elastic form of Eq. avoids problems in the volumetric strain when Λ is much greater than μ, a different problem arises. Because of the relative strain tensor, the integrals in Eqs. can no longer be numerically evaluated by a recursive scheme, where the response at the current time is calculated from the previous time step. Instead, the entire integral histories must be recalculated at each time step. This may not be too traumatic for homogeneous deformations given as classroom exercises, but the storage and integration becomes computationally untenable in large finite element simulations. Since our objective is to develop a constitutive equation that can be used in large-scale, three-dimensional, simulations of actual engineering components, this approach using relative deformation tensors is not currently feasible.None of the paths thus far presented offer a solution to the predicted volumetric inconsistencies for elastic or viscoelastic materials under large deformations well above Tg, where the material response should be nearly isochoric. In all of the previous approaches, we employed one of the traditional deformation measures, i.e., As a final alternative, consider the situation when the free energy is expanded to second order in a generalized strain measure such that the stress is given by Eq. . If the first invariant, Ix, of this generalized strain measure, were a function of volume alone, the first Lame' coefficient would now multiply volume strain and no problems concerning the unwanted dilation at finite strains would arise. Examining the class of strain measures given in Eq. , only the logarithmic strain (i.e., n=0) naturally yields Ix=f(Δv). Consequently, we define the Hencky strainas the only acceptable strain measure for use in Eq. to model viscoelastic solids. A difficulty associated with the Hencky strain lies in the complicated procedures required to calculate a logarithmic tensor and the fourth order tensor transforming the conjugate Hencky stress to the second Piola–Kirchhoff stress as given in Eq. . Although these calculations are tedious, they are exact and can be obtained analytically, so we feel this is preferable to a loss of thermodynamic consistency.In summary, we have obtained a thermodynamically consistent formalism by expanding the decaying and equilibrium terms of the free energy in Frechet and Taylor series, respectively. The truncated expansions require us to employ the Hencky strain measure whose first invariant is a function of volume only, which requires calculation of the fourth order tensor, to transform the corresponding conjugate stress, to the second Piola–Kirchhoff stress and eventually to the Cauchy stress for use in the momentum balance. The free energy, stress, and entropy from this approach are given by Eq. If a generalized strain measure were chosen where the first invariant was not a function of only volume, the resulting constitutive equation would produce unphysical volumetric changes for even modest deformations.For viscoelastic liquids, relative strain measures [see Eq. ] are required; however, these relative strain measures must also have a first invariant that is a function of volume only. The relative Hencky strain measureis again the only natural choice. However, this relative strain measure yields integral equations that cannot be solved recursively, requiring recalculation of the entire integral histories at each time step and limiting the utility of this approach in large-scale computations for viscoelastic liquids.Interestingly, the Hencky and relatively Hencky strain rates are identical for irrotational stretches of λk along the Cartesian axes, i.e.,Therefore, the viscoelastic solid and liquid formulations are identical for these types of deformations. Note that these types of deformations include many common characterization tests such as thermal expansion, heat capacity, and uniaxial tension and compression. Also, the viscoelastic volumetric responses (i.e., that due to the first invariant) are identical for any deformation.Finally, the Hencky and relative Hencky rates are identical for small strains, so the linear viscoelastic responses of the solid and liquid formulations are identical. It is in deformations that include a shear contribution that the two formulations differ. In simple shear shows, however, that the Hencky and relative Hencky shear rates differ by less than 3% for engineering strains up to 25%. Thus, even in shear the two formulations are quite similar for deformations typically seen by glassy polymers. Therefore, we will approximate the response of uncross-linked glassy polymers (i.e., liquids) like polystyrene by the solid formulation of Eq. For clarity in the initial presentation, the prefactors, Ψi, in Eqs. initially were assumed to be constants independent of the current temperature and volume. Let us assume, for example, that Ψ4, which is related to the heat capacity, is now not a constant but depends upon the current temperature. The specific entropy is evaluated via the temperature derivative of the free energy as required by the defining equations of Rational Mechanics (i.e., Eqs. ). The entropy now will include both the single convolution integral with respect to temperature that has already been shown in Eq. and a new double integral term, where the coefficient of the double integral term is the temperature derivative of Ψ4. The specific entropy now becomeswhere the last term is new. There are no new integral terms in the stress, associated with the Ψ4 temperature dependence. However, if Ψ4 depended upon the specific volume as well as temperature, new double integral terms would then appear in the stress constitutive equation.To clarify the relationship between the temperature dependent prefactor Ψ4 and measurable physical quantities, examine an instantaneous thermal quench at constant volume from the reference temperature to some temperature T well below Tref. For such a thermal profile, no relaxations occur and the relaxation functions, fi, equal unity. If the temperature dependence in Ψ4 is expanded through quadratic terms about the reference temperature, the change in entropy is given bywhere ΔCv=Cvg−Cv∞ is the difference between the glassy and equilibrium constant volume heat capacities. Consequently,, Ψ4(Tref)=−ΔCvref/Tref; hence, the temperature derivatives of Ψ4 are chosen so that the experimentally measured temperature dependence of the constant volume heat capacity is reproduced over the temperature range of interest.The prefactors of the four double integral terms in the Frechet expansion of the free energy (i.e., Eq. ) can, in general, depend upon temperature and the three invariants of the strain tensor. The full constitutive equations for stress and entropy will therefore contain a number of additional terms that appear in a similar manner to the extra double integral term that appeared in the entropy expression given in Eq. , when Ψ4 is a function of temperature. We will show in the following paper from which the Cauchy stress can be determined via Eq. and the relationship between the Cauchy and Piola–Kirchhoff stress; specifically,The entropy for this specific assumed form of the prefactors is given byand ΔT=T−Tref. The key result of this section is that when nonconstant prefactors are employed, such as temperature dependence of the heat capacity, new relaxation terms appear. These terms are essential in order to maintain thermodynamic consistency.This completes the discussion of the Rational Mechanics of viscoelastic materials. The framework presented so far ensures that the equations are thermodynamically consistent and (via the use of the Hencky strain measure) that isochoric behavior is preserved for materials well above Tg during arbitrarily large deformations. Unfortunately, Rational Mechanics provides no insight into the definition of the viscoelastic shift factor, except that the shift functional is nonnegative. In order to predict the wide range of relaxation data that is observed for polymers in the glass transition region, an appropriate material clock must be developed. This will be the focus of Motivation for the final improvement to the Rational Mechanics framework, the need for a new material clock, will now be presented. What are the requirements of the material clock? First, it must reduce to a form similar to the WLF4 equation for free expansion above the glass transition; specifically,where C1 and C2 are the WLF constants and ΔT=T−Tref. This convenient fitting form is firmly established by a wealth of data for many polymers. For an equilibrium material, the shift factor diverges when ΔT=−C2, which is typically 50 °C below Tg. However, unlike the rubbery WLF prediction, the viscoelastic material clock cannot lead to a divergent shift factor as the material is cooled below the glass transition to arbitrarily low temperatures in free contraction; instead, must ‘level off’ as the temperature decreases far below the glass transition, when the polymer is no longer equilibrated. The clock must depend not only on the current temperature and volume, but also on their respective histories as well to capture important aging effects. Finally, the clock also must be driven by something else that is capable of accelerating relaxations during glassy deformation to produce ‘yield’. Ideally, we would prefer that it be constructed from thermodynamic quantities.Historically, several functional forms for the shift factor have been applied to glassy polymers under deformation, as has already been thoroughly and critically reviewed in Ref. ‘Strain clocks’ have been proposed, where, for example, a function of total strain is added to the free volume clock to accelerate glassy relaxations under deformation , the engineering stress from the first ramp (i.e., 20% engineering strain) is compared to both the second ramp (i.e., an additional 80% strain) and to a separate ramp on a virgin sample to the same 80%. As clearly seen, the strain states are vastly different and yet the three curves are quite similar, emphatically demonstrating that the total strain does not correlate with the polymeric relaxation rates. This conclusion is independent of the exact form of the strain clock and does not require a detailed calculation for any particular model. Any material clock that asserts relaxation rates vary with the total strain is clearly inconsistent with this experimental result.‘Stress clocks’, where a function of stress is added to a free volume-like clock, can also qualitatively predict yield in tension and compression To help justify a new clock, we will first revisit the Adam–Gibbs where N is the number of monomers in the smallest characteristic cooperative region at that temperature, Δμ is the monomeric barrier to transition (which could be a weak function of temperature and density), k is Boltzmann's constant, and R is the relaxation rate. In this physical picture, relaxation rates slow not due to the emergence of deep monomeric energy wells, but to an increase in the number of monomers required to produce a ‘transition’.The approach now rests on defining how N changes with temperature. Adam and Gibbs believed that the potential energy landscape should define the number of cooperative units. They assumed that the ‘configurational’ entropy (arising from consideration of the potential energy only) of the N units in this smallest cooperative region, always equals the minimum entropy required for a transition, For example, at least two states must exist for a transition to occur so Finally, Adam and Gibbs assumed that the local entropy density in the cooperative region was equal to the global entropy densitywhere ηc is the continuum entropy density arising from consideration of only the potential energy. This physical picture of the entropy landscape is consistent with the original view that relaxation rates slow due to an increase in the number of cooperative units rather than an appearance of deep monomeric wells. Note that as the system's configurational entropy increases, the number of repeat units required to allow a transition decreases. At low temperatures where the configurational entropy is small, an extremely large cooperative motion is required. Eqs. where B can be a weak function of temperature and density.The Adam–Gibbs approach allows for more general configurational thermodynamic quantities than configurational entropy. The original development was focused on equilibrated systems that undergo only isotropic free expansion. For systems under more general deformations, we postulate that a different configurational quantity, like the configurational internal energy, may be more appropriate. The Adam–Gibbs key assumption now takes the form of is the configurational energy associated with the N units in the smallest cooperative region, is its lower bound, and Uc is the macroscopic configurational energy density. Hence, N∼1/Uc and fewer repeat units would be required to allow a transition as the configurational energy increases. This leads to the following definition for the shift factor in our material clockwhere B is a material constant and C1 is the WLF coefficient. The similarity between Eq. is clear. B/Ucref=C1 and Ucref is related to the second WLF coefficient C2. Note that B can be a weak function of temperature and density.But what is the exact definition of configurational energy? Adam and Gibbs Using the full specific Helmholtz free energy expression for a viscoelastic solid through double integral terms (i.e., Eq. ) and the entropy for a particular set of temperature and volume dependent prefactors (i.e., Eq. ), the total thermodynamic internal energy, U=Ψ+Tη, is given byAs stated previously, the configurational energy required for the material clock is related to only the potential energy contributions in Eq. . Since changes in the kinetic energy are associated with a change in temperature, the time dependent terms (i.e., U−U∞) must represent potential energy contributions to the total energy, because these terms can change the total internal energy without a concomitant change in temperature. This reasoning is in line with our intuition that the relaxing terms represent changes in the energy due to molecular configurational rearrangements; thus, and includes all the nonequilibrium contributions to the configuration internal energy.Determination of the equilibrium configurational or potential energy is more complicated. Consider first a change in the temperature holding the density constant. In order to determine the potential energy, the contribution of the kinetic energy must be subtracted from the total energy change due to the jump in temperature. Since volume is held constant, no motion of the individual atom would occur in a truly instantaneous quench of ΔT at constant volume, and thus the potential energy would not change. Therefore, the total energy change for this instantaneous quench at constant volume is just the change in kinetic energy. It is not relevant for this theoretical development that this is an impossible experiment to perform in the laboratory. The change in the equilibrium potential energy, therefore, equals the change in total equilibrium energy minus the change in kinetic energy as determined from the total energy due to the instantaneous, isochoric quench. It may seem inconsistent to state both that the relaxing terms in Eq. represent potential energy contributions only and that the energy of the instantaneous quench in temperature represents kinetic energy only. However, the total internal energy given in Eq. also includes the equilibrium contribution, U∞, which contains both a potential and kinetic contribution. These equilibrium terms will exactly cancel the decaying contributions, leaving only the kinetic energy as required.Using ideas outlined above, the total equilibrium energy change in response to this instantaneous, isochoric change in temperature of ΔT is given bywhere ΨT, ΨTT and ΨTTT are various temperature derivatives of the equilibrium free energy response as defined previously in Eq. . The change in total energy due to the instantaneous, isochoric quench (i.e., the kinetic energy contribution) is determined via Eq. , where the relaxation functions are set to unity (i.e., there are no relaxations). Consequently,, the change in equilibrated potential energy is therefore have a similar structure. First, note that only terms containing temperature changes have survived, since we are considering only jumps in temperature at a constant reference density. Second, both equations contain similar Taylor series terms but with differing prefactors. Eq. , which is the total change in equilibrated energy, obviously contains the equilibrated prefactors. In contrast, Eq. , which is the change in equilibrated potential energy, contains prefactors obtained from the relaxing terms associated with the glassy response.Now consider a change only in volume at the reference temperature. Since the temperature is constant, the kinetic energy is also constant, and the total equilibrated energy change is equivalent to the equilibrated potential energy change.where IH=ln(1+Δv) is the first invariant of the Hencky strain tensor and is only a function of the change in volume. Here note that terms in the Taylor series expansion of Ψ∞ as given by Eq. have not survived, since we are considering a change solely in volume. In contrast to the previous behavior of the equilibrated potential energy in Eq. for a change solely in temperature, the equilibrated potential energy for a change solely in volume is given by the fully equilibrated energy.Finally consider a change in both temperature and volume from the reference state. The Taylor series expansion of the equilibrated potential energy will contain terms in temperature change only, volume change only, and cross-terms with both temperature and volume changes. From the previous discussion, it is necessary that the prefactors of terms containing temperature change only be represented by Eq. and that the prefactors of terms containing volume change only be represented by Eq. . It is the cross-terms that are not so clear, but we believe that the two limiting behaviors must be represented by either assigning glassy values to the cross-term prefactors,or alternatively, by assigning the equilibrium values to the cross-term prefactors:To calculate the total configurational energy, the relaxing terms of the total energy in Eq. must be added to the equilibrium terms of either Eq. . The viscoelastic shift factor then can be calculated with Eq. . It will be shown in the following paper that the shift factor computed using the ‘glassy’ limit for the configurational energy as defined in Eq. performs better than that calculated using the ‘equilibrated’ limit defined in Eq. When the configurational energy is evaluated for isobaric, free expansion, the double integral terms associated with the Helmholtz free energy contributions to U=Ψ+Tη are negligible, and the configurational energy very nearly equals the configurational entropy.That is, the postulated configurational energy clock reduces to a configurational entropy clock for free expansion tests if the changes in T are not large; thus, the original ideas of Adams and Gibbs It is worthwhile to examine the various nonequilibrium contributions to the configurational internal energy to see how the viscoelastic character of the shift factor is necessary to obtain quantitative agreement with experimental measurements. There are two single integral terms in Eq. that are associated with viscoelastic entropic contributions to the configurational internal energy. Ψ4 is related to the heat capacity of the glass and f4(t) is the normalized constant volume heat capacity relaxation response to a step change in temperature. Ψ3 is related to the coefficient of thermal expansion of the glass and f3(t) is the normalized thermal expansion relaxation response. During isotropic cooling into the glass at atmospheric pressure, these are the two dominant viscoelastic terms and deserve special attention. Well above Tg they are zero, but as the temperature decreases, the equilibrium contributions to the potential energy cause log(a) to increase such that the rates of relaxation of f3(t) and f4(t) become comparable to the rates of volume and temperature change. When this occurs, the volume and temperature convolution integral terms in the stress constitutive equation (i.e., Eq. ) contribute significantly to the specific volume response of the material. As a result, the slope of the specific volume vs. temperature curve changes, and the glass transition is observed. More directly, the two convolution integral terms in Eq. contribute significantly to the viscoelastic shift function and cause log(a) to be less than its equilibrium value. This gives rise to the characteristic ‘leveling off’ of the shift factor as it enters the glass. If the temperature is held constant in the glassy regime, these terms will continue to relax albeit slowly, thereby causing the nonequilibrium log(a) to relax towards the equilibrium log(a) response; therefore, physical aging is naturally predicted.Next, consider the double integral terms. The double integral terms have only a minimal effect on the stress constitutive equation, and the contribution of most of the double integral terms in the configurational internal energy is also small; however, the second double integral term in Eq. has been found to be critically important for reproducing the observed material response. This term is defined below for each of the three representations defined in Appendix Bwhere the coefficient Ψ2 is related to the glassy shear modulus, and f2(s,0)=f2(0,u) is just the normalized linear shear stress relaxation modulus. The character of the three expressions is fundamentally different. The second expression in Eqs. is essentially the viscoelastic version of the second stress invariant; therefore, this model can be constructed such that it resembles stress clock approaches.By adopting the third expression, we obtain but mixes effective viscoelastic strains with the current strain state.This double integral term is crucial during deformations in the glassy state. Specifically, the relaxation functions are not negligible in the glass; thus, when deformation occurs, the strain rates (e.g., ) in the integral cause the configurational energy to increase, thereby increasing relaxation in the stress constitutive equation. Yield is a consequence of this deformation-induced increase in mobility. Since this term is quadratic in it is positive in tension, compression and shear; thus, yield will occur in all of these three deformation modes as observed experimentally.As we have described, the response during isobaric cooling is governed primarily by the single integral, entropic contributions to the potential energy. Thus, the potential energy clock includes the character of a configurational entropy clock to describe heating/cooling effects with an additional stress accelerator (i.e., the double integral term of Eqs. ) that produces yield. Notice, however, that the stress accelerator of Eqs. needs no adjustable parameter to modify its strength but naturally arises from use of the potential energy. The key features of this material clock are a natural consequence of Rational Mechanics, where thermodynamic consistency demands that if viscoelastic relaxation is included in the stress constitutive equation, it must also be present in all thermodynamic quantities like free energy, entropy, and even potential energy in the material clock.A thermodynamically consistent, nonlinear viscoelastic theory has been developed. Key features of the model include: (i) the use of Rational Mechanics to ensure thermodynamic consistency between all thermodynamic and mechanical variables; (ii) the incorporation of a Hencky strain so that unwanted deformation-induced dilation is not a problem; (iii) the assumption of thermorheological simplicity; and, (iv) the development of a new potential energy clock that naturally includes the effects of temperature, pressure and deformation on the rate of viscoelastic relaxation. In this paper, we have presented a version of the theory that incorporates the features that we have found necessary to produce predictions that match data. A more complete derivation that includes some features that we have not used here (such as higher order terms) will be presented elsewhere In addition, the proposed constitutive model is practical for engineering analyses, having already been implemented in a three-dimensional finite element code at Sandia National Laboratories. It is currently being used for component stress analyses and in programs where detailed material behavior is needed to understand and predict failure mechanisms. Several details of the exact computational implementation of the constitutive model are key to efficient solution. All double integrals are represented by combinations of single integrals as described in . The relaxation functions themselves are expressed as series of exponential relaxation times, where recursive relations are used to determine all integral terms efficiently without explicit integration over the previous history. The fourth order tensor, is computed analytically for the Hencky strain. From the experimental perspective, the material properties needed to populate the constitutive equation for a given material can be readily determined from a small set of common, well-defined experiments as will be discussed in the following paper Finally, although the mathematical formulation may appear complex, the underlying physical assumptions are quite simple: (i) thermodynamics must be satisfied, (ii) the linear viscoelastic constitutive model must be recovered for infinitesimal deformations, (iii) an isochoric (i.e., constant volume) response for large deformations is required in the rubbery state, (iv) the response is thermorheologically simple and (v) the effect of temperature and density on the rate of relaxation is governed by a generalization of the Adam and Gibbs configurational entropy model. All of these requirements are very reasonable if not rigorously required. A mathematically simpler constitutive model will not satisfy one or more of these important features. Thus, we assert that the proposed constitutive model is among the simplest approaches consistent with continuum and polymer mechanics. The fact that the model can describe a complex data set provides a measure of assurance that the modeling approach has captured at least the important features of the underlying physics.We present a brief development of how the Rational Mechanics machinery yields the fundamental constraints on the constitutive equations. Only a nonlinear elastic material will be considered, since the procedure is sufficiently exposed and avoids the additional mathematical complexity needed for the nonlinear viscoelastic case. The starting point of Rational Mechanics is the conservation principles of continuum physics: conservation of mass, momentum, and energy. A common form for the energy balance is given by is the heat flux, and r is the rate of radiative heat generation. The second law of thermodynamics is also required, but it is here that differences of opinion can arise. The Clausius–Duhem form of the second law places the following constraint on the material response: is the entropy flux due to heat transfer and ρr/T is the rate of entropy generation by radiation. Using Eq. to eliminate the radiative heat generation r and with the definition of the Helmholtz free energy, Ψ=U−ηT, the Clausius–Duhem form of the entropy inequality becomesThis is the principle inequality of Rational Mechanics.One now must make an assumption about the independent variables for all constitutive quantities. For the elastic case, we assume that the strain, temperature, and temperature gradient are the relevant variables. The temperature gradient is included as an independent variable so that Fourier's law of heat conduction can be recovered. Thus, the time derivative of the free energy is given byT, and ∇T are all independent variables, the time derivatives of these variables can either be positive or negative with arbitrary magnitude. Consequently, the only way the inequality given by Eq. These are the defining equations of Rational Mechanics for an elastic material as given previously in Eqs. and are an inescapable consequence of the conservation principles of continuum physics and the Clausius–Duhem form of the second law of thermodynamics.We will eventually need to evaluate selected double integral terms; and, more importantly, we will need to determine these relaxation functions from experimental data. We propose a significant simplification for these relaxation functions. The relaxation functions fi(t,s) are symmetric in their arguments {f(t,s)=f(s,t)}, limit to unity in the unrelaxed state, i.e., {f(0,0)=1}, and decay to zero in the fully relaxed state, i.e., {f(0,∞)=f(∞,0)=f(∞,∞)=0}. A general form that satisfies these constraints iswhere the choice of m and Aij determine the strength and shape of the function. We have investigated three specific cases.and then taking the limit based on the magnitude of the ratio (s/t). coefficients in all cases are obtained from experimental data noting that f(t,0)=f(0,s) must reduce to the discrete relaxation function for the integral under consideration.Using these forms of the relaxation function in a typical double integral term (e.g., the second relaxation term in Eq. ), the double integrals simplify to a pair of single integrals; specifically,Our experience based on the material data given in the next paper Thermal degradation of acrylonitrile–butadiene–styrene (ABS) blendsThis work investigates the accelerated thermal degradation of acrylonitrile–butadiene–styrene (ABS) due to aging at elevated temperatures (>80 °C). The impact resistance is shown to decrease dramatically beyond a critical aging time at 120 °C and this reduction strongly depends on surface property modifications during aging. Visual examination of specimen cross-sections after aging, verifies that (dis)colouration is limited to a surface layer, which is characteristic of degradation where oxygen diffusion into the bulk is limited. Degradation is supported by chemiluminescence assessment, which shows a rapid depletion of residual stabiliser within this layer as compared to the bulk polymer. Micro-indentation measurements also indicate that degradation causes an increase in Young's modulus at the specimen surface, which in turn promotes brittle failure. It is proposed that a critical depth of degradation (approximately 0.08 mm) forms on the surface of ABS due to aging. Applied loading initiates microcracks in this degraded layer, which propagate rapidly, causing bulk failure. Absorbance bands from Fourier transform infra-red spectroscopy indicate that surface degradation proceeds by chain scission and cross-linking in the polybutadiene (PB) phase of aged ABS specimens. Cross-linking is also supported by positron annihilation lifetime spectroscopy, which shows a decrease in free volume sites at the surface of aged specimens. Dynamic mechanical thermal analysis also supports the occurrence of cross-linking, as shown by an increase in the glass transition temperature of the PB phase after aging. Although degradation in the styrene–acrylonitrile (SAN) phase is less significant to the reduction in overall mechanical properties of ABS compared to the PB phase, an assessment of SAN copolymer indicates that heat aging decreases impact resistance. The contribution of SAN to the overall mechanical properties of ABS is also reflected by aging ABS specimens at temperatures just below the glass transition of the SAN phase (∼112 °C). The mechanism of thermal degradation is shown to be non-Arrhenius and governed by diffusion-limited oxidation. The long-term impact strength of ABS at ambient temperature is extrapolated from short-term data at elevated temperatures. As temperature and aging time influence degradation, it is proposed that at ambient service temperatures (40 °C), the degradation mechanism differs to that at elevated temperatures, and comprises both surface and bulk polymer degradation effects.The durability of acrylonitrile–butadiene–styrene (ABS) polymers is important in many applications and depends on composition, processing and operating conditions, environmental weathering, heat aging and installation damage. The availability of a durability prediction model for ABS would allow material types to be selected according to their expected environmental and operating conditions, and would significantly reduce the risk of in-service failures.Specific microstructural aspects of a polymer often facilitate thermal oxidation. In the case of ABS, hydrogen abstraction by oxygen is thermodynamically favourable due to the presence of tertiary substituted carbon atoms in the PB phase. The presence of sufficient thermal energy activates hydrogen abstraction to initiate oxidation, and accelerates the overall process of degradation. After periods of exposure to heat and oxygen, the mechanical properties of ABS such as impact strength and elongation to break, deteriorate as a consequence of this polymer degradation, inducing premature failure In this study, thick specimens of stabilised ABS were aged at elevated temperatures in order to study the effects of heat aging on mechanical properties. The study does not consider environmental effects such as the migration and leaching of stabiliser that may occur in typical outdoor infrastructure applications.The ABS polymer morphology under investigation was comprised of a SAN-graft-PB bimodal matrix with dispersed ungrafted PB polymer.Commercial ABS pipe resin and pure SAN copolymer, in the form of impact (conforming to AS 1146.1 Instrumented impact analyses were performed on ABS and SAN specimens using a Radmana impact tester, in accordance with AS 1146.1 To investigate aging effects on polymer microstructure, FTIR spectroscopy was used to identify the characteristic infra-red absorptions of vinyl groups in as-produced ABS and carbonyl and hydroxyl absorptions due to the oxidation species of ABS. Using a 20 μm thick microtomed cross-section of aged ABS (672 hours), local regions (100 × 100 μm) were analysed at the edge of the section (corresponding to the specimen surface) and within the centre of the microtome (corresponding to the bulk polymer). Optical transmission mode was used with microscope stage control and imaging to allow visual identification of the section. Positron annihilation lifetime spectroscopy (PALS) analysis was conducted on ABS aged at 90 and 120 °C, and SAN aged at 90 °C to investigate the effects of heat aging on the molecular level packing features of the polymer (e.g. free volume). Analyses were conducted in the bulk of aged ABS by removing the surface layer (0.7 mm) by mechanical milling prior to assessment. An Atlas CL 400 chemiluminescence instrument was used to investigate the depletion of residual stabiliser in ABS after aging. Both surface and bulk samples (20 μm) from an ABS specimen aged at 120 °C were analysed at an isothermal temperature of 180 °C in oxygen, after a nitrogen pre-phase. The onset of polymer degradation, or the oxidation induction time (OIT), due to stabiliser depletion was recorded for each sample. shows the variation in impact strength with aging time for ABS specimens that were notched after aging at 90 and 120 °C. As shown, impact strength is effectively independent of aging time at 90 °C, with only a slight loss was observed at 120 °C. In contrast, however, the impact strength of unnotched ABS specimens aged at 120 °C decreases rapidly after an aging time of 168 h, as shown in where the initial impact value of 245 kJ/m2 decreases to a plateau of approximately 30 kJ/m2. Since notching after heat aging removes the surface structure of a specimen, the results in indicate that failure under impact is critically dependent on the condition of the surface layer in ABS. The influence of surface degradation on bulk mechanical properties is evident by comparing the tensile properties of ABS specimens with their surfaces intact and removed. The tensile elongation energy to break for aged ABS specimens after the aged surface is removed (0.7 mm) is restored to that of an as-produced specimen., ABS specimens aged at 120 °C develop a brown colouration at the polymer surface. A cross-sectional examination of these specimens reveals that the colouration of the polymer is limited to the surface even after heat aging for several months. This reinforces previous research by Wolkowicz and Gaggar , the transition in impact resistance corresponds to a surface layer depth of approximately 0.08 mm. Further exposure does not reduce impact strength, even though the depth of the coloured layer continues to increase to 0.3 mm after 672 h. This critical layer depth of 0.08 mm is also in agreement with the lower end of previously reported values between 0.07 and 0.2 mm The effects of aging on the surface mechanical properties of ABS were investigated using a microhardness tester. Microhardness assessment measures the force required to impress an indenter to a fixed depth through the specimen surface. The measured force is proportional to the surface elastic modulus of ABS after correcting for local plastic deformation. Whilst shows that the elastic modulus of ABS aged at 90 °C does not differ significantly from that of as-produced ABS (1500 MPa), the modulus of ABS aged at 120 °C increases by 300% to 4500 MPa. The initial decreases in elastic moduli shown in are due to specimen surface imperfections. It is proposed that this increase in local modulus after aging is associated with cross-linking and reductions in free volume in the surface layer (discussed later), and is consistent with the decrease in impact resistance shown in Many researchers attribute the colouration of ABS to thermal aging, during which radical scavengers are thought to couple with peroxy radicals formed during degradation reactions showing chemiluminescence peaks for ABS polymer aged at 120 °C for 168 h. The voltage recorded in chemiluminescence assessment is directly related to the light intensity created by decaying hydroperoxides in degradation reactions shows FTIR spectra from the surface (upper spectrum) and bulk (lower spectrum) of an ABS specimen after aging at 120 °C for 168 h. As shown, the spectrum from the bulk polymer is similar to a trace that is commonly observed for as-produced ABS. There is no observed change in the characteristic infra-red absorptions of ABS and no evidence of chemical degradation products. In contrast, however, spectral changes are clearly observed at the surface of the aged polymer, in particular the carbonyl peak at 1724 cm–1, indicating changes in chemical structure associated with oxidation.ABS consists of a bimodal polymer system in which non-grafted polybutadiene particles are dispersed in a SAN–graft–polybutadiene matrix phase. Whilst the loss of mechanical properties in ABS is often attributed to thermo-oxidative degradation in the PB phase only, thermo-oxidative degradation and physical aging can also occur in the SAN phase Thermal degradation of the PB phase in ABS is initiated by hydrogen abstraction from carbon atoms in an α position to unsaturated bonds. The abstraction generates radicals which, in the presence of oxygen, lead to the formation of carbonyl and hydroxyl products. Following a reaction scheme proposed by Shimada and Kabuki . The absorbance bands at 966.92 and 911.43 cm–1 correspond to the trans C=C unsaturation (vinyl) in polybutadiene, and the 1,2 butadiene terminal vinyl C–H band respectively. The surface spectrum for aged ABS shows a significant decrease in these bands, indicating chemical changes in the PB microstructure, which are probably attributed to chain scission and cross-linking. Furthermore, degradation of the PB phase at the surface forms hydroperoxide species, as indicated by the carbonyl (1724.4 cm–1) and hydroxyl (3473.49 cm–1) absorbances in the spectrum shows PALS results from the surface of an ABS specimen aged at 120 and 90 °C together with SAN copolymer aged at 90 °C. Whilst the relative free volume fraction, indicated by oPs intensity (I3), for ABS and SAN aged at 90 °C is approximately constant, ABS aged at 120 °C shows a significant reduction in I3 with increasing aging time. As shown in , the relative size of free volume sites, represented by oPs lifetime (τ3) also decreases with increasing aging time for ABS specimens aged at 120 °C. In contrast, (τ3) for ABS or SAN aged at 90 °C does not change significantly. The decreasing values of I3 and τ3 are interpreted as a reduction in free volume due to cross-linking reactions in the PB phase. The presence of free radicals or other electron or positron scavengers due to the degradation process can also cause a decrease in the value of I3Molecular chain cross-linking will increase the Tg of the PB phase, which can be measured by dynamic mechanical thermal analysis (DMTA). DMTA monitors the phase lag (expressed by the ‘tan δ’ parameter) between an imposed cyclic stress and a recorded cyclic strain as specimen temperature increases. At a temperature T=Tg, a peak in tan δ is attained which corresponds to a transition in molecular conformation from a glassy (rigid) to a rubbery state. It follows that cross-linking in the PB phase will increase Tg, by increasing the thermal energy required to free polymer molecules from additional constraint. As shown in , whilst Tg remains unchanged after aging at 90 °C, aging at 120 °C produces an increase in Tg with increasing aging time. The phase lag between imposed stress and strain (tan δ) can also be written as the ratio between two moduli E″/E′. E″ is often referred to as the ‘loss modulus’, and E′ as the ‘storage modulus’ since they relate to the amount of energy that is dissipated and stored during each loading cycle. Referring back to , the increase in Tg due to cross-linking can also be interpreted by the temperature at which a peak in E″ is attained. For specimens aged at 120 °C, maximum energy dissipation generally occurs at a temperature that increases with aging time. The influence of cross-linking in the PB phase is also evident in the Young's modulus of aged ABS specimens. As shown in , whilst aging at 120 °C produces an increase of 40% in Young's modulus, aging at 90 °C results in only a slight increase. Similar to the impact tests in , this aging-induced increase in Young's modulus also influences failure mode at relatively low loading rates. shows a typical stress–strain curve for as-produced and aged ABS at 90 and 120 °C, from a tensile test at 10 mm/min. Although the ultimate elongation decreases for ABS aged at 90 °C, post-yield drawing is still observed, as indicated by the decrease in load before failure. In comparison, aging of ABS at 120 °C eliminates this load drop and failure occurs in the elastic regime. Clearly, aging (and subsequent cross-linking in the PB phase) increases Young's modulus and results in a ductile–brittle transition in failure mode. This change in failure mechanism may be associated with a decrease in fracture toughness (KIC) of the polymer with aging, as has been observed with PVC In summary, ABS heat stabiliser is depleted with increasing aging time and aging temperature, and chemical degradation in the PB microstructure precipitates mechanical failure. As shown in , the loss of impact strength of unnotched specimens occurs at a transitional aging time that decreases with increasing aging temperature. Similarly, shows that the tensile energy to break properties of ABS aged between 90 and 120 °C, significantly deteriorate as the temperature and time of aging increases, as do the tensile strain properties. The decrease in tensile properties is similar to that reported by Tavakioli The type and extent of thermal degradation in the SAN phase of ABS depends on the proximity of the aging temperature to the glass transition temperature of SAN (Tg ∼113 °C). Aging at temperatures below Tg causes physical aging , the impact resistance of SAN copolymer decreases after aging, with a reduction from 27 kJ/m2 to 14 kJ/m2 occurring after aging at 105 °C for 672 hours. Although this temperature is below Tg for SAN and physical aging may occur, thermo-oxidative degradation is also evident by specimen discolouration When glassy polymers such as SAN are aged at temperatures below the glass transition, they are said to be annealed or physically aged. Physical aging rearranges molecules from an unordered state, formed initially by quench cooling, to an equilibrated or uniform glassy state illustrates that this reordering reduces free volume similar to cross-linking in the PB phase . As expected, the initial slope of the tan δ curve for aged ABS is lower than that for as-produced specimens in the approach to the glass transition of the SAN phase. compares the E″ and tan δ parameters for the SAN phase in as-produced ABS and ABS aged at 90 °C. As shown, whilst there is only a slight increase in Tg, the temperature at which the maximum loss modulus E″ is attained increases significantly with aging time. Similar to the influence of cross-linking in the PB phase, physically aging SAN creates sufficient molecular order to require an increase in thermal energy to overcome constraint. Referring back to , physical aging of the SAN phase can also account for the slight increase in Young's modulus in ABS specimens aged at 90 °C. Previous research also suggests that the aging-induced changes in ultimate elongation () in ABS can be partly attributed to SAN physical aging Although the degradation of ABS and the loss of mechanical properties is strongly temperature dependent, the kinetics of degradation are unlikely to follow an Arrhenius law over a large temperature range. Whilst the Arrhenius equation is often used to model degradation kinetics, the extent of diffusion-limited oxidation (DLO) may change with time and temperature in relatively thick specimens. Time and temperature of aging influence the rate of formation and thickness of the degraded surface layer, and therefore the extent to which degradation may occur in the bulk polymer. At ambient service temperatures, the mechanism of degradation may differ to that at elevated temperatures, comprising both surface and bulk polymer degradation effects. Whilst Clough et al. shows normalised impact strength [impact energy (I)/initial impact energy (Io)] against aging time for aged ABS specimens. By plotting three levels of deterioration (I/Io=0.25, 0.5, 0.75) against the reciprocal of temperature (), it is apparent that for the initial stages of degradation (I/Io=0.75) and at lower temperatures, the curve is not linear. As shown by the slope of the curve, the degradation process requires a different activation energy at this stage compared to later stages. For the Arrhenius relationship to be applicable, the activation energy for the degradation reaction at different stages or levels of degradation must not change. Tensile energy to break results were also found to deviate from Arrhenius behaviour. It appears that two different degradation profiles may exist for thermo-oxidative degradation of ABS. For temperatures below 90 °C, such as the ambient operating temperature (in this case, assumed to be 40 °C), degradation may occur consistently throughout ABS and follow an Arrhenius description. However, it is proposed that DLO effects control degradation at elevated temperatures, with oxidation occurring at the specimen surface.In the absence of a physical degradation law, a simple extrapolation of short-term impact properties at elevated temperatures allows the prediction of long-term impact strength at ambient conditions (). However, as suggested previously, discrepancies may be expected in practice due to the influence of aging time and temperature over the formation rate of a degraded layer. Future work will focus on modelling the kinetics of degraded layer formation, and the use of a fracture mechanics approach to derive relationships between layer thickness and equivalent surface notch depths.Degradation of the elastomeric PB phase in ABS is initiated by hydrogen abstraction from the carbon α to unsaturated bonds, producing hydroperoxide radicals, leading to carbonyl and hydroxyl products. Cross-linking of polymer chains is facilitated by the free radicals that are produced. Thermo-oxidative degradation in the PB phase at the surface causes an increase in polymer density, stress hardening and an increase in modulus. Therefore the contribution from the dispersed PB phase to the overall toughening mechanism of ABS, by localised shear yielding and crazing in the SAN matrix, is greatly reduced. Thermal degradation of the SAN phase in ABS also occurs by physical aging and thermo-oxidative degradation, but has only a minor contribution to the deterioration of mechanical properties in ABS.Thermo-oxidative degradation of ABS is a temperature-dependent process, and mechanical property deterioration is significantly lower at lower aging temperatures. The kinetics of degradation are not typically Arrhenius due to diffusion-limited oxidation effects, and at ambient temperatures both bulk and surface degradation processes exist. A model which accounts for changes in degradation kinetics with temperature is required to predict the ambient mechanical properties of ABS from properties derived after aging at elevated temperatures. Since the most critical degradation process in ABS is thermo-oxidative degradation of the PB phase, performance is dependent on adequate levels and distributions of stabilisers for specific temperature applications.Abnormal texture development in magnesium alloy Mg–3Al–1Zn during large strain electroplastic rolling: Effect of pulsed electric currentSingle pass large strain electroplastic rolling (LSER) was conducted on AZ31 alloy, using pulsed electric current of different current densities, to improve the rollability of magnesium alloy sheet. It was found that TD (transverse direction)-split texture developed in the alloy once the current density exceeded a critical value (in the present case 90 A/mm2). Microstructure and texture analysis reveal that deformed grains, rather than the dynamically recrystallized grains, are the major contributor to the overall TD-split texture. Intragranular misorientation axis (IGMA) and viscoplastic self-consistent (VPSC) modeling were employed to analyze the influence of the pulsed electric current on the deformation mechanisms. Prismatic <a> type geometrically necessary dislocations (GNDs) were found to be dominant in the deformed grains of the specimens with TD-split texture. VPSC modeling indicates that the TD-split texture is likely due to the enhanced prismatic <a> activity caused by the electric current. The special athermal effect of the pulsed electric current on the deformation mechanisms makes LSER a promising technique for texture modification in magnesium alloys for improved formability.Magnesium (Mg) alloys, owing to their low density and high specific strength, have long been regarded as attractive structural materials for the construction of components in weight-critical applications such as modern automobiles. However, due to their poor room temperature formability, the use of wrought Mg alloys is still very limited (In Mg alloys with a hexagonal close-packed (HCP) crystal structure, although multiple slip/twinning modes (such as basal <a> slip, {101¯2}<101¯1> extension twinning, prismatic <a> slip, pyramidal <c+a> slip, {101¯1}<101¯2> contraction twinning) are available, basal <a> slip and {101¯2}<101¯1> extension twinning dominate during room temperature deformation due to their much lower critical resolved shear stress (CRSS) values (). This characteristic of the alloys contributes to the development of strong basal texture in rolled sheet products (). With most of the grains oriented with their c-axis approximately paralleled to the sheet normal direction (ND), the strains in ND become increasingly difficult to be accommodated. This consequently results in more difficulty in subsequent forming operations (). Due to this reason, wrought Mg alloys are usually processed by either hot deformation or warm/cold deformation with repeated intermediate annealing. This inevitably results in high processing cost and low efficiency. Therefore, how to introduce non-basal texture in wrought Mg alloys becomes a critical issue in the light metals community.Adding rare earth (RE) elements to Mg alloys is one possible means of improving their formability through texture engineering. For instance, Agnew et al. reported that yttrium (Y) produced RD-split texture in rolled Mg sheet (). Mackenzie et al. obtained TD-split texture by adding cerium (Ce) to Mg-1wt.%Zn alloy (). Hadorn et al. found that neodymium (Nd) weakened the texture of hot rolled Mg sheets (). Basu et al. achieved an almost random texture in the binary alloy Mg with 1 wt% gadolinium (Gd) through rolling and subsequent annealing (). Stanford et al. reported the ‘RE texture’ (<112¯1> parallel to the extrusion direction) in Mg–RE (RE = lanthanum (La), gadolinium (Gd)) extrusions (). Owing to these exciting observations, numerous researchers have been studying the texture weakening mechanisms in Mg–RE alloys. Hypotheses that have been proposed include variation in stacking fault energy (SFE) (), particle stimulated nucleation (PSN) (), deformation twinning nucleation (DTN) (). Unfortunately, however, the high cost of the RE largely limits the application of Mg–RE alloys. Therefore, recent research has been focusing on microalloying () and magnifying the RE effect by non-RE elements (In addition to RE-alloying, texture modification in RE-free commercial Mg alloys can be achieved by some intelligent processing techniques (or processing routes), such as single roller drive rolling (SRDR) (), equal channel angular extrusion (ECAE) () and pre-rolling along the transverse direction (). These techniques result in textures deviating from the conventional basal type and thus improve the formability of the materials. However, there are significant challenges in scaling up these techniques due to their equipment/process complexity.Electroplastic effect is a phenomenon that the electric current/field reduces the flow stress and increases the ductility of materials (). The effect was first found by Troitskii et al., in 1963 who discovered that a Zn crystal during uniaxial tension exhibited lower flow stress and higher elongation to failure when it was irradiated by electrons from certain directions (). Later, similar phenomena were observed by other researchers on various metals and alloys such as Cd, Pb, In, W, Al, Cu, Ni, Fe, Nb, Sn, Ti etc. (). Subsequently, the concept of electrically assisted processing (EAP) has been proposed, and regarded as a promising processing technique especially suited for metallic materials (). Besides reduced flow stress and improved ductility at low temperatures (), the main advantages of EAP over conventional processing also include springback reduction/elimination and enhanced time/energy efficiency (). The technology has also been used for microstructure () modification in various materials. Recently, Kuang et al. found that electropulse treatment (EPT) was an effective method to modify the texture in Mg alloys due to the special effect of pulsed electric current, which seemed to promote the nucleation of non-basal oriented grains and alters the relative grain boundary mobility (). However, the processing route examined in that study involved repeated small-reduction cold rolling prior to EPT, and is far from optimized for application to texture modification in an industrial setting.In the present work, single-pass large strain electroplastic rolling (LSER) is investigated, offering a more promising prospect to research/industry owing to a high efficiency in terms of both time and energy. The microstructure and texture of the as-rolled sheet were studied via both experiments and simulation. The research aimed at contributing to a deeper understanding of the electroplastic effect in Mg alloys. The results should be able to serve as guidance for future applications of LSER in Mg sheet for texture modification.Commercial AZ31 sheet alloy (3.1 wt% Al, 0.9 wt% Zn, 0.2 wt% Mn, balance Mg) used in this study was received in hot-rolled and fully annealed state. The as-received sheets, 1.6 mm in thickness, were electroplastically rolled to a reduction of ∼62% via single pass using various pulsed electric current densities. The schematic view of the process can be found in reference (). The distance between the anode and the cathode was 200 mm. The sheet samples traveled from the anode to the cathode at a speed of ∼0.8 m/min. The rolls were heated to 100 °C in order to avoid catastrophic failure in the sample rolled without the assistance of pulsed electric current. The parameters of the pulsed electric current including the frequency, the peak current, and the pulse width were monitored using a Hall-effect sensor along with an oscilloscope. The surface temperature right before the sheets entered the rolling mill was measured using a K-type thermocouple attached to an AZ9881K thermometer. The parameters of the pulsed electric current are listed in The macrotexture was measured in RD-TD plane (RD, the rolling direction, TD, the transverse direction) with incomplete pole figures between α = 5°–75° using Cu Kα radiation in the Schulz back-reflection mode on a Bruker D8 Advance diffractometer. A set of 6 measured pole figures ({0002}, {101¯0}, {112¯0}, {101¯1}, {101¯2} and {101¯3}) were used to calculate the orientation distribution function (ODF) in the MTEX tool box (Microstructure analysis was conducted on RD-ND plane (ND, the sheet normal direction) of the samples. For optical microstructure observations, samples were etched in an acetic-picral solution to reveal the grain boundaries. For EBSD analysis, samples were electropolished in AC2 electrolyte (Struers). EBSD mapping was conducted on Phillips FEG-ESEM XL30 equipped with EDAX/TSL Hikari EBSD detector. The step size was 0.2 μm or 0.25 μm depending on the microstructure. Any measured points with confidence index less than 0.1 were excluded from the analysis. Post-processing of the EBSD data was accomplished mostly with TSL OIM 7 analysis software and the MTEX toolbox (Viscoplastic self-consistent (VPSC) crystal plasticity modeling () was utilized to reveal the different deformation modes operated in the conventional rolling and electroplastic rolling. The details of the model are not repeated here for brevity.The viscoplastic behavior at local level was described by the means of non-linear rate-sensitivity equation:where ε˙ij and σkl are the deviatoric strain rate and stress, respectively. γ˙0 is a normalization factor. mijs=1/2(nisbjs+njsbis) is the Schmid tensor, where ns and bs are the slip/twinning plane normal and the Burgers vector of slip/twinning system s. τ0s is the threshold stress of slip/twinning system s. n is the inverse of the strain rate sensitivity exponent and was chosen to be 20 in the present study.The input texture was obtained by discretizing the texture of the as-received material into 1000 grains. An ‘effective’ homogenization scheme (neff = 10) was used to account for the grain matrix interaction.Reorientation of the grains by twinning was captured by the Predominant Twin Reorientation scheme (). The Voce empirical hardening rule was employed to describe the hardening behavior of each individual deformation mechanism.where Γ is the accumulated shear in the grain, τ0s, θ0s, θ1s and τ0s+τ1s are the initial critical resolved shear stress (CRSS), the initial hardening rate, the asymptotic hardening rate and the back extrapolated stress for deformation mode s respectively. The actual hardening of each mode is calculated through the following equation at the end of each deformation step.where ΔτCs is the incremental increase in CRSS of slip system s, dτCs/dΓ the change rate of the CRSS of slip system s with respect to the accumulated shear in the grain Γ. Δγs′ is the shear activity in the slip system s′. hss′ are the latent hardening coupling coefficients which empirically account for the obstacles on slip system s associated with slip system s′. In this paper all elements are set to 1 for simplicity. Five deformation modes, basal <a>, prismatic <a>, pyramidal <c+a>, {101¯2} extension twins, and {101¯1} contraction twins, were allowed in the simulation.The dynamic recrystallization (DRX) process was considered in the same way as in Ref. (). A brief introduction of the method is as follows. The stored energy of orientation n is calculated bywhere τns and τ0s are the reference stress and the initial CRSS of slip/twinning system s for orientation n respectively. A is a coefficient and is set to unity in this study. When En reaches a critical value Ecrit (e.g. En≥Ecrit ), nucleation starts in orientation n. The nucleation brings no change in orientation, but the reference stress τns and the accumulative shear Γ are modified in the following waywhere C and D are two coefficients and set to 1 and 0 respectively in the present work. After nucleation, the nuclei growth starts. The growth rate is determined bywhere wn is the weight of orientation n, Eav the weighted average of En of all orientations, B a growth rate coefficient.(a) shows the optical microstructure of the as-received AZ31 sheet. The alloy is in fully annealed state. The measured grain size distribution is displayed in (b). By fitting the experimental histogram with lognormal distribution (), the arithmetic mean grain size is determined as 37.1 μm.From the {0002} and {101¯0} recalculated pole figures given in (c), it is seen that the alloy is strongly textured with basal poles of most grains being nearly aligned with ND, presenting a maximum intensity of 10 multiples of random distribution (MRD). shows typical optical microstructure of the AZ31 alloy after LSER. It can be seen that for the sample rolled without pulsed electric current (ER0) and samples rolled assisted by small-density pulsed electric current (ER5 and ER7), twins and shear bands (and even cracks for ER0) characterize the microstructure. Further inspection into (a), (b) and (c) reveals distinction between ER0, ER5 and ER7 samples. While broad and wavy shear banding dominate the microstructure of ER0 and ER5 samples, two sets of twin clusters/shear bands (aligned ±∼30° with RD) are interwoven with each other in ER7 sample giving a carpet-like structure. Increasing the peak current density to 90 A/mm2 results in qualitative change in the microstructure. As found in (d), small grains formed along the shear bands, grain boundaries and within deformation twins, indicating the occurrence of DRX. This is not seen in the ER0, ER5 and ER7 samples. However, except for being decorated by recrystallized grains, the microstructure of ER10 sample and that of ER7 sample are largely similar. Further increase in current density leads to larger recrystallized grains and higher recrystallization fractions. In addition, instead of shear band nucleation, ER15 and ER18 samples display typical necklace structure together with “ductile” bands. The bands are composed of one or several layers of equiaxed recrystallized grains and aligned within ±15° with RD.The textures of the alloy after LSER are presented in . The ER7 sample displays a biased RD-split basal texture with a maximum intensity of 14.0 MRD. Noteworthy is that two relatively weak TD-split components are also observed in the {0002} pole figure, which might be related to {101¯1} contraction twins and/or {101¯1}−{101¯2} double twins. The {101¯0} pole figure of ER7 indicates two texture components: one orients with <101¯0>//RD and the other with <112¯0>//RD. Similar to the qualitative change in optical microstructure which took place when the peak current density reaches 90 A/mm2, increasing the peak current density from 67 A/mm2 (ER7) to 90 A/mm2 (ER10) also results in dramatic variation in the texture. The two RD-split components are weakened (∼40% decrease in maximum intensity) and slightly rotated around ND, making the line connecting the two peaks aligned at ∼30° with RD. Moreover, the <112¯0>//RD component disappears, leaving <101¯0>//RD being the sole texture component. This texture change is likely due to the occurrence of recrystallization (). ER15 shows a TD-split basal texture similar to that observed in many RE-containing Mg alloys after recrystallization (). The angular spacing between the two peaks is ∼23°. Further raising the current density to 165 A/mm2 (ER18) leads to wider split (∼40°) between the two peaks. This tendency for increasing split is supported by the concentration of {101¯0} poles near RD and the disappearance of the six-fold symmetry in the {101¯0} pole figure.Rotating the basal pole away from ND direction is the major task of texture engineering in Mg alloys. With tilted orientations (no matter the tilt is toward RD or TD), the alloy would be more favorable for basal slip, which results in enhanced room temperature formability and reduced plastic anisotropy. For RE-containing Mg alloys, neither RD split/spread texture or TD split/spread texture is uncommon. The former is usually seen in the as-rolled sheets while the latter is often reported to develop during annealing (). Recently, Estrin et al. obtained TD-split basal texture in RE-free Mg–4Li–1Ca, which might indicate that Li and Ca may have similar effect as RE in terms of texture formation (). For conventional commercial Mg alloys such as AZ series, sheets showing off-basal texture with single peak tilted toward RD has been obtained using differential speed rolling (). More recently, double peak RD-split texture was also reported in AZ31 subjected to electropulse treatment (). TD-split/spread basal texture, however, to the authors’ knowledge, has not been found in either as-rolled or as-annealed state for any of the conventional Mg alloys.Since the final texture is influenced by both deformation and DRX, it is necessary to investigate whether the major contribution in the present study is made by the former or the latter. For this purpose, complete EBSD data for each sample was divided into two subsets (deformed grains and the DRXed ones) based on a criterion considering grain size, aspect ratio and grain orientation spread (GOS). The microstructure and texture information of the deformed grains and DRXed grains are available in . It can be seen that DRXed grains in all DRXed samples (ER10, ER15 and ER18) exhibit weak basal textures, which are quite different from the overall texture (displayed in ). Regarding deformed matrix, however, it is not difficult to find its strong resemblance to the overall texture development except for slightly higher intensities. This seems to suggest that the deformation texture, rather than the recrystallization texture, is a decisive contributor to the overall abnormal (TD-split) texture. Therefore, the following discussion will focus on the reasons to the development of TD-split deformation texture in ER10, ER15 and ER18 specimens.For the same initial texture, the final deformation texture is known to be determined by the operating deformation modes (the selection of activating slip/twinning systems and their corresponding shear amount) which are influenced by the composition, crystal structure, and deformation conditions such as deformation temperature, strain and strain rate. For Mg alloys with a c/a ratio (1.624) approximately equals to 1.633, the dominant slip system in conventional rolling is basal <a>, which generally results in sharp ideal basal texture in the as-rolled plate. In this sense, the abnormally developed TD-split/spread texture in the deformed matrix of ER10, ER15 and ER18 samples could indicate something special in the deformation modes. Based on the fact that TD-split/spread basal texture is frequently seen in Ti and Zr alloys (both with c/a ratios smaller than 1.633) where prismatic <a> slip and basal <a> slip are primary deformation modes (), it is suggested that the activity of prismatic <a> slip in AZ31 is enhanced by the pulsed electric current.Transmission electron microscopy (TEM) is generally considered as the most direct, reliable and convincing technique to reveal dislocation and twinning activity and thus was employed by many researchers in their studies of deformation mechanisms. In TEM studies, however, it is difficult to obtain statistical information from large domains of grains. Due to this shortcoming, intragranular misorientation axis (IGMA) analysis based on EBSD technique, which yields statistically sufficient data, has become popular in the field of dislocation content analysis, especially in alloys of low-symmetric crystal structure (Prior to discussing the present IGMA results, it is necessary first to provide a brief introduction of this method. The lattice rotation caused by slip system s with a slip plane normal nˆs and a slip direction bˆs (both nˆs and bˆs are unit vectors) can be expressed asWhere δγs is the amount of simple shear of the slip system s. The magnitude of rs→ represents the amount of rotation about that axis while the direction of the vector rs→ is the axis about which the crystal rotates and is called Taylor axis (). Through this definition, it can be seen that the Taylor axis lies in the slip plane and is perpendicular to the slip direction. displays the Taylor axes corresponding to individual slip systems. In reality, it is known that grains rarely deformed by a single slip system, alternatively they deform mostly via a combination of multiple slip systems. Thus, the net rotation of the grain is the sum of the rotation caused by each individual slip system:where rs→ is the rotation vector of slip system s. Thus, by simply matching the Taylor axis of a known slip mode to the experimentally measured IGMA, one could determine the dominant geometrically necessary dislocation (GND) in the specimen. For example, a measured IGMA close to <0001> is indicative of the GND of prismatic <a> type, since <0001> is the Taylor axis of the slip mode. It is important to point out that IGMA analysis does not provide information on which dislocation carried the strain during deformation, rather, it only tells the content of GNDs that were ‘left behind’ after the deformation.The IGMA distributions of the deformed matrix of specimens rolled assisted by pulsed electric current with different current densities are also displayed in . ER7 sample shows almost homogeneous IGMA distribution, whereas ER10, ER15 and ER18 samples demonstrate increasingly higher tendency for IGMA to concentrate near <0001> crystal direction. This phenomenon indicates that an abrupt change of GND content occurred at 90 A/mm2 and that the presence of prismatic <a> type GND enhances with increasing current density. This trend correlates well with the aforementioned variation in texture presented by the {0002} pole figures (abrupt change at 90 A/mm2), strongly suggesting the close link between the enhanced presence of prismatic <a> type GND and the formation of TD-spread/split texture.To further verify the correlation between the GND content and the TD-spread/split basal texture, deformed grains in ER10, ER15 and ER18 samples were respectively partitioned into three subsets according to their orientation relationship with ND. Then, IGMA was measured for each of these groups separately. The nomenclature and the criterion for the data partition are as follows: (1) ND group: grains with c-axis aligned within 15° with ND; (2) RD group: grains with c-axis tilted more than 15° toward RD; and (3) TD group: grains with c-axis tilted more than 15° toward TD. The results are similar in ER10, ER15 and ER18 samples, and shows an example of ER10. It can be seen that both ND and RD group exhibit relatively homogeneous distribution of IGMA. For TD group, however, unequivocal concentration near <0001> is observed. This therefore undoubtedly confirms the link between the TD-tilted orientations and the dominant prismatic <a> type GND content.Since the IGMA analysis only gives information of the ‘left-behind’ GND content. VPSC was employed to reveal the effect of pulsed electric current on the slip/twinning activities during the deformation. The material parameters (including the Voce hardening parameters τ0s,τ1s,θ0s,θ0s and the DRX parameters A, B, C, D) were determined via fitting the experimental stress-strain curves. The parameters in the paper by Walde et al. were used as an initial estimate (). The experimental and the simulated curves for through-thickness uniaxial compression at a temperature of 300 °C and a strain rate of 1 s−1 are given in (a). The incorporation of the DRX code enables us to consider the softening of the material and thus to describe the deformation behavior to a true plastic strain of −0.9. The resulting simulated texture after compression ((b)) was also checked for accuracy. As given in (b), the simulated as-compressed texture basically maintains a basal type (though shows some split in RD), which is expected and agrees with the literature ((c), one can see that the simulated texture matches well with the experimental one ((c)). The optimal hardening parameters and DRX parameters obtained are listed in Since hardening parameters and recrystallization parameters reflect material properties and therefore should not be a function of strain path or loading state, the parameters determined above were used to simulate the conventional rolling process at 300 °C. The simulation was carried out to the same true plastic strain as in the LSER (−0.9). The simulated final rolling texture is presented in (a). It shows an obvious split in RD direction. This seems unusual, because the commonly observed texture in RE-free Mg alloy sheet is of typical basal type. The discrepancy should be due to the different plastic strains employed in the present study and in the literature. In the current simulation, the sheet was rolled to a true plastic strain of −0.9, while the majority of the experimental texture data available comes from sheets rolled to a true strain of no more than −0.3. In reality, beyond a strain of −0.3, unstable flow often occurs and shear bands often help carry some strains in the c-axis direction. During simulation, in contrast, the code does not take the shear bands into consideration and, as always, no matter how large the necessary stress is, the “RD split contributor”, either <c+a> or CTW (or both) has to be activated to accommodate the strains in the c-axis direction. This explanation is validated by checking the simulated texture at a series of rolling strains. The results clearly show that the split does not appear until a true plastic strain of −0.3 is reached (the pole figure corresponding to ε=−0.3 is given in (b)). This critical strain (−0.3) also agrees with the slip/twinning activity variation. It can be seen that the activity of <c+a> slip is negligible at early stages of deformation while the deformation mode becomes the second important one when strain exceeded ∼−0.3.As mentioned, the pulsed electric current is supposed to have two distinct effects on the materials. One is the thermal effect (Joule heating) and the other the athermal effect. The former one increases the temperature of the sheet and thus affects the deformation in the same way as conventional furnace heating, whereas the latter influences the deformation by athermal energy input. In the present study, the maximum temperature of ER18 sample was measured to be 307 °C. Therefore, the thermal effect of the current should be almost the same with that in conventional rolling at 300 °C. To reveal the athermal effect, the Voce parameters of different deformation modes are adjusted accordingly from those shown in to get the texture of ER18. Although pulsed electric current is proved to influence the recrystallization of Mg (), the DRX parameters for both conventional rolling and LSER are assumed to be the same for simplicity in the present simulation. Trials indicate that in order to have the TD split as in ER18, the slip resistance of prismatic <a> slip has to be significantly decreased while those of other slip/twinning modes are either kept unchanged or moderately decreased. The optimum Voce parameters for ER18 are given in . The simulated texture that is the most similar to that experimentally observed in the ER18 sample ((c). By comparing the slip/twinning activity in conventional rolling ((d)), we can find that, instead of basal slip, the prismatic <a> slip is predominant in the latter rolling process. This confirms the conclusion from IGMA in the previous section. The connection between TD-split basal texture and the prismatic <a> activity has been reported by Pochettino et al. in Zr alloy using VPSC (). Their research indicates that a prismatic <a> activity higher than 60% is necessary for the formation of TD-split texture. This agrees well with the slip/twinning activity for ER18 as shown in One thing the readers might pay attention to is in the much lower CRSS of prismatic <a> slip than that of basal slip. The authors think this might not reflect the intrinsic properties of the material and should be treated with caution. The reason for this is that the CRSS of the prismatic <a> slip could be underestimated due to the deficiencies of the present simulation. (1) As mentioned above, since the unstable flow is not taken into account in the present model, the <c+a> activity is largely overestimated. The <c+a> slip, as a ‘RD split contributor’, significantly hinders the development of TD-split texture. Consequently, only by reducing the Voce parameters of prismatic <a> slip to very low values would it be possible for the <c+a> activity to be suppressed and thus the TD-split texture to form in the simulation. (2) The dynamic recovery (DRV) process, which is an important process in EAP, is not considered by the model. Due to the much higher stacking fault energy (SFE) of the prismatic plane than that of the basal plane, the DRV is much more intense on the prismatic planes. This means that in order to have the TD-split texture in the simulation, the DRV on prismatic planes needs to be compensated by further reducing the Voce parameters of prismatic <a> slip to unrealistic values.Though the CRSS obtained via VPSC, as mentioned above, might not reflect the real case, the slip/twinning activities, which are directly related with the grain rotations, should be reliable, otherwise we could not come up with a better explanation for the TD-split texture observed. However, it still remains to be explained how the prismatic <a> activity could be rendered so easy by the electric current.Based on conventional electroplastic theory, the pulsed electric current applied on the materials can be viewed as a special non-equilibrium energy input. Besides the thermal effect which plays the same role as conventional furnace heating, the interaction between the electrons and the dislocations certainly makes a difference (athermal effect). The interaction was originally proposed as a force that electrons exerted on the dislocations (electron wind) (). Though various mathematic expressions based on different assumptions are given by researchers, they basically agree that the intensity of the ‘wind’ is positively correlated to the electron density and electric current density. Later, Conrad et al. () proposed that electric current promotes the dislocation motion through a process similar to thermal activation, and that the increase in strain rate produced by electric current can be expressed asln(ε˙jε˙)=ln(ε˙0,jε˙0)−[ΔUj∗−ΔU∗kT]+(Aj∗−A∗)τ∗bkT+Aj*FewkTwhere ε˙0=(Nd*bAν*/M)exp(−ΔS∗/k) with Nd* the mobile dislocation density, b the Burgers vector, A the area of the slip plane swept out per successful thermal fluctuation, ν∗ the frequency of vibration of the dislocation segment involved, ΔS∗ the entropy of activation, k the Boltzmann's constant and M the Taylor orientation factor. ΔU∗ is the activation energy, A∗ the activation area and τ∗ the effective applied stress, Few the electron wind force. Subscript j indicates the parameter with current, no subscript that without (). Through experimental evaluation, they concluded that the contributions from electron wind Aj*Few/kT and from the decrease in the thermal obstacle −[ΔUj*−ΔU*/kT]+(Aj*−A*)τ*b/kT are negligible and the effect of drift electrons is mostly on the pre-exponential term ln(ε˙0,j/ε˙0) (). Since the Burgers vector b is determined by dislocation type and the Taylor factor M is only related to the grain orientation, both of them are not affected by the pulsed electric current. Hence, the observed higher sensitivity to electric current exhibited by prismatic <a> slip should lie in the remaining four parameters: Nd*, A ,ν∗ and ΔS∗. All these four parameters are closely related to the structure and glide behavior of the dislocations. However, up till now, all studies on these two topics in HCP metals rely primarily on computer modeling. The results so far depend strongly on the assumptions taken and often do not agree well with the existing experimental results (). This prevents us from further understanding the enhancement of prismatic <a> activity. Nonetheless, the following hypotheses are proposed based on the existing knowledge.The c/a ratio, because of its direct influence on the close pack geometry (close packed plane and direction), is the very first factor that one might come up with when discussing the slip modes in HCP materials. Reduced c/a ratio was believed to be the reason for enhanced prismatic <a> slip in Mg–Li alloys (), though this hypothesis has been recently proven untrue for Mg–Y alloys by synchrotron radiation experiments (). Is it possible that the c/a ratio of the alloy is reduced when the electric current/field is applied? New experiments are needed to answer this question in future studies.Besides, some recent modeling results attribute the increased activity of non-basal slip in some Mg alloys to the variation in SFE. For instance, Yasi et al. suggest that increasing the I2 stacking fault energy leads to increased ease for the cross-slip of <a> dislocation onto prismatic planes (). Wang et al. suggested that in Mg–3Al–3Sn, the lower unstable SFE of 1st order pyramidal <a> slip than that of the prismatic <a> slip leads to the replacement of the latter by the former to be the 2nd primary slip system (next only to basal slip) (). Sandlobes et al., on the other hand, elucidate the correlation between decreased I1 stacking fault energy and the enhanced pyramidal <c+a> slip (). It is well known that the SFE is determined by the interaction between the adjacent layers. The latter is affected by the charge density distribution, therefore, the electric current/field is likely to be able to change the SFE of the alloy (). Although this impact of the electric current/field on SFE has not been directly reported in literature before, the possibility of such an effect was indeed suggested by the changed frequency of Σ3 boundaries in some cubic metals (). In addition, Brade et al. recently conducted electrically assisted uniaxial tension on three face centered cubic (FCC) materials with different SFEs (). They only observed electroplastic effect in the material with the highest SFE. This could also indicate different effects of the electric current on slip planes with different SFEs (in the present case, basal planes and prismatic planes). Nonetheless, the influence of the electric current/field on the SFE still needs to be verified and would be a subject of future study in the electroplasticity research community.Large strain electroplastic rolling (LSER) was performed on AZ31 sheet. The influence of pulsed electric current density on the microstructure and texture has been investigated. From the experimental and modeling results, the following conclusions can be drawn.TD-split texture can be produced in AZ31 sheet using LSER via only one pass, which would enhance formability of the Mg alloy. The basal poles start to split along TD once the peak current density reaches a critical value of 90 A/mm2 (sheet temperature 133 °C due to Joule heating).The deformed grains, rather than the DRXed grains, are the major contributor to the overall TD-split texture. The formation of the TD-split texture in the deformed grains is accompanied by the dominance of prismatic <a> type GND.VPSC simulation suggests that the TD-split texture is likely due to the enhanced prismatic <a> activity caused by the pulsed electric current. However, the exact mechanisms by which the electric current makes the prismatic <a> slip so easily activated still remains unclear.Within the current density range studied in the present research (peak current density: 0–165 A/mm2; equivalent current density: 0–17.9 A/mm2), a higher current density results in more significant split of the basal poles toward TD.Due to the ease of operation and high efficiency, LSER is a promising technique for texture modification in Mg sheet for improved formability.A 4-wheel independent drive rover is designed, engineered and built. The rover suspension is made to cope with the adverse conditions of all terrains making it reliable to navigate and explore an alien planet like Mars. The robotic arm is also designed and manufactured with six degree of freedom which is useful to assist the rover for multiple purposes such as to collect the soil samples for testing, to open and close the taps or valves and also to assist the astronaut for picking up tools like screwdriver, cutting player, etc. during an emergency.Effect of LCF on HCF crack growth of Ti-17An improved understanding of fatigue crack growth phenomena applicable to titanium engine disks was developed through complimentary experimental and analytical investigations of Ti-17. The effect of low cycle fatigue (LCF) on the high cycle fatigue (HCF) threshold and rate of crack propagation was studied. A simplified variable-amplitude spectrum, consisting of high-R cycles, corresponding to HCF loading, and periodic R=0.1 cycles, corresponding to LCF loading, was used to demonstrate a load-interaction effect. When the ratio of HCF to LCF cycles was 100 or more the fatigue crack growth lifetimes were significantly lower than predicted using linear damage summation methods assuming no load-interaction effect. Thus, it was concluded that the LCF cycle accelerated the fatigue crack growth rate of subsequent HCF cycles, even when closure was concluded to be negligible. A phenomenological model was formulated based on hypothesized changes in the propagation resistance, KPR, and fit to the test data. The model confirmed that the periodic LCF cycles increased fatigue crack growth rates of subsequent HCF cycles.Fatigue crack growth (FCG) predictions have been an essential element of the life-management system for turbine engine components since the Retirement for Cause program The bulk of variable-amplitude fatigue crack growth studies have focused on spectra with overloads, compressive underloads, or a combination of both Ti-17 is the commercial name for Ti–5Al–2Sn–2Zr–4Mo–4Cr (wt%). It is an alpha-beta titanium alloy designed primarily for application as a fan and compressor disk material and is typically used in a beta-forged or beta-annealed microstructural condition due to the increased toughness achievable . The microstructure consisted of approximately 45% acicular alpha (darker phase) and 55% transformed beta. The material had yield and ultimate tensile strengths of nominally 1150 and 1200 MPa, respectively, comparable to values achieved by Redden All FCG tests were conducted on an MTS servo-hydraulic test frame in general accordance with ASTM test specification E647-95a Baseline, steady-state FCG behavior was evaluated at three stress ratios, R=0.1, 0.4, and 0.7 and at a frequency of 40 Hz. The FCG response was determined from two separate test-control procedures as described by Russ As expected, the three data sets layered as a function of R, . As R increases ΔKth decreases and growth rates at the same applied ΔK increase. A tried methodology to converge FCG rate data as a function of R is to plot the data versus ΔKeff instead of ΔKapplied, where ΔKeff=Kmax−Kop. To accomplish this requires a measure of closure or the crack opening load, Pop. Closure is commonly determined from clip gage or back-face strain measurements as the point in the load–displacement trace where the curve deviates from a linear fit to the upper portion. provides representative load–displacement traces from the R=0.1 test as the crack extended. As noted by Russ the loads have been normalized and displacements adjusted based on the range and sequentially incremented for ease of presentation. Note that Pop increased continuously as a function of crack length, eventually reaching a level as high as 0.25Pmax. However, the FCG rates near the beginning and end of the test, when K was similar but the crack length was very different, were comparable. Therefore, attempts to converge the R dependence of the FCG rate curves using closure analysis of this type were abandoned.Crack propagation load measurement (CPLM) tests, similar to those outlined by Lang . A series of load blocks, each consisting of 400,000 cycles at 40 Hz, was applied with each successive block being stepped by nominally 0.1 MPa√m. ΔK was maintained at 2.5 MPa√m, with the exception at R=0.8. The load was stepped until the crack was observed to grow, resolved by an increase in the DCEP voltage. After the test KCG and KNG, the stress intensities of the load blocks when the crack began to grow and the last one where the crack did not grow, were determined. KPR was then calculated fromwhere ΔKT is the intrinsic threshold. Unfortunately, an independent measure of ΔKT was not available. Instead, an initial estimate based on ΔKth for R=0.7 was used. Ultimately, ΔKT served as an additional variable to make subtle improvements to the fit to the FCG rate data as explained in , where KPR, normalized with respect to Kmax of the precrack, is plotted as a function of R during precracking, along with a polynomial fit to the data.Subsequently, KPR was employed in a methodology to account for the R-dependence of the FCG rates. In an effort to collapse the three steady-state FCG rate data sets, (KPR)SS, from the fit in log(dadN)=C1arctanh{C2[log(ΔKeff)+C3]}+C4to represent all three data sets data with one characteristic curve. An iterative process was used to obtain a best fit to the three data sets with ΔKT included as an additional parameter to further minimize any variance between the data and the output of Eq. shows the effectiveness of this approach in collapsing the FCG rate data for R=0.1, 0.4, and 0.7, and the ability of the hyperbolic arctangent fit to adequately describe the entire data set including the threshold region.These results effectively establish the relevance of applying the experimentally determined measure of KPR in the characterization of steady-state FCG behavior. In a more philosophical argument, KPR can be viewed as a state variable—a function of the state of the crack tip and governed by the loading history. For the relatively simple case of steady-state loading conditions, KPR was shown to be merely a function of stress ratio and Kmax. If R is known and maintained, KPR can be determined from the empirical approach used to generate the plot in . KPR can then be utilized in the prediction of the progression of the crack with relative success.To investigate load-interaction phenomena a simple sequence was created consisting of a number of high-R HCF cycles followed by lone R=0.1 LCF cycles. The HCF cycles were applied at R of either 0.7 or 0.4. A schematic of the simple sequence is provided in , where the terms baseline (BL) and underload (UL) are used interchangeably to describe the HCF and LCF cycles, respectively. The FCG tests were conducted at a constant maximum load, Pmax. This spectrum resembles the simple sequences performed in the AGARD study The null hypothesis predictions were built using a linear summation procedure as described in detail by Russ . The crack growth increment for one block of cycles was calculated based on the number of HCF cycles times the high-R growth rate plus the growth from an individual LCF cycle. The crack size was then updated and another increment of growth determined. The experimental results for the RBL=0.7 tests and comparison to model predictions are provided in . The ratio of the predicted to experimental cycles, Npred/Nexp, was used as a measure of model accuracy. In previous modeling efforts Based on the lack of correlation between the damage summation model with no load-interaction effect and the experimental results, a phenomenological model was developed that incorporated a load-interaction effect. It was hypothesized that the underload from the LCF cycle effectively disturbs the steady-state condition of the HCF cycles, temporarily lowering the propagation resistance, and thus accelerating the crack growth rate for some number of cycles. displays how KPR was theorized to continuously change as a function of the number of cycles following the LCF cycle. KPR is shown to initially drop well below the steady-state value and subsequently increase with each additional cycle until the steady-state condition is reestablished. The LCF cycle is again applied, and ΔKeff of the LCF cycle was assumed smaller than if the steady-state value of KPR were assumed.An analytical approach was developed which captures the essence of the hypothesized effect, but was easier to implement. shows a schematic of the simpler form applied to 1000 HCF cycles. The model incorporated six multiplication factors, BLMF1 through BLMF5 and ULMF. The baseline multiplication factors (BLMF#), applied to the HCF cycles following each LCF cycle, effectively increased ΔKeff for a prescribed number of cycles based onwhere (KPR)SS represents the steady-state value of KPR as calculated from the fit in and # can be replaced with 1, 2, 3, 4, or 5. Since the LCF cycle is theorized to increase ΔKeff—resulting in acceleration of the FCG rates of the HCF cycles—the BLMFs were assumed to have to be greater than or equal to one. When the BLMF equals one the steady-state condition is realized. In a similar fashion the ULMF was used to modify ΔKeff for each LCF cycle based on the ULMF would have to be less than one, effectively reducing ΔKeff and the predicted FCG rate of each LCF cycle.This approach builds the growth rate for each block in a cycle-by-cycle fashion, effectively determining an individual growth rate for each baseline cycle within the block and eventually using Eq. to calculate the growth rate for the block being considered.Ideally, if NBL was 1000 there would be 1000 different growth rates, (da/dN)BL,i, one for each HCF cycle. In the procedure implemented in this study up to five effective stress intensity ranges were considered for a given load block; BLMF1 was used in calculating (da/dN)BL for the first 10 HCF cycles after the underload, BLMF2 for HCF cycles 11 through 100, BLMF3 for HCF cycles 101 through 200, BLMF4 for HCF cycles 201–500, and BLMF5 was used for HCF cycles greater than 500. Using this approach the multiplication factors were used to fit the FCG rates observed in the simple sequence tests The models and the FCG data were compared using both FCG rate curves as well as crack length versus blocks (a–B) curves. Comparisons for selected RBL=0.7 tests and the models with and without load-interaction effects are shown in . It is clear that the solid curves, representing the load-interaction model, better represent the data for each test condition. It should be stressed that the multiplication factors in the load-interaction model, , were selected to best fit the FCG rate data shown. Thus, the term prediction would be a misnomer. However, the fact that the load-interaction model significantly improved the ability to describe the test results, while remaining consistent with the theorized effect was interpreted as additional evidence that a load-interaction effect existed. The model captured the theorized effect in both an appropriate manner and magnitude.Applying the phenomenological model with the load-interaction proved very effective in increasing the ability to accurately describe the a–B curves. A typical example is provided in , the number of loading blocks to grow each crack from the initial crack length, ai, to the final crack length, af, was predicted within a factor of 1.4 for all cases. In order to achieve this better representation of the actual crack growth, the model increased the FCG rate of the high-R HCF cycles and decreased the FCG rate of the R=0.1 LCF cycle through the use of the appropriate multiplication factors. The model also predicted an increase in the percentage of the total crack extension attributed to the HCF cycles, tabulated in the last two columns of . It was apparent that at the higher NBL values the contribution of each LCF cycle to the overall crack extension was minimal. However, the LCF cycle was responsible for accelerating the damage attributed to subsequent HCF cycles. Thus, the existence of the LCF cycle was a significant detriment to the overall FCG process. In other words, an acceleration of the FCG rates of the HCF cycles and a slight decrease in the FCG rate of the LCF cycle evinced the theorized load-interaction effect.The results of the tests just described, and the modification to the damage-summation model necessary to predict the results, demonstrated that a load-interaction effect existed for the simple variable-amplitude FCG tests performed. However, the model did not address physically what occurred at the crack tip to induce the load-interaction effect. It merely acknowledged that if KPR was used in the description of ΔKeff, and if KPR of the HCF cycles was decreased as a result of the LCF cycle, it was possible to capture the observed effect on the FCG behavior of Ti-17. To date, the chosen approaches used to capture load-interaction effects have relied primarily on closure phenomena. However, it was concluded that closure was not present at high-R. Therefore, a reliance on closure to describe the load-interaction effect observed in this work was not feasible. Lang Both experimental and analytical investigations were conducted of the fatigue crack growth load-interaction effects applicable to titanium turbine engine components. Steady-state FCG tests were performed to establish nominal behavior of the Ti-17 material and to develop a FCG rate equation applicable for a range of stress ratios. Simple variable-amplitude load spectra were employed, which consisted of high-R HCF cycles and lone LCF cycles at a minimum load near zero. The number of HCF cycles was varied from 10 to 1000 to systematically interrogate the effect of the R=0.1 LCF cycles on the FCG rates of the subsequent high-R HCF cycles. The following conclusions were drawn from this study.Closure was not observed in the R=0.4 or 0.7 steady-state FCG tests. Therefore, capturing the stress-ratio dependence of T-17 using closure-based approaches was not feasible.The relevance of applying the propagation resistance in the definition of the effective crack driving force, ΔKeff=Kmax-KPR, was demonstrated by its ability to capture the stress-ratio dependence of the three steady-state FCG tests at R=0.1, 0.4, and 0.7. A single hyperbolic arctangent equation was fit to the FCG rate data of the three tests, and the a–N curves of the Pmax-constant portions of the tests were predicted with very good agreement.The simple variable-amplitude tests demonstrated that a load-interaction effect existed. The LCF cycle effectively increased the FCG rate of subsequent high-R HCF cycles. When the number of HCF cycles was 100 or more the higher FCG rates of the HCF cycles led to significantly lower FCG lifetimes than predicted based on a damage-summation model assuming no load-interaction effect.A phenomenological load-interaction model was introduced that simulated theorized changes in KPR resulting from an R=0.1 LCF cycle. The output from the model, incorporating a simplified load-interaction effect, was made to mimic the experimental data from the simple variable-amplitude tests. In doing so, all a priori constraints were maintained, and anticipated relationships observed. Thus, it substantiated the experimental observation that a load-interaction effect existed as an acceleration of the FCG rates of the high-R HCF cycles and deceleration of the FCG rate of the R=0.1 LCF cycle.Comparative study on resistance and displacement based adaptive output tracking control strategies for resistance spot weldingTo achieve high quality resistance spot welds (RSW) through intelligent manufacturing of automotive body structures, adaptive control for RSW processes becomes imperative and has gained global attention. Tracking ideal process signal profiles predetermined offline is a commonly-used strategy to realize the adaptive control of RSW and is found effective in compensating for the adverse impacts caused by shunting and electrode wear. However, the validity of this control strategy under other types of disturbance is still unproven. In this article, typical abnormal welding conditions, i.e. initial sheet gaps and edge proximity, were designed to test the performance of output tracking control strategies for steel RSW. Two types of process signals including dynamic resistance and electrode displacement were tracked to approach the referenced signal profiles through the adjustment of weld parameters. Comparative studies were conducted based on metallographic analysis and mechanical tests to validate the two tracking control strategies. Results demonstrated that gap and edge proximity conditions decreased the nugget diameter and the tensile-shear strength of the welds. When satisfactory tracking of the reference resistance signal was realized under these two types of abnormal conditions, weld properties are further degraded, owing to a reduction in welding current. On the contrary, tracking the electrode displacement signal mitigated the negative influences from the disturbance through increasing welding current properly. When the disturbance was too severe, the displacement-based strategy raised the risk of expulsion. The findings suggest that tracking the dynamic resistance signal is not optimal for adaptive control of steel RSW, while tracking the electrode displacement signal is more effective. However, to achieve a more satisfactory adaptive control result, the output tracking control should be used in combination with expulsion suppression techniques.Resistance spot welding (RSW) is widely employed as a joining method in steel auto body assembly due to its high efficiency and low cost []. During the RSW process, two or more metal sheets are pressed together using a pair of electrodes and a welding current is passed through the stack-up, generating heat according to Joule’s Law. After only milliseconds, the metal sheets are locally melted and a molten nugget is formed at the faying interface. Typically, a modern steel car body contains 4000 ∼ 6000 spot welds, and their quality notably affects the safety and reliability of the automobile structure []. However, the wear of the machine parts and assembly variation in fast-paced mass production inevitably produce some abnormal welding conditions such as electrode wear, shunting, initial sheet gaps, and edge proximity which can dramatically reduce the consistency of the welds []. Expulsion and undersized welds are two of the most common weld defects caused by these disturbances, which generally lead to significant degradation of the weld quality and strength []. Therefore, it is a vital and critical task for automotive manufacturers to control the consistency of weld quality.Traditional means of RSW quality assurance primarily rely upon on the offline adjustment of weld parameters, according to the test results of manual inspection. However, this method rarely meets the increasing demands for lean production due to high costs and low efficiency and is not in alignment with the transformation to intelligent automobile manufacturing []. Therefore, adaptive control which achieves real-time adjustment of weld parameters at the moment when welds are produced is considered a promising way to replace the traditional solution and has gained global attention []. Significant effort has been directed to developing a high-performance controller that can compensate for the adverse impact caused by abnormal welding conditions.Since the formation and growth behavior of the weld nugget is only visible through destructive inspection, it is common to use RSW process output signals to indirectly infer weld quality []. Based upon the assumption that a similar process signal should result in a comparable weld property, the output tracking control, tracking to the predetermined ideal process signal which is obtained in advance, is proposed as a widely used adaptive control strategy for RSW process. Won et al. [] established a real-time feedback weld controller to adjust the welding current based upon the difference between measured and desired dynamic resistance signals. After that, other process signals including electrode displacement [] were also utilized as tracking targets to realize on-line adaptive control. Likewise, the technique of output tracking control was applied to trajectory planning and seam tracking of arc welding []. In recent years, this technology has significantly progressed due to the development of machine learning and automatic control. The controller has become much more sophisticated as the control algorithm has upgraded from traditional proportional-integral- differential (PID) algorithm [] to intelligent control methods such as fuzzy logic [], which enable perfect tracking of process signals. For instant, Liu et al. [] constructed a neuro-fuzzy-based human intelligence model and implemented it as an intelligent controller in automated welding process, which robustly controlled the process to a desired penetration state.The effects of adaptive tracking control on weld quality of RSW have been studied under various test conditions. Robert et al. [] used electrode displacement as the feedback signal to adjust power input and found the approach could obtain quality welds despite abnormal conditions including electrode wear, poor fit-up, and surface contamination. Zhang et al. [] developed a neuro-fuzzy system to adaptively regulate weld parameters based on the measured electrode displacement signal and observed that the system could minimize the harmful influence of electrode wear on weld quality and improve electrode life. Yu et al. [] proposed a modified control system based on a resistance tracking strategy. The heating power was adjusted after the resistance signal peak to maintain the total input energy at the same level as the reference weld. The results demonstrated that the method significantly attenuated the adverse effects caused by shunting and oscillation of force and could be applied to steels of different strength levels.Although these adaptive controllers could achieve satisfactory weld quality under some specific welding conditions, limitations in application remain []. The correlation between the process signal and the nugget size is complex and still not fully understood. When abnormal welding conditions occur, electrode contact during the RSW process could change, leading to a different nugget growth behavior []. Using the similarity of process signals to represent comparable weld performance for different welding conditions has not been demonstrated. On the other hand, there are many factors other than the weld nugget that influence these process signals []. The desired signal obtained from one series of experiments may not be generally desired under other weld conditions []. Therefore, the effectiveness of the output tracking control under other types of abnormal welding conditions requires further study.Edge proximity and sheet gaps are universal abnormal welding conditions in the manufacturing of automotive body structures that could potentially result in assembly variation and affect weld quality []. However, little research has focused on output tracking control under these types of disturbances, and the validity of an adaptive control strategy for these abnormal conditions is still unknown. In this article, these two conditions are taken into consideration to verify the validity of output tracking control for steel RSW. Process signals including dynamic resistance and electrode displacement were used to track the referenced profiles through the adjustment of weld parameters, and comparative studies on weld properties were performed based on post-weld metallographic analysis and mechanical testing []. To guarantee the reliability of the study, weld parameters and disturbance of different levels were included and the performances of tracking the two process signals were separately explored.The sheet metal used in this work was 0.8 mm-thick bare DP590 steel which is widely used in vehicle body manufacturing. The chemical composition and mechanical properties of this material are provided in , respectively. Two identical ballnose electrodes comprised of CuCrZr with a tip diameter of 6.0 mm were used to perform a series of welding experiments. Weld parameters (welding current, welding time, and electrode force) are listed in The experiments were carried out on a robotic RSW system consisting of a FANUC R2000iB robot, an OBARA C-type servomotor-driven welding gun, and a MEDAR 6000 s medium-frequency direct-current (MFDC) welding controller, whose inverter frequency was 1 kHz. The maximum welding current of the system was 16 kA and the maximum electrode force was 5.5 kN. displays the servo gun equipped with the sensors employed in this work for process monitoring of RSW. A MEATROL Rogowski coil of 0.5% accuracy class was placed on the fixed shank of the welder to measure the welding current I. Two twisted-pair cablings were mounted to the ends of the upper and lower electrodes to measure the secondary voltage V. Then, the dynamic resistance signal could be obtained according to Ohm’s law, i.e.where Rg refers to the basic resistance of the gun.Meanwhile, the moving-electrode displacement Sm was monitored using a HEIDENHAIN linear encoder installed to the slider of the linear guide; the nominal resolution of the encoder was 0.5 μm. A KISTLER surface strain sensor was mounted on the fixed shank to indirectly measure the electrode force F. The sensor transmitted the force‐proportional surface strain into a standard voltage signal and could achieve a measurement precision of 2% FSO (Full Scale Output). Finally, all output signals of the sensors were collected using a data acquisition device at a sampling rate of 500 kSPS (sample per second) and then displayed on a standard personal computer. Since the sensor installation locations were far away from the electrodes, the sensors were not intrusive to the welding process.], it was observed that the fixed‐electrode displacement Sf is proportional to the electrode force F, and the relative distance S between the two electrodes can be calculated through the subtraction of the two displacement measurements, i.e.where Kg stands for the equivalent stiffness of the fixed electrode arm, and it can be simply obtained through a fast calibration. shows the specimen dimensions and schematic diagrams for the standard condition (ST) and two different abnormal welding conditions: edge proximity (EP) and initial sheet gaps (IG). The specimen was 130 mm in length and 38 mm in width. ST refers to the condition where the spot weld was at the center of the overlapping area and no disturbance occurred. EP represents a spot weld formed close to the edge of the longer side of the metal coupon. Two degrees of edge distance c were selected for the experiments, i.e. c =6 mm (named EP6) and c =3 mm (named EP3). Specimens used to simulate IG utilized two pieces of insulated shims placed symmetrically between the two metal sheets at 40 mm lengthwise spacing. Likewise, two degrees of gap height h were chosen, i.e. h =1 mm (named IG1) and h =2 mm (named IG2). In total, five different welding conditions were included in this study. No combination of abnormal welding conditions such as EP + IG were considered in this study.In order to evaluate weld properties, metallographic observation and mechanical tests were conducted, as shown in . The welds were cross-sectioned in two orthogonal directions and fabricated into metallographic specimens. Specimens were polished to a surface finish of 0.05 μm using aluminum oxide and etched using a 4% nital solution. The nugget diameter was measured in the metallographic cross-section at the faying interface and the average nugget diameter D was calculated based on the measurement of both cutting directions, i.e.where DLen and DWid refer to the lengthwise and widthwise nugget diameters, respectively. Three welds were evaluated for each condition to improve measurement accuracy. Tensile-shear tests were performed at a pull rate of 3 mm/min on a SUNS universal tensile testing machine. Shims with the same thickness as the DP590 sheets were used at the coupon ends to minimize the bending stress inherent during the test. Similarly, three welds were tested for every condition to improve repeatability. (a) shows the metallographic cross-sections in the lengthwise and widthwise cutting directions of the welds performed under ST and EP conditions. Obviously, the nugget diameters in two cutting directions are similar under ST condition. However, the widthwise nugget diameter was observed to be larger than the lengthwise nugget under EP condition, indicating that the occurrence of EP condition would produce an asymmetric nugget. (b) and (c) show the average nugget diameter produced using a welding current of 6 kA and 8 kA, respectively. Compared with the ST condition, the EP condition had a slight impact on nugget diameter but significantly decreased the tensile-shear strength of the weld, especially the peak load. This adverse effect was more pronounced in EP3 than in EP6, refer to (d)∼(f). Since the nugget diameters of ST and EP conditions were similar, the difference of tensile-shear strength can be attributed to the weld position. When the weld is at the centerline in the widthwise direction the tensile load is symmetric. However, an extra torque would be induced when the EP condition was present, creating asymmetric loading. This extra torque increased the shear stress and thus decreased the failure load of the weld. As a result, the measured peak load for the EP condition was lower than that for the ST condition.Similarly, the comparison between ST and IG conditions are provided in . The IG condition also produced directionality of the nugget diameter with a nugget that exhibited a lengthwise diameter larger than the widthwise diameter. Moreover, the occurrence of IG condition caused a small reduction in the nugget diameter, which lead to a minor decrease of the peak load. Based upon experimental results, the presence of abnormal welding conditions (EP and IG) negatively influenced weld properties; the more severe the abnormal condition, the greater the impact on weld properties.The process signals (dynamic resistance and electrode displacement) under ST and IG conditions are shown in . Lower resistance and displacement signal nominal values were noted for the IG condition compared with the ST condition, and the reduction of signal was greater in the IG2 than in the IG1 condition. Furthermore, a lower resistance signal translates to reduced heat input, refer to (c) and (f), which produces a smaller, thinner weld nugget (see ] demonstrated that the relative distance between the two welding electrodes (i.e., the electrode displacement) has a significant correlation to the growth of the nugget thickness. Accordingly, the displacement signal of IG condition was lower than that of the ST condition because the welding electrodes in the IG condition were at a greater relative distance to one another. In addition, the resistance and displacement signals of the IG2 condition at 8 kA exhibited a sudden drop which corresponded to the occurrence of internal expulsion (IX) []. During internal expulsion, a portion of the molten nugget is ejected from the weld. In this study, internal expulsion decreased heat input by 18% and significantly reduced nugget size, refer to Compared to the ST condition, the occurrence of the EP condition also produced a reduction of resistance and displacement signals which resulted in a decrease in heat input and corresponded to decreases in nugget diameter and peak loads, refer to . However, this reduction in signal value was not as severe as that of IG condition except when internal expulsion occurred. Overall, the presence of abnormal welding conditions (EP and IG) reduced the process signal and the magnitude of this reduction correlated with the magnitude of abnormality in welding condition.In this paper, the reference signals of output tracking control are the dynamic resistance signal and electrode displacement signal of ST condition under weld parameters listed in . As previously discussed, the presence of EP and IG welding conditions reduced the dynamic resistance signal relative to ST condition. It was observed that decreasing the welding current raises the dynamic resistance signal as shown in (a) and (b). Thus, it was feasible to manifest a resistance tracking control (RTC) that would make the resistance signal of an abnormal condition resemble reference signal by reducing the current proportionately when the resistance signal of the abnormal condition was lower. (c) demonstrates the parameter adjustment procedure of the RTC strategy. First, a reference signal RR(t) is established under ST condition with a constant current control (CCC) mode. Then the welding time T is divided into N periods equally and the welding current of each period is modified manually in turn to make the measured resistance signal R(t) approach the reference signal. The adjusting principle is that the welding current would be increased if the measured resistance signal is higher than the reference signal and vice versa. The RTC strategy would be accomplished until the resistance signal of every period approached the reference signal.Similarly, the presence of abnormal welding conditions would also decrease the displacement signal; increasing the welding current could increase the displacement signal of the abnormal condition, as described in (a) and (b). A displacement tracking control (DTC) strategy could be achieved by an appropriate current adjustment, as described in (c). However, the regulation principle of DTC is opposite that for RTC as the weld current is increased when the measured displacement signal S(t) is lower than the reference signal SR(t).To verify the validity of the RTC strategy, experiments were conducted under four abnormal welding conditions (EP6, EP3, IG1, and IG2) with different current levels and control modes (CCC and RTC). shows the electric current and dynamic resistance signals of different welding conditions using CCC and RTC strategy with the resistance signal under the ST condition and CCC mode chosen as the reference signal (the dotted line). As shown, the measured resistance signal under the RTC mode approximately overlaps the reference signal by reducing the welding current. Theoretically, the weld created under RTC mode should possess similar weld properties as that of ST condition even if abnormal conditions were present., it can be observed that the peak load and nugget diameter of RTC do not increase but actually decrease compared with those of CCC under abnormal conditions, indicating that the weld quality of welds produced using RTC were worse than that of CCC. The results differ from the common belief that a similar process signal should result in comparable weld properties and is due to the fact that there was a reduction in welding current in order to track the reference resistance signal. From , it can be confirmed that the heat input of RTC is significantly smaller than that of CCC regardless of the welding condition. Since the heat input of the RSW process has a fundamental correlation with nugget size, the reduction in heat input will exacerbate the adverse influences of the abnormal welding conditions and degradation of weld quality. Therefore, the RTC strategy is not an appropriate method to control weld quality under EP and IG abnormal welding conditions.Similar experiments were performed to assess the validity of the DTC strategy, refer to . The displacement signal under the ST condition with CCC mode was selected as the reference signal (the dotted line). As shown, the measured displacement signal under EP6 and IG1 conditions approached the reference signal through an increase in weld current. As a result, the nugget diameter and peak load of EP6 and IG1 conditions with DTC mode surpassed those of CCC mode. For IG1 condition, the weld performance exceeded that of the reference weld, refer to . This behavior was caused by an increase in welding current in order to track the reference signal which led to a concurrent increase in the weld heat input, refer to . Consequently, the nugget size and the tensile-shear strength improved, implying that the DTC mode could attenuate the adverse effect on weld quality caused by the EP6 and IG1 conditions.However, the application of these strategies to severe abnormal welding conditions (EP3 and IG2) was more complex. When the welding current was relatively small (e.g., 6 kA) and no internal expulsion occurred, the increase of welding current could increase the nugget diameter and reduce the difference between the measured and desired displacement signals, refer to the results of ‘EP3 + 6.9 kA’ and ‘IG2 + 7.9 kA’ in . But when expulsion occurred due to further increases of welding current (see ‘EP3 + 7 kA’ and ‘IG2 + 8 kA’ in ), the displacement signal heavily deviated from the reference signal, and the nugget diameter shrank. Moreover, when the welding current was large (e.g. 8 kA), the increase of welding current corresponding to the DTC mode would enlarge the expulsion intensity by increasing the expulsion frequency or by producing expulsion earlier during the weld schedule, resulting in a larger deviation from the reference signal and a further reduction in the nugget diameter, refer to the results of ‘EP3 + 9 kA’ and ‘IG2 + 9 kA’ in . Therefore, tracking of the reference displacement signal could not be accomplished under severe abnormal welding conditions. Although DTC strategy improved weld properties when used with slight abnormal welding conditions (EP6 and IG1), it exhibited a propensity for expulsion and a degradation of weld performance under the severe abnormal conditions (EP3 and IG2). In summary, DTC strategy should only be considered as a candidate for adaptive control of the RSW process when used in combination with expulsion suppression techniques.In this paper, abnormal welding conditions including edge proximity (EP) and initial sheet gaps (IG) were used to study the performance of two adaptive control strategies, i.e. resistance tracking control (RTC) and displacement tracking control (DTC). Based upon metallographic analysis and mechanical tests under different experimental conditions, the following conclusions can be drawn:The existence of EP and IG conditions created asymmetric weld nuggets with directionality in the lengthwise and widthwise dimensions respectively, which negatively impacted joint strength.RTC strategy decreased the welding current in the presence of EP and IG conditions and resulted in a reduction of heat input. Accordingly, this produced a reduction of nugget diameter and tensile-shear strength but this decline in properties was greater than that of welds produced using constant current control (CCC) mode. This demonstrates that the RTC strategy is not appropriate for weld quality control under the EP and IG welding conditions.DTC strategy was successful with slight EP and IG conditions by enhancing the heat input which improved nugget size and tensile-shear strength. However, when the abnormal welding conditions were too severe, the use of the DTC strategy increased the internal expulsion. To provide satisfactory quality control, expulsion suppression techniques must be included with the DTC strategy.Even if the dynamic resistance or electrode displacement signal of the weld under EP and IG conditions was adjusted to coincide with that of the standard condition, significant differences in the nugget size and tensile-shear strength were observed between abnormal conditions and standard condition. This phenomenon requires further study.The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.Finite element analysis of the effect of bending stiffness and contact condition of composite bone plates with simple rectangular cross-section on the bio-mechanical behaviour of fractured long bonesThe relationship between mechanical stimuli and curing tissue generation and development should be quantitatively evaluated to better understand the effective healing of bone fractures. In this paper, finite element analyses were carried out to estimate the interfragmentary strain distribution at the fracture site of a tibia according to the bending stiffness and contact conditions of composite bone plates with simplified rectangular cross-section, and polymeric porous layers at the contact area. We found that a composite bone plate with polymeric porous layers provided positive effects on callus generation at the fracture site, and effectively reduced the contact stress at the contact area.Long bones of the lower limbs, such as the tibia and femur, directly sustain body weight; therefore, these bones can be easily fractured by a traffic accident or a fall. In general, when a diaphyseal fracture of a long bone occurs, prosthetic devices such as bone plates are applied to the fracture site as shown in . Existing commercial bone plates are usually made of a stainless steel or titanium alloy. These conventional metals have an excessive modulus relative to human bones, which brings about stress-shielding effects when the bone plate is applied to the fractured bones. In addition to this functional defect, they produce excessive contact stresses at the bone contact surfaces that are known to cause problems in blood circulation, resulting in bone necrosis in the worst case The goal of this study was to examine the bio-mechanical feasibility of a simple and small-sized composite bone plate for providing an effective mechanical environment for callus generation. We introduce simply-shaped composite bone plates with appropriate mechanical properties to enhance the bio-mechanical function of the plate. The mechanical behaviour of the fractured bones to which the composite bone plates were applied was evaluated using finite element analysis. It is expected that the simple shape of a bone plate could be easily fabricated, and that the reduced surface area of the plate could alleviate the foreign body reaction when the plate is inserted in the human body. The interfragmentary strain and contact stress between the plate and bones immediately after surgery were calculated by controlling the bending stiffness and the shape of the composite bone plates. These results were compared to a result from a case involving a commercial stainless steel bone plate. The commercial finite element code ABAQUS v6.7 (Dassault Systèmes Simulia, USA) was used for the calculations.Human bones are composed of cortical bones and trabecular bones. Cortical bones are formed of dense and hard tissue with an anisotropic material property in the longitudinal and circumferential directions. In contrast, trabecular bones are sparse and weak, and are regarded as an isotropic material from a macroscopic point of view . Several types of carbon/epoxy prepregs (a plain weave fabric and a unidirectional prepregs) were used for the composite bone plates. The bio-compatibility of carbon/epoxy composites as a bone plate has been examined by other researchers . In this study, the dimensions of the composite bone plates with a simple rectangular cross-section were determined by considering existing commercial stainless steel bone plates a). In our bone model, a human tibia was simplified as a composite rod of circular cross-section, as shown in b. The bone fracture was a transverse fracture with a 1 mm fracture gap in the centre part of the tibia. The shapes and dimensions of all the components, including the screws, are illustrated in a). First, the second moment of inertia of the cross-section of the existing bone plate with respect to the centre of the tibia was calculated using Eqs. and the parallel axis theorem, as shown in . A plate width of 5 mm was used to reduce the contact area with the bone. As a result, the height of the plate was determined to be 3.2 mm. The shape and dimension of the screw fastening part was determined by considering the existing bone plate. This model was used as a basis for our new design of the simple composite bone plate (Type-1, see a). To reduce contact stress between the plate and bones, another type of composite bone plate (Type-II) that has a polymeric porous layer (PVC foam, 0.3 mm-thick) at the bottom of the screw-fastened parts was developed, as shown in b. This layer reduced the contact stress not only at the screw-fastening area but also at the other region of the plate bottom by generating a small gap between the bone and the plate. Finally, a composite bone plate that was reinforced in the centre region (Type-III) to enhance the bending stiffness of the bone plate was developed, as shown in c. The specifications of each composite bone plate are summarized in , including the type of material and the bending stiffnesses.Because bone plates are fastened to bone surfaces by screws, contact stresses are generated in both of the contacting bodies. To calculate the contact stresses, contact elements at the interface between the bone plate and bone surface were used in the finite element model. The friction coefficient (μ |
= 0.4) between the bone and the plate was determined by referring to Enzler’s bio-mechanical study a). The finite element analysis was performed in two discrete steps. The first step (STEP-I) modelled the screw-fastening process by contact analysis, and the second step (STEP-II) modelled the loading process, whose initial state was based on the results of the first step analysis The external loads generated by muscles near the fractured tibia produced bending deformation at the plate-bone assembly. During this behaviour, some interfragmentary strain at the fracture gap was generated. As previously mentioned, interfragmentary strain strongly affects the generation and development of curing tissues, especially during the early bone healing process. The deformation pattern and definition of the interfragmentary strain are shown in b. Using finite element analysis, interfragmentary strains were calculated for various applications of composite bone plates to the fractured bone. The calculated results (see a) showed that the Type-I composite bone plate (USN125, [0]2nT), which has the same bending stiffness as the previous composite plate, experienced high interfragmentary strains relative to the previous composite plate. Moreover, the Type-II composite bone plate (USN125, [0]2nT) with polymeric porous layers showed extremely high strains at a level that represents a serious problem in bone stabilization. The hatched box in a represents the most appropriate range of the interfragmentary strain (2–10%) for the generation of calluses proposed by Perren a. On the other hand, the Type-III composite bone plate, which was reinforced at the centre of the plate, had an interfragmentary strain distribution similar to the case of the previous composite plate. This was caused by the enhanced structural bending stiffness and by the contact behaviour of the reinforced plate centre generating additional resisting forces against the bending deformation of the plate-bone assembly (see b). This effect was proved by the contact stress analysis described in the next section.Even though the composite bone plates (Types-I and -II plates made of USN125, [0]2nT) had the same bending stiffness as the previous composite plate, their bending behaviour was quite different due to the different contact conditions. The previous composite plate had the largest contact area among all the composite bone plates because it had the same shape as the existing metal bone plate. On the other hand, the Type-I composite bone plate had a reduced contact area, and the Type-II composite bone plate had the smallest contact area, caused by the polymeric porous layer (see b). This layer diminished resisting forces against the bending behaviour of the structure. The Type-III composite bone plate had the highest bending stiffness due to stiffer material and the structural reinforcement in the centre region. This generated less interfragmentary strain, similar to the previous composite plate. On the other hand, the stainless steel bone plate provided strains that were too low to contribute to effective bone healing (see a). This result implies that the interfragmentary strain was affected not only by the bending stiffness but also by the contact condition.. On the other hand, the Types-II and -III composite bone plates significantly reduced the contact stresses at bone surfaces due to the polymeric porous layers that prevented the plate surfaces from contacting the bones directly, as shown in c and d. The Type-II bone plate showed an extremely low level of contact stress, and at the fracture site there was no contact stress because there was no contact at this region (see c). It was also found that all the plates except the Type-II composite plate had almost the same level of maximum contact stress at the fracture site, as shown in e. The average contact stress was reduced when the composite bone plates were applied to the fracture site. One notable thing from this analysis was that for the Type-III composite bone plate there was almost no contact stress around the screw holes, which reduced the possibility of the screw pull-out phenomenon due to bone resorption during healing process This paper examined simple composite bone plates with a rectangular cross-section using the results from a preliminary study On ductile fracture initiation toughness: Effects of void volume fraction, void shape and void distributionThis paper studies the effects of the initial relative void spacing, void pattern, void shape and void volume fraction on ductile fracture toughness using three-dimensional, small scale yielding models, where voids are assumed to pre-exist in the material and are explicitly modeled using refined finite elements. Results of this study can be used to explain the observed fracture toughness anisotropy in industrial alloys. Our analyses suggest that simplified models containing a single row of voids ahead of the crack tip is sufficient when the initial void volume fraction remains small. When the initial void volume fraction becomes large, these simplified models can predict the fracture initiation toughness (JIc) with adequate accuracy but cannot predict the correct J–R curve because they over-predict the interaction among growing voids on the plane of crack propagation. Consequently, finite element models containing multiple rows of voids should be used when the material has large initial void volume fraction.Ductile fracture of many structural materials is a result of void nucleation, growth and coalescence. In practical applications, the J-integral value at the initiation of crack growth, JIc, is often used as an important parameter to characterize the toughness of ductile materials. Micromechanics analysis of the fracture process makes it possible to link the macroscopic fracture toughness and the microstructure of the material. Two types of the mechanism-based approaches have been proposed in the literature to predict fracture toughness. In the first approach, voids are considered implicitly by using continuum damage material models, e.g., the Gurson–Tvergaard model (). This approach is attractive for simulation of extensive crack growth because detailed modeling of each individual void is avoided (). However, the primary disadvantage of this approach is that a precise constitutive model for characterization of the void-containing material behavior during the ductile fracture process is needed.In the second approach, voids are considered explicitly and modeled using refined finite elements. A distinct advantage of this approach is the exact implementation of void growth behavior. In order to establish crack advance, a failure criterion for the ligament between a void and the crack tip is required. proposed that coalescence occurs when the size of the ligament between the crack tip and the void becomes equal to the vertical diameter of the void. suggested that as soon as the spacing of neighboring voids becomes equal to their length, a slip plane can be drawn between the voids and the localized plastic flow causes ligament failure. proposed that void linkage occurs when the longest axis of the void is of the order of magnitude of the mean planar neighbor spacing. put forth a method to determine the onset of void coalescence by conducting unit cell analysis. Coalescence (internal necking) occurs when the macroscopic deformation of the unit cell shifts to a uniaxial straining state.The disadvantage of the explicit approach is due to computational limitations, only a limited number of voids can be included in the crack tip region. The published literatures in this area are mainly two-dimensional. examined the interaction between the crack tip and a cylindrical void under the plane strain, small scale yielding (SSY) conditions and estimated the fracture initiation toughness using the coalescence models by analyzed the growth of a single cylindrical void ahead of a blunt crack tip in the single-edge-notch bending specimens with different crack lengths. investigated the effect of crack tip constraint on void growth under mixed modes I and II loading. In these studies, only a single void is presented in the crack tip region. More recently, considered a row of six cylindrical voids ahead of the crack tip of the compact tension and center-cracked tension specimens and discussed the effects of specimen geometry, crack length and specimen size on the J–R curve. investigated two distinct mechanisms for ductile crack initiation and growth, the void by void growth mechanism and the multiple void interaction mechanism, by considering a row of cylindrical voids ahead of a plane strain, SSY crack tip. They found that transition of the two mechanisms is primary governed by the initial void volume fraction. For materials having smaller initial void volume fraction, interaction occurs only between the crack tip and the nearest void and crack growth follows a void by void mechanism. For materials with larger initial void volume fraction, simultaneous interaction of multiple voids ahead of the crack tip occurs both during initiation and subsequent crack growth.Published literatures on three-dimensional analysis of ductile fracture process using the explicit approach are relatively limited. investigated the influence of void arrangement on the macroscopic deformation and softening behavior of a unit cell and found that the 3D plane strain model containing a spherical void is stiffer than the 2D plane strain model having a cylindrical void. studied the effect of particle clustering on void damage rates in ductile failure of an aluminum alloy. They assumed a regular distribution of clustered particles and carried out a series of unit cell analyses. To predict fracture toughness, the void-containing cells need to be included in the crack tip region. studied the interaction of a spherical void and the crack tip. Their results suggest that the void grows faster towards the crack tip direction than in the crack opening direction, revealing strong interaction between the growing void and the crack tip. They also demonstrated that the initially spherical void grows much slower than the initially cylindrical void. by considering a row of spherical voids ahead of the crack front in the 3D SSY model. Similar to the results obtained by Tvergaard and Hutchinson, Kim et al. found that transition from the void by void growth mechanism to the multiple voids interaction mechanism is controlled by the initial void volume fraction. Using results of a systematic unit cell analyses as the material failure criterion, they also presented a procedure to predict fracture initiation, subsequent crack growth and the J–R curve.Most of the previous 3D analyses assume voids having spherical shape initially and consider only a single void or a single row of voids ahead of the crack tip. These analyses over-simplify the material microstructure and failure process. Many processed materials, such as rolled plates, have non-spherical voids and the void spacing is not uniform in all directions. Besides void volume fraction, void shape, void orientation and void distribution also have strong effect on the material failure mechanism and thus the fracture toughness. These issues are examined in this paper.This study considers the mode I, SSY problem, i.e., the plastic zone size is assumed to be small comparing to the geometric dimensions of the specimen. In ductile metals, voids often nucleate at relatively low stress levels due to fracture or decohesion of the large inclusions. For the purpose of analysis, voids are assumed to be present in the material at the onset of loading. (a) shows a periodical distribution of voids in the plane of crack propagation. In an attempt to rationalize fracture behavior, a local coordinate system is set up such that the x-axis represents the crack propagation direction, y-axis represents the crack opening direction and z-axis represents the thickness direction. Considering the existence of symmetry about the crack plane, only half of the region needs to be modeled. Except near free surfaces the deformation in the thickness direction can be assumed periodically symmetric. Neglecting the free surface effect allows us to apply the periodic boundary conditions and consider half of the void spacing distance in the thickness direction only, (b). Boundary conditions on the symmetry planes normal to z-direction are prescribed aswhere uz represents the displacement component in z-direction, tx and ty represent the components of surface traction in x and y directions respectively.Three types of initial void shapes, spherical shape, prolate shape, and oblate shape, are considered. shows the geometrical representation of the voids. The prolate and oblate voids are assumed to be axisymmetric about the y-axis and an initial aspect ratio is defined as W0 |
= |
R0y/R0x. Therefore, W0 |
= 1 corresponds to the spherical shape, W0 |
> 1 corresponds to the prolate shape, and W0 |
< 1 corresponds to the oblate shape.Several initial void arrangements are considered in this study. In (a), only one row of voids are included in the model. The voids are directly ahead of the crack tip. The spacing between adjacent voids is X0, which is the same as the distance from the first void to the crack tip. In (b), two rows of void are considered. The distance between the two rows is Y0 and a parameter λ0 is defined as the ratio of the void spacing in y-direction to the void spacing in x-direction, λ0 |
= |
Y0/X0, measuring the relative void spacing. In (c), voids in the second row are shifted towards the x |
= 0 plane by a distance of X0/2.To resolve the crack tip deformation field and enhance convergence of the nonlinear iterations, the finite element mesh contains an initial root radius at the crack tip. Previous studies have shown that the influence of initial root radius becomes negligible if it is sufficiently small comparing to the void spacing. Here the initial root radius of the crack tip is taken to be 0.01X0. Numerical solutions are generated by imposing displacements of the elastic, asymptotic mode I field (plane strain) on the outer circular boundary. In this study, the radius of the outer circular boundary is taken to be 10,000X0 to assure the small scale yielding conditions being satisfied. The displacements at the outer boundary are given byux=1+νEr02πKIcosθ22-4ν+2sin2θ2,uy=1+νEr02πKIsinθ24-4ν-2cos2θ2,where KI represents the mode I stress intensity factor, (r, |
θ) denote the crack tip polar coordinates, and r0 is the radius of the outer circular boundary of the SSY model. Loading of the SSY model proceeds by imposing displacement increments on the outer boundary according to the asymptotic fields.(a) shows a typical finite element mesh of the SSY model. Close-up of the crack tip region is shown in (b) and (c) for finite element modeling containing one row and two rows of voids respectively. A typical mesh containing two rows of voids consists of 18,000 twenty-node, isoparametric, brick elements (86,000 nodes) with reduced integration.It is natural to consider the material ahead of the crack tip as an array of unit cells with each unit cell containing a void at its center. The ratio of the void volume to the volume of the cell (including the void) defines the void volume fraction of the material. For models having a single row of voids as shown in (b), the cells are labeled with increasing numbers starting from the crack tip. For models having two rows of voids, e.g., (c), the cells are labeled using indices ij, where i refers to the row number and j refers to the position from the x |
= 0 plane, i.e., cell 21 refers to the first cell from the crack tip on the second row.The material chosen for this study obeys the power-law hardening (true) stress–strain relationwhere E |
= 200 GPa, σ0 |
= 600 MPa, ν |
= 0.3, and N |
= 0.1, which is representative of structural steel having an intermediate strength and moderate strain hardening. The stress–strain relation is implemented in ) suggest that there exist two failure mechanisms, single void growth mechanism and multiple voids interaction mechanism. The single void growth mechanism is explained by the interaction of the crack tip with the nearest void and the subsequent advance of the crack tip from one void to the neighboring void. The multiple voids interaction mechanism is described by the simultaneous interaction of multiple voids positioned on a plane ahead of the crack tip both during initiation and stable crack growth. Here we examine the effects of the initial relative void spacing, void pattern, void shape and void volume fraction on the void growth and material failure mechanisms.We start with the model containing only one row of five voids ahead of the crack tip as illustrated in (a). Three void shapes, spherical (W0 |
= 1), prolate (W0 |
= 4) and oblate (W0 |
= 0.25), with different values of initial void volume fraction, f0=(4/3)πR0x2R0y/(X02Y0), are considered in the analyses. Here the initial shape of the unit cells ahead of the crack tip is assumed to be cubic, i.e., X0 |
= |
Y0. In , the ratio of the void volume to its initial value, V/V0, is plotted as a function of crack tip loading, J/(X0σ0), for each of the five voids ahead of the crack tip. The trend remains the same for all six cases considered here, (a)–(f). For f0 |
= 0.001, only the first void from the crack tip has significant volume increase as J/(X0σ0) increases, which manifests the single void growth mechanism. As f0 increases, the interaction among voids becomes important, which in turn elevates the void growth rate. As a result, the ductile failure mechanism transits from single void growth to multiple voids interaction. When f0 |
= 0.005, several voids grow almost simultaneously as J/(X0σ0) increases. These results agree with the general conclusion drawn by compares the growth rate of the first void as a function of J/(X0σ0) for the six cases. It can be seen that both initial void shape and void volume fraction have strong effect on the rate of void growth. For the same initial void volume fraction, the oblate void grows faster than the spherical void and the spherical void grows faster than the prolate void, i.e., the void growth rate decreases with W. For the same initial void shape (W0), the void which has a larger initial volume fraction exhibits a faster growth rate, i.e., the void growth rate increases with f0.To demonstrate the significance of void interaction on material failure. The results obtained using the model containing a row of five voids are compared with the results of a model containing only one void ahead of the crack tip. As an example, we consider the spherical void shape. (a) compares the growth rate of the nearest void from crack tip between the single void model and the model containing five voids and (b) compares the reduction of the ligament between the crack tip and the nearest void between the two models. The comparison are made for two initial void volume fractions, f0 |
= 0.001 and f0 |
= 0.01. For the case of low initial porosity, the two models do not reveal any noticeable difference in void growth rate and the reduction rate of the ligament between the crack tip and the nearest void. However, there is a noticeable difference for the case of high initial porosity. Interaction among multiple voids elevates the void growth rate and accelerates the failure process. Therefore, the finite element model must include sufficient number of voids when the failure mechanism is due to multiple voids interaction. Earlier studies, e.g., , often consider a single void ahead of the crack tip. These studies under-predict void growth in high porosity materials and over-estimate the fracture toughness.Previous studies only consider a single row of voids positioned on the plane of crack propagation. But in real materials voids also exist off the plane of crack propagation. We first consider the void pattern shown in (b) with equal initial void spacing in x and y directions, i.e., λ0 |
= 1. For the prolate and oblate voids, the initial aspect ratios are taken to be 4 and 0.25 respectively. shows the ratio of the void volume to its initial value, V/V0, as a function of crack tip loading, J/(X0σ0), for several voids ahead of the crack tip. For f0 |
= 0.001, only the first void directly ahead of the crack tip experiences significant growth as J increases, which demonstrates the single void growth mechanism. The growth rates of the voids in the first row hardly show any difference from the results obtained using models containing only one row of voids. This suggests that void interaction between different rows is negligible. It is worth noting that the second void in the second row (void 22) has noticeable volume increase.For f0 |
= 0.005, the first two voids in the first row experience significant volume increase as J increases, (b), (d) and (f). This is different from the results shown in (b), (d) and (f), where all five voids grow almost simultaneously. Clearly, the presence of the voids in the second row decreases the growth rates of the voids in the first row. In contrast to the model containing only one row of voids where the material above the void-containing cells is dense, for the model containing two rows of voids, the material surrounding the first row cells is porous. The stress triaxiality in porous materials cannot reach as high as in dense materials. Therefore, the void growth rate is smaller in the model containing two rows of voids. The decrease in growth rate is more significant as the distance from the void to the crack tip increases. Consequently, one should expect that the predicted J–R curve to be steeper using models containing two rows of voids than using models containing only a single row of voids.To demonstrate the transition to multiple voids interaction mechanism, we consider a model containing two rows of spherical voids with f0 |
= 0.01. As expected, multiple voids grow concurrently as J increases, . Similar results can be obtained by considering prolate and oblate voids.In summary, the presence of the second row voids has negligible effects on the growth of the voids on the plane directly ahead of the crack front when the initial void volume fraction is small and the void growth mechanism is void by void. Consequently, for computational simplicity, it is sufficient to include only one row of voids in the finite element model when the f0-value is small. However, as f0 increases, the effects of the second row voids become more and more significant. Their presence delays the transition of the fracture mechanism from void by void growth to multiple voids interaction. Therefore, for large f0-values, the finite element should include multiple rows of voids.In above calculations, the void spacing is assumed to be equal in all three directions, i.e., the unit cells are cubic. This assumption is not valid for some materials, e.g., the rolled plate where the void spacing is shorter in the thickness direction. In this subsection the effect of relative void spacing is examined. The finite element models used in Section containing two rows of voids are modified such that λ0 takes different values. For the case of λ0 |
= 1.5, the V/V0 versus J/(X0σ0) curves for several voids ahead of the crack tip are shown in , where the initial void shape is prolate with W0 |
= 4. When f0 |
= 0.001, the results displayed in (c): only the void directly ahead of the crack tip exhibits significant growth. However, when f0 |
= 0.005, the results are quite different: (b) displays a multiple voids interaction mechanism in contrast to the void by void growth mechanism shown in (d). Therefore, an increase in λ0-value intensifies the interaction among neighboring voids and reduces the f0 value at which transition from the void by void growth mechanism to the multiple voids interaction mechanism occurs.For the case of λ0 |
= 2/3, the V/V0 versus J/(X0σ0) curves are shown in for the oblate voids with an initial aspect ratio of 0.25. Comparing with the results shown in (e) and (f), the smaller λ0-value reduces the interaction among the first row voids and delays the occurrence of the multiple voids interaction mechanism.(c) is considered, where voids in the second row are shifted towards the x |
= 0 plane by a distance of X0/2. The spherical void shape is used in the demonstration and the void spacing is assumed to be equal in x and y directions, i.e., λ0 |
= 1. shows the V/V0 versus J/(X0σ0) curves for three initial f0 values, 0.001, 0.005, and 0.01. For f0 |
= 0.001, only the first void directly ahead of the crack tip experiences significant volume increase as J increases. For f0 |
= 0.005, the second and third voids start to show significant volume increase at higher J levels. As f0 increases to 0.01, multiple voids in the first row grow almost concurrently. These results are the same as those shown in . Change of void pattern by shifting the positions of voids in the second row has negligible effect on the grow rates of voids in the first row.Macroscopic crack initiation is said to have occurred upon coalescence of the growing voids with the crack tip. Several mechanistic observations have been put forth to explain void coalescence. Coalescence can occur through shear band formation, or through formation of “void sheets”, or through impingement of neighboring voids. It is very difficult to implement these coalescence mechanisms directly to the numerical model. As a viable alternative, a critical ligament reduction ratio has been introduced to indicate the onset of void coalescence (, the critical ligament ratio cannot be taken as a constant. The dependencies of the critical ligament reduction ratio on the macroscopic stress state of the representative material volume, the initial void shape and void volume fraction, and other factors can be obtained by conducting a series of unit cell analysis., the material in the crack tip region can be considered as an array of cells. Each cell is a representative material volume containing a void at its center. The macroscopic stresses and strains of the cells in the SSY model are computed as follows:In above equations, Σij represent the macroscopic stress components, σij represent the (true) stress components of the matrix, V is the volume of the cell including the void, Eij represent the macroscopic (true) strain components, inc is the index for a load increment and ninc is the total number of increments for a given load. The macroscopic strain increments ΔEij are calculated from the displacement increments Δui as ΔEij=12V∫S(Δuinj+Δujni)dS. The cell volume V is computed as V=∫Sx1n1dS, where S is the outside surface of the cell with ni being the components of the normal vector of S. These values are evaluated using the finite element integration scheme (). The macroscopic effective stress (Σe), hydrostatic stress (Σh), and effective strain (Ee) are given byΣe=12Σxx-Σyy2+Σyy-Σzz2+Σzz-Σxx21/2,Σh=13Σxx+Σyy+Σzz,Ee=23(Exx-Eyy)2+(Eyy-Ezz)2+(Ezz-Exx)21/2.Since the deformed shape of the cells in the SSY model is symmetric about the y and z planes, the macroscopic shear stress/strain components are all zero and are not included in Eq. To characterize the macroscopic stress state of the cell, the following stress ratios are introduced studied the effects of the triaxial stress state and found that the stress triaxiality ratio T alone cannot uniquely characterize the effect of macroscopic stress state on void growth and coalescence. The Lode parameter should be used to distinguish different stress states having the same stress triaxiality ratio. Defining showed that a cell when subject to the same stress triaxiality ratio would tend to react differently when θ is different. The stress triaxiality ratio along with the parameter θ can be used to specify stress state. shows the variation of Σe, T, ρ1 and θ as the increase of applied load J for five cells ahead of the crack tip in the model containing a single row of voids as shown in (a). Here the initial void shape is prolate (W0 |
= 4) and the initial void volume fraction is f0 |
= 0.005. As expected, the stress triaxiality ratio T and the parameter θ are not constant during the loading history. The triaxiality ratio increases with applied J in the plastic deformation region of the cell. A sudden increase in triaxiality ratio occurs due essentially to collapse of the cell and rapid drop of Σe. The parameter θ also increases with the applied load. This is because the stress ratio in the thickness direction becomes larger as the applied J increases. Interestingly, the macroscopic stress ratio ρ1 for each cell remains almost a constant after macroscopic plasticity occurs. Similar results are obtained when different void shapes and initial void volume fractions are considered. compares the variations of Σe, T, ρ1 and θ of the first cell in models containing two rows of voids with different void arrangements. The prolate void shape (W0 |
= 4) with initial void volume fractions of 0.001 and 0.005 is considered here. The trends are similar to those shown in and it seems that the relative void spacing (λ0) does not have a significant effect on the macroscopic stress state of the first cell.Considering the material composed of void-containing cells, failure of the ligament between neighboring voids corresponds to the process of internal necking. Coalescence (internal necking) will occur when the macroscopic deformation of the cell shifts to a uniaxial strain state (). To utilize this idea, we consider the representative material volumes subjected to the loading conditions similar to the cells in the SSY models discussed in the previous section. A one-eighth symmetric finite element mesh of the unit cell containing an initially spherical void at its center is shown in (b) display the resultant deformed shape of the model. Displacement boundary conditions are prescribed on the outer surfaces of the cell. The displacement component in z-direction is constrained on the face normal to the z-axis. The displacement components are specified on the faces perpendicular to the x-axis and y-axis incrementally using the procedure developed by so that the macroscopic stress ratio ρ1 |
= |
Σxx/Σyy remain constant during the loading history. Details of how to prescribe the boundary conditions can be found in Variation of the deformed cell width in x-direction with the macroscopic effective strain of the cell, shown in (a), reveals the shift to uniaxial straining. Here results for three initial void shapes (W0 |
= 0.25, 1, 4) are presented. The cells are initially cubic (λ0 |
= 1) and the initial void volume fraction is f0 |
= 0.005. The macroscopic stress ratio ρ1 is taken as 0.54. (b) shows the macroscopic effective stress versus effective strain for the cell displaying the macroscopic softening. The circles in represent the onset of uniaxial straining mode, i.e., void coalescence.At the onset of void coalescence, the ligament reduction ratio, defined as the ratio of the current ligament length (shortest distance between two adjacent voids in x-direction) to the initial ligament length, can be calculated. The critical ligament reduction ratio is denoted as χc. To determine χc, we conduct unit cell analyses for the cases of various initial relative void spacing, void shape, void volume fraction and different values of the macroscopic stress ratio ρ1. It is found that χc depends on λ0, W0, f0, and ρ1. An increase in either initial void volume fraction, or void aspect ratio, or applied stress ratio tends to increase χc. lists the χc-values for different cases.Using the critical ligament reduction ratios obtained in Section and the SSY models described in Section , the fracture initiation toughness can be predicted. Here the fracture initiation is defined as when the first void coalesces with the crack tip. To determine the onset of fracture initiation, it is necessary to estimate the macroscopic stress ratio ρ1 of the ligament between the crack tip and the first void. However, it is difficult to calculate the ligament stress ratio ρ1 directly. We approximate the ligament ρ1 value by extrapolation using the macroscopic stress ratios calculated for the first two cells. With the ρ1 of the ligament estimated, the χc-values in can be used to determine the applied J at which the first void coalesces with the crack tip. This applied J value can be regarded as the fracture initiation toughness (JIc). Simple linear interpolation is used when the ρ1-value of the ligament is different from the ρ1-values listed in Using the above approach, the variation of JIc with the initial relative void spacing, void pattern, void shape and void volume fraction can be predicted. (a) shows the predicted dependence of JIc on the initial void volume fraction using the finite element models containing two rows of spherical, prolate (W0 |
= 4), and oblate (W0 |
= 0.25) voids. In general, the value of JIc increases as f0 decreases. For the same value of f0, JIc is highest when the initial void shape is prolate and lowest when the initial void shape is oblate, i.e., JIc increases with W0. The difference in predicted JIc for different void shapes becomes less significant as f0 becomes large. Change of the void pattern by shifting the positions of the voids in the second row (indicated by the cross symbol) does not result in noticeable difference in JIc.(b) compares the predicted JIc-values using models containing two rows of voids with those using models containing one row of voids. No noticeable difference is observed between the predicted JIc-values using the two models. This rationalizes the approaches used in the previous studies, e.g., , where only one row of voids are included in the finite element model. However, it is very important to point out that the single row void model over-predicts the growth rates of voids other than the first one when f0 is large (see the results in Section (c) demonstrates the effect of relative void spacing on the fracture initiation toughness. The results show that the fracture toughness decreases with λ0. This is easy to understand because larger λ0-value means shorter relative void spacing in the x-direction and thus earlier coalescence of the voids with crack tip.As defined previously, JIc is determined as the applied J-value when the reduction of the ligament length between the first void and the crack tip reaches the critical ratio χc. The χc for the first ligament varies with the initial relative void spacing, void pattern, void shape and void volume fraction. If we collect the χc-values for all the cases presented in (a)–(c) and plot χc versus the corresponding JIc-value for each case, we can reveal a trend of JIc decreasing with the increase of χc, (d). It is interesting to note that the relationship between JIc and the χc-value for the first ligament can be approximated by a straight line.It is worth noting that in this study coalescence is defined as the onset of internal necking. Most engineering materials contain more than one populations of inclusions and/or second phase particles. Due to localized plastic deformation between the enlarged voids and between the void and the crack tip, small particles in the ligaments will nucleate secondary microvoids. Rapid growth and coalescence of secondary voids will accelerate the ligament failure process. Nucleation and growth of secondary microvoids are not taken into account in this study, and therefore, the critical ligament reduction ratios determined above can be regarded as the lower bound values and the fracture toughness values predicted using those critical ligament reduction ratios are the upper bound values for the material.The method described in previous sections can be used to analyze the anisotropy of a rolled steel plate. analyzed these experiments using a continuum damage model, the Gologanu–Leblond–Devaux model. Their numerical predictions are in reasonable agreement with the experimental results.Here we use the discrete void model presented in previous sections to analyze the anisotropy of fracture initiation toughness of the C–Mn steel in TL and SL directions. Since we are only interested in predicting JIc, the computational models used in these analyses contain only one row of five voids directly ahead of the initial crack. summarizes the parameters characterizing the material and the comparison between the measured and predicted fracture initiation toughness (JIc). The difference in toughness between the two directions of propagation is well captured, and the predicted and measured fracture toughness values are in reasonable agreement given the uncertainty in identifying the void spacing.In this study, effects of the initial relative void spacing, void pattern, void shape and void volume fraction on ductile fracture toughness are analyzed using three-dimensional, small scale yielding models, where voids are assumed to pre-exist in the material and are explicitly modeled using refine finite elements. Based on our detailed analyses, the following remarks can be made.(1) Our analyses re-affirm the two distinct void growth mechanisms put forth by , i.e., void by void growth mechanism for materials containing small initial void volume fractions and multiple voids interaction mechanism for materials containing large initial void volume fractions. Our results reveal that, besides the initial void volume fraction, other factors also affect void growth mechanism when the initial void volume fraction is large. Voids deviated from the crack growth plane reduce the interaction among voids on the crack growth plane and delay the transition from void by void growth mechanism to multiple voids interaction mechanism. Increase of λ0 (relative void spacing) intensifies the interaction among neighboring voids and facilitates the transition from void by void growth mechanism to multiple voids interaction mechanism. Change of the void distribution pattern by shifting the positions of second row voids does not affect the growth rates of voids in the plane of crack propagation. Our results also show that when other parameters are the same, the oblate void grows faster than the spherical void and the spherical void grows faster than the prolate void.(2) A critical ligament reduction ratio (χc), determined from unit cell analysis, is introduced to denote material failure and it is found that χc varies with the initial relative void spacing, void pattern, void shape and void volume fraction. The fracture initiation toughness (JIc) is determined as the applied J-value when the reduction of the ligament length between the first void and the crack tip reaches the critical ratio χc. Our results reveal that JIc increases with decreasing f0. For the same value of f0, JIc is highest when the initial void shape is prolate and lowest when the initial void shape is oblate. Existence of the second row voids and change of void pattern do not result in noticeable difference in JIc. However, the initial relative void spacing has significant effect on JIc. These results can be used to explain why various degrees of fracture toughness anisotropy are observed in industrial alloys.(3) Previous studies often use finite element models containing a single row of voids. Our analyses suggest that these simplified models are sufficient when the initial void volume fraction remains small. When the initial void volume fraction is large, these simplified models can predict JIc with sufficient accuracy but cannot predict the correct J–R curve. In order to predict the J–R curve for a material having large initial volume fraction, the finite element model should include multiple rows of voids.Magnetron plasma mediated immobilization of hyaluronic acid for the development of functional double-sided biodegradable vascular graftThe clinical need for vascular grafts is associated with cardiovascular diseases frequently leading to fatal outcomes. Artificial vessels based on bioresorbable polymers can replace the damaged vascular tissue or create a bypass path for blood flow while stimulating regeneration of a blood vessel in situ. However, the problem of proper conditions for the cells to grow on the vascular graft from the adventitia while maintaining its mechanical integrity of the luminal surface remains a challenge. In this work, we propose a two-stage technology for processing electrospun vascular graft from polycaprolactone, which consists of plasma treatment and subsequent immobilization of hyaluronic acid on its surface producing thin double-sided graft with one hydrophilic and one hydrophobic side. Plasma modification activates the polymer surfaces and produces a thin layer for linker-free immobilisation of bioactive molecules, thereby producing materials with unique properties. Proposed modification does not affect the morphology or mechanical properties of the graft and improves cell adhesion. The proposed approach can potentially be used for various biodegradable polymers such as polylactic acid, polyglycolide and their copolymers and blends, with a hydrophilic inner surface and a hydrophobic outer surface.Cardiovascular disease (CVD) accounts for the death of more than 18 million people a year, with vascular atherosclerosis being the cause in ~65% cases One of the promising strategies for solving the problem of graft deficiency is the development of artificial vessels based on bioresorbable polymers with an ability to stimulate regeneration of a blood vessel in situ To address this problem, several basic strategies have been proposed, including pore size gradient of graft in the direction of luminal surface-adventitia In this work, we propose a two-stage technology for processing vascular graft from PCL, which consists of 1) plasma treatment in a reactive magnetron discharge with the sputtering of a titanium target in a nitrogen atmosphere and 2) subsequent immobilization of hyaluronic acid on its surface.Plasma treatment enables the formation of a gradient coating in the bulk of the porous graft material, characterized by a negative concentration gradient of ions of the sputtered target from the external to the internal surface of the graft, while maintaining the graft structural integrity and mechanical strength Current strategies to reduce aging in plasma polymer film include enhancement of the degree of cross-linking, plasma coating architecture engineering, e.g. gradient structure and post-plasma grafting to decrease the number of reactive sites To obtain stabilized hydrophilic surfaces, a vascular graft modified in a plasma of a reactive magnetron discharge was kept in an aqueous solution of hyaluronic acid (HA) due to its high biocompatibility Hence, proposed strategy makes it possible to obtain a polycaprolactone vascular graft with a stable superhydrophilic outer surface (from the adventitious side) while maintaining high hydrophobicity of its inner surface (luminal surface).Prior to modification, the electrospun PCL vascular grafts were placed in a vacuum at 10-2 Pa to remove the residual solvent. The scaffolds were modified by a DC magnetron sputtering technique. The metal target was a chemically pure (99.99%) titanium (Ti) placed under a nitrogen atmosphere (N2). The modification parameters were set as follows: the discharge power 20, 45, 75, 105, 135 W), operating pressure in the chamber of 0.7 Pa (99.99% N2 gas), magnetron-to-target distance of 40 mm, magnetron area of 240 cm2 and 4 min modification time. To avoid excessive rise of the sample temperature during the plasma treatment, the modification time was divided into one-minute treatments, each separated by a one-minute cooldown period. The maximum chamber temperature during the whole process was 39 °C. Hereafter, ‘top’ refers to plasma-treated side of the electrospun PCL vascular grafts; ‘bottom’ refers to the untreated side of the electrospun PCL vascular grafts.The plasma-treated samples were placed in a 0.1 wt% aqueous solution of HA (Mw = 2 × 106 g/mol, Sinopharm Chemical Reagent Co., China) for 30 min. The samples were washed with distilled water and dried at room temperature. The designed HA concentration (0.1 wt%) maximises the biocompatibility of the scaffolds without evoking an additional immune response The morphology of the samples was investigated by SEM on an ESEM Quanta 400 FEG instrument (FEI, USA). Prior to the investigation, samples were coated with a thin gold layer by the magnetron sputtering system (SC7640, Quorum Technologies Ltd., UK). The fibre diameter was determined from SEM images captured in five fields of view using ImageJ 1.38 software (National Institutes of Health, USA). The average diameter was determined from at least 60 fibres.The wettability of the samples was characterised by depositing 3 μL drops of polar liquid (water and glycerine) at different positions on the samples in a Krüss EasyDrop contact-angle measurement system and capturing the images one minute after depositing the drops. All data are represented as the averages and standard deviations of the measurements taken at five different spots on the surface of the respective sample.XPS measurements were carried out in an Escalab 250Xi machine (Thermo Fisher Scientific Inc., UK) equipped with a monochromatic AlKα radiation source (photon energy: 1486.6 eV). The spectra were acquired in constant-pass energy mode at 100 eV for the survey spectrum and 50 eV for the element core-level spectrum. The spot size of the X-ray beam was 650 µm, and the total energy resolution was approximately 0.55 eV. Investigations were carried out at room temperature in an ultrahigh vacuum (with pressure of the order of 1 × 10−9 mbar; in the electron–ion compensation system, the Ar partial pressure was 1 × 10−7 mbar). The library of the reference XPS spectra, including the atomic registration sensitivity factors, was available in the Advantage Data System provided by the instrument manufacturer. The peaks were deconvoluted by Avantage software (Thermo Fisher Scientific Inc., UK) set to Shirley background subtraction followed by peak fitting to Voigt functions with an 80% Gaussian and 20% Lorentzian character. Each XPS experiment included 2 replicates. Each time, the measurements were done at least at 2 different locations.The mechanical properties of the samples were investigated by uniaxially stretching five pieces of each sample in a tensile testing machine (Instron 3369; Illinois Tool Works, USA) with a 50 N sensor. The traverse speed was set to 10 mm/min.Given that the produced grafts had two different surfaces, the cell adhesion to each of them was studied. To do that, cells were separately cultured on the plasma-treated HA-coated side of the graft (further referenced as PCL-HA top) and the opposite, non-modified side of the graft (further referenced as PCL-HA bottom). Non-treated PCL graft samples were used as a control (Control samples).Cell adhesion was studied using human multipotent mesenchymal stem cells (MMSCs) harvested from subcutaneous adipose tissue of healthy donors. All experiments were performed according to the Declaration of Helsinki within an approval of Ethics Committee of the Almazov National Medical Research Centre (no. 12.26/2014; December 1, 2014). Written informed consent was obtained from all the subjects before the fat tissue biopsy. Adipose-derived human multipotent MSCs had the following phenotype: CD19-, CD34-, CD45-, CD73+, CD90+, CD105+, as confirmed by flow cytometry (GuavaEasyCyte8; MerckMillipore, Darmstadt, Germany) and monoclonal antibodies (Becton Dickinson, Franklin Lakes, NJ, USA). Cells were cultured in α-MEM medium (PanEco, Moscow, Russia) supplemented with 10% fetal calf serum (HyClone Laboratories, Inc., Logan, UT, USA), 1% L-glutamine, and 1% penicillin/streptomycin solution (Invitrogen, Waltham, MA, USA) in 37 °C and 5% CO2. Samples of the grafts were cut in 12 × 8 mm pieces and placed in the wells of a 24-well plate. Cells were seeded on top of the scaffold at a density of 5 × 104 cells per well, and co-cultured with the material for 72 h in above mentioned conditions. The experiment was performed in triplicates. After 72 h, the samples were transferred to the new wells of the plate, washed in PBS and fixed in a 4% solution of paraformaldehyde (PFA) for 10 min. The cells adhered to the sample surface were permeabilized by 0.05% Triton X-100 followed by rinse with PBS, blocked with 10% goat serum solution in PBS for 30 min at room temperature and incubated with an anti-vinculin antibodies (Thermo Fisher Scientific, Waltham, MA, USA) at 1:200 dilution for one hour. After three PBS washes, the cells were incubated with AlexaFluor 568 goat anti-mouse IgG (H + L) anti-bodies (Invitrogen, Waltham, MA, USA) at 1:1000 dilution for one hour at room temperature in the dark. The cells were washed thrice with PBS (5 min each) and stained with 4′,6-diamidino-2-phenylindole (DAPI) for visualization of the nuclei.For the quantitative and qualitative analysis, the adherent cells were imaged using Axiovert inverted fluorescence microscope (Zeiss, Germany) equipped with a Canon camera. Samples were placed between the two glass slides and ten different fields of view were captured at magnifications of ×10 and ×40 for each replicate. Quantitative analysis was performed by analysing images taken at a ×10 magnification (counting the nuclei of cells stained with DAPI), while qualitative analysis was performed using images taken at ×40 magnification (assessing the morphology of cells by stained cytoskeleton). To prevent flotation of the scaffolds on the medium samples were cut slightly oversized to the culturing wells. This allowed to tightly fix the samples at the bottom of the wells. The position of the samples was controlled during the cultivation process.Statistical analysis of physico-chemical data was performed in GraphPad Prism, version 8.00 for Windows (GraphPad Software, La Jolla California, USA) using Kruskal–Wallis test and Student’s t test. The data are shown as mean (SD) ± standard deviation (SD). Statistical analysis of biological data was performed using the non-parametric Mann-Whitney U test. The data are presented as arithmetic mean (Mean) ± standard error (SE). Differences were considered significant at the p < 0.05 level.SEM images of the PCL vascular grafts before and after plasma treatment are shown in The untreated electrospun PCL vascular graft consists of randomly intertwined cylindrical fibres with an average diameter of 2.0 ± 0.4 μm ((a)). As the treatment current increased from 20 to 45 A, the mean fibre diameter slightly decreased due to plasma etching and thermal impact. At a plasma treatment of 75 A, the fibres were melted in crossing areas and fibre surface was wrinkled. The formation of wrinkles and cracks on the fibre’s surface could be caused by the difference in the elasticity of the polymer base and thin TiON coating whose thickness increases with an increase of the treatment time. An increase of discharge power leads to the overall melting of the fibres (The mechanical properties are essential for effective long-term implantation of tissue-engineered scaffolds. To evaluate plasma treated electrospun electrospun PCL vascular grafts, mechanical properties such as tensile strength, elongation and Young’s modulus was measured. Results of the investigation of electrospun electrospun PCL vascular grafts mechanical properties are presented in As the current increased in the plasma treatment, Young’s modulus and tensile strength improved, especially in the 75–135 W range. This trend can be attributed to thermal fibre bonding within the scaffold, which enhances the tensile properties of the electrospun PCL scaffolds by thermally induced shrinkage and molecular chain relaxation of the amorphous regions Lowering the discharge power of the DC magnetron sputtering and applying the HA treatment slightly enhanced Young’s modulus without changing the morphological properties (The effect of the plasma treatment and HA immobilisation on the wettability of the PCL nonwoven material was examined through contact-angle measurements. The results of the wettability investigation are presented in The contact angle was measured on water and glycerol droplets. The initial PCL scaffolds were hydrophobic with a water and glycerol contact angle of 122.4 ± 3.7° and 129.4 ± 2.6°, respectively (). Immediately after the DC plasma treatment at 20, 45 and 75 W, the top side of the scaffold was fully wetted while the bottom side remained hydrophobic (). At the higher power (105 and 135 W), the bottom side became more hydrophilic: the contact angles of glycerol and water on the bottom sides were below 60° (). To demonstrate the differences in wettability of modified samples a highly viscous polar liquid, glycerol, was used.To investigate the wettability changes over time, the contact-angle measurements of the DC plasma-treated samples were repeated after 3 days, 7 days and 6 weeks (). A hydrophobic recovery of plasma treated surface of PCL scaffold was observed. For instance, the contact angle on the top side of the samples treated with 20 and 45 W increased to 38.4° and 41.8°, respectively, after 3 days and to 103.5° and 83.2°, respectively, after one week. However, no significant changes appeared on the bottom sides (). The hydrophobic recovery was retarded on samples treated at a higher power (). After 6 weeks, all plasma treated samples exhibited hydrophobic properties on both sides, and the contact angles increased towards their original values (The HA immobilisation prevented hydrophobic recovery and the top side exhibited superhydrophilic properties throughout the study period (Figs. 2 and S7). As the treatment power increased, the bottom surface became more hydrophilic, as observed on plasma-treated scaffolds without additional modification. This effect is demonstrated in the video file in the Considering the wettability, morphological features and mechanical properties of the produced materials, 20 W was determined as the most suitable discharge power for modifying the electrospun PCL vascular grafts by the DC plasma treatment.The effects of plasma treatment and HA immobilisation on the chemical composition and bonding states of the electrospun PCL vascular graft material were elucidated by XPS. shows the atomic ratios and relative areas of the functional groups calculated by deconvoluting the C1s and N1s peaks.The atomic constituents of the PCL vascular grafts give rise to unique spectral peaks in the photoelectron spectrum (e.g. the C1s peak), informing the bonding states of the PCL atoms. In the XPS survey spectrum of PCL vascular grafts peaks corresponding to C1s and O1s could be found. The C1s spectra of the control PCL vascular grafts were consistent with those reported in previous studies In the XPS survey spectrum of the fresh plasma treated PCL vascular grafts peaks corresponding to C1s, N1s, Ti2p and O1s could be found. The plasma treatment altered the shape of the C1s peak, increasing the C3 and C4 components and decreasing the C2 component. Peaks of amine and/or amide moieties (NH2, H2NC = O, 399.6 eV) and protonated/hydrogen bonded amine (NH3+, 402.3 eV) In the Ti2p spectrum of the plasma-treated PCL vascular grafts three components corresponding to TiO2 were observed: Ti2p1/2 at 464.2 eV, Ti2p3/2 at 458.3 eV, and satellite peak at 471.5 eV (The deposition of TiO2 instead of TiN after the plasma treatment could be explained by the following reasons. It is known that hydrophobic surfaces can adsorb water The mechanism of the coating deposition on PCL-based grafts surface is probably similar to one described for polylactic acid Thus, the interaction with nitrogen would result in amine and amide groups as was found by XPS. The incorporation of oxygen originating from water could give ether [–C–O–C–] as well as hydroxyl moiety [–(C–OH)–]. Indeed, the XPS study showed increased C3 and C4 components and decreased C2 component in the C1s spectra of the plasma treated PCL grafts. The oxidation of amino groups generated on the PCL surface to oximes and amides can also proceed resulting in increased oxygen content on the PCL surface.After the plasma treatment, the polar side groups formed on the PCL grafts surface probably contributed to the hydrophilicity improvement of the scaffold, as observed in the contact-angle measurements (). The XPS spectra and elemental composition of the plasma treated PCL grafts bottom were like those of PCL control graft, indicating that modification did not occur on the other side (In the plasma-treated PCL vascular graft stored for six weeks, the component intensities of the C1s peak differed from those of the freshly treated PCL graft and resembled those of the control graft. The C3 component was decreased and the C2 component was increased comparing to freshly treated PCL graft. In N1s spectra the intensity of the component corresponding to amine and/or amide moieties decreased, whereas the protonated amine component intensity increased (). Also, the decreased C/O ratio observed (The changes in chemical bonding states and element content as well as hydrophobic recovery () of the scaffold surface after six weeks could be explained by the mechanisms of polymer “aging” described earlier in After HA immobilisation on the plasma treated PCL grafts surface, the intensity of the C3 components of the C1s line increased (). The intensities of peaks corresponding to amine/amide moieties and protonated amino group () increase and decrease, respectively. This can be explained by the presence of [–(C–OH)–], [–C–O–C–], and [-C(O)-N] moieties in the HA chemical structure and is consisted with previous reports ) confirm HA immobilisation on the PCL surface. Comparing to O1s spectra of plasma treated PCL large high-energy component corresponding to several groups including amide (531.6 eV), ether (532.6 eV), hydroxy (532.9 eV), carbonyl (532.3 eV) and acetal (533.1 eV) increases in the O1s spectra of the PCL-HA graft (The largish proportion of polar groups in the HA structure contributed to the long-term superhydrophilicity of the PCL-HA surface (HA could bind to the PCL surface either covalently or by physical adsorption. It is known that if polymer is treated at vitreous state, the generated radicals and ions have reduced mobilities and can be trapped. Such trapped species could have a long lifetime. Thus, when a plasma treated graft was immersed in aqueous HA solution, the radicals remaining on its surface could interact with both water and HA resulting in covalent bonding of HA to PCL. However, the obtained data showed no strong evidences to support the described mechanism. On the other hand, partially protonated amino groups appear on the PCL surface after the plasma treatment. These groups can form a network of hydrogen bonding with hydroxy and carboxy groups of HA favouring HA physical adsorption on the graft surface. The elucidation of HA binding mechanism to plasma treated PCL graft surface is a complex issue that requires further additional studies.The XPS spectra and elemental composition of the PCL-HA grafts bottom were like those of PCL control scaffold, indicating that HA was not immobilized on the other side (To assess the influence of plasma treatment and HA immobilisation on biological properties of the fabricated materials, we cultured human adipose-derived MSCs on the superhydrophilic and hydrophobic surfaces of the scaffold () and evaluated the number of adhered cells and their morphology (The fluorescence microscopy results have shown that all studied samples supported MSC attachment and growth in vitro. While some MSCs adhered to the surface of the material, the others started migrating into the bulk of the graft. On the control PCL sample as well as on the PCL-HA bottom sample isolated spindle-shaped cells were mostly found (). The number of MSCs on the surface of the PCL control scaffold was significantly lower compared to other samples (p < 0.0001) (). It indicates a low functional activity of cells and a weak interaction of cells both among themselves and with the surface of untreated PCL graft Plasma treatment of the outer surface of the PCL vascular graft and subsequent immobilization of HA increased the number of adherent cells on the outer surface (PCL-HA top) by more than 65% compared to the surface of the untreated sample (). MSCs cultured on PCL-HA top samples were connected via numerous cytoplasmic bridges forming syncytium (g-i). Improved cell adhesion to the surface of PCL-HA top samples is most likely caused by hydrophilicity of the scaffold surface due to introducing the oxygen- and nitrogen containing polar groups via plasma treatment with subsequent immobilisation of HA. It is known that HA mediates various cellular events in vivo including adhesion and morphogenesis, mainly though the interaction with the cell surface receptor CD44 The number of cells on the inner surface of the PCL of the vascular graft (PCL-HA bottom) is less than on the outer surface (PCL-HA top), but significantly higher than on the untreated graft (). Probably, the reason for this is a low number of functional groups on the inner surface of the vascular graft and hydrophobicity, which prevents the immobilization of hyaluronic acid as well as cell adhesion.Improved cell adhesion, cells functional state and intercellular interaction observed in vitro on the outer surface of the modified graft (PCL-HA top), could result in increased endothelization rate from adventitia in vivo. Moreover, insignificant changes in the structure, mechanical properties and chemical composition of the luminal surface PCL of the graft will allow to maintain a low probability of blood clots inherent to grafts of this type Thin porous PCL-based double-sided grafts were fabricated by DC plasma treatment with subsequent immobilisation of HA. The plasma treatment changed the surface wettability of the PCL-based scaffolds from hydrophobic to superhydrophilic. However, after six weeks, the surface restored its hydrophobicity. The immobilised HA not only conferred high biocompatibility, but also stabilised the superhydrophilic surface against hydrophobic recovery.The proposed modification does not significantly affect the mechanical properties of the electrospun PCL grafts. The fabricated scaffolds with a hydrophilic outer surface that ensures proper endothelization and hydrophobic inner surface that prevents early loss of mechanical integrity and lead to gradient of wettability and degradation rate. Plasma scaffold surface modification followed by immobilization of hyaluronic acid improves cell adhesion.Overall, the use of DC magnetron plasma with appropriate modification parameters and subsequent HA immobilisation is a simple and scalable method for producing thin double-sided PCL vascular grafts with on hydrophilic and one hydrophobic side. This approach can potentially be used for modification of grafts from biodegradable polymers such as PCL, PLA, PGA and their copolymers and blends.Valeriya Kudryavtseva: Conceptualization, Investigation, Methodology, Writing - original draft, Writing - review & editing. Ksenia Stankevich: Investigation, Methodology, Writing - original draft. Anna Kozelskaya: Funding acquisition, Writing - original draft, Writing - review & editing. Elina Kibler: Investigation. Yuri Zhukov: Investigation. Anna Malashicheva: Investigation, Resources. Alexey Golovkin: Methodology, Investigation, Writing - original draft, Writing - review & editing, Resources. Alexander Mishanin: Investigation, Validation. Victor Filimonov: Validation, Resources. Evgeny Bolbasov: Conceptualization, Methodology, Supervision, Writing - original draft. Sergei Tverdokhlebov: Project administration, Resources.The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.Supplementary data to this article can be found online at The following are the Supplementary data to this article:Analysis of damage localization in composite laminates using a discrete damage modelDamage localization around stress raisers and material defects in laminated composites is studied using a discrete damage mechanics model augmented by a fiber damage model. The proposed formulation captures the damaging behavior of plates with initial defects and stress raisers such as holes, including damage initiation, evolution, and ultimate fracture of the specimen. It also helps explain the reduction of stress concentration factor when matrix and fiber damage develop. The state variables are the crack density and the fiber failure damage. The formulation is implemented as a material model in Abaqus applicable to laminated composite plates and shells. Material defects are simulated by inserting an initial crack density in a small region of the specimen. Stress raisers are simulated by an open hole. The predictions are shown to be insensitive to mesh density. Further, damage localizes near stress raiser and material defects, thus numerically demonstrating the objectivity of the proposed model. Qualitative and quantitative comparisons with experimental data are presented.The development of models able to reproduce damage evolution and its effect on the stress and strain fields of a laminate is of great importance in order to reduce the number of tests necessary for the certification of a structural element. Modeling the progressive failure of composite laminates is a complex task, due to the interaction of several failure mechanisms. This problem cannot be considered completely solved There are several methodologies to model the failure of composite materials which have been used in the scientific literature, as failure criteria (usually stress based) or continuum damage mechanics (CDM) models. The two methods are often combined, using failure criteria to predict the onset of the damage and a CDM model to predict its evolution Failure criteria provide information about the onset of damage but not about its evolution, so that for composites that experience damage evolution prior to ultimate failure, these criteria are not sufficient. Several phenomenological criteria have been developed, many of which consider several failure mechanisms such as fiber breakage, fiber buckling, and matrix cracking, CDM models homogenize the damage by reducing the stiffness using a second- or fourth-order phenomenological damage tensor by fitting the evolution of damage variables with an evolution equation. These models require parameters that are difficult to determine experimentally. Other problems of this methodology are their mesh-size dependence and the difficulty to describe the local effects of the stress redistribution on the damage zone An alternative to these methodologies is discrete damage mechanics (DDM)Among the models based on this methodology, Matrix cracking appears in laminae subjected to transverse tensile load and/or inplane shear. However, A model meant to analyze damage evolution must be able to predict damage localization. The damage of a laminate usually begins at points of stress concentration, such as material defects or stress raisers (e.g., holes, fillets, etc.). Many authors use the problem of a laminate with a hole to validate damage models due to the complex stress fields close to the hole caused by stress concentration and anisotropy of the material Due to the complexity of the phenomenon, damage evolution in laminates is usually studied using numerical models In this work, the applicability of the DDM methodology To study damage localization, the discrete damage model Matrix cracking results in a set of parallel cracks, which can be represented by the crack density λ in each lamina. The crack density is defined as the inverse of the distance between two adjacent cracks (number of cracks per unit length, ). The model is formulated on a representative volume element (RVE), defined as the volume enclosed by the mid-surface and the top surface of the laminate t, the surface between two consecutive cracks 2l, and a unit length parallel to the cracks. The cracks occupy the entire thickness of the lamina, since all cracks are parallel to the fiber direction and practical designs avoid thick laminae. A unit length is chosen because it is assumed that the crack propagates along the fiber direction a distance much larger than the ply thickness. In coupons under tensile load, cracks propagate from one edge to the other of the specimen (about 25 mm). Therefore, and to afford an analytical solution, the state of damage on neighboring elements along the fiber direction is not considered in the solution. The accuracy of such assumption can be assessed only indirectly by the quality of measurable macroscopic response such as the failure load in Since the objective is to calculate the laminate stiffness reduction due to cracks, it suffices to work with the average thickness of the variables. Moreover, the model assumes a linear variation of interlaminar shear stress in the z-direction on each lamina The model uses an uncoupled activation function This model has been validated experimentally for the same material used in this work with several stacking sequences Fiber failure onset is estimated by the maximum stress criterion (MSC), which can be written in the customary form g⩽0, as followswhere g⩽0 represents the undamaging domain, F1t,F1c are the longitudinal tensile and compressive strength of the unidirectional lamina, and 〈σ1〉=(σ1+|σ1|)/2. The equal sign is retained so that the return mapping algorithm (RMA) can be made to converge to g=0.Use of a failure criterion like this in nonlinear analysis is not advisable because it is strongly mesh dependent. The elastic energy stored in the volume associated to the Gauss point where the criterion is satisfied is suddenly released. This volume is proportional to the size of the element, which introduces strong mesh dependency. Without regularization, the sudden change of stiffness makes it very difficult to converge to an equilibrium solution. Therefore, a regularized degradation model is necessary.The stochastic fiber strength is represented by a Weibull distribution where m is the Weibull modulus, e is the basis of natural log, and the effective stress σ̃1 is calculated in term of the longitudinal stress aswhere 〈σ1〉 is used to assure that only tensile stress is used in the calculation. To prevent recalculation of damage during unloading, the damage is updated only if the effective stress exceeds the hardening threshold g1t, which is a state variable. In other words, the undamaging domain isWhen g>0, the damage is updated with Eqs. and the threshold is updated to g1t=〈σ̃1〉, where 〈x〉 is the McAuley operator that returns only the positive part of the argument; that is, returns x if x>0 and 0 if x⩽0. Although the updating of g1t represents hardening in effective stress space, nominal (Cauchy) stress softening can be seen by virtue of Eq. Longitudinal tensile failure is brittle and, under load control, failure occurs suddenly with little accumulated damage. Also, localization results in rapid failure even when the boundary is under displacement control. provides a mechanistic regularization model that, while helping achieve numerical convergence of the structural analysis software, assures that the peak of the longitudinal stress–strain curve coincides with F1t. The only material properties needed are the lamina longitudinal tensile strength F1t and the Weibull modulus m of the fibers. When experimental data for the Weibull modulus is not available, m can be interpreted as a numerical regularization parameter, with the advantage that experimental values of m for similar materials can be used to realistically bracket the values used. For example, a compilation of experimental values of Weibull modulus for a broad variety of composite materials is available in A square plate of a×b=25×25mm2 made from a glass/vinylester laminate (Fiberite/HyE 9082Af) was studied. The stacking sequence selected is [0/908/0/908/0]. The mechanical properties of the materials are shown in In this work, the model is implemented in Abaqus/Standard by programming a user subroutine UGENS . First a laminate with a defect in the center of the plate, and second a laminate with a hole located also at the center of the plate. These cases are used to study the objectivity of the model, i.e., sensitivity of the result to mesh refinement and localization of damage near the initial damage and the stress raiser.For both problems, plate with a defect and plate with stress raiser, three discretizations are used to analyze global response and damage evolution as a function of mesh density. The plate is square and symmetry conditions are used to reduce the model to a quarter plate (). The plate is subjected to a uniform applied displacement at x=a/2, thus simulating a uniform applied strain.The plate with 1.25 mm radius hole is discretized as seen in with a combination of S4R and STR165 elements, using 72, 299, and 1498 elements, respectively. For the plate with a defect, the three discretizations are uniform, using 144, 625, and 2500 S4R elements, respectively.A defect is simulated by inserting an initial crack density (0.09 mm−1) at the center of the plate (i.e., at the corner of the discretization with symmetry boundary conditions). The rest of the plate is assigned a low value of crack density (0.02mm-1) to seed the model for possible damage initiation. The plate with a hole has no defect; only uniform seed damage is used.The lack of influence of mesh density on the global response of the plate is corroborated by the load–displacement curves for both problems studied. As it is shown in , the difference between the results from the three discretizations is negligible for both problems.Also, the influence of mesh refinement was analyzed in relation to crack-density evolution. The evolution of crack density in the element with initial damage is shown in (a). The evolution of crack density in the element at the edge of the hole (x=0,y=r) is shown in (b). It can be seen that the differences are negligible.Similar mesh independence can be corroborated at all locations on the plate. From these observations, it can be concluded that the results provided by the proposed formulation are mesh independent.For a specimen with a defect, the load–displacement curve in (a) is linear up to 0.47% applied strain, at which point damage propagates quickly to the whole plate. It can be seen how the residual modulus of the laminate drops significantly at this critical point. After that, the load–displacement curve is almost linear until 2.17% applied strain, where fiber damage begins. for a plate with a defect, showing contour plots of crack density in the 90-deg center cluster of the laminate. Damage propagates outward from the defect, perpendicularly to the load direction, and displaying a peanut shape. Crack density in the load direction is always, in every image, lower than in the rest of the plate because the area near the defect is protected in the load direction by the stress reduction caused by the defect. In the last image, at 0.523% strain, matrix damage has propagated to the entire plate.The magnitude of crack density used to simulate the defect has no significant bearing on the results. Besides 0.09mm-1, two other values are used to see if they affect the results: 0.045mm-1, and 0.135mm-1, while the seed crack density in the rest of the plate is kept at the same value, 0.02mm-1.A delay in the onset of crack-density growth occurred when the initial value applied to the central element of the laminate was increased, continuing later on the same curve in all cases. Therefore, it can be concluded that the value of the initial crack density influences only the initial levels of applied strain but has no effect on the subsequent evolution of this parameter.The load–displacement responses are identical, irrespective of the value of crack density used to simulate the defect as long as the value is higher than the seed damage used in the rest of the plate (). Since load–displacement reflects correct stress redistribution and accumulated damage, all derived predictions, such as ultimate strength, will be also insensitive to the particular value of damage used for representing a defect.The study of damage localization for a plate with a hole is described in this section, analyzing two hole radii: 1.25 and 4.17 mm. Linear load–displacement is observed in (b), almost up to the ultimate load. Matrix damage starts at the edge of the hole at 0.16% applied strain (b), but it does not affect the laminate moduli because the damage is localized near the edge of the hole. Only when matrix damage extends over the entire plate, at about 0.49% strain, it is possible to observe a reduction in modulus in (b). Soon after that, fiber damage begins to take place, which quickly leads to laminate failure. Due to the stress concentration caused by the hole, damage localizes near the edge of the hole rather than suddenly propagating to the whole plate as in the case of the plate with a defect (described in Section ). Therefore, the reduction of laminate modulus is small for a plate with a hole.Prediction of ultimate strength is reported in comparing predicted values of ultimate laminate load and experimental values for [0/±45/907]S T300/1034-C The load–displacementcurve for both radii studied, 1.25 and 4.17 mm, are shown in . The larger hole yields a laminate with lower stiffness and, due to the lower cross-section, lower failure load Crack-density evolution on 90-deg laminae is found for both hole radii at points near the edge of the hole (), where the edge effects are important. Damage starts at a lower strain for a small hole. At the edge of the hole damage onset occurs at 0.16% and 0.19% for 1.25 and 4.17 mm holes, respectively. At the edge of the plate, damage onset occurs at 0.48% and 0.61%, respectively. The plate with smaller hole is stiffer, so it experiences higher stress for the same applied strain; thus, it damages earlier.The presence of a hole causes a stress concentration at the edge of the hole (). The SCF depends on the laminate stacking sequence and the hole radius Variation of SCF at the edge of the hole is shown in as a function of applied strain. SCF starts to decrease at point 2, coincident with the onset of damage. It continues to decrease from point 2 to a minimum at point 3. When the crack density on the elements located on the edge of the hole reaches a value approximately equal to the inverse of the thickness of the 90-deg cluster (0.867mm-1), the SCF reaches a minimum value at the edge of the hole (point 3). When the level of damage in the 90-deg laminae is high, its loading capacity is reduced, so that most of the load is borne by 0-deg laminae. Then, the elements near the hole behave similarly to a unidirectional laminate for which the 90-deg laminae have been discounted. Therefore, the SCF at the edge of the hole slightly increases (between points 3 and 4). When the load is close to causing failure, damage to the fibers causes the SCF to decrease again (point 5)., the calculated SCF for an undamaged laminate is larger for the plate with larger hole radius (SCF = 3.37 for R |
= 4.17 mm vs. SCF = 3.06 for R |
= 1.25 mm). These values are consistent with the results obtained applying the model of Whitney and Nuismer For both hole radii, the model localizes the damage near the hole. Crack-density evolution for both radii is presented in , for the applied strain levels shown in , point 1 through 5. The area with the higher crack density corresponds to the elements near the edge of the hole in the perpendicular direction to the load application. Damage decreases stiffness and causes a redistribution of stresses to the sides of the plate. In both cases damage propagating perpendicular to the load application has a peanut shape, with the edge perpendicular to the applied load. For the applied strain level that causes the greatest reduction of SCF (point 3 in ), damage is highly focused on points near the edge of the hole for both diameters ((3)). As strain increases, damage extends to a larger area. The plates with smaller and larger radii differ in that for the larger hole radius, crack density reaches the edge of the plate before ultimate fracture, but for the smaller radius it does not The load–displacement response and crack density evolution of various laminates featuring initial damage or stress raisers is completely insensitive to mesh density () and it has a significant effect on the SCF of stress raiser cases (The global effect of a point defect is independent of the crack density chosen to simulate the defect (), which is important because it frees the analyst from having to make such choice.The SCF at the edge of the hole decreases quickly as damage develops in that zone (). Once damage is fully developed, the laminate behaves like a unidirectional laminate, similarly to having discounted the 90-deg laminae. Then, SCF decreases again as a result of fiber damage, but that quickly results in ultimate fracture of the laminate (The correlation between notched strength predicted by the present model and experimental results is comparable to those achieved with other methodologies, and even better in some cases (). Therefore, the model may be considered as being validated.The combination of DDM to predict matrix cracking and a Weibull-controlled fiber damage model results in a formulation with the minimum number of additional material properties (namely intralaminar fracture toughness and Weibull modulus), that is able to predict damage onset, evolution, and laminate ultimate strength. The formulation works well, independently of mesh refinement, when incorporated into a classical, displacement-based finite element formulation. Implementation as a UGENS in Abaqus allows for the analysis of laminated plates and shells as long as the laminate stacking sequence is symmetric.Influence of a radial gap between clamped bolt and transducer’s component elements on the wide-band characteristic of multi-resonant Tonpilz longitudinal vibrating piezoelectric transducersIn multi-resonant Tonpilz longitudinal vibrating piezoelectric transducers with a clamped bolt of which both ends are screwed directly into their head mass and tail mass, the influence of a radial gap between clamped bolt and transducer’s component elements on the transducer’s wide-band characteristic was illustrated through detailed qualitative and simulation analysis. In order to verify the validity of analysis, samples of triple resonant Tonpilz longitudinal vibrating piezoelectric transducer was fabricated and measured in both cases that there exists a radial gap between clamped bolt and their elements and there never exists. The 1st intrinsic longitudinal mode created due to the radial gap between clamped bolt and transducer’s component elements is coupled with longitudinal modes of the transducer to affect several characteristics of transducer including wide-band. Therefore, it was recommended that design and fabrication should be taken not to be created preferably a radial gap between clamped bolt and transducer’s component elements in multi-resonant Tonpilz longitudinal resonant piezoelectric transducer.A wide-band sonar transducer that can transmit complex signals can be said to be a core of wide-band sonar in processing sonar signals and keeping frequency sensitive. Wide-band sonar transducers include various types of transducers such as transducer with multi-matching layers In DRTs there are the following structures; those vibrating longitudinally with mechanical series arrangement of inactive stiffness elements such as aluminum rod or complex polymer materials stacked between central mass element and head mass element and active driving elements such as piezoelectric ceramic beam or magnetostrictive rod stacked between central mass one and tail mass element TRT is an extension of DRT added one series resonator (mass-spring system) to create a higher order of longitudinal resonant mode.Generally Tonpilz transducers are designed by using equivalent circuit model and finite element model, both of which offer good agreement with measured results TRT with mechanical series arrangement of head mass element and tail mass element, two central mass elements, two inactive stiffness elements made of an aluminum or complex polymer, piezoelectric stack (consisted of 12 piezoelectric ceramics) was designed by equivalent circuit model and finite element model and measured In these single- and multi-resonant Tonpilz longitudinal vibrating piezoelectric transducers a clamped bolt is used to mechanically interconnect elements of transducers and prevent expansion crack of piezoelectric ceramics when operating at high power.Changes in characteristic quantities of piezoelectric ceramics according to mechanical stress applied by the clamped bolt were studied Meanwhile, the bolt’ longitudinal vibration couples with Tonpilz piezoelectric transducer’s vibrating modes and offers influence on characteristics of the transducer such as wide-band.3D coupled vibration for longitudinally polarized piezoelectric ceramic stack was studied theoretically and numerically However, because 3D coupled vibration between transducer’s vibration and bolt’s one in multi-resonant Tonpilz longitudinal vibrating piezoelectric transducers with a piezoelectric stack polarized longitudinally is very complex, theoretical study on the above coupled vibration is difficult.Therefore, the influence of longitudinal piston vibration of prestress bolt on its operating characteristics in the multimode broadband Tonpilz transducer using both longitudinal vibration mode and flexural one above a radiating surface was studied by numerical simulation in However, clamped bolts considered in literatures are those of which one end is screwed tightly into the transducer’s head mass and other end into a small nut laid behind tail mass. Therefore, in the transducer’s resonance mode corresponding to bolt’s longitudinal mode the bolt behaves as a metal rod of which one end is fixed and other end is free. They don’t have effect on the transducer’s wide-band characteristic, because these longitudinal mode frequencies are in the upper place of the basic operating frequency band of the transducer. Furthermore, there were neither experimental verification on the above considerations nor measurements to avoid this longitudinal mode in literatures.If there exists a radial gap between a clamped bolt and transducer’s component elements in multi-resonant Tonpilz longitudinal vibrating piezoelectric transducers with the clamped bolt of which both ends are screwed into head mass and tail mass of the transducer, there will be the 1st intrinsic longitudinal mode of the bolt, thus affecting their wide-band characteristics. This will be considered qualitatively and proved through computer simulation and practical measurements in our paper. As a result of this, it is recommended to design and fabricate not to be preferably a radial gap between clamped bolt and transducer’s component elements in multi-resonant Tonpilz longitudinal resonant piezoelectric transducer.Generally Tonpilz piezoelectric transducers use clamped bolts for mechanical coupling of transducer’s component elements and stable operation of piezoelectric ceramics. At this point, one end of clamped bolt is screwed rigidly into head mass of transducers and other end is screwed into a small nut behind tail mass or directly into tail mass. The triple resonance Tonpilz piezoelectric transducer that clamped bolt is directly screwed into tail mass is shown in , if a radial gap between the bolt and transducer’s component elements exists, a mode that longitudinal displacements at head and tail masses are nearly zero and is the maximum in the centre of the bolt is one of the possible mechanical intrinsic modes of the transducer, as shown in simulation The transducer used in simulation is a triple resonant Tonpilz piezoelectric transducer that is a mechanical series arrangement of a steel tail mass, a piezoelectric ceramic stack, a steel central mass, an aluminum disc, a steel central mass, an aluminum disc, and an aluminum head mass. The ceramic stack is composed of 12 epoxy-bonded piezoelectric PZT-4 ceramic rings that are 5 mm thick and 25 mm diameter and are electrically connected in parallel. A diameter of radiating face and height in head mass are 35 mm and 30 mm, respectively. Also, diameter and height of tail mass are 25 mm, 15 mm and diameters and heights of central masses 25 mm, 15 mm, 30 mm, 10 mm, respectively. And both aluminum discs are diameter of 20 mm and height of 15 mm. This transducer has total length of 156 mm and total mass of 382 g, which is shown in Here, the clamped bolt is made of iron with effective length (length of part except screwed parts) of 115 mm and radius of 4 mm.In this case, it can be seen that the bolt behaves like a metal rod fixed at both ends. When its cross- section is very small than effective length, the 1st longitudinal mode (length direction resonance) frequency of the bolt, f1 can be calculated as follows where l,E and ρ are length, Young’s modulus, density of the bolt respectively.Calculating the 1st longitudinal mode frequency by Ep. , it is 22.859 kHz, which is nearly coincident with 2nd vibration mode frequency (22.045 kHz) of the TRT considered above; a relative error of frequency shift on these frequencies is about 3.69%. Accordingly, it can be found that transducer’s mode in 2nd vibration mode is decided by the 1st longitudinal mode of the clamped bolt.Comparing with numerical result (simulation result of ANSYS program based on FEM) on longitudinal resonant frequency of the rod in case of considering its cross section, Eq. For example, simulation result for the 1st longitudinal resonant frequency of the steel bar fixed at both ends with length of 100 mm and diameter of 8 mm (Young’s modulus: E = 217 × 109 N/m2, density: ρ = 7850 kg/m3) is 26.394 kHz, as shown in On the other hand, calculating the 1st longitudinal mode frequency by Eq. (cross-section of the bar is ignored) is 26.288 kHz, which leads to about 0.4% of error over simulated value., consider the influences of the 1st longitudinal mode frequency of the clamped bolt on wide-band characteristics of single resonant and multi-resonant Tonpilz transducers. As shown in Eq. , the 1st longitudinal mode frequency of the bolt is in inverse proportion to bolt’s effective length.For example, the effective bolt length of single resonant longitudinal vibrating Tonpilz piezoelectric transducer with 15 kHz of operating frequency in However, in multi-resonant Tonpilz piezoelectric transducers the bolt’s effective length sometimes is longer, so that the 1st longitudinal mode frequency is in operating frequency band this mode has an effect on transducer’s wide-band characteristic.For example, in longitudinal vibrating triple resonant Tonpilz piezoelectric transducer that 1st, 2nd, and 3rd mode frequencies were designed at 15 kHz, 25 kHz, and 37 kHz, respectively a mass-stiffness ratio of the tail mass and stiffness elements should be satisfied with m/k≈10−10 in order to get optimization wide-band characteristic where K, E, A and l are stiffness coefficient, Young’s modulus, cross-section and length of stiffness element, respectively.Using PZT-8 and G10 (E≈3.96×109, pa) materials with small Young’s modulus for stiffness elements in In order to quantify the above qualitative analysis, coupled vibration characteristics between clamped bolt’s 1st longitudinal mode created due to a radial gap and multi-resonant Tonpilz piezoelectric transducer’s mode for the transducer shown in was simulated by using specific program ANSYS 17.0.This transducer was designed for 1st, 2nd and 3rd resonant frequencies of 15 kHz, 25 kHz and 37 kHz respectively and for the relative bandwidth Δf/fc (ratio of bandwidth to center frequency) of 1.When a clamped bolt doesn’t have any effective length since there is no radial gap between it and transducer’s component elements, analysis results of the first three vibration modes in sequence for the transducer were shown in As shown in the above Figures, there is no 1st intrinsic longitudinal mode of bolt and the transducer performs piston vibrations in three vibration modes with maximum virtual vibrating displacements of head mass (radiating surface).When clamped bolt has effective length of 115 mm since there is a radial gap between it and the elements of transducer, analysis results of the first four coupled vibration modes were shown in , virtual vibrating displacements on radiating surface of head mass at the 2nd vibration mode (resonant frequency of 22.045 kHz) because of clamped bolt ‘s 1st longitudinal mode were very small.Virtual vibrating displacements on radiating surface at different modes also are smaller than in case that there is no radial gap and resonant frequencies differ largely from design frequencies, resulting in relative errors of 6.3%, 12.8% and 11.1% respectively.For the above two cases, (without a gap and with a gap) the harmonic analysis results of relative vibrating velocity (dB unit) on the radiating surface except loss are shown in As shown in figures, the maximum relative vibrating velocity (dB unit) on the radiating surface with a gap is 8.8243 dB smaller than that without a gap. (−25.2329 dB without a gap, −34.0572 dB with a gap).The bandwidth with a gap also is 4 kHz narrower than that without a gap when observing at −70 dB level less than the maximum velocity value (28 kHz at −70 dB level without a gap, 24 kHz at −70 dB level with a gap). Moreover, in case that there is a gap, the vibrating velocity has a trough at frequency of 22.5 kHz, that is, in neighbor of 1st longitudinal mode frequency of clamped bolt, so that continuity of bandwidth is not allowed and the wide-band characteristic becomes worse.The 1st longitudinal mode of bolt also affects the characteristics of input admittance and harmonic analysis results of admittance-frequency response characteristic for above two cases are shown in , the input admittance of transducer with a gap has a peak in neighbor of 1st longitudinal resonant frequency of clamped bolt. But at this frequency vibrating velocity of the radiating surface has the minimum value as shown in because most of mechanical energy produced by a piezoelectric effect of piezoelectric ceramic stack is transferred into longitudinal resonant energy of the bolt.Also admittance values in resonant frequencies without a gap in no consideration of loss are about twice larger than those with a gap.Besides, the 1st longitudinal mode of clamped bolt occurs at different frequencies depending on effective length of the bolt, so that its influence on transducer’s wide-band characteristic varies.For example, in case of 70 mm effective length of clamped bolt for the above transducer, the first four vibration modes are shown in The 1st longitudinal mode frequency of clamped bolt with 70 mm of effective length is 37.785 kHz and it is in neighbor of the 3rd mode frequency (34.346 kHz), so that as shown in , the 3rd and the 4th resonant modes aren’t piston vibrations with the maximum virtual vibrating displacement of radiating surface due to the influence of 1st longitudinal mode of the bolt. In both cases of being a gap and no gap with mechanical loss coefficient of 0.01, simulation results of input admittance response and vibrating velocity response of radiating surface according to the frequency were shown in As shown in Figures, input admittance responses are similar to each other, but in vibrating velocity responses, relative velocity values without a gap are about 1 dB larger than those with a gap while the relative bandwidth is also wider (1 without a gap and 0.936 with a gap at −64.5 dB level less than the maximum velocity value, respectively).As a result, if the clamped bolt has 1st intrinsic longitudinal mode due to a certain radial gap between transducer’s component elements and the bolt, wide-band characteristics become worse and vibration velocity and input admittances are also smaller.In this paper in order to verify theoretical and simulation analysis, a transducer with a gap (clamped bolt’s effective length of 115 mm) and that without a gap were fabricated and input admittances and transmitting voltage response (TVR) were measured and analyzed.The transducer without a gap were fabricated to be the same as that with a gap by inserting 0.35 mm of insulating tetrafluoride resin ring in thickness and filling up epoxy resin into a gap.The first three vibration mode simulation results of the transducer fabricated without a gap were shown in , vibration modes of transducer fabricated without a gap using epoxy resin have no 1st intrinsic longitudinal mode of clamped bolt, such as shown in vibration modes (in ) of a transducer without a gap ideally. And all three subsequent vibration modes have piston vibration with the maximum virtual vibrating displacement of the radiating surface.Only resonant frequencies decrease by 0.51 kHz, 0.51 kHz and 1.25 kHz, respectively. shows input admittance–frequency responses of transducers measured in air by using LCR meter (Label: KC-605 LCR METER42Hz-5MHz) and simulation data with mechanical loss coefficient of 0.01., there was no influence of the 1st longitudinal mode of clamped bolt in the transducer without a gap. Only resonant frequencies measured because of fabrication, assembly and measurement errors and disagreement of material properties used in both cases became less (1 kHz degree) than those simulated. Admittance values measured and simulated at resonant frequencies were nearly the same as 3.09%, 8.65%, and 4.38% for relative errors, respectively.In transducer with a gap, however, there was influence of the 1st longitudinal mode of clamped bolt as shown in b) of , so that a peak of input admittance was developed between the 1st and the 2nd modes.Resonant frequencies measured in the same way as measured without a gap became less (1 kHz degree) than those simulated. Admittance values measured and simulated at resonant frequencies were nearly the same as 4.13%, 6.61%, 7.6539%, and 6.1% for relative errors, respectively.In order to measure transmitting voltage response (TVR) Test transducers were set in depth of 1.5 m from water surface and signal radiated from them were received through a standard receiver 8016 far from 1 m of horizontal distance.TVR of transducers measured and simulated under water were shown in , TVRs measured and simulated at resonant frequencies are nearly the same as 1.38%, 1.4%, 1.5% without a gap and 0.69%, 0.77%, 0.7% with a gap for relative errors, respectively. The relative bandwidth also is nearly the same as 3.34% without a gap and 3.2% with a gap for relative errors, respectively.The discrepancies between measurements and simulations are attributed to model assumptions, fabrication and assembly processes, disagreement of material properties used in both cases, and measurement errors. Lower material losses in simulation sharpened the resonances. The lower level and weak resonance in TVR at the third resonance were attributed to the poor epoxy resin joint coupling and imperfect machining of the prototype parts. Difference of material properties used in measurement and simulation, measurement errors such as a reading error also were factors of these discrepancies.The measured relative bandwidth of transducer without a gap is 0.89 at 6 dB level (0.5 time or −6 dB level of the maximum TVR value).TVR of the transducer with a gap, however, has a trough at 20 kHz, that is, in neighbour of 1st longitudinal mode frequency of the clamped bolt, so that the bandwidth of transducer becomes discontinuous, then relative bandwidth in the lower part and the upper part being 0.48, 0.35 respectively.In addition, TVR values with a gap in all frequencies are smaller than that without a gap.Consequently, clamped bolt’s 1st longitudinal mode created due to a radial gap between it and transducer’s component elements has a serious effect on characteristics of transducers.Therefore, in design and fabrication of wide-band multi-resonant Tonpilz piezoelectric transducers structural measurements should be taken not to be created a radial gap between transducer’s component elements and a clamped bolt.In this paper, the influences that a clamped bolt’s 1st longitudinal mode created due to a radial gap between it and transducer’s component elements affects several characteristics of multi-resonant Tonpilz longitudinal vibrating piezoelectric transducer such as wide-band have been considered.In multi-resonant Tonpilz longitudinal vibrating piezoelectric transducers several mass and stiffness (active and inactive stiffness) elements are used. If Young’s modulus of stiffness elements is large, geometric size of transducers increase so that effective length of clamped bolt screwed at both ends also increases. In this case, the frequency for 1st longitudinal mode of clamped bolt created due to a radial gap between it and the transducer’s component elements is in operating frequency band. Thus, the 1st longitudinal mode of clamped bolt couples with longitudinal modes of the transducer so that it has a serious effect on characteristics of the transducer such as wide-band.In transducer with a clamped bolt screwed rigidly into head mass and tail mass at both ends, longitudinal modes of the transducer coupled with the 1st longitudinal mode of the bolt have been qualitatively analyzed.Characteristics of TRTs designed with 15 kHz, 25 kHz, and 37 kHz for the 1st, 2nd and 3rd resonant frequencies, respectively without a gap and with a gap between transducer’s component elements and clamped bolt have been simulated by using a specific program, ANSYS.Also, samples of TRTs illustrated were fabricated and its input admittances in air and TVR under water were measured.Simulations and measured results placed the qualitative analysis that the 1st longitudinal mode of the clamped bolt affecting several characteristics of multi-resonant Tonpilz longitudinal vibrating piezoelectric transducer such as wide-band is resulted from a radial gap between the bolt and transducer’s component elements on high quantitative and scientific bases.As a result of the above, it was recommended that design and fabrication should be done not to produce a radial gap between clamped bolt and transducer’s component elements in multi-resonant Tonpilz longitudinal resonant piezoelectric transducers.The author declare that there is no conflict of interest.A SANS investigation of the irradiation-enhanced α–α′ phases separation in 7–12 Cr martensitic steelsFive reduced activation (RA) and four conventional martensitic steels, with chromium contents ranging from 7 to 12 wt%, were investigated by small angle neutron scattering (SANS) under magnetic field after neutron irradiation (0.7–2.9 dpa between 250 and 400 °C). It was shown that when the Cr content of the b.c.c. ferritic matrix is larger than a critical threshold value (∼7.2 at.% at 325 °C), the ferrite separates under neutron irradiation into two isomorphous phases, Fe-rich (α) and Cr-rich (α′). The kinetics of phase separation are much faster than under thermal aging. The quantity of precipitated α′ phase increases with the Cr content, the irradiation dose, and as the irradiation temperature is reduced. The influence of Ta and W added to the RA steels seems negligible. Cold-work pre-treatment increases slightly the coarsening of irradiation-induced precipitates in the 9Cr–1Mo (EM10) steel. In the case of the low Cr content F82H steel irradiated 2.9 dpa at 325 °C, where α′ phase does not form, a small irradiation-induced SANS intensity is detected, which is probably due to point defect clusters. The α′ precipitates contribute significantly to the irradiation-induced hardening of 9–12 wt% Cr content steels.Martensitic steels with 7–12 wt% Cr are candidates for the internal structures of future generation nuclear reactors (such as fusion or advanced high temperature reactors) or spallation sources, because of their remarkable resistance to swelling and of their adequate mechanical properties (tensile, impact and creep resistance up to 550 °C) It is in this context that a large programme has been undertaken at CEA to compare the behaviour under irradiation and long thermal aging of a great variety of conventional and RA martensitic steels, in order to optimize their performances. Previous publications in the Journal of Nuclear Materials refer to the effect of chemical composition on the physical metallurgy and mechanical behaviour of as-received materials ), the small-angle neutron scattering (SANS) technique is in some specific cases more powerful than transmission electron microscopy (TEM) for studying precipitation at the nanometer scale in ferromagnetic martensitic/ferritic steels, either because of a more favourable contrast between precipitates and matrix (e.g. α–α′ phase separation), or when the particles are very small (<5 nm). In addition, SANS examines a much larger volume of material than TEM.The aim of this work was therefore to study using SANS the evolution of microstructure in RA and conventional martensitic steels after neutron irradiation, focussing on the α–α′ phase separation, and to define the role of chemical composition, of irradiation conditions and of dislocation density.The results obtained will be compared to those published recently on the thermally-aged materials Numerous investigations performed since the 80s have shown that several phases precipitate in conventional martensitic/ferritic steels during irradiation between 400 and 550 °C (see review by Maziasz The unmixing of the ferrite below 600 °C into two isomorphous b.c.c. phases, one Fe-rich (α phase) and the other Cr-rich (α′ phase), is an important feature of the binary Fe–Cr equilibrium phase diagram (see ), where it is observed in thermally-aged alloys with Cr content between 10 and 90 at.%. The phase separation occurs at the nanometer scale and induces hardening of the binary solid solution In fact, it is difficult using TEM to detect and study the α–α′ phase separation, because of the very weak electron scattering contrast between Fe and Cr; for example, the total α′ content deduced from TEM results in the above-quoted F17 steel thermally aged at 450 °C α–α′ phase separation has been studied in detail by SANS in thermally aged ), in the nucleation-growth regime of isolated α′ droplets, two length scales are necessary to characterize the system: the mean distance between precipitates, and the size of precipitates The chemical compositions of the studied steels are listed in . Nine materials have been investigated:four RA martensitic steels, the ‘LA series’ from AEA-Culham, UK a tungsten-stabilised low Cr content RA martensitic steel of Japanese origin (supplied by JAERI): F82H (7.5Cr–2W);and four conventional martensitic steels: EM10 (9Cr–1Mo) and HT9 (12Cr–1Mo–0.5W) (supplied by Aubert & Duval and Sandvik respectively), T91 (9Cr–1Mo(VNb)) and MANET II (10.5Cr–0.5Mo–1Ni).The details of the initial metallurgical states and irradiation conditions for each material are given in All alloys were austenitized at high temperature, then tempered at 750/800 °C. The LA materials, T91 and EM10 alloys have been 10% cold-rolled; this mechanical treatment has been shown to induce more stable impact properties after thermal aging in the case of Fe–9Cr–1Mo(V,Nb) martensitic steels The microstructure, precipitation state and mechanical behaviour of the initial materials and their evolution after thermal aging are detailed in previous publications All materials except LA12TaLC were irradiated in the OSIRIS reactor at Saclay under PWR conditions (325±10 °C, average doses of 0.7 and/or 2.9 dpaThe neutron scattering experiments were performed at the Laboratoire Léon Brillouin (CEA-CNRS), Saclay, on the PAXY small-angle instrument ) range from 0.3 to 1.6 nm−1 (q=4πsinθ/λ, where 2θ is the scattering angle). Measurements have been made at room temperature, under a saturating magnetic field (=2 T) perpendicular to the incident neutron beam direction, in order to separate the magnetic and nuclear scattering cross-sections.The samples for SANS experiments, cut and mechanically polished in the hot cells of CEA/Saclay and NRG Petten, were in the form of platelets of 5×5×e mm3, with a thickness e which has been chosen between 0.1 and 0.5 mm in order to reduce their radioactivity. The measured transmission values (≅95% for e=0.5 mm and λ=0.6 nm) showed that multiple scattering corrections were negligible.Initial data treatment including correction, normalisation, and calibration, has been described by Cotton In the case of ferromagnetic materials, the SANS intensity is the sum of two contributions, a nuclear and a magnetic, which depend respectively on the difference in composition and in magnetisation between particles and the matrix. In terms of cross-section, the SANS intensity can be written as:where fp is the precipitated atomic fraction, and includes a size distribution function h(R), which was usually taken as one (or the sum of several) symmetric normalized Gaussian distribution(s). is the interference term between precipitates, which is not negligible for a precipitated fraction larger than 0.01; it is described by the structure factor for a liquid-like arrangement of hard spheres, calculated analytically by Ashcroft and Lekner where b is the nuclear (nucl) or magnetic (mag) mean scattering length in the precipitates (p) or in the matrix (m), and vp,mat is the mean atomic volume of the precipitates (p) and of the matrix (m). α is the angle between the magnetisation of the sample and the scattering vector As the magnetic moments are aligned parallel to the field , the magnetic scattered intensity is zero in this direction and maximum in the perpendicular direction. In order to use this anisotropy, we have considered separately the scattered intensities obtained in these two directions (⊥ and ∥ to ). Some information about chemical composition can be deduced from the ratio between these two quantities, called ‘A ratio’. For homogeneous particles, the A ratio depends on the chemical composition, magnetisation and atomic density variations between precipitates and the matrix, and is given by:The A ratio value for Cr precipitates in α-Fe matrix is equal to 2.03. To calculate A for α–α′ phase separation in the Fe–Cr b.c.c. solid solution, one has to take into account the variation of the average magnetic moment of the system, μ, with the Cr atomic concentration CCr: μ=2.20–2.39 CCr in Bohr magneton units from the present work). Depending of different authors, the limits of the α+α′ b.c.c. miscibility gap at 325 °C have been extrapolated as –97 at.% respectively. The corresponding A ratio value is 2.09–2.13. This A value is only weakly composition dependent: for example, it is equal to 2.35 for α′ with the composition Cr–13Mo–8Fe–3Si in wt% measured by Gelles and Thomas In the initial, non-irradiated state, all materials show a strong SANS signal; this is mainly due to the M23C6 carbide particles formed after quench and tempering, as confirmed by the A ratio value, which is between 3 and 4 for all the samples, at small q values.After irradiation, most samples (except those with the lowest Cr content, LA13Ta and F82H, irradiated 0.8 dpa at 325 °C) show a supplementary SANS signal, mainly observed at large q, with a smaller A ratio (≈2 in most cases) than measured in the non-irradiated state; this shows that a new nanometer-sized precipitation has formed under irradiation. An example of the fit performed on this irradiation-induced signal is shown in . The structural information obtained from the data analysis is summarised in . Experimental results for each material are detailed below.For Cr-rich alloys, irradiation by fast neutrons induces a large increase in the SANS intensity at q⩾0.5 nm−1 (see We shall consider in detail a typical example, the behaviour of the RA steel LA4Ta (Fe–11Cr–0.7W). (open symbols), the increase in SANS signal between the sample irradiated 0.7 dpa at 325 °C and the as-received sample shows a broad maximum around q∼1.1 nm−1. This behaviour indicates a spatial periodicity in the composition of the material, with a characteristic length 2π/q∼5–6 nm, similar to that observed in the thermally aged Fe–Cr solid solution Considering the SANS results obtained in the same LA4Ta steel after thermal aging (previously published in Ref. ), temperature close to that of irradiation. This increase, which is not compatible with a metal-diffusion governed process, given the mean displacement of Fe atoms for a duration of 10 000 h is ≈3×10−11 m at 325 °C, cannot be associated to the precipitation of α′, but rather to a slight rearrangement of the microstructure. Consequently, in order to characterise more precisely the α–α′ phase separation, we used the sample thermally aged 10 000 h at 350 °C as the reference of the irradiated 0.7 dpa. In this case, the best fit leads to modified average radius and volume fraction of the α′ particles, 1.15 nm and 0.008 respectively, corresponding to a number density N=1.3×1024 cm−3.On the other hand, a further increment of SANS intensity from 2000 to 10 000 h out-of-pile aging of LA4Ta is observed at higher temperature (400 °C) for q⩾0.8 nm−1, with A=2; this is consistent with the precipitation of α′ particles of mean radius 1.5 nm and a total volume fraction of 0.0015 shows directly the enhancement of α–α′ phase separation by neutron irradiation.The effect of irradiation dose is evident in the LA4Ta material: after 2.9 dpa at 325 °C, the SANS intensity is higher than after 0.7 dpa (see ), due to an increase in α′ volume fraction. For this long irradiation, the SANS intensity is so high that the effect of the reference sample on data analysis is weak; comparable precipitate parameters have been found with the non-irradiated sample (see fit in ) and with the thermally aged one taken as reference: R=1.2 nm, close to the mean radius observed after the lowest irradiation dose (0.7 dpa), fp=0.038, and N=3.4×1024 cm−3 (2–3 times larger than after 0.7 dpa).In fact, the data on the low-Cr content F82H steel discussed below () show that the contribution of point defect clusters to the SANS intensity is small, but not negligible. Considering the case of the LA4Ta alloy irradiated 2.9 dpa at 325 °C, if we assume in a first approximation that this contribution is the same than estimated for F82H irradiated in similar conditions, the precipitated α′ fraction reevaluated after correction is found slightly smaller, 0.034 (instead of 0.038) (see The two other Cr-rich materials, HT9 (Fe–12Cr–1Mo–0.5W) and MANET II (Fe–10.5Cr–0.5Mo), show also a large increase of the scattered intensity after 0.7 dpa at 325 °C, but, contrary to LA4Ta, the SANS intensity increment is a monotonically decreasing function with increasing q and shows no maximum. Also, the A ratio, close to 3 in both cases, is higher than calculated for α–α′ phase separation. Therefore, these materials probably contain another irradiation-induced precipitated phase; this could be M6X carbonitride, which has been observed in irradiated HT9, but only above 400 °C When the chromium concentration of the steel is reduced, the intensity of the radiation-induced SANS signal decreases. For a Cr content of 9 wt% (e.g. in LA12LC), only a weak effect is seen after 0.7 dpa at 325 °C. In these samples, the evidence of α–α′ unmixing could only be demonstrated by irradiation performed at lower temperature and higher dose (250 °C, 2.4 dpa). The effect of irradiation temperature is shown for LA12LC in , where the SANS curves are compared after irradiation at a dose of 2.4 dpa, at two temperatures, 250 and 400 °C. For the irradiation performed at lower temperature (250 °C), one observes a marked increase in the SANS profile at large q. The A ratio value, 1.9±0.3, corresponds to α′ precipitates. In order to obtain a good fit to the experimental data, it was necessary in this case to use a bimodal size distribution. The mean radii deduced from the fit are 3.2 and 1.2 nm, and the corresponding volume fractions 0.0014 and 0.006, for the large and small particles respectively. In fact, the contribution attributed to large size particles may rather be the result of a change of the precipitates (carbides) already present in the initial state; consequently, it is hazardous to consider this population as α′ precipitates. In the following, for the discussion, we shall take into account only the small size distribution. On the other hand, after 2.4 dpa at higher temperature (400 °C), the radiation-induced increase of SANS intensity is weak; the A ratio being equal to 1.9±0.3, this variation can be interpreted by the formation of α′ particles (fp=0.001) with a mean radius of 0.9 nm.For the T91 ‘conventional’ steel (Fe–8.5Cr–1Mo) irradiated 0.7 dpa at 325 °C, the increase of SANS intensity is weak. Nevertheless, the corresponding A ratio (1.8±0.3) suggests irradiation-induced α′ precipitates, with a mean radius of 1.3 nm and a volume fraction of about 0.001.For the EM10 N&T material (Fe–9Cr–1Mo) irradiated 0.7 dpa at 325 °C, the A ratio at high q (1.8) is also in agreement with α′ precipitation. But the A ratio at low q being equal to 3, this suggests that another phase is precipitated together with α′; this might be the same one than observed after thermal aging at 400 and 450 °C (A=3.5±0.6), tentatively attributed to M2C (M=83Cr–12Fe–5Mo in at.%) carbides For the materials with the lowest Cr content (F82H and LA13Ta), the SANS signal is unchanged by the irradiation at 325 °C up to 0.7 dpa. In the case of F82H, a weak increase in the scattered intensity is observed after 3.4 dpa at 325 °C; the A ratio value (=1.2) does not correspond to α′ precipitation, but is in agreement with vacancy clusters (A=1.4). These could be small cavities or dislocation loops. Assuming the simple case of spherical cavities, their average radius would be 1.1 nm, their number density 3×1023 m−3 and the corresponding volume fraction 0.002. Assuming the case of small edge dislocation loops, the analysis of the SANS data in the frame of the formalism developed by Seeger and Rühle The effect of initial cold-work was considered, by comparing two samples of conventional EM10 steel (Fe–9Cr–1Mo) irradiated 0.7 dpa at 325 °C, either after 10% cold-work, or without cold-work (normalised and tempered). The difference between the two measured SANS curves was found to be weak, with a somewhat lower number of larger α′ precipitates in the case of the CW material (see Another aim of the present study was to detect possible effects of the alloying elements Ta and W introduced in RA steels, on stability under irradiation. The effect of Ta is weak, as the SANS curves for LA12TaLC and LA12LC are very similar: a small increase after irradiation at 400 °C, and a sharper variation at 250 °C corresponding to a bimodal size distribution; the volume fractions of the small and large particles are slightly larger for LA12TaLC than for LA12LC, but within the error bars (see ). On the other hand, no difference was observed between LA13Ta and F82H, which differ by their W content, respectively 3 and 2 wt% W; but in these samples subsequent analysis (see ) shows that the Cr content remaining in the ferritic matrix is only 7 at.%, which is below the threshold value for α–α′ phase separation.In order to quantify the effect of Cr content on α–α′ phase separation, it is necessary to estimate the Cr concentration in the matrix, which differs from the nominal content because of the presence of Cr-rich precipitates, in particular of carbides. This was done in two ways. On the one hand, ‘THERMOCALC’ software also contains, for each irradiated steel, the volume fractions of α′ phase deduced from SANS analyses. The determination of the α′ volume fraction depends on its chemical composition (supposed to be 95%Cr–5%Fe) and on the assumption that the irradiation-induced variation of the SANS intensity at q⩾0.5 nm−1 is only due to α′ precipitates. In the case of LA4Ta irradiated 2.9 dpa at 325 °C, correction has been made for the contribution of point defect clusters to the SANS intensity (see The SANS results show that the Cr-rich α′ phase appears under irradiation at 325 °C in materials with a Cr concentration in the ferritic matrix equal to or larger than 8.0 at.%. The α′ volume fraction deduced in the EM10 alloy seems overestimated compared to those obtained for the LA12TaLC and LA12LC alloys, of close Cr content; this is possibly due to another precipitated phase in EM10 contributing also to the scattered intensity at high q.Except in the EM10 alloy, the volume fraction of α′ measured after 0.7 dpa at 325 °C in the studied martensitic steels is found to increase monotonically with the Cr content in the matrix (At 325 °C, the Cr threshold concentration in the ferritic matrix for α–α′ phase separation, calculated from the α′ fraction precipitated at large dose in LA4Ta (0.034 at 2.9 dpa), when the defect cluster contribution to the SANS intensity is taken into account and assuming that the saturation is reached (see at.% Cr. At 400 °C, this threshold concentration is probably close to the Cr content in the ferrite matrix of the LA12TaLC and LA12LC steels, where the precipitated α′ fraction is very weak (≈0.001): at.% Cr. At the lowest irradiation temperature (250 °C), the volume fraction of α′ in LA12TaLC and LA12LC, 0.006–0.0075 (weaker than the value ∼0.015 calculated from %), shows that in this case saturation has not been reached at 2.4 dpa.The above results show clearly that the precipitation kinetics under irradiation are much faster than during thermal aging. For the studied materials (Cr content <20 wt%), in both cases the α′ phase forms by classical nucleation and growth process. Two mechanisms can be involved: either (i) a simple irradiation-accelerated mechanism, where the point defect supersaturation allows the rapid achievement of equilibrium much faster than in out-of-pile conditions at the same temperature; or (ii) an irradiation-induced mechanism, where the coupling between migrating point defects and solute atoms (Cr) can induce a non-equilibrium state, and in particular modify the composition range of existing phases.An argument for the second mechanism is that weak binding between vacancies and Cr atoms, consistent with a solute size-effect response, has been shown experimentally in the Fe–Cr solid solution by the observation of Cr-depletion near the voids and near the grain boundaries after irradiation The oversized Cr solute atoms exchange preferentially with vacancies and flow in opposite direction to vacancy flow; this will increase the local Cr concentration far from vacancy traps and therefore the tendency for α′ precipitation. A recent study has also shown that coupling between fluxes of point defects and solute Cr atoms can explain in a coherent way α′ precipitation kinetics under electron irradiation and under simple thermal aging in Fe-10 to 25 wt% Cr Direct evidence for the irradiation-induced mechanism could be a modification of the α+α′ miscibility gap boundary, compared to the thermodynamic equilibrium case. However, the Cr threshold concentrations in the ferrite for α′ precipitation under irradiation estimated from SANS data, CCrα≈7.2 at.% Cr at 325 °C and 8.3 at.% Cr at 400 °C, are not significantly different from the out-of-pile values obtained from THERMOCALC modelisations of the binary Fe–Cr equilibrium phase diagram (see Despite the above arguments for vacancy-Cr bonding, our experimental observations are consistent with a simple irradiation-accelerated mechanism.However, the above discussion should be taken with caution. Indeed, it has been shown experimentally, and justified theoretically from electronic structure arguments, that a chemical short-range order inversion occurs as a function of concentration in the Fe–Cr binary solid solution: from α–α′ clustering for chromium contents CCr⩾10 at.% to tendency to form an ordered compound for CCr⩽5 at.% Cold-work pre-treatment seems to increase the coarsening kinetics of irradiation-induced precipitates in the EM10 conventional steel. A faster growth of α′ precipitates in EM10 CW steels could be due either (i) to faster Cr transport due to pipe diffusion along dislocations, or (ii) to irradiation-induced Cr segregation far or close to dislocations acting as vacancy traps (depending on the sign of the chromium-vacancy interaction, see It seems that in the RA steels (i.e. the LA series), the unique precipitated phase under irradiation is α′. But the four conventional steels which contain significant Mo and Ni contents present some departures from this simple behaviour: a larger A ratio for HT9 and MANET II, or a total precipitated fraction (for T91 and especially EM10) larger than expected from the curve depicted in . This could be due either to a second (unidentified) precipitated phase, or (in the case of T91 and EM10) to a role of Mo in the threshold criterion for α′ precipitation.On the other hand, the influence of other alloying elements added in the RA steels (e.g. Ta and W) was not detected. An analysis of the effect of Ta atoms would require a knowledge of their distribution between MX carbonitrides and solid solution in the ferritic matrix For the weakest chromium content alloys F82H (7.47 wt% Cr) and LA13Ta (8.4 wt% Cr) irradiated respectively 0.7 and 2.9 dpa (for F82H) and 0.7 dpa (for LA13Ta) at 325 °C, the precipitation of α′ phase is not observed. For the higher irradiation dose (2.9 dpa), we probably detected the formation of very small vacancy clusters in the F82H steel.The microstructure of the W-stabilised F82H steel after neutron irradiation has been studied by TEM by several authors Our results concerning the absence of α′ precipitation are in agreement with those of Kohno et al. Concerning vacancy clusters, Kohno et al. On the basis of these TEM observations, it is likely that the defects detected by SANS (after 2.9 dpa at 325 °C) are the black dots observed by Schäublin and Victoria By TEM on F82H irradiated 2.5 dpa at 250 °C: φBy SANS on F82H irradiated 2.9 dpa at 325 °C, assuming spherical cavities: φ=2.2 nm, N=3.0×1023 m−3; assuming edge dislocation loops: φ=2.1 nm, N=2.2×1026 m−3.Obviously, the number densities N obtained by TEM and SANS are in much better agreement if the form factor of the defects is assumed to be cavity-like rather than dislocation loop-like. A reconciliation between these viewpoints might be that the majority of the black dots are very small non-collapsed vacant disks.This defect cluster contribution to SANS probably exists in all samples after long irradiation, but is weak compared to the α′ precipitates scattering when it occurs. Indeed, we have recently observed by TEM black dots and dislocation loops in the LA12LC sample irradiated 0.7 dpa at 325 °C ). This is close to the value (0.032) calculated from the equilibrium phase diagram of , for complete unmixing at 325 °C of the binary solid solution Fe89.8Cr10.2 (Cr content of the ferritic matrix in LA4Ta, see ) in a two-phase mixture of α and α′ containing respectively 7.5 and 95 at.% Cr. Therefore, saturation of the precipitation has probably been reached in this case. New SANS analysis performed on this material after a higher irradiation dose should confirm this point., whatever their Cr and α′ phase contents, display a large increase (Δσirr) of yield stress, ranging from 100 to 275 MPa (with the only exception of LA13Ta, 0.7 dpa at 325 °C, where Δσirr≅50 MPa). No simple correlation appears between Δσirr and chemical composition or microstructural parameters.The relationship between α–α′ unmixing and the mechanical properties of thermally-aged Fe–Cr binary alloys has been studied by Triki et al. When correcting for the content of precipitated α′ phase (which occurs in the yield stress increase as fp1/2) and for small variation in the average precipitate radius given in , the calculated α′ contribution to the hardening of our steels ranges from 10–20 to 100 MPa. This is always smaller than the measured increase of yield stress, but represents a significant part of it in the case of the Cr-rich materials: for example, ≈60–65 MPa (calculated for an α′ volume content fp=0.008) compared to measured Δσirr values of 100 MPa for LA4Ta (RA steel) and 260 MPa for HT9 (conventional steel) irradiated 0.7 dpa at 325 °C. However, the main hardening contribution seems to be due to radiation-induced point defects clusters, as the largest yield stress increase at 325 °C was found in the F82H irradiated 2.9 dpa, where we observed a vacancy cluster (but no α′) contribution to the SANS (see The large increase of yield stress observed for the LA12LC and LA12TaLC steels, when decreasing the irradiation temperature from 325 to 250 °C, contains very likely a contribution due to the corresponding enhancement of α′ precipitated fraction reported in the present work (). The detailed analysis of the mechanical properties evolution in relationship with SANS data (including samples irradiated at higher dose) will be the subject of another paper.The Small-Angle Neutron Scattering technique under applied magnetic field has been shown to be very powerful in the study of nanoscale precipitation in ferromagnetic martensitic steels.The Cr-enriched b.c.c. α′ phase has been shown to precipitate at the nanometer scale under neutron irradiation at temperatures as low as 250 °C, when the Cr content of the ferritic matrix is ⩾8% at.The volume fraction of precipitated α′ phase increases with Cr content, with the irradiation dose, and as the irradiation temperature is lowered. Up to 2.9 dpa, these parameters have only a weak influence on the average size of α′ precipitates (radius ≈1.3 nm). The Cr threshold concentration in the ferrite for α–α′ unmixing is lower than thought previously; it is estimated to be 7.2 at.% Cr at 325 °C and 8.3 at.% Cr at 400 °C.The precipitation kinetics under irradiation are much faster than during thermal aging. Our data are in qualitative agreement with a simple irradiation-accelerated mechanism, but do not completely exclude a more complex irradiation-induced mechanism, suggested by literature data on the binary Fe–Cr solid solution.In low-Cr content alloys, the SANS signal after irradiation at a dose of 2.9 dpa suggests a small contribution due to vacancy clusters. This is probably related to the black dots observed by TEM. TEM and SANS data are in reasonable agreement if the defect form factor is assumed to be cavity-like.Mo (and perhaps Ni) contained in conventional steels, could induce a secondary (unidentified) precipitation and/or play a role in α–α′ phase separation. The influence of Ta and W added in the RA steels was not detected.Cold-work pre-treatment increases slightly the coarsening of irradiation-induced precipitates in the EM10 conventional steel irradiated 0.7 dpa at 325 °C; this observation needs to be confirmed on other materials.No simple correlation appears between the irradiation-induced hardening Δσirr of the studied materials and their chemical composition or microstructural parameters. Δσirr seems to be mainly due to point defect clusters, but for the Cr-rich materials a significant contribution comes from the precipitation of α′ phase.thermo-plastic/viscoplastic damage coupledPublished by AMSS Press, Wuhan, China Acta Mechanica Solida Sinica, Vol. 24, No. 3, June, 2011 ISSN 0894-9166 ATHERMO-PLASTIC/VISCOPLASTICDAMAGE MODEL FORGEOMATERIALS Hui Zhou 1 Dawei Hu 1,2 Fan Zhang 2,3 Jianfu Shao 2 ( 1 State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071,China) ( 2 LML, UMR 8107, CNRS, University of Lille I, Villeneuve d’Ascq, France) ( 3 School of Civil Engineering and Architecture, Hubei University of Technology, Wuhan 430068,China) Received 24 March 2010, revision received 9 January 2011 ABSTRACT A thermo-plastic/viscoplastic damage coupled model was formulated to describe the time independent and time dependent behaviors of geomaterials under temperature effect. The plastic strain was divided into instantaneous plastic strain and creep plastic strain. To take temperature effect into account, a temperature variable was introduced into the instantaneous and creep plastic behavior descriptions and damage characterization, and a linear thermal expansion law was used in constitutive equation formulation. According to the mechanical behavior of rock salt, a specific model was proposed based on the previous model and applied to Avery rock salt, in which the numerical results obtained from our model had a good agreement with the data from experiments. KEY WORDS thermo-plastic/viscoplastic damage coupled, mechanical model, geomaterial, rock salt I. INTRODUCTION The use of underground space is increasing in many engineering fields, such as geological storage, power station and mining. The long term mechanical behavior is the essence of the safety and stability analysis during design and construction, therefore, many constitutive models were proposed to describe the time independent and time dependent behavior. The research work was started by Kelvin [1] and Bingham and Durham [2] , their models were very practical but lack physical mechanism. Within the framework of viscoplasticity there exist two major concepts: the Duvaut-Lions [3] and the Perzyna [4] format. In the Duvaut-Lions format, the concept of closest-point projection of the stress onto a static yield surface is introduced and the direction of viscoplastic flow is then determined by the difference between the current stress and the closest-point projection. In the Perzyna format, the direction of the latter one due to its simplicity. Some specific models were also developed to describe the viscoplastic flow is generally determined by the gradient of a plastic potential function calculated at the current stress point. Many constitutive models [5–9] were proposed based on mechanical behavior of geomaterials [10–18] . In these models, the creep deformation was often divided into three stages, the transient, steady and accelerating stage, and each stage was defined by its own function according to their time scales. Corresponding author. E-mail: [email protected] Project supported by the National Natural Science Foundation of China(NSFC) (Nos. 10772190, 50979104 and 51009132). · 196 · ACTA MECHANICA SOLIDA SINICA 2011 Furthermore, many structural parts are exposed to varied ambient temperature for a long time. It is necessary to formulate the constitutive models to adequately describe short term and long termmaterial behavior with temperature effect. The aim of this paper is to formulate a thermo-plastic/viscoplastic damage coupled model, in which the three stages were considered uniformly through the creep plastic characterization and the temperature effect was also taken into account. §II will formulate the general framework of the thermo-plastic/viscoplastic damage coupled model, including instantaneous plastic and creep plastic characterization, damage characterization and thermal characterization. In §III, after the review of the mechanical behavior of rock salt with different stresses and temperatures, a specific model will be proposed to describe its mechanical behavior. §IV will explicate the identification methods of parameters, and show the numerical result of time independent and time dependent behavior under temperature effect. Finally, some conclusions will be given in §V. II. GENERAL FRAMEWORK In this section, we will present the framework of the thermo-plastic/visocoplastic damage coupled model. The small strain assumption is adopted and only isotropic materials are considered in the present work. 2.1. State Variables and State Laws Considering a given loading condition with constant stresses and temperatures, we assume the strain in geomaterials starts with instantaneous strain, which can be either elastic or elastic-plastic, and increases with time. The typical creep curve generally has three stages: the transient, steady and accelerating stage(see Fig.1). Fig. 1. Schematics of a creep curve. The so-called ‘superposition concept’ is adopted here. It is assumed that the elastic behavior is time independent and exhibits inelastic deformation in both short term and long term scale. The plastic deformation is composed of two parts: instantaneous plastic strain ε p ij (which is time independent in short term) and creep plastic strain ε vp ij (which is time dependent and includes the deformation during the transient, steady and tertiary creep stages in long term). Therefore, the total strain rate is combined with the following three parts ε˙ ij =ε˙ e ij +ε˙ p ij +ε˙ vp ij (1) where a dot denotes the rate with respect to time. Using T representing the absolute temperature, we can write Helmholtz’s free energy ψ per unit volume in the following form: ψ = ψ e (ε ij − ε p ij − ε vp ij ,T,ω)+ψ p (γ p ,T,ω)+ψ vp (γ vp ,T,ω)(2) where ω represents damage variable, which may be either scalar or second-order tensor although a scalar variable is used in present study for the sake of simplicity; γ p and γ vp denote the isotropic hardening variables of instantaneous plastic and creep plastic deformation, respectively. The decomposition of Eq.(2) is similar to Eq.(1), and corresponds to the assumption [19] that the elastic response does not Vol. 24, No. 3 Hui Zhou et al.: A Thermo-plastic/Viscoplastic Damage Model for Geomaterials · 197 · depend on the internal hardening variables γ p and γ vp . The function ψ p (γ p ,T,ω)andψ vp (γ vp ,T,ω) represent the locked strain energy of instantaneous plastic and creep plastic deformation, respectively. The elastic strain energy can be written as follows: ψ e = 1 2 ( ε ij − ε p ij − ε vp ij ) : C ijkl (T,ω):(ε kl − ε p kl − ε vp kl )(3) where the fourth-order tensor C ijkl (T,ω) denotes the elastic stiffness of a damaged material at different temperature. The standard deviation leads to the following expression of the stress tensor σ ij : σ ij = ∂ψ e ∂ε e ij = C ijkl (T,ω):(ε kl − ε p kl − ε vp kl )(4) Under the assumption of isotropic material, the effective elastic stiffness tensor of damaged material has the following form following Hill’s notation: C ijkl (T,ω)=2G (T,ω)K ijkl +3K (T,ω)J ijkl (5) where G (T,ω)andK (T,ω) are the shear and bulk modulus of damaged material at different temper- ature, respectively. The two symmetric fourth-order tensors J ijkl and K ijkl are defined by: J ijkl = 1 3 δ ij δ kl ,K ijkl = I ijkl − J ijkl ,I ijkl = 1 2 (δ ik δ jl + δ il δ jk )(6) where the second-order tensor δ ij denotes the unit tensor and I ijkl is the symmetric fourth-order unit tensor. Noting that for any second-order tensor E ij ,weobtainJ ijkl E kl = 1 3 (trE kl ) δ ij and K ijkl E kl = E ij − 1 3 (trE kl ) δ ij , which are the spherical and deviatoric part of E ij , respectively. The thermodynamic forces associated with the internal variables are given by Y p ij = − ∂ψ e ∂ε p ij = C ijkl (T,ω):(ε kl − ε p kl − ε vp kl )=σ ij (7) Y vp ij = − ∂ψ e ∂ε vp ij = C ijkl (T,ω):(ε kl − ε p kl − ε vp kl )=σ ij (8) s = − ∂ψ ∂T = − ( ∂ψ e ∂T + ∂ψ p ∂T + ∂ψ vp ∂T ) (9) Y ω = − ( ∂ψ e ∂ω + ∂ψ p ∂ω + ∂ψ vp ∂ω ) (10) η p = − ∂ψ p ∂γ p (11) η vp = − ∂ψ vp ∂γ vp (12) where s is the entropy per unit volume, Y ω denotes the thermodynamic conjugated force of damage variable, the thermodynamic forces η p and η vp conjugate to the internal variables γ p and γ vp , respectively. The Clausius-Duhem inequality then takes the form − Ë™ ψ − Ë™ Ts− Y ω ω˙ + σ ij ε˙ ij ≥ 0 (13) which expresses the non-negative mechanical entropy production. Finally, the rate form of the consti- tutive equation can be easily written as σ˙ ij = C ijkl (T,ω)ε˙ e kl + ∂C ijkl (T,ω) ∂T ε e kl Ë™ T + ∂C ijkl (T,ω) ∂ω ε e kl ω˙ (14) 2.2. Instantaneous Plastic and Creep Plastic Characterization The instantaneous plastic strain rate is characterized through the determination of instantaneous plastic yield function, plastic hardening law and plastic flow rule. Since the instantaneous plastic volu- metric deformation is an important feature of geomaterial, it seems necessary to use a non-associated · 198 · ACTA MECHANICA SOLIDA SINICA 2011 plastic flow rule which can better capture volumetric compressibility and dilatancy. Usually the instan- taneous plastic yield criterion and plastic potential can be respectively expressed by a scalar valued function of stress tensor, temperature variable, damage variable and thermodynamic force associated with hardening variables f p (σ ij ,η p ,T,ω) ≤ 0,g p (σ ij ,η p ,T,ω) ≤ 0 (15) The instantaneous plastic flow rule as well as the loading-unloading condition is described as follows: ε˙ p ij = Ë™ λ p ∂g p (σ ij ,η p ,T,ω) ∂σ ij (16) f p (σ ij ,η p ,T,ω)=0, Ë™ λ p ≥ 0,f p (σ ij ,η p ,T,ω) Ë™ λ p = 0 (17) Like the instantaneous plastic characterization, the creep plastic yield criterion and plastic potential can be expressed as follows: f vp (σ ij ,η vp ,T,ω) ≤ 0,g vp (σ ij ,η vp ,T,ω) ≤ 0 (18) The creep plastic flow rate may be determined in either the Perzyna format or the Duvaut-Lions format and will be given in detail in §3.2. 2.3. Damage Characterization Damage kinetics is determined by the pseudo-potential of dissipation, which is a scalar valued function of thermodynamic damage force Y ω . For the sake of simplicity, the pseudo-potential can be considered as an indicative function defined by the damage criterion, which may be expressed in the following general form: f ω (Y ω ,ω)=Y ω − r (ω) ≤ 0 (19) The function r (ω) represents the current damage energy release threshold at a given value of damage. Equation (19) also states that damage hardening is only associated with ω. However, similar to the thermodynamic damage force Y ω depending on instantaneous plastic and creep plastic deformation and temperature variables (see Eq.(10)), the damage evolution is then controlled by the instantaneous plastic flow, creep plastic flow and temperature. A normal dissipation scheme is employed here and the damage evolution rate is then determined by ω˙ = Ë™ λ ω ∂f ω ∂Y ω = Ë™ λ ω (20) The damagemultiplier Ë™ λ ω is a positive scalar obtained fromthe loading-unloading condition according to the Kuhn-Tucker relations f ω (Y ω ,ω)=0, Ë™ λ ω ≥ 0,f ω (Y ω ,ω) Ë™ λ ω = 0 (21) Specifically, in the case of elastic damage loading without instantaneous plastic and creep flow (ε˙ p ij =0andε˙ vp ij = 0), the damage consistency condition requires the rate of damage multiplier should satisfy the following equation: Ë™ λ ω = ∂Y ω ∂ε e ij ε˙ e ij + ∂Y ω ∂T Ë™ T r ′ (ω) = − ( ∂C ijkl (T,ω) ∂ω ε e kl ) ε˙ e ij + ∂Y ω ∂T Ë™ T r ′ (ω) (22) 2.4. Thermal Characterization The non-negative thermal entropy production is expressed by −q i T, i /T ≥ 0 (23) where q i is the heat flux vector. As usual, this inequality is fulfilled by relating q i to the temperature gradient T, i via Fourier’s law: Vol. 24, No. 3 Hui Zhou et al.: A Thermo-plastic/Viscoplastic Damage Model for Geomaterials · 199 · q i = −λT, i (24) where λ is the coefficient of thermal conductivity. The energy balance equation is obtained from the thermodynamic principles and expressed as follows: T sË™ = −div (q i )+r + ψ 1 (25) where r denotes the rate of internal heat production and ψ 1 the intrinsic dissipation due to plastic deformation and damage evolution. The thermal expansion of material is characterized by the linear thermal expansion σ˙ ij = C ijkl (T,ω)ε˙ e kl − α t Ë™ Tδ ij (26) where α t is the coefficient of thermal expansion. III. SPECIFIC MODEL FOR ROCK SALT Rock salt is extensively used as the host rock in underground engineering, such as high-level nuclear waste and petrol storage; moreover, the time and temperature effects are probably more pronounced in rock salt than in any other geomaterials. Hence, the present model was applied to rock salt and a specific model was developed to describe the time independent and time dependent behavior of rock salt under temperature effect. Many investigators [20–24] have conducted short term and long term tests on rock salt under different stress and temperature condition and obtained some understanding on the mechanical behavior of rock salt: (1) the elastic deformation of rock salt subject to deviatoric loading constitutes a very small fraction of the total deformation, and the elastic constants show weak dependence on pressure and temperature; (2) the plastic deformation in short and long term is strongly influenced by the ambient temperature and stress; (3) an increase in deviatoric stress causes increased plastic deformation; whereas an increase in confining stress tends to reduce plastic deformation and the damage induced by microcrack growth; (4) ductile crystalline plasticity is more temperature sensitive than is the brittle intercrystalline microcracking; and (5) higher deviatoric stress and temperature produce higher rates of creep deformation. According to the mechanical behavior, the functions in instantaneous plastic, creep plastic, damage and thermal modeling are determined and written in the following specific forms. 3.1. Instantaneous Plastic Modeling Due to the fact that the instantaneous plastic yield condition of rock salt is strongly dependent on confining stress, a curve-type yield surface is then necessary. In the present model, the following function is proposed. f p (σ ij ,η)=q − g (θ) η p (T,γ p ,ω)R c ( C s + p R c ) m ≤ 0 (27) p = − σ kk 3 ,q= √ 3J 2 ,J 2 = 1 2 s ij s ij ,s ij = σ ij − σ kk 3 δ ij θ = 1 3 sin −1 [ 3 √ 3 2 J 3 (J 2 ) 3/2 ] ,J 3 =dets ij (28) where p, q and θ are the mean stress (compressive mean stress is taken as positive), deviatoric stress and Lode’s angle, respectively; R c is a normalized coefficient which is equal to the uniaxial compression strength at room temperature; C s represents the coefficient of material cohesion at room temperature; and the parameter m defines the curvature of yield surface. Note that when m = 1, the yield function reduces to the classical Drucker-Prager yield function. The function g (θ) in Eq.(27) is introduced to describe the dependency of yield function on Lode’s angle in the deviatoric plane. The following function proposed by Willam and Warnkee [25] wasusedinthismodel g(θ)= 2(1−R 2 )cos ( θ + pi 6 ) +(2R− 1) · [ 4(1−R) 2 cos ( θ + pi 6 ) +5R 2 − 4R ] 1/2 4(1−R 2 )cos 2 ( θ + pi 6 ) +(2R− 1) 2 (29) · 200 · ACTA MECHANICA SOLIDA SINICA 2011 The advantage of the this function is that the failure surface is unconditionally convex for 0.5 ≤ R ≤ 1. For the sake of simplicity, we took g (θ) = 1 in the numerical simulation in the present work. The function η p (T,γ p ,ω) is responsible for the instantaneous plastic hardening which is related to the temperature and internal hardening variables. Furthermore, brittle deformation, which induced by microcrack and being the significant part of the total inelastic deformation, occurs primarily at low confining stress and temperature. The mechanism leading to brittle deformation is essentially attributed to the progressive destruction of interlocked structure. Here a damage variable ω is introduced into instantaneous plastic hardening law to describe the material softening. Therefore, the instantaneous plastic hardening law is an increasing function of the internal hardening variable γ p but a decreasing function of the damage variable ω. Based on the laboratory data in triaxial compression tests, the following function is proposed η p (T,γ p ,ω)=(1− ω) [ η p 0 +(η p m −η p 0 ) γ p b p (T)+γ p ] (30) where η p 0 and η p m denotes the initial yield threshold and ultimate value of plastic hardening law, re- spectively. The parameter b p (T ) controls the rate of plastic hardening which is temperature sensitive. After investigation of the temperature effect on instantaneous plastic flow, b p (T ) can be written as follows: b p (T)=b p 0 exp [a p (T − T 0 )] (31) where b p 0 denotes the value at a given temperature T 0 and a p a constant which describes the temperature effect on the parameter b p (T ). The internal hardening variable γ p is taken as the equivalent plastic shear strain which is defined by dγ p = √ 2 3 de p ij de p ij + √ 2 3 de vp ij de vp ij , de p ij =dε p ij − dε p kk 3 δ ij , de vp ij =dε vp ij − dε vp kk 3 δ ij (32) where de p ij andde vp ij are the deviatoric parts of instantaneous plastic and creepplastic strains, respectively. In order to complete plastic modeling, a plastic potential should be defined. For most geomaterials, a non-associated plastic flow rule is generally used. For instance, under given confining stress, the plastic volumetric strain rate may exhibit a transition from compressibility to dilatancy with the increase of applied deviatoric stress. The rate of plastic dilatancy decreases when confining stress increases. For a triaxial compression test at low confining stress, plastic dilatancy may occur at the beginning of plastic yielding. However, under higher confining stress, plastic compressibility is produced in the first stage before entering into the dilation domain. If confining stress becomes high enough, no plastic dilation is produced and the whole plastic domain is compressive, which induces pore collapse or grain crushing. Based on these evidences and inspired by the plastic model proposed by Pietruszczak et al. [26] ,the following plastic potential is used: g p = q − (η p − β p )(p+ C s R c ) (33) where the parameter β p defines the transition point from the compressibility zone (η p <β p ) to dilatancy zone (η p >β p ). Fig. 2. Illustration of yield and failure surfaces, plastic potential surface and boundary between compressibility and dila- tancy. Vol. 24, No. 3 Hui Zhou et al.: A Thermo-plastic/Viscoplastic Damage Model for Geomaterials · 201 · In Fig.2, a schematic illustration of the initial yield surface(peak surface), failure surface(yield sur- face), potential surface and compressibility-dilatancy transition line is shown in the p-q plane. 3.2. Creep Plastic Modelling In long term scale, the creep plastic strain gradually becomes significant after the first term of instantaneous plastic flow. Based on the previous work [18] , evolutions of instantaneous plastic yield surface and creep plastic loading surface are related to the same internal variables, for instance, the equivalent plastic shear strain γ p . The creep plastic loading surface is described by the same mathematic function as that of instantaneous plastic flow, which is given by f vp (σ ij ,η)=q − g (θ) η vp (T,γ p ,ω)R c ( C s + p R c ) m ≤ 0 (34) The function η vp (T,γ p ,ω) defines the current plastic hardening evolution of creep plastic loading surface and it takes the form η vp (T,γ p ,ω)=(1− ω) [ η vp 0 +(η vp m −η vp 0 ) γ p b vp (T)+γ p ] (35) Similar to η p 0 and η p m in Eq.(30) for instantaneous plastic hardening function, η vp 0 and η vp m define the initial threshold and ultimate value of creep plastic deformation. For the sake of simplicity, it is assumed that the value of η vp 0 is equal to the one of η p 0 . The parameter b vp (T ) which is similar to the parameter b p (T ) in Eq.(31) controls the kinetics of creep plastic hardening and takes the following form: b vp (T)=b vp 0 exp [a vp (T − T 0 )] (36) where b vp 0 denotes the value at a given temperature T 0 , a vp is a constant which describes the temperature effect on parameter b vp (T ). For creep plastic potential, the same function as that for instantaneous plastic potential is used as follows: g vp = q − (η vp − β p )(p+ C s R c ) (37) In the present work, the overstress concept of Perzyna is adopted, so the creep plastic rate can be determined by ε˙ vp ij = γ (T ) 〈 f vp R c 〉 n ∂g vp (σ ij ,η p ,T,ω) ∂σ ij (38) where 〈x〉 =(x+ |x|)/2 is the Macauley bracket, the fluidity coefficient γ (T ) is the function of tem- perature and takes the following form according to experimental data from creep tests at different temperature γ (T)=γ 0 exp ( − Z RT ) (39) where γ 0 is the value of fluidity at a reference temperature, Z the activation energy, and R the universal constant of perfect gas which takes the value of R = 8.31441 kJ ·mol −1 ·K −1 . 3.3. Damage Modelling In this section, a physical approach based on experimental evidence is employed to determine the damage evolution law. A number of experimental data on geomaterials revealed that the shear strain under compressive stress is the main cause of the damage evolution at different temperature. Mazars [27] has proposed a damage criterion for the concrete by taking an equivalent tensile strain as the driving force for damage evolution. For the rock salt, according to many experiment studies, the brittle deformation resulting frommicrocracking is coupled with plastic deformation in low confining stress and it appears to be much less temperature sensitive than plastic deformation. Therefore, it is assumed that the damage evolution is temperature independent, and the equivalent plastic shear strain γ p is chosen as the driving force of damage evolution f ω (γ p ,ω)=ω − ω c [1− exp (−b ω γ p )] ≤ 0 (40) · 202 · ACTA MECHANICA SOLIDA SINICA 2011 where b ω is a model parameter which controls the damage evolution rate; ω c is the ultimate damage value which determines the residual strength of material. For the rock salt, the influence of confining stress on the strain hardening and softening response decreases rapidly as confining stress increases and the strain softening almost occurs under confining stress only between 0 and 3.4 MPa. Hence, ω c is defined as a function related to confining stress σ m . ω c = ω 0 c exp (−a ω σ m ) (41) where ω 0 c represents the ratio between residual strength and peak strength in uniaxial compression test; a ω is a model parameter which describe the effect of confining stress on ultimate damage. In Eq.(41), the ultimate damage value ω c is gradually decreased with the increase of confining stress σ m .Figure 3 shows the evolution of instantaneous and creep yield surface due to material damage ω. Fig. 3. Evolution of yield surface due to material damage. Based on micromechanical analysis corresponding to a dilute distribution of penny shaped micro- cracks [28,29] , the effective elastic properties can be expressed as functions of damage variables. For isotropic materials, the bulk modulus and shear modulus of damaged material are affected by induced damage independently. However, geomaterial is essentially subjected to compression-dominated stresses. In this case, microcracks are closed and crack growth is related to frictional sliding along crack surface. According to mechanical analysis, the induced damage mainly affects the shear modulus of material. Therefore, the following relations are adopted for the determination of effective elastic properties of damaged material K (T,ω)=K 0 (T ),G(T,ω)=(1− κω)G 0 (T ) (42) where K 0 (T)andG 0 (T ) are the bulk modulus and shear modulus of intact material at different tem- perature, respectively. The coefficient κ characterizes the damage effect on the elastic shear modulus and its value is related to the elastic properties of solid matrix [27] . However, since a macroscopic mod- eling is concerned here, the value of κ should be determined from relevant experimental data showing progressive degradation of elastic modulus with damage evolution. For the sake of simplicity, we will take κ = 1 in this paper. 3.4. Thermal Modelling Some researchers have studied the influence of temperature on the elastic constants of rock salt. Their results show that the stiffness of rock salt decreases a little as temperature increases. Young’s modulus E in uniaxial compression tests at a rate is bounded by [20,30] −0.04 ≤ dE dT ≤−0.016 GPa/K (43) Frost and Ashby [31] reported that the change in the shear modulus G of rock salt to be dG dT = −0.01 GPa/K (44) Pfeifle et al. [20,30] can discern no temperature dependence for the Poisson’s ratio over the repository range of temperature. Therefore, it may be assumed that the elastic constants of rock salt are constant over the repository range of temperature. Then Eq.(5) can be updated as C ijkl (T,ω)=2(1− ω)G 0 K ijkl +3K 0 J ijkl (45) Vol. 24, No. 3 Hui Zhou et al.: A Thermo-plastic/Viscoplastic Damage Model for Geomaterials · 203 · IV. NUMERICAL SIMULATIONS The proposed model is applied to the Avery rock salt in this paper. Firstly, the identification of parameters will be presented based on the experimental data [22] . Secondly, the numerical result will be compared with the data of time independent and time dependent tests at different temperatures. Finally, the mechanical behavior in long term creep test under varied temperature will also be predicted. A numerical implementation algorithm is developed for the simulation of such tests. The general outline of the integration algorithm for the kth loading step can be summarized as follows: (i) calculate the trial elastic part of the total strain with the following known quantities: σ k−1 , ε k−1 , γ p k−1 , ω k−1 and T k−1 ; (ii) set j = 1 and start iterative loop; (iii) give an incremental total strain ∆ε k and temperature ∆T k , perform an elastic prediction of stresses as follows: ε e k,j = ε e k,j−1 + ∆ε k ,σ k,j = C (T k,j−1 ,ω k,j−1 ):ε e k,j with ε e k,0 = ε e k−1 , T k,0 = T k−1 and ω k,0 = ω k−1 ; and update the parameters related to temperature; (iv) check the plastic yield condition f p j−1 ≤ 0 in the absence of damage evolution and determine plastic multiplier if plastic flow occurs; then calculate ∆ε p j ; (v) check the plastic yield condition f vp j−1 ≤ 0 in the absence of damage evolution and calculate ∆ε vp j if creep plastic flow occurs; (vi) calculate γ p j , then check the damage evolution f ω j ≤ 0 and determine the updated damage state; (vii) if ω k,j − ω k,j−1 >e(e is a small positive tolerance coefficient), then j = j + 1, go to (iii); (viii) update values of quantities: σ k = σ k,j , γ p k = γ p k,j and ω k = ω k,j . 4.1. Time Independent Behavior at Different Temperature The two elastic constants E,ν and eight parameters R c ,m,C s ,η p 0 ,η p m ,b p 0 ,a p ,β p in instantaneous plastic modeling and three parameters ω 0 c ,a ω ,b ω in damage function may be determined from classic triaxial compression tests with different confining stress at different temperature. The elastic parameters E and ν are obtained from the linear part of stress-strain curve at the first stage of loading at room temperature, their values for Avery rock salt are E =30.6GPa,ν=0.38. R c represents the strength of uniaxial compression under room temperature and here takes the value of R c =12MPa. The procedure of identification of other parameters is described as follows: at a given temperature, all parameters are first identified from triaxial compression tests using loading-unloading cycles under different confining stress. Then, the evolution of certain parameters with temperature variables is determined from triaxial tests at different temperatures. The parameters m and C s related to the failure surface can be identified by fitting the relation between peak deviatoric stress q peak =max(σ 1 −σ 3 )and mean stress at room temperature (see Fig.4). η p 0 ,η p m and b p 0 are obtained by plotting the plastic hardening function η p versus the equivalent plastic shear strain γ p at room temperature. For the temperature related variable a p , we can plot failure curves at different temperature (see Fig.4), then the value of a p is determined by fitting the relation between failure curves and temperature. In addition, the parameter β p can be identified by identifying the stress point at the transition from plastic contractile to dilatant volumetric strain (see Fig.5). Finally, ω 0 c ,a ω can be obtained by studying the evolution of the ratio of the residual strength and peak strength at room temperature with different confining stress. According to the unloading–reloading cycles in triaxial compression test, the evolution of damage variable is calculated by the degradation ratio of effective elastic modulus; consequently, the parameter b ω , characterizing damage evolution kinetics, may be determined by plotting damage variable (identified from elastic modulus degradation) versus the equivalent plastic shear strain γ p . The values of parameters for time independent behavior of Avery rock salt are given in Table 1. After identification of above parameters, the numerical result is compared with short time triaxial compression tests under different confining stress at T = 100 â—¦ C (Fig.6) and T = 200 â—¦ C (Fig.7), respectively. Table 1. The values of parameters for time independent behavior of Avery rock salt Instantaneous plastic parameters Damage parameters mC s η p 0 η p m b p 0 a p β p ω 0 c a ω b ω 0.45 3.5 0.2 0.88 0.002 0.026 0.7 0.5 0.7 10 · 204 · ACTA MECHANICA SOLIDA SINICA 2011 Fig. 4. Failure curves of rock salt as a function of tem- perature (after Wawersik [20] and Langer [23] ). Fig. 5. Compressibility-dilatancy boundary of rock salt in the p-q plane (after Cristescu [21] ). Fig. 6. Short term triaxial compression test with different confining stress at T = 100 â—¦ C. Fig. 7. Short term triaxial compression test at T = 200 â—¦ C. In Figs.6 and 7, we can find that our numerical results are in good agreement with the experimental data in describing the stress and temperature effects in short term triaxial compression test. The volume strain of Avery rock salt is not available in present literatures, while in order to exhibit the model performance in volumetric strain evolution, the result from numerical simulation is given in Fig.8. Figure 8 shows that the volumetric strain undergoes compaction-dilatancy transition and confining stress greatly influences the amount of dilatancy. 4.2. Time Dependent Behavior under Stable Temperature There are nine parameters R c ,m,C s ,η vp m ,b vp 0 ,a vp ,β p ,γ 0 and Z in creep plastic modeling, among which R c ,m,C s and β p have been determined in previous section, other parameters η vp m ,b vp 0 ,a vp ,γ 0 and Z can be identified by creep tests at different temperature. Like the identification of parameters η p m and b p 0 , η vp m and b vp 0 are obtained by plotting the curve of the creep plastic hardening function η vp vs. the equivalent plastic shear strain γ p at room temperature. Finally, the parameter a vp can be determined by fitting the relation between the creep plastic hardening function η vp and temperature variable. In Vol. 24, No. 3 Hui Zhou et al.: A Thermo-plastic/Viscoplastic Damage Model for Geomaterials · 205 · Fig. 8. Prediction of volumetric strain in short term triaxial compression test with confining stress of 3.4 MPa and 20 MPa at T = 100 â—¦ C. addition, γ 0 and Z are obtained by fitting the curve of the creep deformation rates versus temperature. The values assigned to these parameters are listed in Table 2. Using above parameters, the presentmodel is applied to the creep tests under different stress condition at different temperature. Table 2. The values of parameters in creep plastic modeling Parameters η vp m b vp 0 a vp γ 0 (s −1 ) Z (kcal · mol −1 ) Value 1.2 0.21 0.038 2.0 × 10 −8 30 Fig. 9. Uniaxial creep tests at T = 100 â—¦ C. From Fig.9, we can find the creep deformation behavior is strongly influenced by the axial stress and the proposed model can well describe this character. Figure 10 shows that increasing temperature greatly increases the inelastic deformation in the creep test of rock salt under a given stress condition, and there is a good agreement between numerical result and experimental data. · 206 · ACTA MECHANICA SOLIDA SINICA 2011 Fig. 10. Triaxial creep tests (confining stress 15 MPa, deviatoric stress 15 MPa) at different temperature. 4.3. Time Dependent Behavior under Varied Temperature There is little laboratory study on the time dependent behavior of Avery rock salt under varied temperature in present literatures. Therefore, we will only present numerical results in the following part. For the rock salt, temperature has a weak influence on both the thermal expansion coefficient α t and heat conduction coefficient λ; while for the sake of simplicity, their values are assumed as constants over the temperature range 25-125 â—¦ Candtakenasα t =40×10 −6 K −1 and λ=5Wm −1 K −1 , respectively. Fig. 11. Prediction of deformation behavior of triaxial creep test under varied temperature and constant temperature. Figure 11 shows the numerical results of triaxial creep test under varied temperature. During the process of creep deformation, the confining stress and deviatoric stress were kept at σ 3 =5MPaand σ 1 − σ 3 = 5 MPa, respectively, and the temperature was increased from 25 â—¦ C to 150 â—¦ Cwithastep of 25 â—¦ Ceach1.296× 10 6 s. The deformation under the same confining pressure and deviatoric stress at a constant temperature of 25 â—¦ C (dash line) was also predicted in order to study the temperature effect. The numerical result reveals that the deformation rate is greatly increased with the increase of temperature. Vol. 24, No. 3 Hui Zhou et al.: A Thermo-plastic/Viscoplastic Damage Model for Geomaterials · 207 · V. CONCLUSIONS A thermo-plastic/viscoplastic damage model was formulated to describe the time independent and time dependent behaviors of geomaterial with the consideration of temperature effect. The plastic deformation was divided into two parts: instantaneous plastic strain and creep plastic strain. The two parts were independently described by their own yield functions, hardening functions and potential functions, which are in the same form. The temperature variable was introduced into the hardening functions of both instantaneous plastic and creep plastic modeling in order to consider temperature effect. The damage evolution was described by an indicative function including temperature variable. And a linear thermal expansion law was used in the formulation of constitutive equations. A specific model was proposed according to the mechanical behavior of rock salt under the stress and temperature effect. We employed the Drucker-Prager type yield function in instantaneous plastic and creep plastic modeling, and introduced the temperature variable into hardening function to depict the temperature effect on plastic flow. The equivalent plastic shear strain is chosen as the driving force for damage evolution in our model. At last, the proposed model was implied to Avery rock salt, from which we can figure out that numerical results obtained from the model had a good agreement with the corresponding experimental data in triaxial compression tests and creep tests at different temperatures, and the deformation behavior under varied temperature was also predicted. References [1] Kelvin,L., Elasticity. In Encyclopedia Britannica (9th edition), Baynes,T.S. (edited.). London: Adam & Charles Black, 1878. [2] Bingham,E.C. and Durham,T.C., The viscosity and fluidity of suspensions of finely divided solids in liquids. American Chemical Journal, 1911, 46: 278-297. [3] Duvaut,G. and Lions,L.J., Inequalities in Mechanics and Physics. Berlin: Springer, 1962. [4] Perzyna,P., Thermodynamic theory of viscoplasticity. New York: Academic Press, 1971. [5] Valanis,K.C., A theory of viscoplasticity without a yield surface. Archives of Mechanics, 1971, 23: 517-551. [6] Miller,A.K., An inelastic constitutive model for monotonic, cyclic, and creep deformation. ASME, Journal of Engineering Materials and Technology, 1976, 98: 97-105. [7] Bodner,S.R. and Merzer,A., Viscoplastic constitutive equations for copper with strain rate history and temperature effects. ASME, Journal of Applied Mechanics, 1978, 100: 388-394. [8] Liu,M.C.M. and Krempl,E., A uniaxial viscoplastic model based on total strain and overstress. Journal of Mechanics and Physical of Solids, 1979, 27(5-6): 377-391. [9] Lemaitre,J. and Chaboche,J.L., Mechanics of Solid Materials. Cambridge: Cambridge University Press, 1998. [10] Munson,D.E. and Dawson,P.R., Salt constitutive modeling using mechanism maps. In: First Conference of Mechanical Behaviors of Salt, Pennsylvania State University, 1984. [11] Langer,M., Rheological behavior of rock masses. In: Proceedings of 4th International Congress on Rock Mechanics, Montreux, 1979, 3: 29-96. [12] Cristescu,N., Damage and failure of viscoplastic rock-like materials. International Journal of Plasticity, 1986, 2(2): 189-204. [13] Dahou,A., Shao,J.F. and Bederiat,M., Experimental and numerical investigations on transient creep of porous chalk. Mechanics of Materials, 1995, 21(2): 147-158. [14] Jin,J. and Cristescu,N., An elastic viscoplastic model for transient creep of rock salt. International Journal of Plasticity, 1998, 14(1): 85-107. [15] Maranini,E. and Yamaguchi,T., A non-associated viscoplastic model for the behaviour of granite in triaxial compression. Mechanics of Materials, 2001, 33(5): 283-293. [16] Shao,J.F., Zhu,Q.Z. and Su,K.. Modeling of creep in rock materials in terms of material degradation. Computers and Geotechnics, 2003, 30(7): 549-555. [17] Pietruszczak,S., Lydzba,D. and Shao,J.F., Description of creep in frictional materials in terms of microstruc- ture evolution. Journal Engineering Mechanics, 2004, 130(6): 681-690. [18] Zhou,H., Jia,Y. and Shao,J.F., A unified elastic-plastic and viscoplastic damage model for quasi-brittle rocks. International Journal of Rock and Mining Sciences, 2008, 45(8): 1237-1251. [19] Lubliner,J., On the thermodynamic foundations of non-linear solid mechanics. International Journal of Non-Linear Mechanics, 1972, 7(3): 237-254. [20] Wawersik,W.R. and Hannum,D.W., Mechanical behavior of New Mexico rock salt in triaxial compression up to 200 â—¦ C. Journal of geophysical research, 1980, 85(b2): 891-900. · 208 · ACTA MECHANICA SOLIDA SINICA 2011 [21] Cristescu,N. andHunsche,U., Time Effects in RockMechanics. NewYork: JohnWiley& Sons Incorporation, 1998. [22] Senseny,P.E., Hansen,F.D., Russell,J.E., Carter,N.L. and Handin,J.W., Mechanical behavior of rock salt: phenomenology and micromechanisms. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 1992, 29(4): 363-378. [23] Langer,M., Geotechnical investigation methods for rock salt. Bulletin of IAEG, 1982, 25: 155-164. [24] Udo,H. and Andreas,H., Rock salt—the mechanical properties of the host rock material for a radioactive waste repository. Engineering Geology, 1999, 52(3): 271-291. [25] Willam,K.J. and Warnkee,E.P., Constitutive model for the triaxial behavior of concrete. In: Proceedings of International Association of Bridge and Structural Engineers Seminar on ‘Concrete Structures Subjected to Triaxial Stresses’, Paper III-1, Bergamo: ISMES, 1975, 174-186. [26] Pietruszczak,S., Jiang,J. and Mirza,F.A., An elastoplastic constitutive model for concrete. International Journal of Solids and Structures, 1988, 24(7): 705-722. [27] Mazars,J., Application de la me’ canique de l’endommagement non lineaire et a la rupture du beton de structure. Ph.D. Thesis, University Paris 6, 1984 (in French). [28] Mura,T., Micromechanics of defects in solids (2nd ed). Dordrecht: Martinus Nijhoff, 1987. [29] Nemat-Nasser,S. and Hori,M., Micromechanics: Overall Properties of Heterogeneous Materials. Amster- dam: North-Holland, 1993. [30] Pfeitle,T.W., Mellegard,K.D. and Senseny,P.E., Preliminary Constitutive Properties for Salt and Nonsah Rocks from Four Potential Repository Sites. Report ONWI-450, prepared by RE/SPEC Inc. for Office of Nuclear Waste Isolation, Battelle Memorial Institute, Columbus, OH, 1983. [31] Frost,H.J. and Ashby,M.F., Deformation-mechanism Maps. New York: Pergamon Press, 1982. copper with strain rate history and temperature effects. ASME, Journal of Applied Mechanics, 1978, 100: 388-394. [8] Liu,M.C.M. and Krempl,E., A uniaxial viscoplastic model based on total strain and overstress. Journal of Mechanics and Physical of Solids, 1979, 27(5-6): 377-391. [9] Lemaitre,J. and Chaboche,J.L., Mechanics of Solid Materials. Cambridge: Cambridge University Press, 1998. [10] Munson,D.E. and Dawson,P.R., Salt constitutive modeling using mechanism maps. In: First Conference of Mechanical Behaviors of Salt, Pennsylvania State University, 1984. [11] Langer,M., Rheological behavior of rock masses. In: Proceedings of 4th International Congress on Rock Mechanics, Montreux, 1979, 3: 29-96. [12] Cristescu,N., Damage and failure of viscoplastic rock-like materials. International Journal of Plasticity, 1986, 2(2): 189-204. [13] Dahou,A., Shao,J.F. and Bederiat,M., Experimental and numerical investigations on transient creep of porous chalk. Mechanics of Materials, 1995, 21(2): 147-158. [14] Jin,J. and Cristescu,N., An elastic viscoplastic model for transient creep of rock salt. International Journal of Plasticity, 1998, 14(1): 85-107. [15] Maranini,E. and Yamaguchi,T., A non-associated viscoplastic model for the behaviour of granite in triaxial compression. Mechanics of Materials, 2001, 33(5): 283-293. [16] Shao,J.F., Zhu,Q.Z. and Su,K.. Modeling of creep in rock materials in terms of material degradation. Computers and Geotechnics, 2003, 30(7): 549-555. [17] Pietruszczak,S., Lydzba,D. and Shao,J.F., Description of creep in frictional materials in terms of microstruc- ture evolution. Journal Engineering Mechanics, 2004, 130(6): 681-690. [18] Zhou,H., Jia,Y. and Shao,J.F., A unified elastic-plastic and viscoplastic damage model for quasi-brittle rocks. International Journal of Rock and Mining Sciences, 2008, 45(8): 1237-1251. [19] Lubliner,J., On the thermodynamic foundations of non-linear solid mechanics. International Journal of Non-Linear Mechanics, 1972, 7(3): 237-254. [20] Wawersik,W.R. and Hannum,D.W., Mechanical behavior of New Mexico rock salt in triaxial compression up to 200 â—¦ C. Journal of geophysical research, 1980, 85(b2): 891-900. · 208 · ACTA MECHANICA SOLIDA SINICA 2011 [21] Cristescu,N. andHunsche,U., Time Effects in RockMechanics. NewYork: JohnWiley& Sons Incorporation, 1998. [22] Senseny,P.E., Hansen,F.D., Russell,J.E., Carter,N.L. and Handin,J.W., Mechanical behavior of rock salt: phenomenology and micromechanisms. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 1992, 29(4): 363-378. [23] Langer,M., Geotechnical investigation methods for rock salt. Bulletin of IAEG, 1982, 25: 155-164. [24] Udo,H. and Andreas,H., Rock salt—the mechanical properties of the host rock material for a radioactive waste repository. Engineering Geology, 1999, 52(3): 271-291. [25] Willam,K.J. and Warnkee,E.P., Constitutive model for the triaxial behavior of concrete. In: Proceedings of International Association of Bridge and Structural Engineers Seminar on ‘Concrete Structures Subjected to Triaxial Stresses’, Paper III-1, Bergamo: ISMES, 1975, 174-186. [26] Pietruszczak,S., Jiang,J. and Mirza,F.A., An elastoplastic constitutive model for concrete. International Journal of Solids and Structures, 1988, 24(7): 705-722. [27] Mazars,J., Application de la me’ canique de l’endommagement non lineaire et a la rupture du beton de structure. Ph.D. Thesis, University Paris 6, 1984 (in French). [28] Mura,T., Micromechanics of defects in solids (2nd ed). Dordrecht: Martinus Nijhoff, 1987. [29] Nemat-Nasser,S. and Hori,M., Micromechanics: Overall Properties of Heterogeneous Materials. Amster- dam: North-Holland, 1993. [30] Pfeitle,T.W., Mellegard,K.D. and Senseny,P.E., Preliminary Constitutive ProA thermo-plastic/viscoplastic damage model for geomaterialsA thermo-plastic/viscoplastic damage coupled model was formulated to describe the time independent and time dependent behaviors of geomaterials under temperature effect. The plastic strain was divided into instantaneous plastic strain and creep plastic strain. To take temperature effect into account, a temperature variable was introduced into the instantaneous and creep plastic behavior descriptions and damage characterization, and a linear thermal expansion law was used in constitutive equation formulation. According to the mechanical behavior of rock salt, a specific model was proposed based on the previous model and applied to Avery rock salt, in which the numerical results obtained from our model had a good agreement with the data from experiments.thermo-plastic/viscoplastic damage coupled Available online at www.sciencedirect.com ScienceDirect Materials Today: Proceedings 5 (2018) 7168–7173 www.materialstoday.com/proceedings 2214-7853 Selection a (IMME17). IMME17 Experimental studies on Drilling of 410 Stainless Steel drilling experiments were conducted on SS410 to analyze the effect of drilling parameters and machining environment on surface roughness and machining time. Experiments were conducted in a jig boring machine with HSS twist drill of 10 mm on 5 mm thickness SS410 plate. Drilling tests were carried out as per Taguchi’s L9 array using three cutting speeds (Vc: 11, 16 and 21 m/min) and three feeds (f: 0.02, 0.05 and 0.08 mm/rev) using three different cutting fluids (castor oil, kerosene and coconut oil). Surface roughness (Ra) values were decreased from 7.326 to 2.423 µm with increase in 'Vc' and decrease in 'f' and machining time was increased from 7.4 to 31.5 seconds with increase in 'Vc' and 'f'. Optimum process parameters were identified and verified experimentally to improve the machinability of the SS410. Coconut oil medium gave good machinability results at higher 'Vc' and 'f'. © 2017 El Selection d Manufac Keywords: *Correspo kjayakum sevier Ltd. All rights reserved. and/or Peer-review under responsibility of International Conference on Emerging Trends in Materials an turing Engineering (IMME17). A. Jayagantha, K. Jayakumarb,*, A. Deepaka, K. Pazhanivelc aB.E Final year student, Department of Mechanical Engg, SSN College of Engineering, Chennai-603110, India bAssociate Professor, Department of Mechanical Engg, SSN College of Engineering, Chennai-603110, India cProfessor and Director, Department of Mechanical Engg, ARS College of Engineering, Chennai - 603209, India Abstract Stainless Steel 410 (SS410) is 12% chromium martensitic steel which can be heat treated to obtain wide range of mechanical properties. Due to its strong corrosion resistance and excellent combination of toughness and strength, it is used for furnace parts, micrometer parts, steam turbine buckets, blades, pump parts, petrochemical equipments, etc. However, during machining of SS410, difficulties are there due to its low thermal conductivity, high ductility and work hardening which can be minimized by using suitable machining parameters and cutting fluids. Therefore, now a day’s research works have been focused towards machinability of SS410 through different machining processes. Among different machining processes, many works have not been started on drilling of SS410. Drilled holes are used to receive screws, bolts, shafts, steam pipes, fitting of furniture and equipment. By considering the above applications and challenges in machining of this material, © 2017 Elsevier Ltd. All rights reserved. nd/or Peer-review under responsibility of International Conference on Emerging Trends in Materials and Manufacturing Engineering SS410 material; Drilling studies; Surface roughness; Machining Time. nding author [email protected] A. Jayaganth et al. / Materials Today: Proceedings 5 (2018) 7168–7173 7169 1. Introduction SS410 is a corrosion and heat resistant chromium stainless steel and is widely used in aerospace industries as bearings, water valves, pumps, gas turbines, compressor components, surgical tools, nuclear applications, fasteners, etc. But machinability is low for SS410 material because of low thermal conductivity, high ductility, high strength and hardness (HRC above 40), high fracture toughness and rate of work hardening [1]. From the year 2000 onwards researchers have started working in machining of SS304, SS304L and SS410 by different machining processes due to its increase in application. Initially machining of the material by conventional turning process has been started. Senthilkumar et al. [2] carried out experimental studies on effect of cutting speed on different tool wears through turning process using alumina based ceramic cutting tools. They concluded that Ti[C, N] mixed alumina ceramic cutting tools which exhibits slightly higher tool life than zirconia toughened alumina ceramic cutting tool. Apart from machining parameters, cutting fluids are also affects the machining of the material. Dry machining is ecologically desirable and it will be considered as a requirement for manufacturing enterprises in the near future but the friction and adhesion between chip and tool tend to be higher, which causes higher temperature, wear rate and consequently, shorter tool life. To overcome the difficulty in dry machining, wet machining is an alternate method which has the advantage of making better part quality with less tool wear by improving the tribological characteristics of the work piece–tool–chip system. Galanis et al. [3] investigated the effect of dry and wet machining on turning of SS422 material and found that surface roughness and tool temperature were decreased in wet machining than dry machining with increase in machining parameters. In 2014, Jitendra Thakkar and Mitesh I Patel did review on turning studies on SS410 with respect to the effect of machining parameter on surface roughness and material removal rate (MRR) [4]. In order to achieve good surface finish, cylindrical grinding study of SS410 has been carried out by changing grinding parameters and coolant flow rate for getting high surface finish of 0.28 µm at optimum process parameter [5]. Recently, coated carbide insert tool has been used to turn the SS410 effectively for getting high surface finish by varying machining parameters. From the study, lowest surface roughness (Ra) was obtained at medium spindle speed, lowest feed and depth of cut [6]. Due to difficulty in conventional machining of SS410, unconventional machining processes came in to manufacturing industries to get high surface finish and accuracy especially for complex shaped products. Mount et al. [7] machined variety of stainless steels including SS410, SS316, Jethete (J), Duplex (D) and Super Duplex (SD) using Electro Chemical Machining (ECM) process with chloride and nitrate electrolytes. Later electrical discharge machining (EDM) and wire EDM (WEDM) processes came in to market to machine SS410 for getting complex shaped product with reasonably good tolerance. Viral B. Prajapati et al. [8] conducted EDM studies on SS410 by copper electrode and EDM oil as dielectric fluid. They concluded that increase in pulse on time and discharge current increased the MRR and Ra and increase in pulse off time decreased the Ra and MRR. WEDM study has been carried SS410 material using brass wire electrode to understand the effect of WEDM parameters on MRR and surface finish [9]. Major findings from the work were increase in Pulse on time and wire tension increased the MRR and Ra. At the same time, increase in pulse off time and wire feed decreased MRR and Ra. From the above literature, machining studies on SS410 is still in preliminary investigation stage and it needs a detailed study especially in effect of drilling parameters and cutting environment of machinability of SS410. Hence, the present work is focussing on drilling studies on SS410 at different cutting speed, feed in three different coolant and its machinability analysis in terms of surface roughness and machining time. 2. Material and machining arrangement Semi circular disc shaped SS410 workpiece of ∅80 mm and 5 mm thickness was used for drilling experimentation and the drilled workpiece at different condition is shown in Fig. 1. Composition of the SS410 used in the present study in weight percentages are Fe (approximately 84%), <0.15% C, 11.5-13.5% Cr, >0.75% Ni, <1.0% Mn, <1.0% Si, <0.04% P, <0.03% S. HSS drill bit with ∅10 mm and 118° semi cone angle has been used as drill tool. Mitsui seiki of Japan make Jig boring machine, with model number of A12207 was used for drilling experiments and Fig. 2 shows the closed view of the experimental setup. The machining conditions selected for drilling of SS410 are taken from literature and process parameters with their levels are given in table 1. 7170 A. Jayaganth et al. / Materials Today: Proceedings 5 (2018) 7168–7173 Table 1. Process parameters and their levels. Process parameters Level 1 Level 2 Level 3 A: Cutting speed (rpm or m/min) 350/11 500/16 650/21 B: Feed (mm/rev) 0.02 0.05 0.08 C: Cutting fluid Castor oil Kerosene Coconut oil Fig. 1. Drilled workpiece. Fig. 2. Drilling of SS410. Fig. 3. Measurement of surface roughness. Since 3 factors and 3 levels are involved, Taguchi’s L9 orthogonal array experimental design is used to minimize the number of experiments. The workpiece with nine drilled through holes and coolant supply around the drill bit is shown in Fig. 2. After machining, Mitutoyo make SJ-210 portable contact type surface roughness tester was used to measure the average surface roughness (Ra) values. Ra values were measured at three different locations on the drilled hole inner surface by rotating the workpiece at random angle. Measurement conditions used were cut off length (λc) 0.8 mm and number of sample (n) as 5 using diamond tip of ∅5 μm and the arrangement for Ra measurement is shown in Fig. 3. Workpiece is oriented in different height and angle with the help of slip gauges. Average of three Ra values was considered as a result for each experiment. Machining time during drilling was recorded using a normal digital clock. Table 2 lists the experimental run and obtained results. Table 2. Experimental run and results. Sl No Speed (RPM) Feed (mm/rev) Coolant Machining Time (Sec) Avg Ra (µm) 1 350 0.03 Castor oil 31.5 2.814 2 350 0.05 Kerosene 24.03 7.326 3 350 0.08 Coconut oil 13.96 4.343 4 500 0.03 Kerosene 24.45 6.612 5 500 0.05 Coconut oil 15.14 3.624 6 500 0.08 Castor oil 9.34 3.665 7 650 0.03 Coconut oil 14.42 2.423 8 650 0.05 Castor oil 10.61 4.139 9 650 0.08 Kerosene 7.4 2.683 Drilled hole at optimum condition Detector with stylus tip Drill bit Coolant supply A. Jayaganth et al. / Materials Today: Proceedings 5 (2018) 7168–7173 7171 3. Results and Discussion 3.1. Effect of process parameters on surface finish Surface quality of the machined workpiece surface is dependent on cutting conditions as well as type of coolant used and it plays critical role in functioning as well as fatigue life of the product. The graphs (Figs. 4 and 5) were obtained from Design Expert (DX7) software from experimental results. Fig. 4a shows the response of surface roughness against cutting speed. The surface finish is improved when increase in cutting speed because of easy removal of chips from the drilled hole and continuous chips without built-up edge (BUE) was formed at higher cutting speed which results in improvement of surface finish [10]. C: Coolant (1. Castor oil: 2. Kerosene: 3. Coconut oil) Fig. 4. (a-c). Effect of process parameter on surface roughness. From the experimental results, the surface roughness increased with increase in feed which is in line with the classical theory of metal cutting (Fig. 4b). Also, friction between workpiece and drill bit interface increased when increase in feed, which eventually increases the cutting temperature near the cutting zone. Hence, the shear strength of the composite material reduces and the material behaves ductile fashion and SS410 is gummy in nature, which makes the chips to detach from the workpiece is difficult, thereby increasing the surface roughness [11]. During machining, cutting fluids are mainly used for cooling, lubrication and removal of heat and chips. The viscosity of coconut oil (80 cP) is more than that of kerosene (1.64 cP) and less than castor oil (650 cP), which favours easy flow of cutting fluid on all interior area of the hole at minimal oil condition [10]. This enables the drop of friction between the tool and work piece, and easy removal of heat developed at the interface thus caused effortless removal of chips and improved the surface finish than other coolant. From the slop of the graphs (Fig. 4(a-c)), feed is taken as least affecting parameter on surface roughness in the present study. 350 500 650 2.4 3.65 4.9 6.15 7.4 A: Speed (rpm) S ur fa ce r ou gh ne ss ( m ic ro n) g ( ) 0.02 0.05 0.08 2.4 3.65 4.9 6.15 7.4 B: feed (mm/rev) S ur fa ce r ou gh ne ss (m ic ro n) g ( ) S ur fa ce r ou gh ne ss ( m ic ro n) g ( ) 1 2 3 1.25228 2.78921 4.32614 5.86307 7.4 7172 A. Jayaganth et al. / Materials Today: Proceedings 5 (2018) 7168–7173 3.2. Effect of process parameters on machining time C: Coolant (1. Castor oil: 2. Kerosene: 3. Coconut oil) Fig. 5. (a-c). Effect of process parameter on machining time. Figure 5. (a-c) shows the variation of machining time with respect to process parameter. MRR is mainly depends on machining time (MT) and decrease in MT increases the MRR and production rate of an industry. From Fig. 5(a-c) it is found that increase in drill rotational speed and feed decreased the machining time with faster removal of chips from the drilled hole and tool interface. Machining time was observed as low in coconut oil compared to castor oil and kerosene. This is due trouble-free flow of coconut oil inside the hole due to its moderate viscosity which enhanced thermal conductivity at tool chip interface and also washed the chips in faster rate [10]. In case of kerosene, due to its lower viscosity, it sparkled out without removing chips during machining before reaching to the full depth of the hole. Due to high viscosity of castor oil, it may take more time to reach drill and work interface thus reduced the faster chip removal and increased the machining time. 3.3. Selection of optimal machining parameters From the response graphs shown in Figs. 4 and 5, it was identified that high speed (A3: 650 rpm), high feed (B3: 0.08 mm/rev) and coolant (C3: Coconut oil) are the optimal process parameter combination for getting low surface roughness and machining time. The A3-B3-C3 combination is not present in the L9 OA and the confirmation experiment (Indicated in Fig. 1) was carried at this level and the corresponding surface roughness value obtained as 2.312 µm and machining time as 4.71 seconds which were lower than all 9 conditions. 350 500 650 7 13.25 19.5 25.75 32 A: Speed (rpm) M ac hi ni ng T im e (S ec ) g ( ) 0.02 0.05 0.08 7 13.25 19.5 25.75 32 B: feed (mm/rev) M ac hi ni ng T im e (S ec ) g ( ) M ac hi ni ng T im e (S ec ) g ( ) 1 2 3 7 13.25 19.5 25.75 32 A. Jayaganth et al. / Materials Today: Proceedings 5 (2018) 7168–7173 7173 4. Conclusions Drilling experiments on SS410 were carried out using HSS drill bit with different speed, feed and cutting fluid. The major conclusions arrrived are:  The use of coconut oil showed considerable improvement in surface finish as compared to kerosene and castor oil due it its viscosity property.  Reduced machining time was observed for coconut oil environment compared to kerosene and castor oil.  From the slope of the response graphs, drilling speed and type of cutting fluids were observed as the major influencing factors for surface roughness. The machining time is influenced by drill rotational speed and feed. References [1] K. Chandrasekaran, P. Marimuthu, K. Raja, A. Manimaran, Indian J. of Engg. and Materials Sciences. 20 (2013) 398–404. [2] A. Senthil Kumar, A. Raja Durai, T. Sornakumar, J. of Materials Processing Tech. 173 (2006) 151–156. [3] N. I. Galanis, D. E. Manolakos, N. M. Vaxevanidis, International Conference on Manufacturing Engineering. (2008) 91–98. [4] Jitendra Thakkar, Mitesh I Patel, Inter. J. of Engineering Research and Applications. 4 (2014) 235–242. [5] V. Saravana Kumar, Hridul Pavithran, K. V. Sachin, Inter. J. of Engg. Research and Tech. 3 (2014) 503–506. [6] B. R. Dabhi, K. M. Viramgama, Inter. J. for Scientific Research and Development. 4 (2016) 478–481. [7] A. R. Mount, P. S. Howarth, D. Clifton, J. of the Electrochemical Society. 150 (2003) D63–D69. [8] Viral B. Prajapati, H.G.Rajput, Jitendra J. Thakkar, Sachin P. Patel, Inter. J. for Scientific Research and Develop. 2 (2014) 12–16. [9] Bijender Singh, Inter. J. of Enhanced Research in Science Technology and Engineering. 4 (2015) 296–304. [10] A. Jayaganth, A. Deepak Mani, K. Jayakumar, Applied Mechanics and Materials. 852 (2016) 273–278. [11] Swapnagandha S. Wagh, Atul P. Kulkarni, Vikas G. Sargade, Procedia Engineering. 64 (2013) 907–914. 018) 7168–7173 3.2. Effect of process parameters on machining time C: Coolant (1. Castor oil: 2. Kerosene: 3. Coconut oil) Fig. 5. (a-c). Effect of process parameter on machining time. Figure 5. (a-c) shows the variation of machining time with respect to process parameter. MRR is mainly depends on machining time (MT) and decrease in MT increases the MRR and production rate of an industry. From Fig. 5(a-c) it is found that increase in drill rotational speed and feed decreased the machining time with faster removal of chips from the drilled hole and tool interface. Machining time was observed as low in coconut oil compared to castor oil and kerosene. This is due trouble-free flow of coconut oil inside the hole due to its moderate viscosity which enhanced thermal conductivity at tool chip interface and also washed the chips in faster rate [10]. In case of kerosene, due to its lower viscosity, it sparkled out without removing chips during machining before reaching to the full depth of the hole. Due to high viscosity of castor oil, it may take more time to reach drill and work interface thus reduced the faster chip removal and increased the machining time. 3.3. Selection of optimal machining parameters From the response graphs shown in Figs. 4 and 5, it was identified that high speed (A3: 650 rpm), high feed (B3: 0.08 mm/rev) and coolant (C3: Coconut oil) are the optimal process parameter combination for getting low surface roughness and machining time. The A3-B3-C3 combination is not present in the L9 OA and the confirmation experiment (Indicated in Fig. 1) was carried at this level and the corresponding surface roughness value obtained as 2.312 µm and machining time as 4.71 seconds which were lower than all 9 conditions. 350 500 650 7 13.25 19.5 25.75 32 A: Speed (rpm) M ac hi ni ng T im e (S ec ) g ( ) 0.02 0.05 0.08 7 13.25 19.5 25.75 32 B: feed (mm/rev) M ac hi ni ng T im e (S ec ) g ( ) M ac hi ni ng T im e (S ec ) g ( ) 1 2 3 7 13.25 19.5 25.75 32 A. Jayaganth et al. / Materials Today: Proceedings 5 (2018) 7168–7173 7173 4. Conclusions Drilling experiments on SS410 were carried out using HSS drill bit with different speed, feed and cutting fluid. The major conclusions arrrived are:  The use of coconut oil showed considerable improvement in surface finish as compared to kerosene and castor oil due it its viscosity property.  Reduced machining time was observed for coconut oil environment compared to kerosene and castor oil.  From the slope of the response graphs, drilling speed and type of cutting fluids were observed as the major influencing factors for surface roughness. The machining time is influenced by drill rotational speed and feed. References [1] K. Chandrasekaran, P. Marimuthu, K. Raja, A. Manimaran, Indian J. of Engg. and Materials Sciences. 20 (2013) 398–404. [2] A. Senthil Kumar, A. Raja Durai, T. Sornakumar, J. of Materials Processing Tech. 173 (2006) 151–156. [3] N. I. Galanis, D. E. Manolakos, N. M. Vaxevanidis, International Conference on Manufacturing Engineering. (2008) 91–98. [4] Jitendra Thakkar, Mitesh I Patel, Inter. J. of Engineering Research and Applications. 4 (2014) 235–242. [5] V. Saravana Kumar, Hridul Pavithran, K. V. Sachin, Inter. J. of Engg. Research and Tech. 3 (2014) 503–506. [6] B. R. Dabhi, K. M. Viramgama, Inter. J. for Scientific Research and Development. 4 (2016) 478–481. [7] A. R. Mount, P. S. Howarth, D. Clifton, J. of the Electrochemical Society. 150 (2003) D63–D69. [8] Viral B. Prajapati, H.G.Rajput, Jitendra J. Thakkar, SacExperimental studies on Drilling of 410 Stainless SteelStainless Steel 410 (SS410) is 12% chromium martensitic steel which can be heat treated to obtain wide range of mechanical properties. Due to its strong corrosion resistance and excellent combination of toughness and strength, it is used for furnace parts, micrometer parts, steam turbine buckets, blades, pump parts, petrochemical equipments, etc.However, during machining of SS410, difficulties are there due to its low thermal conductivity, high ductility and work hardening which can be minimized by using suitable machining parameters and cutting fluids. Therefore, now a day’s research works have been focused towards machinability of SS410 through different machining processes. Among different machining processes, many works have not been started on drilling of SS410. Drilled holes are used to receive screws, bolts, shafts, steam pipes, fitting of furniture and equipment. By considering the above applications and challenges in machining of this material, drilling experiments were conducted on SS410 to analyze the effect of drilling parameters and machining environment on surface roughness and machining time. Experiments were conducted in a jig boring machine with HSS twist drill of ∅10 mm on 5 mm thickness SS410 plate.Drilling tests were carried out as per Taguchi’s L9 array using three cutting speeds (Vc: 11, 16 and 21 m/min) and three feeds (f: 0.02, 0.05 and 0.08 mm/rev) using three different cutting fluids (castor oil, kerosene and coconut oil). Surface roughness (Ra) values were decreased from 7.326 to 2.423 µm with increase in 'Vc' and decrease in 'f' and machining time was increased from 7.4 to 31.5 seconds with increase in 'Vc' and 'f'. Optimum process parameters were identified and verified experimentally to improve the machinability of the SS410. Coconut oil medium gave good machinability results at higher 'Vc' and 'f'.Deformation behavior in nanocrystalline copperQuestions remain as to the mechanisms of deformation in metals with grain size in the nanocrystalline range. Calculations [e.g., 1] indicate that Hall-Petch behavior should cease when the average grain size in a specimen drops below ∼10–20 nm. The reasoning behind the calculations is based on the idea that very small grains cannot sustain dislocation pile-ups at the observed yield stresses In an attempt to examine deformation processes directly by TEM, in situ straining experiments were carried out on nanocrystalline copper. The results of the first of these tests are reported in this paper.Copper samples were made at Argonne National Laboratory by inert gas condensation During the experiment a displacement rate of 0.1 μm/sec was maintained by the straining stage. The sample was viewed in a Philips CM30 TEM operating at 300 kV in the bright field mode. Images of the deforming sample were captured on a video camera operating at 30 frames/second.The structure of the nanocrystalline Cu sample used in the deformation experiment is shown in It is evident that the distribution of grain sizes is very broad, ranging from about 20–500 nm, with the majority lying between 50–80 nm. Because of grain overlap it is difficult to detect grains much smaller than 20 nm. Overlap also often makes it difficult to determine the position of the boundaries of a given grain. It can be seen that many of the grains contain twins. Although the straining stage maintained a constant displacement rate, deformation in the sample was highly non-uniform. The original perforation was quite round with a notch at one point on its periphery that probably was the source of the branching cracks that formed as the straining proceeded. Plastic deformation was most apparent in the regions of high stress concentration near a crack tip. As the crack tip moved away the stress relaxed and deformation events became less evident.The video tape shows rapid and repeated changes in contrast in individual grains during straining. This activity is most pronounced in the larger grains and in regions that are likely to be experiencing high stress concentrations. Sudden contrast changes were seen in grains as small as 30 nm in size. An individual grain or cluster of grains could show rapid and frequent changes in contrast while no evidence was seen of activity in the immediate neighborhood. The intense contrast changes usually ended abruptly, with occasionally neighboring grains then taking over. Rapidly forming and reforming networks could be seen in the larger grains undergoing deformation. The behavior observed in the larger grains almost certainly is the result of dislocation activity.The contrast changes in grains containing twins show that interfaces between twinned segments can act as strong barriers to dislocation motion. For example, in a grain containing one twin in the center, the outer segments underwent vigorous contrast changes while the twinned region remained quiescent. Parallel arrays of roughly equally-spaced dislocations were seen in grains as small as 50 nm. In some instances these move away from an apparent source and into another grain but little evidence was seen of compressed spacing between the dislocations in a portion of the array that would indicate a pile up at a barrier.In situ experiments in a TEM straining stage have the advantage of providing direct evidence of deformation mechanisms. However deformation processes in a thin, electron-transparent foil may differ significantly from those in the bulk material. Further, in most cases it is necessary to thin the foil to perforation, so that the stress and strain in the sample produced by the straining stage is highly non-uniform. Because of the broad dispersion in grain size in the present samples and thus spatial variability in ease of deformation, together with the lack of constraint perpendicular to the foil, deformation probably is somewhat localized to ribbon-like areas throughout the sample. The video tape of the present in situ straining experiment presents strong evidence of dislocation activity in the larger (∼100 nm) grains. Sudden contrast changes are seen in grains down to at least 30 nm in size that also are likely to be the result of dislocation motion. The higher frequency of deformation activity in the larger grains is consistent with a dislocation process, whereas grain boundary sliding mechanisms are predicted to be more important as grain size diminishes In situ straining experiments in foils of nanocrystalline Cu viewed in the TEM and recorded on video tape show that deformation occurs in grains ∼100 nm in size by a dislocation activity. Evidence is seen for a similar deformation mechanism in grains down to 30 nm in size. Below this size overlap problems make viewing of individual grains difficult. Parallel arrays of dislocations have been observed in grains as small as 50 nm in size. There is no evidence for grain boundary sliding or rotation in the range of grain sizes that could be studied in the present experiment.Mechanical properties of open-cell rhombic dodecahedron titanium alloy lattice structure manufactured using electron beam melting under dynamic loadingThe compressive behavior of Ti-6Al-4 V lattice structures with rhombic dodecahedron unit cells is investigated at four different strain rates. Quasi-static compressive experiments are conducted by an electronic universal testing machine with a strain rate of 10−3/s, while Split Hopkinson pressure bar(SHPB) tests are used for achieving higher deformation rates about 1000/s. The loading processes are recorded by a digital and high speed camera respectively for all tests in order to determine the failure modes at different strain rates. The nominal stress-strain relationship is curved afterwards. The results show that the peak stress exhibits certain dependence on the loading rate for the structures with smaller unit cells. The deformation modes are found to be unchanged in quasi-static and SHPB experiments. All specimens are deformed with a shear band along 45°plane firstly. Finite element(FE) model is established based on the specimen with 5 mm unit cell size by 3D reconstruction from X-ray tomography to take the surface quality of struts into account. Afterwards, numerical analysis is conducted by LS-DYNA to simulate the specimens impacted at different velocities. The FE results which can be employed to make some useful predictions are partly consistent with the experimental data. Then the crushing behavior of Ti-6Al-4 V lattice structure is analyzed by the rigid-power-law hardening(R-PLH) model, and the critical velocities for deformation mode transition are predicted.Cellular materials have attracted much attention in the applications in energy absorption, heat exchange and light weight load-bearing, due to their excellent properties combining light weight, outstanding mechanical behavior and low thermal conductivity The mechanical behavior of metallic foams and periodic lattices under quasi-static loading has been widely reported. Deshpande Some contradictory conclusions have been reached about whether the strain rate effects exist in aluminum foams. Deshpande and Fleck The above studies were all performed on stochastic cellular materials with non-uniform cells and shapes. In the past several years, some investigation on periodic cellular lattices has also been conducted. Lee The purpose of the present study focuses on the compression behavior of Ti-6Al-4 V lattice structures at different loading rates. Observations of their deformation evolution at four different strain rates are reported here. Electronic testing machine is adopted for quasi-static tests and Split Hopkinson bar is used for higher loading velocities. A digital high-speed camera is used to reveal the failure modes. Two groups of design configurations with different unit cell sizes are taken into consideration. Finite element method(FEM) considering the surface quality of struts is adopted to reveal the stress distribution and deformation evolution of the lattice structures with 5 mm cell size under different impact speeds. Afterwards the critical velocities for the deformation transition are predicted by one-dimensional shock wave model(R-PLH model).The specimen used in the present study was titanium alloy lattice structure with a cell shape of rhombic dodecahedron (shown in ) manufactured by electron beam melting (EBM) process. It should be noted that according to the criteria proposed by Deshpande The WDW-300 electronic universal testing machine was used for quasi-static compression tests. The specimens were placed between two hardened steel circular platens and compressed at a strain rate of 10−3/s. The nominal stress could be calculated via |
where F is the force measured by the load cell and A0 is the initial cross-sectional area of the lattice specimen. The displacement of the platen could be measured with high precision by the photoelectric encoder attached to the platen, which transformed the rotation of screw into pulse to be recorded by the computer. The displacement resolution was 0.001 mm and the relative error of the measured displacement could be within the range of ± 0.5%. The average nominal strain in specimens could be obtained by |
where δ is the measured displacement from the load starts to increase, and L0 is the initial length of the specimen along the loading direction. A normal digital camera “Canon PowerShot SX240 HS” was positioned in front of the set up to capture the macroscopic deformation of specimens during compression, which could help to correlate the deformation evolution in the specimen with specific strains. As the quasi-static experiments were performed with a low strain rate, the related acquisition frequency was 30fps. is a typical experimental set up for testing dynamic behavior of materials at nominal strain rates ranging from 100/s to 104/s, which has also been used for cellular materials In order to obtain the full-field displacements of the lattice specimens under high loading rates impact, a high-speed camera FASTCAM SA5 was adopted to capture the deformation evolution in the material. The time interval between two continuous images was set to 10μs. The camera was triggered by a speedometer designed by using an infrared reflection type photoelectric sensor, which could be meanwhile applied to measure the speed of the projectile. When the projectile moved through the speedometer, the camera began to record the compression process synchronously and the whole deformation history of the specimen was obtained. In our experiments, three groups of velocities as 13 m/s, 18 m/s and 23 m/s were adopted to obtain different strain rates. As the strength of Ti-6Al-4 V is much higher than that of nylon 110, which may cause damage to the pressure bars during the process of dynamic experiments, two nylon gaskets with a thickness of 1 mm were added between the specimens and bars to protect the bars from being destroyed.Based on the hypothesis of elastic wave in the input and output bar with the force balance at the both interfaces of the specimen, the loads and velocities imposed on the both ends of the specimen were calculated by: |
{Finput=EBAB(ɛi(t)+ɛr(t))vinput=C0(ɛi(t)−ɛr(t))Foutput=EBABɛt(t)voutput=C0ɛt(t) |
where EB is the Young's modulus of the pressure bars, AB is the cross-section area of the bars. C0 represents the sound speed in pressure bars. εi(t), εr(t) and εt(t) are the incident wave, reflected wave and transmission wave collected by the strain gage installed on the pressure bars respectively. The change in length of the specimen could be computed from |
According to the one-dimensional simple-wave theory and homogeneous assumption, the nominal stress, nominal strain and strain rate could be calculated by the following formula: |
{σ(t)=Finput+Foutput2A0=EBAB2A0(ɛi(t)+ɛr(t)+ɛt(t))ɛ(t)=Δll=C0l∫0t(ɛi(τ)−ɛr(τ)−ɛt(τ))dτɛ.(t)=dɛ(t)dt=C0l(ɛi(τ)−ɛr(τ)−ɛt(τ))As nylon is a kind of viscoelastic material, the attenuation and dispersion during the wave propagation in the bars can't be ignored |
where ω is the angular frequency. The constitutive equation of viscoelastic rod is |
Defining the propagation coefficient γ satisfies γ2=−ρω2/E*(ω), then the general solution of wave equation can be derived as |
where P(ω) and N(ω) are the Fourier transform of strain wave transmits along the forward and negative direction of the rod at the x |
= 0 position respectively. The particle velocity v(x, ω) and normal load of the cross section σ(x, ω) can be expressed as |
{v(x,ω)=−iωγ[P(ω)e−γx−N(ω)eγx]σ(x,ω)=−ρω2γ[P(ω)e−γx−N(ω)eγx]The relation between propagation coefficient γ(ω), attenuation coefficient α(ω) and phase velocity C(ω) satisfies Thus, once P(ω), N(ω) and the propagation coefficient of the viscoelastic rod are determined, the strain history at any position can be obtained. Due to this, the propagation coefficient of the bars was determined by experiments according to Bacon's method . It can be concluded that the specimens were in an equilibrium state during the impact process. More than three tests were conducted for each strain rate, and the average values obtained from the experimental results with good repeatability were taken as the final results.The Ti-6Al-4 V rhombic dodecahedron lattice specimens in quasi-static experiments were compressed at a speed of 0.9 mm/min. The nominal stress-strain curves are given in . The curves are comprised of three stages: elastic region, plateau region and densification region. This is similar to the curves of common metallic foams and honeycombs. Significant stress drop is observed when the stress reaches the first peak value (In the present paper, this value is defined as yield strength or collapse strength). The decrease is resulted from the fracture of struts which is described in the subsequent section. Another interesting phenomenon which should be noticed is that although the specimens of the second design configuration have higher porosities, they exhibit higher strength than those of the first design configuration. The reason may be owing to the surface damage or the different material properties of Ti-6Al-4 V struts with different section dimensions when fabricated by EBM. This problem will be explored thoroughly in our future work.The SHPB experiments were performed at average velocities from 13 m/s to 25 m/s, and the relevant deformation rate was 700-1300/s. lists the stress-strain curves with good repeatability of all the different experiments. The average stress versus strain curves of both design configurations at different loading rates are plotted in that the stress-strain curves exhibit slight difference with quasi-static results. The specimens experience an initial elastic region before the loadings reach the first peak value, which drop sharply afterwards due to the prime failure in the lattices. Then the stress begins to increase again as the remaining intact parts start to bear the load. Unlike the aluminum foams, the stress-strain curves of the current specimens show certain of oscillation due to the progressive collapse and fracture of the lattice material, which is related to the brittleness of the Ti-6Al-4 V matrix material. exhibits the comparison of stress-strain curves under different strain rates between the two design configurations. It can be concluded that the lattice structure with 5 mm unit cell show higher yield strength than the lattice structure with 3 mm unit cell.(In this manuscript, the compressive strain is regarded to be positive by default). It can be noticed that the deformation modes of the tested specimens are similar to those compressed quasi-statically. All the specimens are initially damaged along the diagonal and destroyed layer by layer afterwards. The failure mode transition of the tested lattice structures is discussed in the following section.As a lot of macro defects such as nonuniform section size, unmelted powder and rough surface exist on the struts which can be observed from , the surface quality must be taken into consideration during the numerical simulation. In fact, the surface quality of struts is a critical factor which concerned the deformation mode of the material exhibits the built 3D FE model containing 3,968,543 hexahedral elements. Then the commercial software LS-DYNA was employed for numerical analysis.Johnson-Cook(JC) strength and failure model The JC strengthen model is expressed as |
σ−=[A+B(ɛp−)n][1+Cln(ɛp.−/ɛ0p.−)][1−(T−T0Tm−T0)m] |
where σ¯=[(3/2)σij′σij′]1/2respects the Mises flow stress, σij′ is the deviatoric stress, ɛp− is the accumulated equivalent plastic strain which is defined as ɛp−=[(2/3)ɛijpɛijp]1/2, ɛp.− is plastic strain rate, ɛ0p.− is a reference strain rate which equals 1/s, T0 and Tm are room temperature and melting respectively. In , the first term describes the strain hardening effect, while the second and third term considers the strain rate and thermal softening effect respectively.The unknown parameters A, B, C, n and m can be determined by experiments. When the experiments are performed at ambient temperature with ɛp.−/ɛ0p.−=1/s, the parameters A, B and n can be obtained. By conducting dynamic and high temperature experiments, C and m can be derived respectively. The detail characterization of JC strengthen model has been presented in , which have been concluded by Biswas from quasi-static and dynamic experiments Johnson-Cook failure model is an energy-based ductile failure criterion. In Johnson-Cook failure model, the fracture of material is determined by the accumulated damage. The ductile failure process can be described by two stages: damage initiation and damage evolution as displayed in The damage initiation is specified by a state variable ω which satisfies 0 ≤ ω ≤ 1. ω = 1 represents the damage initiates. ω can be expressed by a monotonic increasing function related to plastic deformation as |
where Δεp− is the increment of effective plastic strain during an increment in loading. εf−, the plastic strain at the beginning of damage initiation, can be defined in terms of mean stress, strain rate and temperature as ɛf−=[D1+D2exp(D3σ*)][1+D4ln(ɛp.−/ɛ0p.−)][1+D5(T−T0Tm−T0)] |
where σ* is the mean stress normalized by the effective stress as |
here, p is the average of the three normal stresses, D1∼D5 are material constants which can be achieved from experiments on mechanical properties of material. D1,D1 and D3 illustrate that the initial failure strain is affected by the stress triaxiality. The influences of equivalent plastic strain rate and temperature on the initial failure strain are described by D4 and D5 respectively. The adopted parameters for numerical analysis are given in The damage evolution is specified by a state variable D which satisfies 0 ≤ D ≤ 1. The damage initiates when D = 0 and completely fracture occurs when D reaches 1. An energy-based fracture criterion is applied to the FE model in the damage evolution stage. Due to strain localization and dependence of energy dissipation with mesh sizes, the use of the damage relation shows strong mesh dependency. According to Hillerborg's |
where up.−=0 when before damage initiation, and |
when the material is totally damaged. L depends on the element geometry which is defined as the the square root of the element area. A more detail description about the failure model and damage evolution can be referred to In order to simulate the specimens subjected to uniaxial dynamic compression, a moving rigid wall was defined on one side of the model and a fixed rigid wall on the opposite one. The velocity of the moving rigid wall, which was determined by , was kept constant to ensure a constant strain rate during the loading process. The loading process was illustrated in Here, ɛ. is the strain rate while H is the original height of the specimen. In the current study, ɛ.=700,1000,1300/s and H = 15 mm. Thus, the constant velocity equals 10.5, 15 and 19.5 m/s respectively. The nominal stress and strain can be calculated by |
where F is the reaction force of the rigid wall which can be achieved directly in the LS-PrepPost, A respects the corresponding original surface area of the model and t the loading time.The debate about the strain rate sensitivity of cellular materials has been continued for a long time. It has been demonstrated that the strength enhancement of porous media subjected to high speed impact is resulted from the deformation mode transition The initial collapse strengths of the both cell sizes subjected to different loading velocities have been listed in . It can be observed from the above images that the specimens exhibit different strain rate sensitivity. For the samples with 3 mm cell size, the collapse strength increases by 4.6% from 0.001/s to 1000/s, and the increase reaches 17.36% when the strain rate rises to 1300/s. This strain rate strengthening effect is not evident for those of 5 mm cell size, whose strength only increases 3.5% with the strain rate varying from 0.001/s to 1300/s. The strength of lattice structure with 5 mm unit cell is nearly unchanged under dynamic loadings. The reason leads to the strain rate sensitivity may be the strain rate effect of Ti-6Al-4 V or the micro-inertial effect during the loading process. In order to provide more evidence to reveal the actual reason, numerical simulation is conducted by using the same material parameters obtained from Ti-6Al-4 V manufactured by other method. Although the manufacturing process (electron beam melting from Ti-6Al-4 V powder) may cause different mechanical properties compared with the traditionally fabricated Ti-6Al-4 V alloy, but the assumption seems reasonable when the dynamic test data is absent. Due to the restriction of computer, only the lattice structure with 5 mm unit cell is considered to explore the strain rate sensitivity of such specimens, and the comparison between the numerical results and experimental data is listed in . In order to save the calculation time, only the first appearance of decline in stress is taken into account.It should be noticed that the elastic region obtained by FE analysis agree well with the experimental data. The strengths obtained by numerical simulation are slightly higher but also within an acceptable range compared with those got by experiments. The error of the maximum stress between the FE results and experimental data under three different strain rates are 8.59%, 5.85% and 6.51% respectively. The results demonstrate that the adopted parameters of strengthen model is appropriate for calculation. In contrast to the experimental results, the FE curves exhibit smaller initial failure strain and the stress drops more sharply after the peak value is achieved. This is resulted from the inaccurate failure parameters of Ti-6Al-4 V adopted in the FE analysis. Lu etal. gives the comparison of FE results with different strain rates. It can be concluded that the numerical data exhibits no distinction in strength which is quite consistent with the results experimentally obtained.It should be noted that in our experiments and FE analysis, the friction between the lattice strands and the loading plate has been minimized. No face sheet to which the lattice strands can be bonded is adopted here either. Thus, the struts at the both ends are not fixed which allows the strand nodal points to move along the face of the compression platens and the lateral stabilization is deterred. Besides, the deformation of the current lattice structure is dominated by plastic bending instead of buckling. It suggests that inertial effect is not significant in the tested lattices. In other words, the moderate strain rate sensitivity of the Ti-6Al-4 V lattice structure is caused by the strain rate effect of the bulk material.The different strain rate sensitivity between the two lattice structures may be resulted from the different cross-section of struts. As no optimal fabrication process has been determined, the individual struts of the lattice structure are with poor quality which appears as variations in the strut cross-section (). The imperfection factor may reduce the strain rate strengthening effect on the collapse strength. Since the two lattice structures are fabricated in different sizes, the struts are with different imperfections which can lead to different strain rate sensitivity. Another important factor thought to affect the strain rate sensitivity is the cell number of the lattices. Due to the limited number of cells in the lateral direction, the tested results will be influenced by lack of lateral confinement. Especially for the lattice with 5 mm unit cell, only three cells exist along the lateral direction, which is even fewer than those with 3 mm unit cell. The absence of lateral confinement promotes the deformation and failure of the structure, which may result in less strain rate sensitivity of the lattice structure with 5 mm unit cell. Further investigation about the specimens with equal cell number will be conducted so that the edge effect on the strain rate sensitivity crushing can be determined.From the images obtained by high speed camera (), it can be observed that deformation localizations appear along the 45°plane in Ti-6Al-4 V lattice specimens. Cheng (b). However, the deformation mode of Ti-6Al-4 V lattice structure adopted in the present study is different from stainless steel lattice structure. The discrepancy is thought to be caused by the different properties between the two kinds of matrix materials. Detailed images corresponding to the deformed Ti-6Al-4 V lattice structures are given in to provide an insight into the microstructures. Compared with the large plastic deformation of stainless steel struts (In order to get an inner view of the stress distribution in the specimens during deformation, numerical analysis is conducted to simulate the dynamic compressive behavior of Ti-6Al-4 V lattice structures with 5 mm unit cell. The numerical simulation results of deformation evolution under dynamic compression loads with different strain rates for the 3D porous Ti-6Al-4 V lattice model are compared with experimental images in . It can be observed that the deformation mode obtained by numerical simulation is exactly consistent with experimentally observed results. When impacted with a strain rate of 700∼1300/s, the model deforms in a quasi-static mode. It can also be concluded from the Von-Mises stress contour map that the stress is uniformly distributed in the model during the loading process. With the loading proceeding, the struts get to be fractured and a shear band appears along the 45°plane.It should be figured out from the above images that when the model is loaded with a strain rate of 1300/s, a higher extent of localization can be observed at almost the same strain. This phenomenon can be explained by the stress-strain curves in . The experiments with higher strain rates drop earlier than those with lower speed impact. But when the strain reaches 0.07, the stresses with 1300/s begin to rise as the fractured struts are compacted and the remaining intact struts start to bear the loading. However, the stresses of other groups still drop owing to the initial damaged struts have not been ruptured completely.In the previous sections, the dynamic response of Ti-6Al-4 V lattice structures has been discussed by SHPB experiments and numerical simulations. Nevertheless, the “shock wave mode” deformation which collapses from the impact end to the supporting end has not been observed. Some previous literatures have mentioned that the deformation mode is related to the impact velocity[15,47,48]. In order to verify the deformation mode obtained by experiments and FE analysis, predictions of critical velocities to generate different deformation modes is made on the basis of shock wave analysis in this section. The conclusion will also offer some guidance for our future experiments.Some research about the dynamic behavior of metallic foams has been conducted based on shock wave models. A simple one-dimensional shock wave model was adopted by Tan and Reid Considering the material is impacted at a velocity v, a shock wave is generated after the elastic waves and then propagates from the loading end to the support end. The parameters in the regions ahead and behind of the shock front are listed as follows:State in front of the shock front: ρ0, v0 |
= 0, σ0, ε0 |
= 0;State at the rear of the shock front: ρ1, v1 |
= |
v, σ1, ε1.where ρi, vi, σi, εi are density, particle velocity, stress and strain in the two regions respectively. ρ0 is the original density of cellular material and ρ1 refers to the density of the deformed part behind the shock front. According to the conservation condition of mass, moment across the shock front, the basic equations can be achieved: |
Here, φ and φ. denote the Lagrangian position and travelling speed of shock front respectively. The symbol ‘[]’ represents the jump of physical quantities across the shock front. The changes of parameters across the shock front can be expressed as: |
It should be noted that only a rightward travelling front is taken into account for convenience. Assuming the constitutive relation ofthe cellular material satisfies the rigid-power-law hardening (R-PLH) idealization, which is |
where σ0, K and n are the initial yield stress, strength index and strain hardening index respectively. These three material parameters can be obtained by quasi-static experiments. Substituting and the strain behind the shock front can be determined: |
Then the stress behind the front can also be derived: |
The dynamic deformation modes of porous materials have been demonstrated to be dependent on the impact velocities, which can be divided into three categories: a ‘quasi-static’/homogeneous mode, a transition mode and a shock mode {v≤vcr1homogeneousmodevcr1<v<vcr2transitionmodev≥vcr2shockmodeThe first critical velocity vcr1 is determined by the ratio of the stress at the front and back of the front. The corresponding velocity when σ0=0.9σ(ε1) is defined as the first critical velocity , the first critical velocity vcr1 can be obtained as |
The second critical velocity vcr2 is decided by the initial strain behind the shock front. When the initial strain reaches the densification strain εD, the related velocity is defined as vcr2. Substituting ε1=εD into The densification strain εD is achieved by quasi-static experiments. exhibits the quasi-static stress-strain curves of the investigated Ti-6Al-4 V lattice structures and the corresponding idealisation. Firstly, σ0 was set to be plateau stress σpl which could be calculated by σpl=1ɛD−ɛcr∫ɛcrɛDσ(ɛ)dɛ, where εcr and εD denoted the initial collapse strain and densification strain respectively. We used σpl instead of initial collapse stress here is due to the large oscillation of the stress-strain curves. Then and the known parameters σ0, ρ0, and v were adopted to fit the experimental stress-strain curves and the initial values of unknown parameters K and n could be obtained. The difference between experimental data and the equation prediction was minimized by employing a Gauss-Newton iterative algorithm until all the final convergent parameters were reached. All the determined parameters are also listed in the photos. According to , the critical velocities of the tested Ti-6Al-4 V lattice structures can be obtained.As the impact velocities adopted in the SHPB experiments were all below the first critical velocities, the Ti-6Al-4 V lattice structures with different unit cell sizes were deformed in a homogeneous mode. The experimental results match well with the model predictions. In order to further verify the predictions, FE analysis with higher impact velocities was conducted and dichotomy was adopted to determine the critical velocities. The first and second critical velocities of the model with 5 mm unit cell were found to be 36 m/s and 57 m/s respectively. The first critical velocity obtained by the analytical prediction is in excellent agreement with the numerical result, but the second critical velocity obtained by FE analysis is much lower than the theoretical data. This discrepancy may be due to the different densification strain gained by the two approaches. As stated in the prior paragraphs, the material parameters adopted in the numerical simulations seem to be with poorer plasticity than those of the actual Ti-6Al-4 V fabricated by EBM, which may lead to a lower εD. Thus, more precise material constants are urgently needed to achieve better FE results. The deformation evolutions of numerical model under 36 m/s and 57 m/s impact are presented in Different deformation patterns can be observed from the above images. When impacted with a speed of 36 m/s, both shear band along the 45°plane and collapse near the loading end can be found in the model. With the impact velocity is elevated to 57 m/s, the localization is totally distributed near the impact end, and spread to the supporting end in a ‘shock wave’ manner. This deformation transition is similar to the mostly investigated aluminum foams and honeycombs. It should be noted that as apparent localization appears in the model, the definition of strain is not suitable any longer for high speed impact. The ε used here is just the ratio of the rigidwall displacement and the model height to measure the global deformation of the specimen, instead of being regarded as strain.The compressive behavior of titanium alloy lattices under various strain rates is investigated via experiments and numerical methods. Two design configurations of different cell sizes are taken into consideration. Electronic testing machine is used for quasi-static tests and higher loading velocities are performed by using SHPB. A digital camera and a high-speed camera are employed to observe the deformation evolution in quasi-static and dynamic experiments respectively. FEM simulation is implemented to reveal the stress distribution in the specimens and the effect on failure mode caused by the strain rate. Some conclusions have been reached as follows: |
Comparing with the quasi-static experiments, the initial yield stress is found to be more sensitive to the loading rate in SHPB experiments for the specimens with smaller unit cell. The strain rate effect on the collapse strength is thought to be caused by the strain rate effect of Ti-6Al-4 V. The discrepancy of strain rate sensitivity between the two lattice structures may be resulted from the different imperfection of strut and different cell numbers, which needs to be verified by further study.The deformation mode of material does not exhibit a transition when loaded from 0.015 mm/s to 25 m/s. For both design configurations, the collapse is dominated by the bending and fracture of the struts which evolves into shear bands along 45°direction in the specimens.The critical impact velocity for failure mode transition is predicted by the one-dimensional shock wave model (R-PHL model). It is concluded that for the specimens with 3 mm and 5 mm unit cell, the critical velocity for generating the dynamic localization mode is 63.8 m/s and 68.1 m/s respectively.Three-dimensional FE models considering the surface quality of struts based on X-ray tomography are developed to investigate the dynamic compressive behavior of Ti-6Al-4 V lattice specimens. The deformation mode and yield strength obtained by numerical simulation match well with the experimental results and theoretical predictions. However, owing to the different properties of Ti-6Al-4 V alloys between those fabricated by EBM and traditional methods, the stress-strain curves and second critical velocity achieved by FEM are slightly different from experimental results. A more detail research about the mechanical behavior of 3D printing Ti-6Al-4 V alloy will be conducted and complete model parameters will be achieved in our consequent study.Synthesis and electrorheological performance of nanosized composite with polar inorganic compounds► An economical and applicable electrorheological material has been synthesized. ► The starting materials are all low-toxic and economical and facile. ► The static yield stress of a suspension of the material reached 22.8 kPa. ► The relative yield stress reached 12.7–33.3 for the suspensions of 30–50 wt%.To obtain an economical and applicable electrorheological (ER) material, a novel nanocomposite composed of polar inorganic compounds, NH4Al(OH)2CO3, AlO(OH) and (NH4)2SO4, has been synthesized using low-toxic and economical and facile starting materials by a simple chemical reaction process. The experimental result shows that this material has better ER performance. The static yield stress (τy) of the suspension (50 wt%) of the material in silicone oil reached 22.8 kPa at a DC electric field of 4 kV/mm, and the relative yield stress (τr) (the ratio of the yield stresses with to without an electric field) is also higher (12.7–33.3 for different concentration suspensions). The composition, grain size, dielectric and surface properties of the material have been studied by the elemental analyses, X-ray diffraction (XRD), infrared spectroscopy (IR), transmission electron microscopy (TEM), dielectric spectroscopy and determinations of the surface area and surface energy of the material. The influences of the grain size, dielectric and surface properties on ER performance of the material have been discussed.Electrorheological fluids can switch from a liquid-like material to a solid-like material within a millisecond under an external electric field. This rapid and reversible response has potential application in many electrically-controlled mechanical devices which transform electrical energy into mechanical energy, such as clutches, valves, and damping devices All reagents were provided by Beijing Chemistry Reagent Co. (China) and used without further purification.The material was prepared through the following process. First, 600 mL solution (0.25 mol/L) of NH4Al(SO4)2 was dropped slowly into 300 mL solution (2.5 mol/L) of NH4HCO3 under vigorous agitation. The pH value of the solution was adjusted to 8–9 by adding aqueous ammonia drop-wise under stirring. The suspension was stirred for 10 h at room temperature. The sediment was separated and washed with distilled water first and then absolute ethanol, then dried at 60 °C for 4 h. The white product was ground and finally dried in vacuum for 36 h at 50 °C. The material containing NH4Al(OH)2CO3, AlO(OH) and (NH4)2SO4 was thus obtained.The ER experiments were carried out using a Rotary Viscometer (Type HAAKE CV20, Germany) and a circular plate type viscometer that was self-constructed by Lu Kun Quan et al. The German Rotary Viscometer was used for the suspensions of 30 and 35 wt%, it consists of a pair of coaxial cylinders with a 0.545 mm gap in between. The ER fluid was placed in the gap, with the inner cylinder kept stationary and the outer cylinder rotating at preconcerted rates while the apparatus is operating. The measured ranges are 0–3000 Pa for shear stress and 0–300 s−1 for shear rate (γ). The circular plate type viscometer was used for the suspensions of 40, 45 and 50 wt%. Its two parallel plates of 20 mm in diameter were separated with 1 mm gap. The lower plate was driven by a step motor and the upper plate was connected to a torque sensor (AFT1, Mecmesin Ltd., UK). The ER fluid was filled in between two plates that were connected to a high voltage electric source. By measuring the torque the rheological properties of ER fluids were recorded. In this paper, the static yield stresses (τy) of the samples and the shear stresses (τ) of sample 1 and 2 have been determined under different electric field strengths (E, dc field) at room temperature.The particle material was mixed quickly, after water removal, with methyl silicone oil (density ρ |
= 0.64 g cm−3 and viscosity η |
= 480 mPa s at 20 °C) under vigorous stirring, to yield the ER fluid samples. The samples were then put in the gap between the cylinders or two circular plates as soon as possible for ER experiments. The concentrations and some ER data of different ER fluid samples were listed in The IR spectra of the material were recorded with a Nicolet Magna-IR 750 spectrometer at 295 K. XRD analyses of the material were carried out on a Bruker D8 ADVANCE X-ray diffractometer with Cu Kα radiation at wavelength 1.5406 nm in a range of 10–90°. The elemental analyses of the materials were performed on a German Elementar Vario EL instrument.A suspension (40 wt%) of the material was used to investigate its dielectric properties. The capacitance C and dielectric loss tangent (tan |
δ) at room temperature under various frequencies (f) were obtained on a HP4274A Multi-frequency LCR Meter. The dielectric constant (ε) was derived from the measured C according to the conventional relation, ε |
= |
C |
⋅ |
d/(ε0 |
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