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Ac,eff/bd is the ratio of the effective area in tension and Ac,eff is the effective area in tension taken as the lesser of 2.5(h−d)b and (h−x)b/3. ρeq is the equivalent reinforcement ratio. Mk is the characteristic value of moment. Es and Ef are respective Young’s modulus of the steel reinforcement and the FRP plate. d is distance between the tensile steel reinforcements and the top face of the concrete member. fctm is the tensile strength of concrete, and τfm |
= 1.25fctm and τsm= 1.8fctm are the mean bond stress of the FRP plate and the steel. us and uf are the bond perimeter of steel and FRP reinforcement.The approach described above can be further extended to the case with a number of secondary cracks, as shown in . Specifically, if there are N active secondary cracks (where the steel reinforcement has yielded) dividing the debonded region into N−1 zones, the stresses within the ith zone (the zone between the (i−1)th and the ith crack, counted from the elastic zone) can be obtained from the following equations, which are derived from infinitesimal element analysis (for i |
> 1):By substituting ξ with Li, the corresponding tensile stress in FRP and the interfacial shear stress at the ith crack is given by:The crack opening (wi) and spacing (Li) of the ith crack can be calculated according to Eqs. . The tensile stress in FRP is continuous at the ith crack, but the shear stress at the ith crack is discontinuous due to the opening of the crack. When one goes from one side of the ith crack to the other side, the interfacial shear stress is given by:In the calculation, the number of active secondary cracks and the corresponding crack width can be calculated from Eqs. . The total length of the debonded zone (∑Li) will be extended in small increments, and the force at the major crack can be computed. When the debonding passes through the last secondary crack, the force in the FRP plate reaches a maximum value under the major crack. Depending on the opening of the secondary cracks, there are three possible scenarios for the shear stress distribution in the debonded zone: (1) the attainment of maximum force occurs as long as the shear stress drops to zero at the major crack, (2) the maximum force occurs with a zero shear stress region within the debonded zone and (3) the FRP stress reaches its maximum value when the interfacial shear stress has not softened to zero at the major crack. The second scenario occurs when the increase in integrated shear stress in zone 1 (the debonded zone closest to the elastic region) over-compensates for the decrease in the other zones during the propagation of the debonded region. Under this situation, the size of the part of debonded zone which is still carrying shear stress can be determined by: and appropriate boundary conditions, the tensile stress in the FRP plate and the interfacial shear stress can be calculated for the nearest zone to the major crack.With the equations given above, the FRP stress with considering the effect of multiple secondary cracks can be calculated as a function of the total debonded length Ld. By increasing Ld at small increments, the maximum FRP stress/strain can be obtained.To verify the proposed model, eight beams with different configurations () were tested to investigate FRP debonding under a single or multiple cracks along the concrete beam. Each specimen was 2.0 m (length) × 0.15 m (width) × 0.2 m (depth). Two high yield steel bars of 10 mm in diameter were used as the tension reinforcement, and also placed in the compression zone. The yield strength and Young’s modulus of the steel reinforcement are respectively 460 MPa and 200 GPa. Fourteen steel stirrups made of mild steel (8 mm in diameter) were employed to reinforce the beam in shear. The compressive strength of concrete was found to be 42.9 MPa from the tests of a number of concrete cylinders after standard curing. After the beams were hardened, two layers of FRP sheets, 1.7 m in length, 0.22 mm in thickness and 150 mm in width, were bonded to the substrate of the RC beam using epoxy resin. The properties of the FRP and adhesive are illustrated in . Before bonding the FRP sheets, a layer of primer resin was applied on the concrete surface, which was roughened with a needle-gun and cleaned with pressured air. After the primer hardened, the FRP sheets were bonded on the tensile surface of the concrete beam layer by layer until the desired thickness was reached. For each concrete beam, a ‘U’ shape FRP anchor, 100 mm in width, was also bonded to one side of the beam to ensure the occurrence of debonding at the other side of the member, where a number of strain gauges were bonded on the surface of the FRP plate to measure the strains during the loading process.As mentioned above, eight beams with different configurations were tested in the experimental investigation. For the beams B1 and B2, a notch was cut on the tensile side of the section below one of the loading points. The notch was deep enough to go through the steel reinforcements. To avoid the secondary cracks in the shear flexural span, the FRP plates are applied on the sides between the cut section and the support to strengthen the concrete beams. With these beams, the debonding behavior under a single major crack can be experimentally studied. For beams B3 and B4, an additional notch was cut at a section between the loading point and the support. The distance between two notches was 140 mm for B3, and 200 mm for the beam B4. For these beams, FRP plates were bonded to both beam sides between the two adjacent notches to avoid additional cracks to form in this zone. For the beams B5–B8, the tensile steel reinforcements were not cut off. B5 contained two shallow pre-notches, and B7 was pre-loaded to form distributed flexural cracks. The FRP plate was then bonded to the tensile face of the beam after it was fully unloaded. B6 and B8 were normal specimens with no pre-notch or pre-cracks to start with. After testing, all these concrete beams were found to fail in FRP debonding under the presence of one major crack or multiple secondary cracks along the beams. The test results for the various cases were summarized in During testing, four-point bending was performed with clear span of 1800 mm, and loading span of 300 mm between the two loading points. The material test machine with load capacity of 2500 kN was used for testing under constant displacement rate of 0.3 mm/min. Two linear variable differential transformers (LVDT) were used to measure the displacements at the mid-span and one of the loading points. During the loading process, the applied load, as well as readings from strain gauges and the LVDTs, were collected by an automatic data logger.In the experiments, all beams failed by FRP debonding that initiated at the bottom of a crack or pre-made notch. According to the crack patterns, the failure of the specimens may be classified into two distinct categories: (1) FRP debonding under a major crack (the one that induces debonding) in the concrete beams (for B1–B4), and (2) FRP debonding under the presence of multiple cracks along the beams (for B5–B8), as shown in . The load versus displacement curves for all specimens were shown in . For the beams B1 and B2, one notch under the loading point was pre-made and the tensile steel bars at the notched section were cut off. The two beams finally failed by FRP debonding from the bottom of the pre-made notch without additional active cracks along the shear flexural span. For beams B3 and B4, two notches were pre-cut with the original purpose to study debonding under multiple cracks. However, in both beams, FRP debonding initiated from the bottom of the notch that was closer to the plate end and propagated to the free end of the FRP plate. During the growth of the debonded zone, a vertical crack propagated from the top of the notch towards the nearer loading point. Hence, debonding failure in these two beams was essentially occurring under a single major crack. Indeed, according to a, the debonding behavior of beams B3 and B4 are very similar to B1 and B2, and the ultimate load values are also very close. The four beams are therefore classified into the failure mode of FRP debonding under a major crack.For the beams B6 and B8, the tensile steel reinforcement was not cut off at any section. Under external loading, a number of discrete cracks were found to form along the span. According to visual observation as well as the strain gauge readings, four of the cracks (including the pre-made notches) were found to open significantly. The crack closest to the loading point was the major crack inducing the interfacial debonding, while the other three are secondary cracks affecting the debonding behavior. For the beam B5, the pre-made notch closer to the loading point acted as the opening crack to induce interfacial debonding along the FRP-to-concrete interface. Together with the other notch and two additional cracks, total four active cracks were found to open in the half span. Before bonding the FRP plate, the beam B7 was pre-loaded with a number of cracks along the span. After strengthened with the FRP plates, the beam was re-loaded, and four actively opening cracks could be observed along the beam. The spacing between two adjacent cracks is measured to be around 70 mm for the tested beams, with the final failure of FRP debonding under multiple cracks.. The average load capacity for the beams B5–B8 is 77.7 KN, which is much higher than the average value for the beams B1–B4, which is only 27.9 KN. Certainly, the difference is mainly caused by the cutting of tensile steel reinforcements in beams B1–B4. However, a considerable difference in debonding behavior for the two groups of beams can also be distinguished from the effective strain in the FRP plate at failure. The maximum FRP strain of the beams with multiple cracks (which is around 10,000 με) is found to be much higher than that of the beams with only one major concrete crack (which is around 7000 με). The average maximum FRP strain in beams B5–B8 is about 35% over that of B1–B4, indicating that the presence of multiple secondary cracks along the beam will increase the maximum FRP strain.During the loading process, the FRP strains were collected by electrical strain gauges. To illustrate the failure mechanisms of the above two distinct failure modes, the strain variations for B2 and B7 are plotted in a, when there is only one major crack in the shear span, the strain distributions in the FRP plate are similar to those reported for pull-off tests. Initially, before debonding occurs, the strain values are found to decrease drastically from the location of the major crack, where the FRP plate is under the highest axial force. Increasing loading would lead to up-shift of the curve with the same shape. When the applied load is beyond 10 kN, the strains exhibit a more gradual decrease at the vicinity of the crack. Since the slope of the curve represents the rate of strain change along the FRP plate, which is proportional to the interfacial shear stress, a decreasing slope indicates a reduction of the interfacial shear friction in the debonded zone. When debonding propagates further from the notched section, the strain curve tends to form a plateau in the debonded zone, indicating continuous softening along the interface.For the beam B7, which failed by FRP debonding under multiple cracks, the strain distributions at different load levels were plotted in the b. In the figure, the vertical lines indicate the locations of the cracks along the beam. Despite the inevitable ‘noise’ in strain gauge measurements, the strain distributions are clearly different from those for the case with a single major crack. When the applied load is low, the stress distribution does not show a significant increase towards the middle of the beam, where the moment (and hence the force acting on the FRP) should be maximum. Instead, several peak values can be observed along the beam, which is caused by stress concentrations due to the opening of the secondary cracks. The presence of strain peaks is consistent with the situation illustrated in b. Arguably, the significant increase of strain at the location of the fourth crack (at about 420 mm) as well as the observation of rapid drop in strain with distance beyond this crack may indicate that the debonded zone has also extended beyond the fourth crack. The conclusion that can be drawn from the results is clear: before the occurrence of ultimate failure due to unstable debonding, it is possible for the debonding to propagate beyond two or more adjacent cracks. The test results therefore provide additional support that unstable debonding occurs after the debonded zone has extended beyond several cracks, which is the scenario illustrated in To verify the proposed model for debonding under the presence of multiple secondary cracks along the concrete beam, the experimental results are compared with the modeling results in this section. For the calculation, the effective length of the FRP plate is 700 mm from the loading point to the nearer plate end. The material properties for the specimen are used according to . Since the concrete was roughened before the FRP sheets were bonded, the exact thickness of the adhesive is difficult to measure in the tests. Before debonding behavior under multiple cracks can be simulated, the interfacial parameters (τs, τ0 and k) need to be obtained first. These parameters can be determined from the fitting of strain data for B1, which failed by debonding from the bottom of a major flexural crack. In the calculations, the adhesive thickness (h) and the maximum shear stress (τs) along the interface can be obtained by fitting the strain data with theoretical curves determined from the debonding model before initial debonding occurs (a). Additionally, by fitting the strain distributions along the FRP plate after initial debonding, the interfacial parameters of softening rate (k) and residual shear stress after debonding (τ0) can be extracted (b). For the FRP strengthened concrete beams in the present study, the three parameters were found to be: τs |
= 12 MPa, τ0 |
= 1.5 MPa and k |
= 1.0 MPa/m, and the mean thickness of the adhesive is found to be 2.5 mm from the best fitting of experimental data. With the interfacial parameters, the ultimate maximum FRP stress/strain can be determined from the models presented in Section , for cases with or without secondary cracks.For the beams B1–B4, debonding occurs from the bottom of a single major crack and the maximum FRP strain is calculated to be 6937 × 10−6 according to the Eqs. . By using this strain value, the predicted failure load is calculated to be 27.6 kN according to the conventional section analysis for RC beams. These values are compared with the experimental results in and good agreement is obtained. For the beams B5–B8, which exhibited FRP debonding under multiple secondary cracks along the beam, the proposed model in Section is employed to calculate the number of active cracks and the maximum FRP strain at ultimate debonding. Based on the Eqs. , there are three active secondary cracks with spacing of 62.4 mm in the half span of each beam, and the crack widths corresponding to cracks from the plate end to the mid-span of the beam are respectively 0.456 mm, 0.415 mm and 0.373 mm. With these parameters, together with the analytical model for FRP debonding under multiple cracks, the maximum FRP strain is calculated to be 10891 × 10−6. Assuming plane section remains plane, the failure load of the FRP strengthened concrete beam with multiple cracks is calculated to be 76.4 kN, which is again in good agreement with the test results in To illuminate the effect of opening cracks on the debonding behavior, the stress distributions for debonding under a major crack and multiple secondary cracks are shown in . For FRP debonding under a major crack (a), the debonded zone is only about 280 mm in extent. However, for FRP debonding under three active cracks, the maximum stress of FRP occurs after the debonded zone has extended over several flexural cracks and the debonded zone is around 450 mm in length (b). The increasing size of the debonded zone is accompanied by a significant increase in the integrated shear force along the zone and the maximum FRP strain/stress at failure. According to the experimental and theoretical results, the multiple secondary cracks along the beam can have a significant effect on FRP debonding, and should be considered in the calculation of the maximum strain of the FRP plate at failure.In this paper, FRP debonding under the presence of a major or multiple secondary cracks along a concrete beam is systematically studied. The limitations of existing models are first discussed with reference to existing experimental results. Then, a new analytical model, based on the mechanism of delayed softening due to the opening of secondary cracks, is proposed. With this model, the longitudinal stress/strain distribution along the FRP is derived, as well as the interfacial shear stress distribution for a particular load value. To verify the validity of the proposed model, eight FRP strengthened beams with different configurations were tested to study FRP debonding under a major crack and multiple secondary cracks along the beam respectively. According to the experimental results, the maximum strain in the FRP plate for the beams with multiple cracks is about one-third higher than that for the beams with single crack. With the model for FRP debonding under a major crack, the interfacial parameters are extracted by fitting strain data along the FRP plate with the modeling results. By putting these interfacial parameters into the model for debonding under the presence of multiple cracks, the calculated maximum FRP strain/stress at debonding failure is found to be in very good agreement with experimental value. Specifically, the model for FRP debonding under multiple cracks is able to predict the increase in ultimate FRP strain (or stress) relative to the case with a single crack. The potential of the new model for the prediction of the debonding failure in practice is hence demonstrated.Manufacturing and compressive properties of tube-filled metal syntactic foamsIn this research, aluminum tubes were integrated into metal syntactic foam (MSF) to manufacture novel tube-filled foam (TFF) structures. Counter-gravity infiltration casting was used to manufacture TFFs using an innovative single-step process. The density of the resulting TFFs was between 1.79 and 1.91 g cm−3 and thus similar to the density of the surrounding MSF. Microstructural analysis of the interface between tube and MSF indicates no significant chemical reaction. For comparison, MSF reference samples without tubes were produced. Quasi-static compression tests were conducted to determine the mechanical properties of the samples produced. The results indicate that embedded aluminum tubes and surrounding MSF mutually stabilize their deformation. TFF samples compressed with uniform barreling from the center of the sample whereas the deformation of MSFs was localized towards one end of the sample. As a result, TFFs showed superior mechanical performance compared to MSFs.Metal syntactic foams (MSFs) are a subgroup of cellular metals using either porous or hollow particles to introduce porosity into a metallic matrix []. Similar to other classes of metallic foams, MSFs absorb considerable amounts of energy during compressive loading []. This ability gives them the potential to be used as energy absorbers in e.g. automotive and road safety applications [Different studies have investigated the manufacturing and mechanical properties of metal foam-filled tubes (FFTs). In these materials, metallic foams are inserted into metallic tubes using mostly in situ or push fit techniques [] manufactured in situ FFTs using A356/expanded perlite syntactic foams and stainless steel tubes. The characterization of the samples under quasi-static compressive and bending loadings indicated a higher energy absorption of foam-filled structures compared to the sum of the individual components, i.e. foam and tubes. This behavior was related to the interaction effect between the components due to the reduction of the ovalization of the tube and enhancing the properties of the foam under constrained deformation. The axial and lateral compression of metal syntactic foam-filled tubes were also investigated []. It was shown that the direction of compression influenced the energy absorption of FFTs, i.e. the energy absorption of FFTs under axial loading exceeds radial compression. In another study, Duarte et al. [] used powder metallurgy to manufacture aluminum foam-filled thin-walled aluminum tubes. Their results indicated similar deformation modes for quasi-static and dynamic loading conditions. A mixed formation of axisymmetric and non-axisymmetric folds was related to structural imperfections, a low wall thickness to diameter ratio, and the interaction between components. Filling the thin-walled aluminum tube with closed cell-aluminum foam changed the tube deformation from diamond to concertina mode regardless of the foam core density []. The mechanical properties of closed-cell aluminum foam-filled stainless steel tubes at elevated temperatures up to 450 °C were investigated in Refs. []. The results confirmed that FFTs achieve a higher energy absorption in comparison to their individual components under quasi-static compression. At each temperature, the interaction effect between closed aluminum foam core and stainless steel tube decreased the propagation of cracks during the deformation of FFTs in comparison to empty tubes.In the present study, a new configuration of metal foam and tube is introduced. To this end, aluminum tubes are surrounded by ZA27/expanded perlite MSF using counter-gravity infiltration casting. The resulting structure is subsequently referred to as a tube-filled foam (TFF) structure. Conventional MSF samples were produced for comparison. Interestingly, samples of both groups exhibit a similar bulk density. All samples were subjected to compressive testing and superior strength and energy absorption was found for the TFF arrangement.Zinc aluminum alloy (ZA27) and expanded perlite (EP) particles were used as the matrix and filler materials of the MSF, respectively. Two different sample types were produced in this study: i.e. MSFs (without aluminum tube) and TFFs (with embedded aluminum tube).All samples were of a cylindrical shape with an outer diameter of ∼28 mm and a height of ∼36 mm (see b). For the manufacture of TFFs a 6060-T5 aluminum tube with an outer diameter of D=12 mm and a wall thickness of t=1.6 mm was embedded at the center of the samples. According to ASTM-B86-13 [], ZA27 alloy consists of 25–28 wt% Al and 2.00 to 2.50 wt% Cu as the main alloying elements. The density of this alloy is 5.00 g cm−3 []. The chemical composition of 6060 aluminum tube is 0.3–0.6 wt% Si, 0.35–0.6 wt% Mg, 0.1–0.3 wt% Fe and the balance made up by aluminum []. The samples were manufactured using counter-gravity infiltration casting [For the MSFs, the procedure described in Ref. [TFF samples were manufactured using a similar procedure but applying an infiltration pressure of 0.02 MPa. The filling of the graphite mold commenced by inserting an aluminum tube at the center of the mold. The aluminum tube was sealed at both ends using aluminum discs to prevent melt infiltration. Then, EP particles were added to surround the aluminum tube in four batches, each followed by tapping and vibration. After solidification, both ends of the sample were machined to obtain TFFs as shown in All samples were T6 heat-treated. To this end, solution heat treatment was performed at 365 °C for 1hr followed by cold water quenching. Then, the samples were subjected to aging at 140 °C for 24 h. This heat treatment improves the ductility of the ZA27 matrix alloy and thus of both MSFs and TFFs []. The aluminum tubes that were tested individually in this study underwent the same thermal treatment to ensure identical properties to the tubes encased in the TFFs.The interface between ZA27/EP MSF and the embedded aluminum tube was studied using a scanning electron microscope (Zeiss) equipped with Bruker EDS. To this end, a TFF sample cross section was scanned across this interface to analyze the elemental distribution. In order to minimize surface contamination, the sample was not polished after cutting and instead used directly for EDS analysis.In order to analyze the physical properties of the syntactic foams, the volume fractions of ZA27 matrix (φM), EP particles (φP) and voids (φV) were calculated. To this end, the mass of particles (mP) was estimated using the bulk density (ρB) of expanded perlite particles and the volume of the syntactic foam sample (VSF):In MSFs, (VSF) is calculated using the outer dimensions of the cylindrical samples. In TFFs the central volume is occupied by the tube and must therefore be subtracted for the calculation of the foam volume. Furthermore, the mass of the aluminum tube (mt) must be considered. To this end, the tube solid volume (Vt) was calculated using the following equation:where R and r are the outer and inner diameters of the aluminum tube, respectively. The variable ht is the height of tube, which coincides with the sample height. Next, the tube mass (mt) can be obtained using the density of the 6060 aluminum alloy ρAl=2.70 g.cm−3 [By subtracting (mt) from the total TFF samples mass (mtot), the mass of the syntactic foam (mSF) contained in TFFs can be calculated. In MSFs, the sample mass (mtot) is identical to (mSF). Now, the MSF density can be obtained according toIn TFFs, the bulk density (ρbulk) was calculated according to Equation using (mtot) and (Vtot) for each sample.For all samples, the volume fraction of the ZA27 matrix (φM) can be calculated using:The volume fraction of EP particles (φP) was assumed to be a constant and determined using:Based on previous measurements, the bulk density of expanded perlite beds (ρB) and particle density (ρP) were taken to be 0.091 and 0.160 g cm−3, respectively []. Therefore, a constant fraction of φP=56.88% was assumed for all samples. The remaining volume must then be occupied by voids within the sample, i.e.The mechanical properties of the samples were analyzed using quasi-static compressive testing. A 50 kN SHIMADZU and a 240 kN MTS uni-axial testing machine were used for the compression of MSFs and TFFs, respectively. In addition, individual aluminum tubes were compressed using the 50 kN SHIMADZU testing machine.Prior to all compression tests, both ends of the samples were lubricated to minimize friction with compression platens using CRC® 5–56 multipurpose lubricant (CRC Industries, NSW, Australia). During compression, forces were measured using either a SFL-50KN AG load cell for MSFs and aluminum tubes or a 240 kN (type 661.21A-02) unit for TFFs. Displacements were measured as the machine cross-head movement. All samples were compressed at a constant crosshead speed of 1 mm min−1. The measured forces and displacements were then converted to engineering stress (σ) and strain (ε) using the initial sample dimensions. The entire cross section of TFFs (including the tube cavity) was considered in calculation of engineering stress. Stress-strain data was further processed following ISO 13314 [] to obtain the compressive stress corresponding to 1% of strain (1% offset yield stress), the arithmetic mean of stress between 20% and 40% of strain (plateau stress (σPl)), and the corresponding strain at a stress of 1.3×σPl (plateau end strain ePle). Moreover, the energy absorption (W) of the samples was calculated as the area under the compressive stress strain curves up to 50% strain according to:Finally, the energy absorption efficiency was determined using:where σmax is the maximum stress up to 50% of strain.a a SEM image of the interface between ZA27 matrix and aluminum tube is shown. The microscopy clearly indicates the excellent bonding between the aluminum tube and ZA27 matrix. No gaps between the two undeformed components could be observed. In addition, no reaction layer between the aluminum tube and ZA27 matrix is visible. Aluminum and zinc are known to not form intermetallic compounds under the casting conditions of this study []. The absence of significant chemical reaction is further confirmed by the EDS analysis (see b). Starting from the left, the EDS line scan first shows the elemental distribution of the ZA27 matrix, i.e. predominantly zinc and aluminum. At the interface with the aluminum tube (dashed line), the concentration of the zinc rapidly decreases, whereas the concentration of aluminum increases. Interestingly, the concentrations of zinc and aluminum change locally at the interface but are near constant elsewhere. A likely explanation is the interdiffusion between the zinc matrix and the aluminum tube during aging []. The concentration of oxygen and silicon in the ZA27 matrix is negligible. The highest concentration of oxygen is detected at the matrix/tube interface. This peak likely indicates the presence of aluminum oxide that may be formed during casting or was already present on the surface of the aluminum tube prior to casting. At the casting temperature of 535 °C, Al2O3 is more stable than ZnO and hence its formation is more likely []. The peaks of silicon and oxygen at almost similar points suggest a compound consisting of these elements. It is most likely related to the presence of SiO2 (crushed expanded perlite particles) on the surface due to cutting the sample prior to EDS analysis.The physical properties of TFFs and MSFs are shown in . The samples of both groups were manufactured with aspect ratios ((hd) in the range of 1.3–1.6. All samples (both MSFs and TFFs) have a similar bulk density (ρbulk ) (see ). In other words, replacing the central part of the ZA27/EP MSFs with the selected aluminum tubes does not alter the density of the material due to the addition of a large cylindrical void. This is in direct contrast to conventional FFTs, where the addition of a solid outer shell increases the bulk density of the resulting structure. For comparison, the bulk density of a virtual foam filled tube (FFT) structure using a ZA27/EP syntactic foam core with an aluminum tube with the same wall thickness t=1.6 mm and outer diameter of 28 mm is calculated to be 3.79 g.cm−3 using equations . This is almost twice the average bulk density of TFFs in this study., the matrix volume fraction (φM) of the foam section in TFFs is higher than in MSF samples. First, the presence of the central tube may partially counteract the dense packing of particles during vibration. As a result, a larger interstitial volume exists between particles permitting a higher volume of melt infiltration. Second, the tube decreases the available cross section for melt infiltration. The resulting higher casting pressure (0.02 MPa for TFF compared to 0.01 MPa for MSF) supports melt infiltration in TFFs and enables melt penetration into thinner inter-particle channels.The compressive stress-strain curves of three individually tested aluminum tubes are shown in a. Three distinct regions can be identified. In the first predominantly elastic region, the stress rapidly increases. The second region shows only a small stress increase over a wide range of strain and coincides with the localized buckling of the tubes. In the third region, the stress increases abruptly and densification takes place. The stress-strain curves show a ductile deformation of aluminum tubes without any significant stress oscillations. In the second region, the deformation of the aluminum tubes commences with the formation of individual folds in the longitudinal tube direction at various positions (see ε=0.2). No cracks are visible on the outer surface of aluminum tube throughout the entire deformation. This is likely due to the high ductility of the 6060 aluminum alloy. At ε=0.3, global buckling of the tube can be observed (see arrow in b). This buckling occurs towards the center of the aluminum tubes and causes inward folding in a non-axisymmetric mode. It has been shown previously that the deformation of aluminum tubes used for FFTs mostly follows a concertina or mixed mode []. These different deformation modes are likely due to the tube geometry []. In FFTs, a smaller length to diameter ratio (LD) combined with a smaller wall thickness to diameter (tD) ratio changes its deformation towards concertina or mixed modes []. Concertina or mixed deformation modes in FFTs trigger the symmetrical folding of tubes from both walls []. In contrast, the deformation of the empty aluminum tubes shows (see b) the progression of a single fold that leads eventually to global buckling.The stress-strain curves of MSFs and TFFs (see a) show three characteristic regions that resemble the aluminum tube (see a) and are typical for metallic foams: a quasi-elastic deformation identified by a rapid stress increase, a constant stress plateau and finally densification at high strains. For a better comparison, the average stress-strain curve of each group is shown as a bold line in (a). The stress-strain curves of all TFFs are higher than those of MSFs. Inside each group (MSFs or TFFs), samples with a higher bulk density exhibit higher stresses. Both sample types show strain hardening, i.e. stresses gradually increase throughout the compression. This is attributed to the heat treatment of the samples that enhanced the ductility of the ZA27 matrix []. It can be concluded that the MSFs deformation is mostly controlled by the ductile deformation of the ZA27 struts. Due to the low compressive strength of the EP particles [], their contribution to material strength and deformation is considered negligible. Their main function is to introduce the porosity into the MSFs. Due to the similar shape of the stress-strain curves of both groups, it can be inferred that the deformation of TFFs is also mostly controlled by the ZA27 matrix. However, the higher stress values of TFFs in comparison to MSFs indicate a reinforcement-effect of the embedded aluminum tube. Interestingly, integrating the aluminum tube into the MSF not only keeps the bulk density almost constant, but also enhances its overall strength.The deformation of MSFs and TFFs is shown in b and c, respectively. The deformation mechanism of both samples is layer-by-layer collapse. However, this mechanism is localized in different volumes of the two sample types. In MSFs, the initial deformation occurs by barreling of the foam from one end. This trend continues up to higher strains (see dashed lines in b). The same deformation behavior was previously observed for similar MSFs with slightly different densities and is triggered by a density gradient within the samples []. In contrast, the deformation of the TFFs occurs more uniform. The barreling encompasses the entire sample and is not localized in a sub-volume as in MSFs (see dashed lines in c at ε=0.5). This is likely due to the stabilization of the structure by the embedded aluminum tube. In TFF, the deformation of the tube transmits forces to the entire sample resulting in the observed uniform deformation. Similar to FFTs [], an interaction effect between the aluminum tube and metal foam alters the deformation mechanism of the material.The deformation of the aluminum tube is also altered by its interaction with the surrounding foam section of TFFs. shows deformed aluminum tubes at ε=0.5 and ε=0.8 either compressed individually or contained inside the MSF section of TFFs:Tube compression The deformed aluminum tube at ε=0.5 (see a) shows folding at one side of the tube (see arrow in . The deformed aluminum tube at ε=0.8 exhibits a two lobe diamond crushing mode (see a). Based on the dimensions of the aluminum tube (wall thickness to dimeter ratio: 0.13, length to diameter ratio: 3.00) this deformation mode is correctly predicted by Ref. [TFF compression The deformation mechanism of the aluminum tube embedded in the TFF is clearly different. During compression, the aluminum tube forms two symmetrical folds (see b at ε=0.5). It appears that the deformation of the aluminum tube is stabilized due to lateral support by the surrounding MSF. The additional plastic deformation (i.e. the formation of an additional fold) requires additional energy for the compression of the embedded tube. Moreover, the deformed tube at ε=0.5 shows some localized disconnections from the outer MSF (see arrow). Such disconnections are concentrated adjacent to the folds of the tube. This is likely due to tensile forces arising from the relative motion of the buckling tube and the surrounding MSF. The MSF tends to barrel outwards but folding of the tube occurs locally in the opposite inward direction. The compressed TFF at ε=0.8 shows no detachment between aluminum tube and surrounded MSF (see arrows in To analyze the interaction between MSF and embedded tube, shows the averaged force-displacement curves of different sample groups. The green dotted line corresponds to the compression of individual aluminum tubes. The force-displacement curve of the MSF section in TFFs is shown as a red dashed line. This curve is obtained from the compression of the MSF samples and has been rescaled to account for the different MSF cross sections. The black dash-dotted line is obtained by the summation of the aluminum tube and MSF section curves. It therefore corresponds to a parallel compression of these components without any interaction effects. Interestingly, this line is distinctly below the blue line that represents the average force-displacement curve of TFFs. The deviation between these two curves quantifies the interaction effect between tube and MSF. At low displacements (i.e. mostly linear elastic deformation), the two curves coincide, i.e. no interaction is observed. However, at displacements > 1 mm an approximately constant offset between the curves emerges. An explanation for this increased deformation resistance of TFF is the previously described change in the buckling mode of the embedded aluminum tube (see ) and the more uniform global deformation of TFFs (see The mechanical properties of MSF and TFF samples are plotted versus their densities in . Each individual point represents the mechanical property of one sample from a, the 1% offset yield stress of TFFs and MSFs are shown. This parameter is a commonly used strength indicator for metallic foams and separates the predominantly elastic deformation from the plastic deformation in the stress plateau. Distinctly higher stress values are obtained for TFFs compared to MSFs. This quantifies a strengthening effect of the aluminum tubes in TFF under compressive loads.The plateau stress is plotted versus density in b. Analogous to the offset yield stress, a higher plateau stress of TFFs in comparison to MSFs can be observed. As expected, the plateau stress generally increases with density. At the same density, the higher plateau stress of TFF compared to MSF can be related to their different deformation mechanisms. The plastic deformation of the embedded tube and the resulting homogeneous deformation of the surrounding MSF results in a higher resistance to compression, which is further visible as an increased energy absorption (see c). The limited data points suggest some scattering of plateau stress and energy absorption of TFFs. Unlike in MSFs, these material properties do not increase continuously with density. Such scattering could be explained by localized disconnections between aluminum tube and metal foam (see b) which may occur randomly in only some of the samples.The energy absorption efficiency of both groups changes in a narrow band between 0.61 to 0.67 (d). This property appears to be independent of sample type, which can be explained by the similar deformation mechanism of TFFs and MSFs (layer-by-layer collapse). Although there are some differences during their deformation (see b and c) their stress-strain curves show similar strain hardening behavior (see In this study, tube filled foams (TFF) were manufactured and tested for the first time. The following main conclusions can be drawn:The integration of the selected aluminum tubes does not increase the bulk density of TFF (1.79–1.91 g cm−3) compared to MSF (1.79–1.92 g cm−3).SEM analysis shows good bonding between aluminum tube and MSF.EDS shows no evidence of chemical reactions between TFF components.The compressive deformation of TFFs and MSFs occurs in a layer-by-layer deformation mode. In addition, strain hardening was observed for both sample groups.The integration of aluminum tubes in TFF structures stabilizes their compressive deformation. Instead of localized deformation towards one end (MSF) uniform barreling originating at the samples’ center is observed (TFF).Due to lateral support by the MSF section, the buckling mode of the embedded aluminum tubes changes to axisymmetric.Because of this stabilized deformation, the overall strength (initial strength and plateau strength) and energy absorption of TFFs clearly exceed MSFs.Nima Movahedi manufactured the samples, conducted the tests and evaluated the data. He further drafted the first version of the manuscript.Graeme Murch consulted on Materials Science and edited the later versions of the Manuscript.Irina Belova helped with experimentation and data evaluation and proof read the final version of the manuscript.Thomas Fiedler initiated the research, provided ongoing supervision and advice and strongly contributed to the later versions of the manuscript.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.The following is the Supplementary data to this article:Supplementary data to this article can be found online at Three-dimensional singular stress/residual stress fields at crack/anticrack fronts in monoclinic plates under antiplane shear loading► 3D stress intensity-factor/singularity-coefficient for center-crack/anticrack are derived. ► A new mode III crack deflection/bifurcation criterion is derived. ► 3D energy release/absorption rates for crack/anticrack are derived. ► 3D energy release rate on anticrack is derived, using reverse Irwin method. ► Through-thickness variations of normalized stress intensity factors presented.A recently developed eigenfunction expansion technique, based in part on separation of the thickness-variable, is first developed to derive three-dimensional asymptotic stress field in the vicinity of the front of a semi-infinite through-thickness crack/anticrack weakening/reinforcing an infinite monoclinic plate, of finite thickness and subjected to far-field antiplane shear loading. Crack/anticrack-face boundary conditions and those that are prescribed on the top and bottom (free or fixed) surfaces of the anisotropic (monoclinic) plate are exactly satisfied. Five different through-thickness crack/anticrack-face boundary conditions are considered: (i) slit crack, (ii) anticrack or perfectly bonded rigid inclusion, (iii) transversely rigid inclusion (longitudinal slip permitted), (iv) rigid inclusion in part perfectly bonded, the remainder with slip, and (v) rigid inclusion located alongside a crack. The three-dimensional stress intensity factor for a center-crack, and stress singularity coefficients for/on a center-anticrack are then derived by incorporating an extension of the Stroh type approach in the present analysis. Through-thickness distribution of stress intensity factor and stress singularity coefficient for a crack and an anticrack, respectively, is also presented. Additionally, singular residual stress fields in the vicinity of the fronts of these cracks, anticracks and similar discontinuities are also discussed. Hitherto unavailable expressions for three-dimensional energy release and absorption rates for center-cracks and anticracks are derived by using Irwin’s crack closure and Eshelby’s eigenstrain approach, respectively. A heretofore unavailable expression for the energy release rate on the super-rigid inclusion is derived, using an approach which is analogous, in a reverse sense, to Irwin’s crack closure method. Finally, a new mode III crack deflection/bifurcation criterion is also derived. The crack deviation under antiplane shear loading is strongly correlated with the elastic stiffness constant, c45, of the monoclinic single crystal or off-axis composite lamina concerned.constants related to stress singularity coefficient (anticrack) and stress intensity factor (crack), respectivelyelastic stiffness constants of an anisotropic (monoclinic) plate, i, j |
= 1, … , 6,energy release rate on the super-rigid inclusionsymmetric (with respect to z) mode III stress intensity factorantisymmetric (with respect to z) mode III stress intensity factorlocal stress intensity factor at the front of the infinitesimal kink emanating from a cracksymmetric (with respect to z) mode III stress singularity coefficientantisymmetric (with respect to z) mode III stress singularity coefficientcomponents of the displacement vector in x, y, z directionstransverse shear moduli of an orthotropic laminafiber orientation angle in a composite lamina or the principal direction of orthotropy in a monoclinic plateinplane components of the stress tensor in Cartesian coordinatesout-of-plane components of the stress tensor in Cartesian coordinatescomplex variable in the affine transformed complex planeMonoclinic materials occupy an important place in terms of both geological occurrences as well as modern materials technology. Important classes of geological materials with monoclinic symmetry include orthoclase and sanidine classes of alkali feldspars, barium feldspars and hornblende (a complex inosilicate series of minerals), while important synthetic monoclinic materials include anthracene (coal tar byproduct), naphthalene, tartaric acid as well as its salt, potassium tartarate Asymptotic behavior of two-dimensional stress fields at the tips of cracks, anticracks (through slit cracks filled with infinitely rigid lamellas) and homogeneous/bimaterial wedges, has been studied extensively in the literature Strength of anisotropic materials, such as single crystals and fiber reinforced composites, is also adversely affected by residual stresses since a pre-loading has been introduced. For example, residual stresses are developed in silicon wafers during the growth phase as well as in the thermal processing The mathematical difficulties posed by the three-dimensional crack and anticrack problems are substantially greater than their two-dimensional counterparts (to start with, the governing PDE’s are much more complicated). In the absence of the knowledge of the strength of singularity, in regions where the elastic stresses become unbounded, the majority of weighted residual type methods, e.g., the finite elements, finite difference and boundary elements, which are generally employed to solve fracture mechanics problems, encounter overwhelming numerical difficulties, such as lack of convergence, and oscillation resulting in poor accuracy A three-dimensional eigenfunction approach has recently been developed by Chaudhuri and Xie In regards to the antiplane shear problems, Ma and Hour In what follows, a recently developed eigenfunction expansion type method provides a mathematical statement of the problem under investigation. This three-dimensional eigenfunction expansion method is employed to derive explicit expressions for the three-dimensional asymptotic singular stress fields in the vicinity of points located at the fronts of through-thickness cracks/anticracks, subjected to the five combinations of crack/anticrack-face boundary conditions. Three different boundary conditions, free, fixed and lubricated, are applied on the top and bottom surfaces of the monoclinic plate. These expressions for the singular stress fields in the vicinity of the crack/anticrack fronts weakening/reinforcing monoclinic plates, of finite thickness and subjected to antiplane shear loading, presented in Sections . Satisfactions of crack-surface, anticrack-surface as well as other discontinuity-surface boundary conditions for semi-infinite cracks, anticracks and other discontinuities are discussed in Sections , respectively. Explicit expressions for three-dimensional stress intensity factor (resp. stress singularity coefficients) and energy release (resp. absorption) rate (s) for a through-thickness crack (resp. anticrack) are derived in Section discusses a crack deflection criterion for an infinitesimal kink at the front of a macroscopic crack in a monoclinic single crystal and fiber reinforced composite lamina. Appendix presents a similitude analysis comparing the solution for a monoclinic plate with its orthorhombic and isotropic counterparts.The Cartesian coordinate system (x, y, z) is convenient to describe the deformation behavior in the vicinity of a semi-infinite through-thickness crack/anticrack, weakening/ reinforcing an infinite anisotropic plate of thickness, 2 h (). Here, the z-axis is placed along the straight crack/anticrack front, while the coordinates x, y, are used to define the directions along the length of the crack/anticrack and transverse to it, respectively, in the plane of the plate. u, v and w represent the components of the displacement vector in the x, y and z directions, respectively. The anisotropic material investigated here is of monoclinic type with x–y plane serving as the plane of symmetry In what follows, σx, σy and σz represent the normal stresses, while τyz, τxz and τxy denote the shear stresses. εx, εy and εz are the normal strains, while γyz, γxz and γxy represent shear strains. cij, i, j |
= 1, … , 6, denotes the elastic stiffness constants of an anisotropic (monoclinic) plate, given in the form:σxσyσzτyzτxzτxy=c11c12c1300c16c12c22c2300c26c13c23c3300c36000c44c450000c45c550c16c26c3600c66εxεyεzγyzγxzγxy.The three equilibrium equations for a linear elastic solid can be expressed in terms of the displacement components u, v, and w, as follows:c11∂2u∂x2+2c16∂2u∂x∂y+c66∂2u∂y2+c55∂2u∂z2+c16∂2v∂x2+(c12+c66)∂2v∂x∂y+c26∂2v∂y2+c45∂2v∂z2+(c13+c55)∂2w∂x∂z+(c36+c45)∂2w∂y∂z=0,c16∂2u∂x2+(c12+c66)∂2u∂x∂y+c26∂2u∂y2+c45∂2u∂z2+c66∂2v∂x2+2c26∂2v∂x∂y+c22∂2v∂y2+c44∂2v∂z2+(c36+c45)∂2w∂x∂z+(c23+c44)∂2w∂y∂z=0,(c13+c55)∂2u∂x∂z+(c36+c45)∂2u∂y∂z+(c36+c45)∂2v∂x∂z+(c23+c44)∂2v∂y∂z+c55∂2w∂x2+2c45∂2w∂x∂y+c44∂2w∂y2+c33∂2w∂z2=0.The boundary conditions include those at the plate faces and crack/anticrack-side surfaces. The three boundary conditions imposed on the plate faces, z |
= ±h, are given as follows:The boundary conditions at the crack or anticrack-side surfaces are more conveniently expressed in local cylindrical polar coordinates (), which are, for r |
> 0, given as follows:Rigid Inclusion in Part Perfectly Bonded, the Remainder with Slip Mixed or Rigid Inclusion alongside a Crack where ur and uθ represent the components of the displacement in r and θ directions, respectively. σr, σθ, σz represent the normal stresses, and τrθ, τrz, τθz denote the shear stresses, while εr, εθ, εz denote normal strains, and γrθ, γrz, γθz represent the shear strains in the cylindrical polar coordinate system (r, θ, z).The assumed displacement functions for the three-dimensional crack/anticrack problem under consideration are selected on the basis of separation of z-variables. These are as given below where k is a constant, called the wave number, required for a Fourier series expansion in the z-direction. It may be noted that since the separated z-dependent term and its first partial derivative can either be bounded and integrable at most admitting ordinary discontinuities, or the first partial derivative at worst be square integrable (in the sense of Lebesgue integration) in its interval z |
∈ [−h, |
h], i.e., admitting singularities weaker than square root (i.e., z(−1/2+ε), ε |
> 0 being a very small number), it can be best represented by Fourier series yields the following system of coupled partial differential equations (PDE’s):c11∂2U∂x12+2c16∂2U∂x1∂y1+c66∂2U∂y12+c55U+c16∂2V∂x12+(c12+c66)∂2V∂x1∂y1+c26∂2V∂y12+c45V+(c13+c55)∂W∂x1+(c36+c45)∂W∂y1=0,c16∂2U∂x12+(c12+c66)∂2U∂x1∂y1+c26∂2U∂y12+c45U+c66∂2V∂x12+2c26∂2V∂x1∂y1+c22∂2V∂y12+c44V+(c36+c45)∂W∂x1+(c23+c44)∂W∂y1=0,(c13+c55)∂U∂x1+(c36+c45)∂U∂y1+(c36+c45)∂V∂x1+(c23+c44)∂V∂y1+c55∂2W∂x12+2c45∂2W∂x1∂y1+c44∂2W∂y12+c33W=0,The solution to the system of coupled partial differential equations , subjected to the far-field antiplane shear loading, can now be sought in the form of the following modified Frobenius type series in terms of the variable x1 |
+ py1 as follows Here, s represents a fractional exponent, while n denotes an integer in the power series. The combined variable x1 |
+ |
py1 represents an affine transformation in the same spirit as that by Eshelby et al. and equating the coefficients of (x1 |
+ |
py1)s+2n−1 yields the following two recurrent relations:(s+2n+1)(s+2n)(c11+2c16p+c66p2)as+n+c55as+n-1+(s+2n+1)(s+2n){c16+(c12+c66)p+c26p2}bs+n+c45bs+n-1+(s+2n){c13+c55+(c36+c45)p}cs+n=0,(s+2n+1)(s+2n){c16+(c12+c66)p+c26p2}as+n+c45as+n-1+(s+2n+1)(s+2n)(c22+2c26p+c66p2)bs+n+c44bs+n-1+(s+2n){c23+c44+(c36+c45)p}cs+n=0. and equating the coefficients of (x1 |
+ |
py1)s+2n−2 yields the third recurrent relationship:(s+2n+1)[(c13+c55)as+n-1+(c36+c45)pas+n-1+(c36+c45)pbs+n-1+(c23+c44)pbs+n-1]+(s+2n)(s+2n-1){c55+2c45p+c44p2}cs+n+c33cs+n-1=0.For n |
= 0, which represents the first term of an asymptotic expansion, Eq. reduce to the following algebraic equations:(s+1)(c11+2c16p+c66p2)as+(s+1){c16+(c12+c66)p+c26p2}bs+{c13+c55+(c36+c45)p}cs=0,(s+1){c16+(c12+c66)p+c26p2}as+(s+1)(c22+2c26p+c66p2)bs+{c23+c44+(c36+c45)p}cs=0, supplies the following characteristic equationNow, since the separated z-dependent term is, as mentioned earlier, assumed in the form of a Fourier series, and the power series is cut off at n |
= 0, the asymptotic (dominant) form of the assumed displacement functions, which represents the near-field deformation state, can be written as follows:w(x,y,z)=(D¯1cos(kz)+D¯2isin(kz))(ik)s[C¯1{(x-ξy)+iηy}s+C¯2{(x-ξy)-iηy}s],u(x,y,z)=(D¯1isin(kz)+D¯2cos(kz))(ik)s+1[A¯1{(x-ξy)+iηy}s+1+A¯2{(x-ξy)-iηy}s+1],v(x,y,z)=(D¯1isin(kz)+D¯2cos(kz))(ik)s+1[B¯1{(x-ξy)+iηy}s+1+B¯2{(x-ξy)-iηy}s+1],where A¯1,A¯2,B¯1,B¯2,C¯1,C¯2,D¯1 and D¯2 are constants, which are related to the boundary conditions as well as the far-field loading for finite length-scale of a crack or anticrack. The corresponding stress field can easily be obtained from Eq. . It is convenient to express the components of the displacement vector and stress tensor, in terms of the cylindrical polar coordinate system (r, θ, z), by expressingρcos(ψ)=r(cos(θ)-ξsin(θ)),ρsin(ψ)=r(ηsin(θ)),cos(ψ(θ))=cos(θ)-ξsin(θ){(cos(θ)-ξsin(θ))2+η2sin2(θ)}1/2,sin(ψ(θ))=ηsin(θ){(cos(θ)-ξsin(θ))2+η2sin2(θ)}1/2,The general asymptotic form for the displacement components can be written as follows:w(r,θ,z)=rsDb(z)[{(cos(θ)-ξsin(θ))2+η2sin2(θ)}s/2{C1cos(sψ)+C2sin(sψ)}]+O(rs+2),The singular stress field can be obtained from Eq. τxz(r,θ,z)=-rs-1Db(z)s[{cos(θ)-ξsin(θ)}2+η2sin2(θ)](s-1)/2〈C1[(c45ξ-c55)cos{(s-1)ψ}+c45ηsin{(s-1)ψ}]-C2[c45ηcos{(s-1)ψ}+(c55-c45ξ)sin{(s-1)ψ}]〉+O(rs+1),τyz(r,θ,z)=-rs-1Db(z)s[{cos(θ)-ξsin(θ)}2+η2sin2(θ)](s-1)/2〈C1[c44ηsin{(s-1)ψ}]-C2c44ηcos{(s-1)ψ}〉+O(rs+1),The symbol O(…) means “of the order of the quantity (…)”. The corresponding stress field in the cylindrical polar coordinate system can be obtained through use of standard tensor transformation rule as follows:τrz(r,θ,z)=-rs-1Db(z)s[{cos(θ)-ξsin(θ)}2+η2sin2(θ)](s-1)/2〈C1[(c45ξ-c55)cos{(s-1)ψ}cos(θ)+c45ηsin{(s-1)ψ}cos(θ)+c44ηsin{(s-1)ψ}sin(θ)]-C2[c45ηcos{(s-1)ψ}cos(θ)+(c55-c45ξ)sin{(s-1)ψ}cos(θ)+c44ηcos{(s-1)ψ}sin(θ)]〉+O(rs+1),τθz(r,θ,z)=-rs-1Db(z)s[{cos(θ)-ξsin(θ)}2+η2sin2(θ)](s-1)/2〈C1[c44η2cos{(s-1)ψ}sin(θ)-c45ηsin{(s-1)ψ}sin(θ)+c44ηsin{(s-1)ψ}cos(θ)]+C2[c45ηcos{(s-1)ψ}sin(θ)+(c55-c45ξ)sin{(s-1)ψ}sin(θ)-c44ηcos{(s-1)ψ}cos(θ)]〉+O(rs+1),It may be noted that since s or Re s (when s is complex) is positive, all the higher order terms in Eq. vanish as r |
→ 0. The above expressions for stresses and displacements must satisfy the plate surface boundary conditions. They also need to satisfy the boundary conditions on the crack, anticrack or mixed crack-anticrack-side surfaces.Satisfaction of stress-free boundary conditions, given by Eq. , on the plate faces results in the following:The special case of symmetric deformation is obtained as follows:The antisymmetric deformation case can also be obtained in a similar manner, as given below:Satisfaction of fixed or lubricated boundary conditions, given by Eqs. , respectively, on the plate faces results in identical expressions for the displacement and singular stress fields as above. yields the characteristic equation for a crack, which reduces to either contributes to the lowest non-vanishing eigenvalue, s |
= 1/2, in the range 0 < |
s |
< 1, as required by the criterion of locally finite energy. Eq. yields s |
= 0, 1, in which s |
= 0 reproduces the rigid body motion, while s |
= 1 represents the second term in the eigenfunction expansion, which is constant with respect to r (and is related to T-stress would yield the following stress distribution in the vicinity of a crack front:τxz(r,θ,z)=KIII(z)2πr[{cos(θ)-ξsin(θ)}2+η2sin2(θ)]-1/4{ξcos(ψ/2)-ηsin(ψ/2)},τyz(r,θ,z)=KIII(z)2πr[{cos(θ)-ξsin(θ)}2+η2sin2(θ)]-1/4cos(ψ/2),The corresponding stress field in the cylindrical polar coordinate system can be obtained through use of standard tensor transformation rule, given by Eq. , while the transverse displacement component is given byw(r,θ,z)=KIII(z)c44η2rπ{(cos(θ)-ξsin(θ))2+η2sin2(θ)}1/4sin(ψ/2).The above results reduce to their orthorhombic KIIIs(z)=π2C2c44ηDbs(z),KIIIa(z)=π2C2c44ηDba(z),KIIIa thus obtained represents the self-equilibrating stress intensity factor for a semi-infinite crack, resulting in the residual stresses in the anisotropic (monoclinic) plate material. It must be noted that KIIIa cannot be determined by a two-dimensional approximation, which yields constant KIIIs, while both KIIIa and KIIIs, derived by the present three-dimensional approach, are functions of z.It may be noted that in the expression for transverse shear stress, τxz, given by Eq. , under mode III loading, there is a coupling between cos (ψ/2) and sin (ψ/2). Appendix presents a kinematic similitude based comparison of the solution for a monoclinic plate involving a pair of complex roots with its isotropic counterpart, and concludes that the expressions for ρ and ψ look different from their isotropic counterparts. This is especially true for ψ at x |
= 0. The significance of this, albeit being semi-qualitatively and implicitly inferred at this stage (see Section for an explicit quantitative approach for validating this), is that for a monoclinic single crystal or an off-axis fiber reinforced unidirectional composite lamina, the through (0 1 0)[0 0 1] crack (i.e., oriented at θ |
= 0) considered here is, unlike its isotropic counterpart, more likely to deviate from its original plane (0 1 0) and original propagation direction [1 0 0], because of this coupling, which arises from the complex roots. It can then be inferred that the (0 1 0)[0 0 1] is not an easy cleavage system for monoclinic single crystals or off-axis fiber reinforced unidirectional composite laminas with c45 |
≠ 0. For the special case of orthorhombic single crystals or 0o-laminas, c45 |
= 0, which makes the through (0 1 0)[0 0 1] crack (i.e., oriented at θ |
= 0) considered here, like its isotropic counterpart, more likely to propagate in its original plane (0 1 0) and original propagation direction [1 0 0], because of absence of this type of coupling, which arises from the imaginary roots The constant, C2, which is related to the stress intensity factor, KIIIs(z), cannot be determined unless the far-field loading and a characteristic length (e.g., crack geometry) are specified. The stress intensity factor for an infinite plate with a central crack of length, 2a, and subjected to far-field loading, τyz∞, is given by Sih et al. KIII(z)=-22πLimζ3→ζ0(ζ3-ζ0)1/2ϕ3′(ζ3)Db(z),in which the complex variable, ζ3, is defined as follows:where Γ3 is a constant. In polar coordinates, Eq. KIII(z)=Limr→0(2πr)1/2τxz∞1-(a+r){(a+r)2-a2}1/2Db(z)=πaτyz∞Db(z). reduces to its two-dimensional counterpart, given by Sih et al. The corresponding (through-thickness average) energy release rate can now be derived by employing Irwin’s GIII=πaτyz∞24hc44c55-c4521/2∫-hh(Db(z))2dz.GIII,ortho=πaτyz∞24hc¯44c¯55∫-hh(Db(z))2dz,where c¯44=μ23 and c¯55=μ13, in which μ23 and μ13 represent the transverse shear moduli (in the planes normal and parallel to the fiber direction, respectively, in the case of a unidirectional composite lamina). For a cubic material, Eq. is further simplified, and can be written as follows:For an isotropic material, this reduces to the following: reduce to their two-dimensional plate (of unit thickness) counterparts The through-thickness distribution of the energy release rate for a center-crack can also be derived by introducing the thickness-wise partial crack closure method as follows:Gm(z)=πaτyz∞22c44c55-c4521/2∂∂z∫-hz(Db(z))2dz=πaτyz∞22c44c55-c4521/2(Db(z))2.Eq. (53) reduces to the corresponding special cases of orthotropic, cubic and isotropic materials in a manner similar to Eqs. (51), (52) and follow-up discussion thereafter.Supposing that crack forms an infinitesimal kink at an angle, θ¯, from its original plane of propagation, Eq. τyz(r,θ¯,z)=K¯III(θ¯,z)2πr[{cos(θ¯)-ξsin(θ¯)}2+η2sin2(θ¯)]-1/4cos(ψ(θ¯)/2),where K¯III(θ¯,z) represents the local stress intensity factor at the front of the infinitesimal kink. Following the procedure outlined in standard texts, e.g., Anderson K¯III(θ¯,z)=2πrτyz(r,0,z)=KIII(z)2[1+cos(ψ(θ¯))]-1/2[{cos(θ¯)-ξsin(θ¯)}2+η2sin2(θ¯)]1/4,Identifying the angle θ¯ as the angle of fiber orientation in a unidirectional composite or the principal direction of orthotropy (i.e., c¯45=0), Eq. K¯III(θ¯,z)=2KIII(z)[1+cos(θ¯)]-1/2cos2(θ¯)+c¯55c¯44sin2(θ¯)-1/4,where c¯44=μ23 and c¯55=μ13. The crack will deflect in the direction, θ¯, ifK¯III(θ¯,z)KIII(z)=2[1+cos(θ¯)]-1/2cos2(θ¯)+c¯55c¯44sin2(θ¯)-1/4>1,θ¯>0,c¯44c¯55=μ23μ13=R^>sin2(θ¯){1+cos(θ¯)}2{2+cos(θ¯)+cos2(θ¯)}{2-cos(θ¯)-cos2(θ¯)}=F^(θ¯),θ¯>0. yields the characteristic equation for an anticrack, which reduces to eitherEq. (59a) contributes to the lowest nonvanishing eigenvalue, s |
= 1/2, in the range 0 < |
s |
< 1, as required by the criterion of locally finite energy. Eq. (60a) yields s |
= 0, 1 (see Section ). The minimum root (eigenvalue), contributing to the singular stress field, s |
= 1/2, is the same as that for its orthorhombic τxz(r,θ,z)=SIII(z)2πr[{cos(θ)-ξsin(θ)}2+η2sin2(θ)]-1/4[ηcos(ψ/2)+ξsin(ψ/2)],τyz(r,θ,z)=SIII(z)2πr[{cos(θ)-ξsin(θ)}2+η2sin2(θ)]-1/4sin(ψ/2),It may be noted that in the expression for transverse shear stress, τxz, given by Eq. , there is a coupling between cos (ψ/2) and sin (ψ/2). The significance of this lies in the fact that for a monoclinic single crystal or an off-axis fiber reinforced unidirectional composite lamina, the through (0 1 0)[0 0 1] super-rigid lamellar inclusion (oriented at θ |
= 0) considered here would be reduced (because of negative energy release rate discussed in the next section) via some damage mechanisms, nucleating a crack in the process of doing so. This newly formed through-crack would, unlike its isotropic counterpart, more likely deviate from the original plane (0 1 0) and original length direction [1 0 0] of the super-rigid lamellar inclusion, because of this coupling, which arises from the complex roots (i.e., c45 |
≠ 0). For the special case of orthorhombic single crystals or 0°-laminas, c45 |
= 0, which would likely make the newly nucleated through (0 1 0)[0 0 1] crack (oriented at θ |
= 0), like its isotropic counterpart, more likely to propagate in the original plane (0 1 0) and original length direction [1 0 0] of the super-rigid lamellar inclusion, because of absence of this type of coupling, which arises from the imaginary roots [11] (see also Appendix).The corresponding stress field in the cylindrical polar coordinate system can be obtained through use of standard tensor transformation rule, given by Eq. , while the transverse displacement component is given byw(r,θ,z)=SIII(z)c44η2rπ{(cos(θ)-ξsin(θ))2+η2sin2(θ)}1/4cos(ψ/2).The above results reduce to their orthorhombic SIIIs=π2C1c44ηDbs(z),SIIIa(z)=π2C1c44ηDba(z).SIIIa thus obtained represents the self-equilibrating stress singularity coefficient for a semi-infinite anticrack, resulting in the residual stresses in the material. Like its crack counterpart, SIIIa cannot be determined by a two-dimensional approximation, which yields constant SIIIs, while both SIIIa and SIIIs, obtained by the present three-dimensional approach, are functions of z.The constant, C1, which is related to the stress singularity coefficient, SIII(z), cannot be determined unless the far-field loading and anticrack geometry are specified. The stress components, just ahead of the rigid inclusion front, at θ |
= 0, or y |
= 0, can be written, following the procedure outlined by Li and Ting τyz(a+r,0,z)=Re{c45+(-ξ+iη)c44}ζ3ζ32-a21/2γxz∞Db(z)=0,τxz(a+r,0,z)=-Re{c45+(-ξ+iη)c44}(-ξ+iη)ζ3ζ32-a21/2γxz∞Db(z)=(c44c55-c452)c44a+r{(a+r)2-a2}1/2γxz∞Db(z),Stress singularity coefficient, SIII(z), can now be written as follows:SIII(z)=Limr→0(2πr)1/2τxz(a+r,0,z)=πac44c55-c452c44γxz∞Db(z)=πaτxz∞-c45c44τyz∞Db(z)=SIII,2DDb(z).Forτxz∞=0,SIII(2)(z)=-c45c44πaτyz∞Db(z)=SIII,2D(2)Db(z),which vanishes in the case of an orthorhombic, cubic or isotropic material.Forτyz∞=0,SIII(1)(z)=πaτxz∞Db(z)=SIII,2D(1)Db(z),which is the same as its orthorhombic, cubic and isotropic counterparts. Eqs. reduce to their two-dimensional counterparts, by taking, as before, Db(z) = 1. It may further be noted that the normalization factor SIII(z)/SIII,2D is equal to Db(z), which is the same as its crack counterpart.The through-thickness average energy release rate due to the rigid inclusion can be derived by employing the equivalent inclusion (more popularly known as the eigenstrain) method due to Eshelby where the eigen-strain, γxz∗, can be expressed in terms of the far-field applied shear strain as follows The total increase of the free energy of the system (the elastic energy in the body and the potential due to the applied loading) can then be written as follows:ΔΠ=-ΔW=πa2τxz∞γxz∞2∫-hh(Db(z))2dz=πa2τxz∞-c45τyz∞+c55τxz∞2c44c55-c452∫-hh(Db(z))2dzThe through-thickness average energy release rate can now be obtained as follows:JIII=-∂Π∂(4ah)=-c44SIII,2D(1)SIII,2D(1)+SIII,2D(2)4hc44c55-c452∫-hh(Db(z))2dz,which is hitherto unavailable in the literature, and is, like its (two-dimensional) isotropic counterpart JIII,ortho=-(SIII,2D)24hc¯44∫-hh(Db(z))2dz,which is identical to its cubic counterpart. For an isotropic material, this reduces further to the following: reduces to its two-dimensional plate (of unit thickness) counterpart, given by Wang et al. It is interesting to point out here that Bigoni et al. Unlike its crack counterpart, the stress singularity coefficient on an anticrack or super-rigid inclusion is non-zero. The non-vanishing asymptotic stress component in the interior of a super-rigid thin inclusion, very close to its front at θ |
= 0, or y |
= 0, can be written, following the procedure outlined by Li and Ting τyz(a-r,0+,z)=Re{c45+(-ξ+iη)c44}ζ3ia2-ζ321/2γxz∞Db(z)=c44c55-c4521/2a2rγxz∞Db(z)The corresponding stress singularity coefficient, SIIIInc, can now be derived as follows:SIIIInc=Limr→0(2πr)1/2τyz(a-r,0+,z)=πac44c55-c4521/2γxz∞Db(z)=πac44c55-c452-1/2c44τxz∞-c45τyz∞Db(z).Forτxz∞=0,SIIIInc(2)(z)=-c45c44c55-c452-1/2πaτyz∞Db(z),which, as in the case of its counterpart ahead of the rigid inclusion front, vanishes in the case of an orthorhombic, cubic or isotropic material.Forτyz∞=0,SIIIInc(1)(z)=c44c44c55-c452-1/2πaτxz∞Db(z), reduce to their two-dimensional counterparts, in which case Db(z) = 1. It may further be noted that the normalization factor SIIIInc(z)/SIII,2DInc is equal to Db(z), which is the same for its crack counterpart.The through-thickness average energy release rate on the super-rigid inclusion can be derived, using an approach which is analogous, in a reverse sense, to Irwin’s JIIIInc=-limΔa→012h(Δa)∫0Δa{τxz(Δa-x)+τyz(Δa-x)}w(x)dx∫-hh(Db(z))2dz,in which τxz(Δa |
− |
x) and τyz(Δa |
− |
x) are obtained by setting θ |
= |
π in Eqs. , respectively, while w(x) is determined by setting θ |
= 0 in Eq. . Finally, JiiiInc can be derived as follows:JIIIInc=-limΔa→01+ξ2hc44ηπΔa∫0ΔaxΔa-xdx∫-hhSiiiz2dz=-Sm,2D(1)Sm,2D(1)+Sm,2D(2)4hc44c55-c4521/2∫-hhDbz2dz,which is unavailable in the literature. Eq. reduces to its hitherto unavailable two-dimensional plate (of unit thickness) counterpart, in which case, 12h∫-hh(Db(z))2dz=1. Again, the 3D/2D normalization factors are identical to their crack counterparts, for a given geometry and applied far-field loading.The through-thickness distribution of the energy release rate on a center-anticrack can also be derived by introducing the thickness-wise partial inverse crack closure method as follows:JIIIInc=-limΔa→01+ξc44ηπΔa∫0ΔaxΔa-xdx∂∂z∫-hzSiiiz2dz=-SIII,2D(1)SIII,2D(1)+SIII,2D(2)4hc44c55-c4521/2Dbz2. supplies the same characteristic equation as in the case of an anticrack. supplies the same characteristic equation as in the case of a through crack. It may be noted here that the interface continuity condition, given by Eq. yield the characteristic equation for a mixed crack/anticrack.The two lowest roots (eigenvalues), contributing to the singular stress field, is given by s |
= 1/4, s′ = 3/4, which are identical to their orthorhombic displays two transverse shear moduli of three categories of commonly used unidirectional fiber reinforced (smeared out) composites—polymer matrix composite (PMC), metal matrix composite (MMC) and ceramic matrix composite (CMC) laminas also shows the dependence of crack deflection on transverse shear moduli ratio, R^=μ23/μ13, and fiber orientation angle, θ¯, in these commonly used unidirectional fiber reinforced composite laminas. More specifically, F^(θ¯)/R^ values for θ¯=π/6,π/4,π/3 and π/2, for these fiber reinforced composites are displayed here. It is clear from Eq. that the stress intensity factor for an infinitesimal kink increases with the fiber orientation angle, θ¯. Eq. further show that for a particular composite lamina (i.e., R^=μ23/μ13 fixed), F^(θ¯)/R^ increases with θ¯, implying increase of K¯III(θ¯,z)/KIII(z). Although the above analysis is based on the assumption of smeared out material properties of a two-phase composite lamina, it still correctly predicts crack deflection in a direction parallel to the fiber direction. Secondly, although this analysis may predict crack branching at an angle, θ¯-π/2, the probability of its uninterrupted propagation is quite low because of the physical reality of the presence of fiber matrix interfacial barrier or obstacle. At the most, such branching would lead to crack bridging by fibers facing a matrix crack; see the discussions on crack bridging in Chaudhuri and Chiu displays transverse shear properties of some common naturally occurring (geological) as well as synthetic monoclinic mono-crystalline plates also depicts the dependence of crack deflection and/or bifurcation on the transverse shear stiffness ratio, c¯44/c¯55, and the principal directions (of orthotropy), θ¯, in these monoclinic mono-crystalline plates. Of the six monoclinic single crystals investigated here, crack bifurcation does not occur only in the case of anthracene. For the remaining five single crystals, a mode III crack oriented in the x-direction would most likely bifurcate into two branches, each of which corresponds to one of the two principal directions of orthotropy. The reason for a crack oriented in the x-direction deflecting but not bifurcating in the case of anthracene is the unusually high value of its transverse shear-to-transverse shear coupling stiffness, c45, which results in the relatively low value of the ratio, c¯44/c¯55, which falls short of F^(θ¯) for θ¯ |
= 57.36°. Here, θ¯ is the angle between the initial direction of crack orientation and the principal direction of orthotropy into which the crack would finally deflect. Here are, in essence, two competing mechanisms in play: (i) c¯44/c¯55 in which the transverse shear-to-transverse shear coupling stiffness, c45, sometimes plays a major role, such as in the case of anthracene, and (ii) the turning angle, θ¯. show variations of the normalized stress intensity factor, KIII∗(z)=KIII(z)/KIII,2D, through the thickness of a monoclinic plate, weakened by a through-crack investigated here. For an infinite (in the x-y plane) plate with a center-crack of length 2a, KIII,2D was first given by Sih et al. shows the through-thickness variation of the stress intensity factor for a far-field symmetrically distributed saw-tooth antiplane shear (mode III) load, while its antisymmetric counterpart responsible for singular residual stress field is displayed in . Such types of results are generally unavailable in the literature. Through-thickness variations of the stress singularity coefficients can be represented in a manner similar to their stress intensity factor counterparts as shown in In regards to the energy release rates for a through-thickness crack, when both GIII,3D and GIII,2D are computed for a plate of unit thickness, the normalization factor GIII,3D/GIII,2D is equal to ∫-1/21/2(Db(z))2dz=1/3, corresponding to both the stress intensity factors shown in due to symmetric and antisymmetric saw-tooth loadings, respectively. The same is true for the normalization factors for energy absorption rates for/on a super-rigid inclusion.Finally, the singular residual stress in a metallic (single crystal) monoclinic plate appears to be screw dislocation-like. This can be interpreted as the core of a screw dislocation getting trapped at the crack front. Since for mono-crystalline silicon, the room temperature is significantly lower than what is needed for brittle-to-ductile transition, such a trapped screw dislocation core would not be able to induce large scale plasticity in its neighborhood. It should be possible to measure the out-of-plane displacement component, w, by cutting a cylindrical piece of solid radially (see e.g., Fig. 3.22 of Courtney a. Although silicon is cubic, it behaves like a monoclinic single crystal for a (110)[11¯0] crack. Further research is needed in this regard, especially its experimental verification.In a fiber reinforced composite material, the singular residual stress field can be high enough to cause cracking within the matrix. This microcracking of the matrix can expose the fibers to degradation by chemical attack A recently introduced eigenfunction expansion technique, based partly on separation of the z-variable and in part, on the Eshelby For a monoclinic single crystal or an off-axis fiber reinforced unidirectional composite lamina, the through (0 1 0)[0 0 1] crack (i.e., oriented at θ |
= 0) considered here is, unlike its isotropic counterpart, more likely to deviate from its original plane (0 1 0) and original propagation direction [1 0 0], because of nonzero c45. It can then be inferred that the (0 1 0)[0 0 1] is not an easy cleavage system for monoclinic single crystals or off-axis fiber reinforced unidirectional composite laminas with c45 |
≠ 0.The stress intensity factor for an infinitesimal kink in a smeared out composite material increases with the fiber orientation angle, θ¯. Although the above analysis is based on the assumption of smeared out material properties of a two-phase composite lamina, it still correctly predicts crack deflection in a direction parallel to the fiber direction.Crack deflection and/or bifurcation in monoclinic mono-crystalline plates depends on the transverse shear stiffness ratio, c¯44/c¯55, and the principal direction (of orthotropy), θ¯. The reason for a crack oriented in the x-direction deflecting but not bifurcating in the case of anthracene is the unusually high value of its transverse shear-to-transverse shear coupling stiffness, c45, which results in the relatively low value of the ratio, c¯44/c¯55, Here are, in essence, two competing mechanisms in play: (i) c¯44/c¯55 in which the transverse shear-to-transverse shear coupling stiffness, c45, sometimes plays a major role, such as in the case of anthracene, and (ii) the turning angle, θ¯.For the special case of orthorhombic single crystals or 0°-laminas, c45 |
= 0, which makes the through (0 1 0)[0 0 1] crack (i.e., oriented at θ |
= 0) considered here, like its isotropic counterpart, more likely to propagate in its original plane (0 1 0) and original propagation direction [1 0 0], because of absence of this type of coupling, which arises from the imaginary roots.A through (0 1 0)[0 0 1] super-rigid lamellar inclusion would be reduced via some damage mechanisms, nucleating a crack which would more likely deviate from the original plane (0 1 0) and original length direction [1 0 0] of the super-rigid lamellar inclusion, because of nonzero c45.The 3D/2D normalization factor, which quantifies the z-dependence, is identical for both crack stress intensity factor and anticrack stress singularity coefficient for a given z-dependence of the far-field loading. The same is true for energy release rate for a crack and energy absorption rates for/on a super-rigid inclusion (anticrack).The energy absorption rate on the super-rigid inclusion is different from its counterpart on the mono-crystalline monoclinic plate or a (smeared-out) off-axis composite lamina.KIIIa and SIIIa, obtained here represent the self-equilibrating stress intensity factor and stress singularity coefficient, respectively, for a through-thickness crack/anticrack, resulting in residual stresses in the anisotropic (monoclinic) plate material. It must be recognized that KIIIa (or SIIIa) cannot be determined by a two-dimensional approach, which yields constant KIIIs (or SIIIs), while both KIIIa (or SIIIa) and KIIIs (or SIIIs), obtained by the present three-dimensional approach, are functions of z.In regards to the energy release rates for a through-thickness crack, when both GIII,3D and GIII,2D are computed for a plate of unit thickness, the normalization factor GIII,3D/GIII,2D is the same (and ≠1) for both the symmetric and antisymmetric saw-tooth loadings investigated here. The same is true for the normalization factors for energy absorption rates for/on a super-rigid inclusion.The singular residual stress in a metallic (single crystal) monoclinic plate appears to be screw dislocation-like. This can be interpreted as the core of a screw dislocation getting trapped at the crack front. The mono-crystalline plate material undergoes high shear deformation in the vicinity of a crack or anticrack in the r–z and r–θ planes, which may cause high shear distortion in the lattice, resulting in both bond scission and bond rotation.In a fiber reinforced composite material, the singular residual stress field can be high enough to cause microcracking within the matrix. In addition, the fibers may themselves be sheared by the action of high transverse shear stress gradients, especially in the presence of out-of-plane waviness of fibers.Similarity or dissimilarity of the present asymptotic solution for a monoclinic plate involving complex roots with its orthorhombic and isotropic counterparts, referred to in Section , is presented here. This type of similitude analysis, extensively employed in fluid mechanics, but not generally taken advantage of in fracture mechanics, may be termed kinematic similitude. As has been demonstrated by Chaudhuri (a) Isotropic/Cubic: |
The transverse displacement for an isotropic material Therefore, for an isotropic material when x |
= 0, ψ |
= ±π/2 for all nonzero values of y.(b) Orthorhombic (Solution Involving Imaginary Roots): |
Referring back to Eqs. in which ρ and ψ can be rewritten as follows:Therefore, for an orthorhombic plate with imaginary roots when x |
= 0, ψ |
= |
± |
π/2 for all nonzero values of y, which is in accord with its isotropic/cubic counterparts.(c) Monoclinic (Solution Involving Complex Roots): |
Going back to Eqs. , the transverse displacement can be rewritten in the asymptotic form (i.e., for n |
= 0):in which ρ and ψ can be rewritten as follows:Therefore, for a monoclinic plate with complex roots when x |
= 0,for all nonzero values of y, which differs from its isotropic, cubic and orthorhombic counterparts. For the special case of an orthorhombic single crystal, ξ |
= 0, which reproduces ψ |
= ∓π/2, as shown above.Sesqui-power scaling of plateau strength of closed-cell foamsExperimentally obtained elastic and strength properties of closed-cell foams under compressive loading are frequently characterized by power-law scaling with the relative density of the material, i.e. the volume fraction of the solid material phase. Hereby, the so-called plateau strength, which may constitute the most vital material parameter for moderate deformation states, obtained from experiments on samples with different overall porosities, but same matrix material and pore topology often exhibits sesqui-power scaling. In this paper, this porosity-property relation of closed-cell foam is assessed by: (i) specialization of classical schemes from continuum micromechanics for high porosities, with the so-called differential scheme resulting in sesqui-power scaling of the effective strength; (ii) applying the closed form solution for plastic buckling of a thin spherical shell within the scopes of a unit-cell think-model for closed-cell foams, also giving the power-law scaling exponent as 3/2.The proper understanding of morphology vs. property relations of foams is vital for their application and tailoring. Foams are characterized by high volume fractions of porosity ϕ, i.e. low volume fraction of solid material matrix fm=1−ϕ. In the past, substantial efforts have been undertaken (see, e.g. following the initial linear elastic regime (for reference see, e.g. Fig. 8 in plastic collapse stress σcr,eff refers to the first local maximum in the stress–strain curve;the so-called foam efficiency E is the current adsorbed energy (per unit volume) scaled by the current stress the densification strain εD is determined at the global maximum of efficiency, i.e. the plateau stress σpl,eff is the time-related average (or strain-related average when the experiment is characterized by a constant strain rate) of the stress between plastic collapse and densification σpl,eff=1t(ε=εD)−t(σ=σcr,eff)∫t(σ=σcr,eff)t(ε=εD)σ(t)dt.Aluminum foam shows a pronounced strain-rate dependency of σcr,eff and σpl,eff (for strain rates exceeding ≈5×103s−1 see, e.g. , classical schemes from continuum micromechanics (Mori-Tanaka scheme However, experimental data for various foams (e.g. a wide range of experimental data is available for aluminum foam, see ) suggest an exponent of p≈q≈3/2 for both strength (in terms of the foam plateau stress) and elastic properties.Recently, the sesqui-power scaling of Young's modulus (in compression) of closed-cell foams as Eeff∝(1−ϕ)3/2 was deduced from the elastic solution for a thin spherical shell , the closed form solution for plastic buckling of a thin spherical shell In this paper, the subscript “m” refers to the matrix material, i.e. the solid material phase in foams; the subscript “eff” is related to the engineering properties of closed-cell foams, i.e. to the behavior at the length scale of engineering application.A review of this classical scheme can be found in shows the prediction of effective elastic properties for the porous material for spherical pores and different values of Young's modulus Em and Poisson's ratio νm of the matrix material. For incompressible matrix behavior, i.e. μm=Em/3, the Mori-Tanaka estimate can be written as (see Eq. (7.62) in μeff=M(ϕ)μm=1−ϕ1+2ϕ/3μmkeff=K(ϕ)μm=4(1−ϕ)3ϕμm.In a double-logarithmic diagram, the slopes for highly-porous materials with ϕ→1 are given as ∂[lnμeff]/∂[ln(1−ϕ)]=5/(3+2ϕ)|ϕ→1=1 and ∂[lnkeff]/∂[ln(1−ϕ)]=(1/ϕ)|ϕ→1=1, irrespective of the value of νm,. Hence, the asymptotic behavior for ϕ→1, relevant for foams, can be written asμeff∝M(ϕ→1)μm=(1−ϕ)μmkeff∝K(ϕ→1)μm=(1−ϕ)μm.For von Mises-type behavior of the matrix material, with cm as the cohesion of the matrix material, the domain of admissible macroscopic stress states is given as (see Eq. (7.70) in where Σvol and Σdev denote the volumetric and deviatoric macroscopic stress, respectively. Setting Σvol=0, the effective cohesive strength ceff for ϕ→1 is obtained, considering Σdev=ceff and M(ϕ)=M(ϕ→1) in Eq. (1−ϕ)∝1M(ϕ→1)(ceffcm)2=1(1−ϕ)(ceffcm)2→ceff∝(1−ϕ)cm.Hence, the Mori-Tanaka scheme specialized for ϕ→1 yields the effective elastic properties as μeff∝(1−ϕ)p, keff∝(1−ϕ)p and the effective cohesive strength as ceff∝(1−ϕ)q, i.e. all showing the same power-law scaling with p=q=1.Note #1: The main assumption for the derivation of the Mori-Tanaka scheme is that in a sufficient distance from the inclusions, i.e. the pores in this paper, the strain field can be approximated by the volumetric average of the strain field over the entire matrix material domain. In other words, there has to be a sufficient distance between the pores (i.e. low porosity), a condition not met in highly-porous foams. We have, however, specialized the Mori-Tanaka scheme for ϕ→1. When considering the results in this subsection, this violation of the limits of applicability of the Mori-Tanaka scheme has to be kept in mind.Note #2: The prediction of (i) the composite sphere model (a)). The effective shear modulus following from the generalized self-consistent scheme is given in (a), showing a slope of p=1 for ϕ→1. For the composite sphere model, bounds for the effective shear modulus are given in (b)) coinciding with the prediction of the Mori-Tanaka scheme. Both the upper and the lower bound are characterized by a slope of p=1 for ϕ→1. Hence, the conclusion drawn for the Mori-Tanaka scheme as regards power-law scaling with q=1 for the effective cohesive strength also applies to these two homogenization schemes.The strain localization tensor for a single ellipsoidal pore embedded in an infinitely extended matrix domain, Aϕ∞ (=constant within the pore), is derived from the Eshelby solution The differential scheme represents an infinitesimal formulation of the dilute distribution estimation. Starting with the homogeneous matrix material Cm, the inclusion phase is embedded in the matrix material in infinitesimal steps dϕ. After each of these steps, the behavior of the matrix phase is updated based on the dilute distribution estimation, which in case of an infinitesimal volume fraction of the inclusion phase is exact. At the (n+1)-st step, the behavior of the matrix material is given as Ceffn(ϕn) (see ). An infinitesimal volume dV is removed from the domain, containing ϕndV of the porosity ϕnV. On the other hand, the same volume dV is replaced by porosity. Hence, the (n+1)-st step is characterized by an increase of the porosity towards ϕn+1=ϕn+dϕ with a volumetric balance for the porosity at the (n+1)-st step reading for the (n+1)-st step of the differential scheme, one getsCeffn+1(ϕn+1)=Ceffn+1(ϕn+dϕ)=Ceffn(ϕn)−dVVCeffn(ϕn):[I−Seffn(ϕn)]−1,with dV/V as the considered porosity increase according to Eq. . The differential equation for determination of the effective material tensor is obtained asdCeffn+1(ϕn+1)dϕ=Ceffn+1−Ceffndϕ=−11−ϕnCeffn(ϕn):[I−Seffn(ϕn)]−1,giving Ceffn+1=Ceffn+dCeffn+1. Using the initial condition Ceff(ϕ=0)=Cm, the numerical solution of Eq. for different values of Young's modulus Em and Poisson's ratio νm for the matrix material is illustrated in . For νm=0.2, implying μm=Em/2/(1+νm)=5/12Em and km=Em/3/(1−2νm)=5/9Em, respectively, a closed form solution of Eq. provides a good estimate for the effective behavior for fm<≈0.2, i.e. ϕ>≈0.8 (see ). For incompressible matrix behavior Eq. Using the reasoning applied previously in the scopes of the Mori-Tanaka scheme, i.e. inserting Eq. and setting Σvol=0 and Σdev=ceff finally givesHence, in the scopes of the differential scheme, for ϕ→1, the elastic response is characterized by power-law scaling with p=2; on the other hand, the effective cohesive strength shows sesqui-power scaling with q=3/2.Note #3: This result for p and q coincides with the power-law scaling previously obtained for open-cell foams (see e.g. (b)). The proportionality factor of 5/2=1.12 in the sesqui-power scaling law (Eq. ) is, however, significantly larger than the one given in (a) and (b)) has been treated since the late 1950s, see e.g. Recently, the scaling of Young's modulus (in compression) of closed-cell foams with the sesqui-power of the volume fraction of solid material phase Eeff∝fm3/2Em=(1−ϕ)3/2Em, observed in various experiments on closed-cell foams (see e.g. ), was rationed by employing the elastic solution for the compression of a thin spherical shell , i.e. inserting M(ϕ→1)=(1−ϕ)3/2 into Eq. However, note that the continuum micromechanics based approach for transition from elastic towards plastic material behavior cannot capture buckling-type inelastic behavior as depicted in 1st stage. Flattening of the portion of the shell in contact with the rigid plate.(b)); the cap area undergoes isometric transformation characterized by Gaussian curvature not changing upon transformation (and no associated membrane effects).3rd stage. Non-symmetric buckling of dimple with formation of changing number of lobes.where σy,m denotes the yield stress of the solid material phase and h the shell thickness. Note that the load–displacement relation is independent on the radius of the shell. According to may only provide an approximate solution for displacement δ in the order of the shell thickness. In This geometric/material parameter implies elastic buckling for ξ<1, i.e. the inverted cup remains in the elastic state. On the other hand, for ξ⪢1 a rigid-plastic analysis (as employed in ) is sufficient to characterize the deformation state of the shell Taking the considered spherical half-shell as unit-cell think-model for closed cell foams under uniaxial loading (see (c)), with a volume of the unit cell of V=area×height=(2R)2×R=4R3, the volume fraction of solid phase fm=(1−ϕ) is given aswhere the volume Vm of the thin half-shell (h⪯¡R) is determined from its surface 2πR2. The sought-for effective material with the effective yield stress σy,eff is subjected to the same axial compressive loading conditions (see The effective material is characterized by the same deformation as the considered thin half shell, i.e. inserting the applied force F from Eq. σy,eff=32π(δR)1/2(hR)3/2σy,m=6/πεp1/2fm3/2σy,m=6/πεp1/2(1−ϕ)3/2σy,m,hence, sesqui-power scaling with fm, where the effective plastic strain in the direction of loading, εp=δ/R and h/R=2/πfm (see Eq. For aluminum foam (and probably other metallic foams), the use of a rigid-plastic material model for determination of the sesqui-power scaling relation (Eq. ) is justified by specifying the diagnostic parameter ξ for the solid material phase aluminum as ξ=(h/R)/[(1−νm)σy,m/Em]=(2/πfm)/[(1−νm)σy,m/Em]=(2/π×0.03)/[(1−0.33)×95/70000]=21⪢1, where fm≈0.03 was set to the smallest volume fraction encountered in literature data (), the use of a rigid-plastic material model provides an approximation as ξ=(2/π×0.025)/[(1−0.33)×42/3300]=1.9>1, where fm≈0.025 was set to the smallest volume fraction encountered in the experiments.Note #4: The upper limit for applicability of Eq. were determined at εp=0.002 and εp=0.05, respectively (for the full range of different densities investigated in σpl,eff=6/π0.05fm3/2σy,m=0.31fm3/2σy,m=0.31(1−ϕ)3/2σy,m,with the proportionality factor of 0.31 remarkably close to the one given in (b), 0.54 for small-cell foam, 0.17 for large-cell foam).When considering different types of unit-cells representing open-cell foams, evoking various plastic failure mechanisms when loaded by a far-field stress σy,eff (see , however neglecting buckling), the previously obtained sesqui-power scaling lies within the bounds set by these unit cells, 1<q=3/2<2. Scaling for the bending-dominated strut-based unit cell, q=3/2, coincides with scaling obtained for the thin-spherical shell based unit cell. Bending-dominated strut-based unit cells have previously been used to model open-cell foams (see, e.g. When, on the other hand, considering unit cells loaded by the far-field stress σpr,eff, triggering elastic buckling-type failure (see ), the obtained scaling for initiation of inelastic material response is of power-law type with q=2 for a strut-based microstructure and q=3 for plate-based microstructure. Considering the experimental findings, i.e. q=3/2 (see, e.g. the proportionality limit σpr,eff in may not be relevant for the investigated foams.In this paper power-law type models for the effective behavior of closed-cell foams were sought for / investigated. In the scope of these models, effective elastic properties, i.e. effective shear and bulk moduli μeff and keff, are written as a power-law functions of the moduli of the matrix material μm and km, the solid volume fraction in foam (1−ϕ), with ϕ denoting the porosity, and the exponent p asOn the other hand, the exponent q characterizes the power-law function relating the effective foam strength, i.e. the so-called plateau stress σpl,eff to the yield stress of the matrix material σy,mclassical schemes from continuum micromechanics, i.e. the Mori-Tanaka scheme and the differential scheme were specialized for a matrix/pore morphology and high porosities, leading to power law-scaling relations with p=q=1 for the Mori-Tanaka scheme and p=2, q=3/2 for the differential scheme;the closed-form solution for plastic buckling of a thin spherical shell was employed in the scopes of a unit-cell model for closed-cell foam, resulting in sesqui-power scaling of the effective plateau stress, i.e. q=3/2. The same unit-cell model has previously been employed to derive the sesqui power-law scaling relation for closed-cell foam elasticity, i.e. p=3/2Most experimental campaigns on closed-cell foams showed power-law scaling with exponents of p≈q≈3/2 for both strength and elastic properties. Hence, the unit-cell approach employing the closed-form solution for plastic buckling of a thin spherical shell seems to represent data best within the investigated range of models. One may expect that the obtained sesqui-power scaling relates to realistic foam behavior when the in-situ topology of the cells comes close to the spherical, thin-walled shape as assumed for the definition of the underlying unit cell.On the other hand, the investigated homogenization schemes based on continuum micromechanics do not seem to capture actual foam behavior as they have been developed (and successfully applied, see, e.g. This appendix contains a compilation of experimental data (from the open literature for the case of aluminum foam and by the authors for the case of expanded polystyrene foam) used for model validation, i.e. experimental validation of the power-law scaling of strength (and stiffness).In recent years, the research efforts relating to the understanding of closed-cell metal foams have intensified (see, e.g. shows the experimentally obtained scaling of plastic collapse stress and plateau stress of closed cell aluminum foam with the sesqui power of the volume fraction of the solid material matrix fm3/2). Whereas the strength data in the latter references are characterized by sesqui-power scaling, Young's modulus is rather scaled by p≈2. A similar conclusion was drawn from the data presented in Highly-densified expanded polystyrene (HD-EPS) foam were obtained from compression tests on the source material and on HD-EPS with ρeff≈100 and ≈400kg/m3, respectively. When underlying a density of ρm=1050kg/m3 of the solid material phase in the composite (polystyrene), the volume fraction of the solid material phase, fm=ρeff/ρm, is given as ≈0.025 for the source material, and ≈0.095 and ≈0.38, respectively, for HD-EPS. Uniaxial compression tests were conducted on prismatic specimens (141 samples with width/height=25/50 mm) with a displacement rate of 5 mm/min. Whereas EPS and HD-EPS with fm≈0.095 show a pronounced plateau and densification, HD-EPS with fm≈0.38 is characterized by a shorter plateau, lack of densification, and shear-type failure (see ). This behavior is also reflected by the efficiency and the related densification strain, with the latter only well-defined for EPS and HD-EPS with fm≈0.095 (see ). As opposed to aluminum foam, a distinct plastic collapse stress (local maximum in the stress–strain curve) is not observable in the experimental data (see ). This may be due to the, at least, two-orders-of-magnitude smaller characteristic cell-size (see Fig. 1(a) in Young's modulus of the foam Eeff (MPa) was determined by linear regression of the steepest part of the (initial part of the) stress–strain curves. Underlying elasto-plastic material behavior, i.e. unloading with a slope of Eeff in the stress–strain curve (see ), the proportionality limit σpr,eff was determined for a plastic strain of εp=0.002. As there is no distinct plastic collapse stress and the densification strain only allocable for fm≈0.025 and fm≈0.095, the plateau stress σpl,eff for expanded polystyrene foam in was related to εp=0.05, ensuring that the location of σpl,eff is found within the plateau region of the stress–strain curves (see Acoustic analysis in friction stir welding of 6082 - T6 aluminium alloy platesAcoustic analysis has been performed on the sound recorded during friction stir welding of AA 6082 – T6 aluminium alloy plates. The experiments were performed by taking three process parameters viz., Speed of tool rotation, Traverse speed (Feed), Tool tilt angle. Six experiments were performed by varying the process parameters and sound generated during the welding process is recorded. Frequency – domain analysis was performed on the sound recorded and peak magnitudes were noted. Regression analysis was performed between peak magnitude and Ultimate Tensile Strength (UTS) values of all experimental conditions. From the two graphs plotted between experiment number versus Peak magnitude and experiment number versus UTS it was shown that as the peak magnitude values increase the UTS also increases and it was confirmed from the coefficient of the peak magnitude in the regression equation. The fits as predicted from the regression equation yielded minute residuals and were able to predict the UTS values with good accuracy. A straight line was fitted in the Time – Domain graph obtained from the recorded sound which was half wave rectified and sample points taken for every 5 s. Regression analysis performed on coefficients obtained from straight line equations for all experiments and Feed values show that the Feed values of all experimental conditions can be predicted to high degree of accuracy.Friction Stir Welding (FSW) is a solid state joining process in which a non consumable rotating tool with a specially designed pin and shoulder is inserted into the abutting sheets or plates to be joined and subsequently traversed along the joint line resulting a weld in no melting occurs AA 6082 – T6 Aluminium alloy plates of dimensions 150 × 75 × 5 mm were welded in butt configuration using Friction Stir Welding process using tool having straight cylindrical pin with hemispherical end and with a concave shoulder. Three process parameters chosen are Speed of rotation of the tool, traverse speed and tool tilt angle. The combination of process parameters used for experimental conditions are given in A contact type microphone which is attached to the bed of the milling machine adjacent to fixed plates is used to record the sound generated during the process. The sound was recorded using ‘AUDACITY’ a free digital audio editing and recording application software. The sound was recorded at a sample rate of 48 kHz.Tensile Tests were performed on the friction stir welded plates and the average tensile strengths (Ultimate) were given in Frequency domain analysis is performed on the recorded audio signal. While time – domain analysis shows variation of signal over time, frequency – domain analysis shows how signal’s energy is distributed over a range of frequencies FFT is performed on recorded audio signals and frequency vs magnitude graphs are plotted In each of the frequency domain graphs peak magnitudes and corresponding frequencies are noted and tabulated (see , one by taking experiment number on X – axis and peak magnitude on Y – axis and the secondary graph between experiment number on X – axis and UTS on Y – axis. By comparing the two graphs, it is evident that as the peak magnitude increases the ultimate tensile strength also increases.Regression analysis is performed by taking peak magnitude as predictor and UTS as response. The Analysis of Variance for the Regression is given in the The P value for the term peak magnitude is less than 0.05 indicating the association between the predictor (peak magnitude) and the response (UTS) statistically significant.The coefficient of the predictor (Peak Magnitude) is 0.000309. The average UTS of friction stir welded specimens increases by approximately 0.000309 for every 1 unit increase in Peak Magnitude. The sign of the coefficient of the predictor is positive which indicates as the peak magnitude increases so does the UTS.A single point in the bottom right corner is away from the fitted line plot may be an indication of error in that particular experiment condition. As the fitted line plot is between peak magnitude and UTS, any variables governing the predictor or response such as variation in the sound generated during the welding process due to external factors (vibrations generated due to sudden drop of any object near to the machine may be picked up by the recorder) or variation in speed, traverse or even depth of plunge which may be of manual errors or improper base plate might have caused the location of the single point away from the fitted line.The predicted values of UTS by using regression equation and corresponding residuals are given in A straight line in the normal probability plot indicates the residuals are normally distributed and the randomness and lack of patterns in the versus order plot b indicates that the residuals are independent from one another.The recorded sound was loaded in the MATLAB software and time domain graph (waveform) was generated . The waveform is then passed through high pass filter (Butterworth). Sample points were taken for every 5 s interval to make out the profile of the graph.If a straight line is fitted through the sample points taken in the time domain graph, the straight line equation will beThe terms p1 and p2 are coefficients of the equation of the fitted line. The coefficients for lines fitted to the time domain graphs of all the experiments are tabulated in The P value for the term peak magnitude is less than 0.05 indicating the association between the predictors (Coefficients: p1 & p2) and the response (Feed) statistically significant (see The predicted values of Feed by using regression equation and corresponding residuals are given in the The values predicted from the regression equation are closer to the actual values indicating that the equation is a good fit.The peak magnitude obtained from Frequency – Domain graph, increases with increase in UTS of the sample for that experimental condition.The regression equation generated between Peak magnitude and UTS is able to predict the ultimate tensile strength of the Friction Stir welded plate if the predictor variable (peak magnitude) is given. The R- Sq value of the regression analysis for Peak magnitude versus UTS is 96% indicates the model is a good fit and it is proved by lower residuals as predicted from regression equation.A curve (straight line) is fitted through the sample points in the Time – Domain graph (sample taken for every 5 s) and coefficients are obtained from the straight line equation. When regression analysis is performed on Feed versus Coefficients data, the R- Sq value for the analysis is 98% which is a very good fit and is proved by the low residual values. The regression equation between Feed and Coefficients can be used to predict the feed of a given experimental condition from the recorded sound.Discontinuous deformation analysis based on strain-rotation decompositionThe S-R (strain-rotation) decomposition theorem has an ability to capture strain components and rotation components at the same time. Using the principle of virtual power (VP), in this study, a new formulation independent of specific numerical methods is proposed for the analysis of dynamic large or small deformation. Then, the formulation is applied to the discontinuous deformation analysis (DDA), yielding a new DDA based on the S-R decomposition theorem, abbreviated as SRDDAvp. Compared with the conventional DDA, SRDDAvp adopts a slightly modified basic variables together with the generalized-α method. The analysis of some typical examples indicates that SRDDAvp can naturally overcome the issue of volume expansion, effectively improve the calculation accuracy and, equip DDA with the potential to treat large deformation.The discontinuous deformation analysis (DDA) is a discrete block-based method. In both 2D-DDA and 3D-DDA, the special shape functions and basic variables are employed to make the approximation of displacement field is independent of the shape of block. The effectiveness of DDA in geotechnical problems has been recognized, and extensively applied in the analysis of seismic landslidesDuring the past 20 years, the performance of DDA is enhanced largely. The higher-order DDA improved the deformability of objects simulated by DDA. The post-adjustment method, the displacement-strain modification method overcame the volume expansion of block due to small deformation assumption, and a procedure to mitigate the elastic distortions with large rotation. Some convergence criterions, and the strategy of strengthening the movement trend speeded up the open-close iteration. The augmented Lagrange multiplier method improved the accuracy of contact force. The one temporary spring method handled the indeterminacy of vertex-vertex contact. For 3D-DDA, the contact sub-matrices modified the stiffness matrix. The models of point-to-face and edge-to-edge contact dealt with the various contacts. An algorithm coped with the frictionless vertex-to-face contacts. Another algorithm searched and calculated geometrical contacts. A fast algorithm identified the common plane. A multi-shell cover algorithm detected contacts. A nodal-based 3D-DDA was developed. Moreover, the new contact theory developed by Shi is expected to significantly simplify the difficulties in treating three-dimensional singular contacts.It is worth mentioning that the S-R decomposition theorem is an important result in the field of geometric nonlinearity. By this theorem, the strain and local rotation can be simultaneously and accurately captured. However, a dynamic formulation based on this theorem remains absent. In this study, using the principle of virtual power (VP), a new formulation for dynamic analysis is firstly deduced. The S-R-D-based formulation is independent of specific numerical methods. In other words, it provides an opportunity to develop DDA under the background of the new theory, in which the small strain assumption is no longer needed. Compared with the conventional DDA, a slightly modified displacement function and the generalized-α method are utilized in the S-R-D-based DDA, abbreviated by SRDDAvp, in which the subscript “vp” stands for the principle of virtual power. The results obtained show that SRDDAvp can naturally overcome the issue of volume expansion, effectively improve the calculation accuracy and, equip DDA with the potential to treat large deformation.The S-R decomposition theorem is always associated with the co-moving coordinate description method. The connection between the theorem and the co-moving coordinate has been demonstrated and illuminated in. Here, for completeness, we only touch upon the related concepts and theories.For a deformable body in Euclidean space E3, the following two reference frames are chosen to describe the motion of a body:A global reference system {Xi} (i=1, 2, 3), which is fixed in space.A co-moving coordinate system {xi} (i=1, 2, 3), which is embedded in the deformable body, with its coordinate line allowed to stretch and rotate.In general, the initial reference frame or the initial co-moving coordinate system is chosen as a rectilinear or curvilinear orthogonal system. However, owing to the occurrence of deformation of the considered body, a new curvilinear system may be formed following the deformation. shows the configuration change of a co-moving coordinate system in the two-dimensional case. The situation in the three-dimensional case is similar. Let r and R be the position vectors of a point before and after deformation, and u the displacement vector. Then, the three vectors have the relationshipWe define the basis vectors at a point in the initial co-moving coordinate system byAfter deformation, the basis vectors at the same point change toIn the curvilinear system, any vector can be decomposed with respect to the basis vector gi0. For the displacement u, we haveThen, the following transformation of basis vectors can be obtainedwhere Fij is a linear differential transformation function and can be described aswhere δij is the Kronecker-delta. The covariant derivative uj|i of displacement is expressed aswhere Γikj0 is known as the Christoffel symbol of the second kindΓikj0=12gjl0(∂g0li∂xk+∂g0lk∂xi−∂g0ik∂xl).It should be pointed out that gi0 and gi represent two very important local basis vectors; the stretch and rotation of a deformable body are reflected precisely through the transformation of these vectors.On the other hand, the S-R decomposition theorem states that any invertible linear differential transformation function F yields a unique additive decomposition:where S is a symmetry sub-transformation representing the strain tensor and is positive definite and is called Chen strain, and R is an orthogonal sub-transformation representing the local mean rotation tensor.where Lji is the unit vector of the rotation axis, and uj|i is the displacement gradient. The superscript T denotes the transpose, and the notation “|i” represents the covariant derivative with respect to gi0. And Lji can be written asThe mean rotation angle θ is determined by the following formulasinθ=12(u1|2−u1|2T)2+(u2|3−u2|3T)2+(u1|3−u1|3T)2.In addition, the strain rate Ṡji can be written aswhere Vi‖j is the velocity gradient, and the notation “‖i” represents the covariant derivative with respect to gi, in order to distinguish it from “|i”. It should be noted that, in accordance with the theory of tensor analysis, the corresponding physical components should be adopted in the calculation.Based on the S-R decomposition, the principle of virtual power can be applied to establish the incremental governing equation, with respect to the current configuration. Assuming that the solutions for the static and kinematic variables have been obtained from time 0 up to time t inclusively, and that the subsequent solution for time t+Δt is now targeted. The procedure to obtain the solution for the next required equilibrium position is representative, and can be carried out repetitively until the final state is achieved. At time t+Δt, the principle of virtual power for the deformable body can be expressed by the following equation with respect to the current configuration∫Ωt+Δtσjit+Δtδ(Ṡijt+Δt)dΩ+Winet+Δt+Wpent+Δt−Wextt+Δt=0,where the first term represents the virtual power corresponding to the inner force, and Winet+Δt, Wpent+Δt and Wextt+Δt are the virtual powers of inertia force, constraint force of specified displacement, and the surface and body force, respectively. Ω is the domain of integration. In addition, σji is the stress, and δṠij is the virtual strain rate, with the definitions,where ρ is the material density, Ai is the acceleration, δVi is the virtual velocity, and ki is the penalty number; Δui and Δu˘i represent the undetermined and specified displacement increments, respectively; P˘i is the specified traction, and fi is the force per unit volume. Throughout this paper, the superscript t+Δt implies that the representation refers to the configuration of time t+Δt. It should be noted that Eq. is now specified with respect to the co-moving coordinate system git+Δt.In the incremental interval Δt between time t and time t+Δt, we takewhere Δσji, ΔṠji and ΔV are the undetermined stress increment, strain rate increment and velocity increment, respectively. Linearizing the stress increment yieldswhere σ̇ji is the unknown stress rate in the incremental interval Δt. Thus, Eq. For time t+Δt, the values of all variables have been known at time t, namely,In this study, ρ, ki, and fi are all treated as constant. At an arbitrary time, the specified stress and displacement boundary conditions might be known; however, the configuration of time t+Δt is unknown. Therefore, the following approximation can be employed, and using Eq. Wpent+Δt≈Wpent=∫Γuki(Δui−Δtu˘i)δ(ΔVi)dS.Wextt+Δt≈Wextt=∫ΓPP˘itδ(ΔVi)dS+∫Ωtρfiδ(ΔVi)dΩ,∫Ωtσjitδ(ΔṠij)dΩ+Δt∫Ωtσ̇jiδ(ΔṠij)dΩ+Winet+Δt+Wpent−Wextt=0.This is the so-called the incremental governing equation, where the superscript t means that the description is with respect to the configuration of time t. And Eq. indicates that the virtual work equation Eq. with respect to the configuration of time t+Δt has been transformed to that refer to the configuration of time t. ΔṠij and Winet+Δt are the undetermined variables and will be disposed next.In order to establish the updated co-moving coordinate formulation of the incremental governing equation, the initial co-moving system git0 of time t is chosen as the reference frame of the co-moving coordinate system git at time t. There are three the main purposes of this choice: into an equation with regard to the initial co-moving system git0 at time t.By using the results of the fourth section inFTT=∫Ωtσ¯jiδ(ΔṠ¯ij)dΩ+Δt∫ΩtD¯jlikṠ¯klδ(ΔṠ¯ij)dΩ,where D¯jkil and ΔṠ¯kl are the material tensor and the unknown strain rate increment in the interval Δt with respect to git0, respectively. The bar “−” over a variable indicates that the variable refers to git0.For time t, the velocity vector referring to the co-moving coordinate systems git and git0 can be expressed asSimilarly, the velocity increment vector is given asFor time t+Δt, with respect to the co-moving coordinate systems git+Δt and git+Δt0, which is the initial co-moving system of git+Δt, we can obtainwhere the double bar “=” over a variable indicates that the variable refers to git+Δt0 in order to distinguish with the variable referring to git0. At time t+Δt, on the other hand, the acceleration vectorA=Ait+Δtgit+Δt=(∂Vi∂t+VjVi‖j)t+Δtgit+Δt., owing to fact that the space derivative is not involved, we haveFurthermore, for the second term of Eq. A=Ait+Δtgit+Δt=(∂Vi∂t+VjVi‖j)t+Δtgit+Δt=(∂V¯¯i∂t+V¯¯jV¯¯i|j)git+Δt0=A¯¯it+Δtgi0., the virtual powers of the inertia force Winet+Δt, in reference to git+Δt0, can be expressed asWinet+Δt=W¯¯ine=∫Ωt+Δtρ(∂V¯¯i∂t+V¯¯jV¯¯i|j)δ(ΔV¯¯i)dΩ.With respect to git0 and git+Δt0, the velocity increment vectors can be written asBecause git0 and git+Δt0 are isomorphic, namely,Winet+Δt=W¯¯ine=∫Ωt+Δtρ(∂V¯¯i∂t+V¯¯jV¯¯i|j)δ(ΔV¯i)dΩ.Wextt=W¯ext=∫ΓPP˘itδ(ΔV¯i)dS+∫Ωρfiδ(ΔV¯i)dΩ,where W¯pen and W¯ext are both expressed with respect to the co-moving coordinate system git0. Hence, the incremental governing equation Eq. ∫Ωtσ¯jiδ(ΔṠ¯ij)dΩ+Δt∫ΩtD¯jlikṠ¯klδ(ΔṠ¯ij)dΩ+W¯¯ine+W¯pen−W¯ext=0. exactly express the new formulation, which is based on the S-R decomposition theorem and is described in the updated co-moving coordinate form. The equations will be implemented further in the context of DDA. It should be pointed out that the rotation tensor R does not appear explicitly, and that the inertia item is primarily considered. The above two points are the main differences from the counterpart employed. Moreover, a dynamic analysis cannot be achieved despite the insertion of the inertia item into the static formulation. That is, there seems to be no shortcut to the dynamic formulation.Providing theoretical basis for updating the co-moving coordinate and state variables of material point.In solving the incremental governing equation, the initial co-moving system of each incremental step is required to be reselected and constantly changing. That is, from time t to time t+Δt, the initial co-moving system is given by git0 at time t; whereas, from time t+Δt to time t+Δt+Δt, the initial co-moving system is defined by git+Δt0 at time t+Δt, as shown in . For the case that these initial co-moving systems are all selected to be isomorphic to the rectilinear orthogonal coordinate system that is fixed in the space, a formula for updating the co-moving coordinate of material point has been given bywhere Δui0 is the displacement increment in the git0 system during Δt. As for the stress, the following formula can be adopted:σ¯jit+Δt=σ¯jit+Δtσ̇¯ji=σ¯jit+ΔtD¯jlikΔṠ¯kl.It is worth mentioning that σ̇¯ji is also an objective stress rate in the git0 system. For an isotropic material, D¯jkil=Djkil was proved inDue to the above-mentioned property, when isomorphic coordinate systems are chosen, the Christoffel symbol of the second kind, Γikj0, will vanish from Eq. Therefore, the related deduction and calculation can be simplified considerably. From here on, Γikj0 will no longer be required, unless otherwise noted.Now we start to construct SRDDAvp. In order to facilitate the description, the bar “−” and the double bar “=” over some variables are omitted from this point on, unless otherwise noted. The first-order displacement approximation is adopted for any block. In this study, the following shape function is usedT(x,y)=[10y0−yx−x00y−y0201x−x00y−y0x−x02].Considering an arbitrary shape block B, for any point (x,y) inside the block B, the displacement u can be expressed aswhere db={u,v,θ,εx,εy,γxy}T is the generalized displacement vector of the block. u and v represent translational components of the block. While θ is corresponding to the mean rotation angle in S-R decomposition theorem. εx, εy and γxy are the three Cauchy strain components. Further, the increment displacement Δu can be described aswhere Δdb={Δu,Δv,Δθ,Δεx,Δεy,Δγxy}T is the generalized increment displacement vector of the block. The velocity V can be written aswhere Vb={u̇,v̇,θ̇,ε̇x,ε̇y,γ̇xy}T is the generalized velocity vector of the block. Moreover, the acceleration A can be expressed aswhere Ab={ü,v̈,θ̈,ε̈x,ε̈y,γ̈xy}T is the generalized acceleration vector of the block. On the other hand, to represent the stress and strain of any point inside of the block, the vectors σ(x,y)={σ11,σ22,σ21}T and S(x,y)={S11,S22,2S21}T(refer to Eq. ) can be employed, respectively. The strain rate vector of any point can be expressed as (refer to Eq. and the strain rate increment vector can be written asNoticing the arbitrariness of δ(ΔVb), Eq. can be recast in the following matrix format∫ΩBbTσdΩ+Δt∫ΩBbTDBbVbdΩ+∫ΩρTTTAbdΩ︸Dynamicterms+Wpen−Wext=0,where Δu˘, P˘ and f are the specified increment displacement, specified traction, and force per unit volume, respectively. The penalty matrix k isIt should be noted that the definitions of the stress and the velocity must be employed, with respect to the co-moving coordinate system. Next, the disposition of the dynamic terms in Eq. will be explained based on the generalized-α method, which is an implicit method for dynamic analysis. The velocities and accelerations of the Newmark format at the end of time t+Δt are as follows:The generalized mid-point velocities and accelerations are given byVbt+Δt−αf=(1−αf)γβΔtΔdb−(1−αf)γ−ββVbt−(1−αf)(γ−2β)2βΔttAb,Abt+Δt−αm=1−αmβ(Δt)2Δdb−1−αmβΔtVbt−1−αm−2β2βtAb,where β, γ, αf and αm are the algorithmic parameters and the relationship between them are as follows:After some mathematical manipulations, the incremental governing equation of one block can be written aswhere Kb, Mb and Fb are the stiffness matrix, the mass matrix and the equivalent force vector of the block, respectively. As for the other matrices, such as matrices of normal contact and shear contact and friction force, they are similar to those given by. Once these matrices are obtained, the global control equation can easily be assembled. Up to now, SRDDAvp is established.It should be pointed out that SRDDAvp possesses the ability to capture the deformation and rotation simultaneously, which inherits from the S-R decomposition theorem. Exactly due to this ability, SRDDA can naturally eliminate the volume expansion of blocks.In this section, several classical tests are analyzed to validate the availability and potential of SRDDAvp. In this study, DDA0 signifies the original DDA, DDA1 denotes the enhanced DDA by post-adjustment method to eliminate the volume expansion. It should be pointed out that Eigen, which is a C++ template library for linear algebra, is used to solve the governing equation., Block1 is a long rod with a length of 9.90 m and a high of h=0.05 m; Block2 is a quadrate and its length of side is l=0.20 m. Point A serves as the center of rotation, point B is the centroid of Block2, and the distance between points A and B is given by L=10.00 m. Block1 and Block2 are connected at point B by contact springs with a stiffness of 0.20×1011 MN/m.The simple pendulum falls from a horizontal position. In this example, the gravity of Block2 is the only external force and the mass of the Block1 is not considered. Let the time step size Δ=0.001 s, the acceleration of gravity g |
=–10 m/s2, Young's modulus E =0.20×1011 |
MPa, Poisson's ratio υ=0.25, the penalty parameter P=0.20×1011 |
MPa, and the spectral radius of the generalized-α methodρ∞=1. The total number of calculation steps is 6000, and the open-close iterations are not performed during the simulation. The trajectories of point B are shown in . It should be emphasized that compared with DDA1, the accuracy of the maximum angular velocity of the simple pendulum is improved obviously by SRDDAvp (see ), especially for the motion from right to left. The relative error is reduced to RE=0.1346% from RE=0.2774%, implying that the maximum angular velocity given by SRDDAvp is closer to the theoretical solution.The configuration of a slender rod is shown in . The length and high of the rod are L=10.00 m, h=0.10 m, respectively. Point A is the center of rotation, and point B is the centroid of rod. The slender rod falls freely from a horizontal position. Let the time step size Δ=0.001 s, the acceleration of gravity g =–10 m/s2, Young's modulus E =0.20×1011 |
MPa, Poisson's ratio υ=0.25, the penalty parameter P=0.20×1011 MN/m. And the spectral radius of the generalized-α methodρ∞=1. Under the action of gravity the slender rod starts fall from a horizontal position. The total number of calculation steps is 5000. The trajectories of points B and C are shown in , the accuracy of maximum angular velocity of the slender rod is enhanced significantly by SRDDAvp, especially for the movement from right to left. The relative error is reduced to RE=0.1608% from RE=0.2834%. It would be more meaningful for long time simulation.Now, the propagation of a sine wave is used to verify SRDDAvp. A bar-spring structure () serves as the medium. The bar-spring structure consists of 40 bars and 39 springs. For each bar, the length is given by l=0.25 m and the height is h=0.10 m, respectively. The distance is L=10.00 m between points A and B. Let the time step length Δ=0.005 s, Young's modulus E =0.20×105 |
MPa, Poisson's ratio υ=0.25, the stiffness of springs k=200 MN/m, the penalty parameter P=200 MN/m, and the spectral radius of the generalized-α methodρ∞=1. The total number of calculation steps is 900. Moreover, the weights of bars are ignored and point B is always fixed in the horizontal direction during simulation. The following displacement is assigned to point A.where n is the current calculation step. Eq. indicates that the amplitude A =0.25 and the period T=300 CS of the sine wave. Some results are shown in , DDA0, DDA1 and SRDDAvp all can obtain the accurate period T=300 CS; while at CS =600, see , only SRDDAvp can give the accurate period, namely, T1=299 CS for the first period and T2=302 CS for the second period. For DDA0 and DDA1, the total number of calculation steps corresponding to two periods is shortened to about 573 CS. At CS =900, see , the distinction becomes clearer between the periods given by DDA0, DDA1 and SRDDA. Especially for the first period, the periods obtained by DDA0, DDA1 and SRDDAvp are T1=270 CS, 275 CS and 301 CS, respectively. In general, only for the third period, DDA0 and DDA1 can obtain the acceptable period T3=301 CS and 304 CS, respectively. While, for the all three periods, SRDDAvp can always give the satisfactory period, namely T1=301 CS, T2=302 CS and T3=301 CS. The shorter period means the faster velocity, thus, the velocity of propagation of the sine wave is increased by DDA0 and DDA1.A nunchaku is consisted of two sticks AB1 and B2C, and for each stick the length is L =1.00 m and the high is h =0.10 m, respectively, as shown in . Point A is always fixed in double directions and the stiffness of fixed springs is given by k=0.20×1015 N/m. Points B1, B2 and C are chosen as the three checking points. Moreover, Young's modulus E =0.20×1011 |
Pa, Poisson's ratio υ =0.45, the density of material ρ =2500 kg/m3, and the acceleration of gravity g =−10 m/s2, and the spectral radius of the generalized-α method ρ∞=1. Let the time step length to be Δ=0.002 s, and the total calculation step is 1800. Under the action of gravity the nunchaku falls from a horizontal position.In this example, during the course of the simulation, we want to achieve a scenario that points B1 and B2 are always coincide with each other at any instant. Namely, there is the following constrain between points B1 and B2 at each calculation step.(ΔuB1xΔuB1y)−(ΔuB2xΔuB2y)=TAB1(xB1,yB1)ΔdAB1−TB2C(xB2,yB2)ΔdB2C=(00),where (ΔuB1x,ΔuB1y) and (ΔuB2x,ΔuB2y) the increment displacements of points B1 and B2, respectively. And ΔdAB1 and ΔdB2C are the generalized displacement vectors of the sticks AB1 and B1C, respectively. Additionally, TAB1(xB1,yB1) and TB2C(xB2,yB2) are the shape functions corresponding to the sticks AB1 and B1C, respectively (refer to Eq. ). Introduction of the Lagrange multipliers λ1 and λ2 yields[A12×12T˜12×2TT˜2×1202×2][Δd˜12×1λ2×1]=[F˜12×102×1],As for the matrixes A12×12 and F˜12×1 one can be refer to Eq. (a), it is apparent that the trajectories of points B1 (the blue solid line) and B2 (the red solid line) are coincidence; while there is an offset between two trajectories corresponding to the to-and-fro movement of the nunchaku. Now, let us see (b), for point B1 there is not an offset; however, the trajectories of points B1 (the blue solid line) and B2 (the red solid line) do not overlap, this phenomenon is against the control equation Eq. (c) it cannot be observed that the separation and offset associated with points B1 and B2.The some distances between points B1 and B2 obtained by DDA0, DDA1 and SRDDAvp are list in , as we can see, for DDA0 and SRDDAvp the distance between points B1 and B2 are always equal to zero, this fully complies with the governing equation Eq. . However, for DDA1 the distance is lengthening gradually even though that the angular velocity of the nunchaku is fluctuant and completely regardless of the constraint of Lagrange multipliers on them. shows further the trajectories of point C obtained by DDA0, DDA1 and SRDDAvp. The difference between them is easily observed.A model test example, which is to be conducted, is designed to demonstrate the capability of SRDDAvp to treat large rotation. The model configuration is shown in . On a portion of a rocky slope, a stone, with an initial velocity V0=(2.50 m/s, 0)and angular velocity ω0=-1.00 s–1, is falling under the action of gravity. The first length L1=1.00 m, the second length L2=2.00 m, the third length L3=3.00 m and the forth length L4=5.00 m. The three slope angles are α=60°, β=45° and γ=15°, respectively.Points P3, P4, and P5 are fixed in double directions. Points P1 and P2 are the centroid and a vertex of the stone, respectively. Let the maximum allowable step displacement ratio to be 0.001, the time step size Δ=0.0005 s, Young's modulus E =0.10×108 |
MPa, Poisson's ratio υ=0.35, the acceleration of gravity g =–10 m/s2, the spectral radius of the generalized-α methodρ∞=1 and the penalty parameter is 50E. The total calculation step is 4000. Considering the fact that the volume expansion is mainly caused by the large rotation, thus, we will pay more attention to the angular velocity. Some data are listed , while the trajectories of points P1 and P2 are shown in ), at which the first contact occurs between the slope and the stone, the angular velocities given by DDA0 and DDA1 are nearly equal. By comparison, the angular velocities by SRDDAvp are more accurate. For example, at CS =1000, the relative errors are 0.000203% (DDA0), 0.000203% (DDA1) and 0.000040% (SRDDAvp), respectively.At CS =1382, the stone touches the slope for the first time (see ). Then, at the following calculation step (CS =1383), the angular velocities by DDA0, DDA1 and SRDDA are different, with −10.25734346 s–1, −8.85281450 s–1 and −8.85595077 s–1, respectively. Moreover, due to the issue of volume expansion, the second contact by DDA0 can be observed at CS =1973 (see ); while for DDA1 and SRDDAvp, the second contact occurs at CS =2005. Whereafter, the third contact appears at CS =2912, 2593 and 2821 corresponding to DDA0, DDA1 and SRDDAvp, respectively., for DDA1 and SRDDAvp, the moments of the second contact (CS =2005) is the same. However, at the next calculation step (CS =2006), the angular velocities by DDA1 and SRDDAvp are different, namely −18.34628241 s–1 and −18.55805717 s–1, respectively. While, following the third contact, namely at CS =2913, 2594 and 2822, respectively. The angular velocities by DDA0, DDA1 and SRDDAvp are 2.77335470 s–1, −23.45725286 s–1 and 14.49043619 s–1, respectively., the blue line denotes the trajectory of point P1, the red line denotes the trajectory of point P2. Due to the volume expansion, as shown in (a), we observe that the trajectory by DDA0 is distorted. By comparing (b) and (c), we can see, the times of contact between the stone and the slope (β=45°) are once and twice for DDA1 and SRDDAvp, respectively. Moreover, for the sliding distance of the stone, the results by DDA1 are longer than that by SRDDAvp. In addition, from the zoomed views (zoom1, zoom2 and zoom3) in , several turning points of the trajectories can be observed. The corresponding angular velocities can found in . The potential of SRDDAvp is accordingly demonstrated by the simple example.The S-R decomposition theorem is an important result in the theory of geometric nonlinearity. It has an ability to capture strain components and rotation components at the same time. By utilizing this feature, a dynamics formulation was first deduced through the principle of virtual power. Moreover, the update process for the co-moving coordinate, which is closely related to the S-R decomposition theorem, was proposed. The new formulation is independent of the specific numerical methods. Then, in the setting of DDA, an S-R-D-based DDA, abbreviated as SRDDAvp, was established. Compared with the conventional DDA, the slightly modified basic unknown variables were adopted in SRDDAvp. Some examples have illustrated that SRDDAvp can naturally overcome the issue of volume expansion, effectively improve the calculation accuracy and also equip DDA with the potential to treat large deformation and large rotation.For arbitrary shape three-dimensional block, the shape function becomesT(x,y,z)=[1000z−zcyc−yx−xc000(z−zc)/2(y−yc)/2010zc−z0x−xc0y−yc0(z−zc)/20(x−xc)/2001y−ycxc−x000z−zc(y−yc)/2(x−xc)/20],the increment displacement Δu at any point (x,y,z) can be given byΔu(x,y,z)=(Δux(x,y,z)Δuy(x,y,z)Δuz(x,y,z))=T(x,y,z)Δdb,where Δdb={Δu,Δv,Δw,Δrx,Δry,Δrz,Δεx,Δεy,Δεz,Δγyz,Δγzx,Δγxy}Tis the generalized increment displacement vector. Δrx, Δry and Δrz represent the rigid-body rotation angle increments corresponding to x-, y- and z-axis respectively. Moreover, Δεx,Δεy,Δεz,Δγyz,Δγzx,Δγxy is the six increments strain components. The strain rate increment vector can be chosen asΔṠ(x,y,z)={ΔṠ11,ΔṠ22,ΔṠ33,2ΔṠ32,2ΔṠ13,2ΔṠ21}=BbΔVb,Bb=[000000100000000000010000000000001000000000000100000000000010000000000001].V(x,y,z)=(vx(x,y,z)vy(x,y,z)vz(x,y,z))=T(x,y,z)Vb,A(x,y,z)=(Ax(x,y,z)Ay(x,y,z)Az(x,y,z))=T(x,y,z)Ab,where Vb={u̇,v̇,ẇ,ṙx,ṙy,ṙz,ε̇x,ε̇y,ε̇z,γ̇yz,γ̇zx,γ̇xy}T and Vb={ü,v̈,ẅ,r⃛x,r̈y,r̈z,ε̈x,ε̈y,ε̈z,γ̈yz,γ̈zx,γ̈xy}T are the generalized velocity vector and acceleration vector, respectively. And the penalty matrix k should beThen, the governing equation for three-dimensional case can be easily constructed. And the increment strain is obtained by Eqs. only need to replace Sji, ui|j and θ by ΔSji, Δui|j and Δθ, respectively, readsΔSji=12(Δui|j+Δui|jT)−ΔLkiΔLjk(1−cos(Δθ)),i,j,k=1,2,3,sin(Δθ)=12(Δu1|2−Δu1|2T)2+(Δu2|3−Δu2|3T)2+(Δu1|3−Δu1|3T)2.Compositional effects on the mechanochemical synthesis of Fe–Ti and Cu–Ti amorphous alloys by mechanical alloyingIt is well-known that the mechanochemical processing of binary mixtures of transition metals can induce the formation of nanostructured and amorphous phases. Previous kinetic studies have shown that the phase transformation in binary mixtures follows sigmoidal conversion curves irrespective of the intensity of mechanical processing. The rate of such transformations is instead strongly influenced by both the intensity of the mechanical treatment and the physical properties of the reactant powders. In the present work, we give additional information on the amorphisation kinetics of binary mixtures by exploring the effect of composition on the transformation kinetics. Cu–Ti and Fe–Ti mixtures with composition within the amorphisation ranges have been subjected to mechanical processing at fixed intensity. The kinetics of phase transformation has been followed by quantitative X-ray diffraction. The experimental data obtained demonstrate that the amorphisation rate has a strong dependence on the product of the mass fractions of the single reactants.The mechanochemical synthesis of metallic amorphous alloys usually involves the cold-working of powders or intercalated foils, the grinding of intermetallic compounds and the mechanical alloying (MA) of powders This induced us to perform a systematic investigation on the kinetics of metastable phase formation in transition metal binary mixtures, pointing out its relationships with the processing parameters. The development of both an experimental methodology and a modelling approach, for the measurement of the impact energy and collision frequency on the one hand and for reproducing the dynamics of milling devices on the other, allowed us to quantify the rate of the energy transfer, i.e. the milling intensity, and the total mechanical energy transferred to the powders in the course of the mechanical treatment Experimental evidence suggests that a unique mechanism governs the metastable phase formation independent of the alloy chemistry, the structure of the end products, the milling device, the treatment intensity and the vial charge. Sigmoidal conversion curves are observed, characteristic of solid state reactions proceeding at the interface under the effects of impact-induced mass transport and continuous fresh surface generation. Such effects are expected to depend on the mechanical response of the mixture and its tribological properties. The transformed mass fraction is related to the mechanical energy transferred to the processed powders by simple formal kinetic equations The generalisation of the results mentioned above to stoichiometries other than the one corresponding to 40 at.% in TM represented the next task for our research work in order to gain a deeper insight into the MA processes. Accordingly, in this paper we systematically investigate the amorphization kinetics of homogeneous series of binary mixtures Ti–Fe and Ti–Cu at different stoichiometric compositions. Only compositions yielding a final amorphous phase have been considered.In the course of milling treatments, the rate of phase transformation is essentially controlled by the impact energy, E, and the impact frequency, N. As discussed in detail elsewhere where mb is the ball mass and υimp, the relative velocity of the ball, is determined by the ball and vial velocities at the impact. The collision frequency N has been shown to be almost exactly twice the frequency of the vial motion, i.e. two collisions per cycle occur. The milling intensity, I, and the energy dose, D, are defined as:where t is the milling time. The total energy transferred per unit mass of the powder, i.e. the specific dose, Dm, is then introducedwith mp the mass of the powder batch and t the milling time.The present milling trials were carried out by employing a modified commercial Spex Mixer/Mill mod. 8000, equipped with a variable speed motor allowing to change the frequency of the vial motion from 750 to 1400 cycles min−1. Except in one case, the frequency of the vial motion was kept constant at 850 rev./min. A hardened steel vial was employed with a single stainless steel ball, having a mass of 11.7 g and a diameter of 14 mm, for a powder load of 8 g. Working parameters are summarised in The kinetics of phase transformation was followed by wide angle X-ray diffraction (XRD). The so-called absolute crystalline content of the scattering sample, expressed by the ratio of the integrated intensity under the crystalline lines to the total integrated intensity, has been evaluated according to the Ruland’s method is the correction factor. D(s) is the function describing the so-called isotropic disorder, s the scattering vector defined as (4π/λ) sin θ, with θ and λ, respectively, the scattering angle and the X-ray wavelength. f is the mean square scattering factor and Icr and It are, respectively, the crystalline and the total coherent intensities. so and su are the integration limits set by the experimentally available scattering data. The application of the well-known Rietveld method allowed for the evaluation of the size, L, and the strain content, ε, of the coherent diffraction domains according to the Williamson–Hall method Deconvolution procedures are illustrated in . Hereafter we will refer to the amorphous fraction α, instead of xcr, the two parameters being related as α=(1−xcr).Powders with a 99.99% purity level were used in the milling trials. All the handling procedures were performed inside a glove box with nitrogen, oxygen and residual humidity controlled below 1–5 ppm.Small quantities of powder were sampled from the vial at regular time intervals for routine analysis and scanned over a stick tape on a Siemens diffractometer equipped with a Cu Kα radiation tube. Larger amounts of the milled powder were periodically employed to prepare compacted samples in order to perform more refined X-ray analysis and thus check the reliability of the routine analysis. In this case, the alloying process was restarted from the beginning. Patterns were collected over an angular range of 2θ from 10° to 135°, corresponding to a wave-vector range of about 15–160 nm−1, employing a Rikagu diffractometer equipped with a Mo Kα radiation tube.The two binary systems show similar reactive behaviours, probably due to the relatively similar mechanical ductility of Cu and Fe. Solid solutions are formed, respectively, at low and at high Ti contents. Amorphisation ranges are slightly different and the formation of an amorphous alloy is observed at the compositions Cu30Ti70, Cu40Ti60, Cu50Ti50, Cu60Ti40 and Cu70Ti30 in the case of the Cu–Ti system. In the case of the Fe–Ti system, the occurrence of amorphisation is restricted to the compositions Fe30Ti70, Fe40Ti60, Fe50Ti50 and Fe60Ti40.Both in Fe–Ti and Cu–Ti systems at the different stoichiometries indicated above, the mechanical treatment of the powder mixtures determines the progressive broadening of crystalline peaks and the fading of their intensity. These features are accompanied by the growth of the amorphous halo between the two more intense crystalline signals of Ti and its partners in the mixture, Cu and Fe, respectively. This can be observed in , where the pattern sequence relevant to the mixture Fe40Ti60 is reported.As evident from the figure, no intermetallic compound is formed during the mechanical processing before or concomitantly with the amorphisation process. Irrespective of the system examined and of the mixture composition within the amorphisation range, crystalline equilibrium compounds only become detectable at the latest stages of the transformation process.According to previously reported results, the 2θ position of the maximum of the amorphous phases is strictly connected to the stoichiometry of the amorphous alloys.The positions of the maxima for the Cu–Ti amorphous alloys at different stoichiometry are quoted in as a function of the scattering vector s. As expected, at decreasing the Ti content in the alloy the maximum of the amorphous halo moves toward larger diffraction angles.Quantitative XRD analysis reveals that the amorphous weight fraction α develops according to characteristic sigmoidal trends already observed in previous work as a function of specific milling dose, Dm.The comparison clearly points out that each system, at a given stoichiometry, has a characteristic rate of phase transformation. Such a rate can be approximately quantified by the reciprocal of the semi-transformation dose, Dm,1/2, i.e. the specific milling dose required to obtain α=0.5 and then amorphise half of the powder charge. High reaction rates correspond then to low semi-transformation doses. The semi-transformation doses for the mixtures considered are marked in the figure with the dotted lines.The evaluation of the semi-transformation dose for the mixtures at different composition of both the Cu–Ti and Fe–Ti systems puts into evidence the dependence of the amorphisation rate on the relative concentration of the pure metallic reactants. As shown in , the reciprocal values of the semi-transformation dose for the different mixture compositions arrange according to regular trends.Maximum transformation rates are detected at compositions Cu40Ti60 and Fe40Ti60. Cu-based mixtures, however, always show higher reaction rates than the Fe-based ones. It is worth noting that such behaviour is not immediately recognisable from a simple inspection of the X-ray pattern sequence, as that quoted in , usually employed to describe mechanical alloying progress.A more realistic, even if still phenomenological evaluation of the amorphisation rate can be performed by fitting the sigmoidal conversion curves, as those reported in , with mathematical functions able to suitably describe the kinetic behaviour. A general analytical expression has been developed in previous work to account for the observed kinetics. All the details regarding the assumptions of the kinetic modelling have been already presented and will not be discussed here The quantity kD, expressed in g kJ−1, has to be regarded as the kinetic constant of the amorphisation process. As can be noted from is able to satisfactorily interpolate the experimental points pertaining to mixtures at different composition.The rate constants for the amorphisation process of all the mixtures considered are quoted in as a function of the atomic fraction of the metal partner of Ti in the mixture, xCu and xFe, respectively.A comparison of the data reported in these figures with the ones reported in immediately reveals that the rate constants kD follow the same trends of the semi-transformation dose, with maximum values still occurring at a Ti content of xTi=0.6 for both the Cu–Ti and Fe–Ti systems. are best-fitted by a continuous line which expresses the product between the mass fractions χTi, χCu and χFe of Ti and its partners in the mixtures. In particular, it is possible to correlate the kinetic constant for the amorphisation process, kD, and the mass fractions of atomic species, χ, through the following equation:where M is the metal partner of Ti and η is a proportionality constant equal to 0.435 g kJ−1 and 0.251 g kJ−1 for the Cu–Ti and the Fe–Ti mixtures, respectively.The results obtained extend our previous findings on the amorphisation processes of binary mixtures. Even if experimental trials are limited at the moment to only two binary systems, it seems that the behaviour observed could be of a general concern. XRD analysis show that the formation of the amorphous phase always follows sigmoidal transformation curves characteristic of the so-called topochemical reactions occurring at the interfacial regions between two solid phases The composition of the initial mixture deeply affects the transformation rate, which is found to be proportional to the product between the mass fractions of the reactant powders. Finding a rationalisation for this particular feature is not trivial. Actually, such evidence gives partial support to the hypothesis that the amorphisation reaction is controlled by the formation of surface of intimate contact between the reactants. Indeed, under such circumstances the reaction rate should be proportional to the area of interface and this, in turn, to the so-called dispersion of reactant phases, i.e. the intermixing of the solid phases. Even if it is extremely difficult to make experimental measurements of dispersion, and then of the surface area of interfacial regions, it is possible to correlate it with the total surface area of single reactants. It is indeed reasonable to assume that the dispersion is proportional to the product between the total surface areas of reactant phases, expressing the probability of contact between the different species where fL is the shape factor, Vsp the specific volume and L the average crystallite size of the component under consideration. The total surface area S is then defined aswhere m is the total mass of a given reactant in the mixture. Under the assumption that the reaction rate is proportional to the dispersion of reactants, the kinetic constant of the amorphisation process should be proportional to the product between the total surface areas of reactants, so thatwhere SA and SB are the total surface areas of reactants A and B, while mA and mB represent their mass in the mixture. Since the specific surface areas can be considered as characteristic quantities of each chemical species subjected to mechanical processing, it appears thatin agreement with the experimental observation. This suggests that the opening of new surfaces and the consequent formation of a reactive interface could be actually considered as the rate-determining step of the alloying reactions. However, many other factors are expected to play a role in the processes of co-deformation and comminution such as the mechanical strength and the stiffness of the processed phases. It seems therefore necessary to develop a coherent framework able to relate the microscopic mechanisms operating at reactant interfaces under impact conditions and the mechanical properties of the powders.The experimental evidence collected in the present work provides an additional piece of information on the kinetics of amorphisation of binary mixtures of transition metals. Compositional effects of the powder mixtures are restricted to the rate of phase transformation, while the basic mechanism of transformation is not affected. As in mixtures at a fixed stoichiometry and undergoing mechanical processing at different intensity, the formation of the amorphous phase follows well-defined sigmoidal curves which are possible to reproduce through a phenomenological kinetic model. As shown in previous work Maximum credible damage of RC bridge pier under bi-directional seismic excitation for all incidence anglesExtending conventional Park-Ang damage index, a simple yet rational damage measure applicable for bi-directional seismic shaking has been evolved. Comparing predicted damage states of a bridge pier in terms of the evolved damage index to the component strains, efficacy of the proposed index has been verified. Quantitative assessment of damage of a bridge pier demonstrates substantial magnification in damage due to bi-directional shaking relative to uni-directional companion. This amplification in damage may further vary with incidence angles. Unfortunately, conducting true bi-directional analysis is challenging and implications of incidence angles are elusive yet. Collectively investigating these two crucial issues, present study identifies suitable orientations/ incidence angles in terms of pure ground motion parameters. Maximum credible damage of bridge pier under bi-directional shaking may be estimated by combining damage (30% rule) due to less-rigorous uni-directional analysis conducted in the most preferred orientation defined by the ratio of energetic length of components. The present investigation thus appears to evolve a rational damage index and a novel strategy to quantify the same for bridge pier under bi-directional seismic excitation.One of the major steps for seismic design and performance assessment involves a rational and quantitative estimation of damage. Assessment of damage generally requires nonlinear response history analysis of structure under a consistent suite of ground motions. In the event of an earthquake, a structure is excited by a pair of horizontal components along with vertical and rotational components. In many design contexts, however, two horizontal components appear to be primarily ‘deleterious’. This has motivated a number of researchers to evaluate structural damage under a pair of horizontal components.Numerous studies under bi-directional loading have been made computationally that, among others, employed plasticity and multi-surface plasticity Parallel to the recognition of additional damaging potential of bi-directional shaking, it may be worthwhile to note that, during real earthquakes, accelerograms are recorded by pre-installed sensors. Thus the pair of horizontal components of a real record available for engineering analysis are arbitrarily oriented in relation to the structure. However, components of accelerogram so recorded are often applied along the principal axes of the structure and is deficient in itself In practice, responses computed by independently applying two translational components along the principal axes of structure are often combined by one of the standard combination rules Foregoing review on the implications of (a) bi-directional shaking and (b) incidence angles motivate the authors to evolve a simple yet holistic strategy to suitably estimate maximum credible damage (MCD) that may be caused under bi-directional excitation (real situation) for all admissible incidence angles from a relatively simple uni-directional analysis conducted for appropriate incidence angle. Prior to approaching to this goal for RC bridge pier, attempt has been made to evolve a suitable damage index for both uni-directional and bi-directional loading – another important outcome of the current investigation. Thus the orientation-dependent strategy for damage assessment evolves through:Formulating a rational damage index under bi-directional seismic shaking and comparing its performance with other state-of-the-art indices.Exploring amplification of damage due to bi-directional excitation and estimating MCD covering all incidence angles.Identification of the incidence angle in terms of pure ground motion parameters, where damage under uni-directional shaking may be the most proximate (most preferred orientation) and least proximate (least preferred orientation) to MCD under bi-directional shaking, i.e., MCD induced by bi-directional shaking may be best/worst predicted from simpler uni-directional analysis.Achieving improved estimates of MCD by applying standard combination rules in the preferred orientations so evolved.Thus, to establish a simple yet practical framework for damage assessment, a suitable damage index is first evolved in the following section.Damage indices represent physical deterioration of RC element to a quantifiable degree. A specific level of damage beyond which residual load-carrying capacity is perceptively low is defined as ‘failure’. Any damage parameter is usually defined as the damage value normalized to that at failure and thus a damage index ideally lies between zero and unity.It may be physically intuitive that the displacement amplitude required to cause failure depends on number of cycles of loading. Hence, the use of ‘ductility ratio as a sole measure of damage is not sufficient’ Suitability of a chosen damage parameter, in the words of Hancock and Boomer It may thus be contended that response at any stage of loading may depend on the past load history and hence cumulative parameter may be a more precise index of damage especially for structural material that experience degradation in stiffness and strength under cyclic loading (e.g., reinforced concrete). From the exhaustive list of damage parameters as documented elsewhere where Δm: maximum deformation under earthquake, Δu: ultimate deformation under monotonic loading (taken as ∼3.8% of pier height due to reasons explained later), Fy: yield strength, dE: incremental dissipated hysteretic energy and β: a nonnegative parameter that may be empirically estimated from structural parameters Damage assessment at component level appears to have been favored in the recent literature Estimating component strain and peak displacement under bi-directional loading is generally straightforward from a fiber-based model. However, majority of the studies using DIPA have considered uni-directional excitation (subsequently denoted as DIPA(U) for distinction) and a suitable definition of DIPA under bi-directional loading, denoted herein as DIPA(B), is scarce. It may hence be essential to suitably define Park-Ang index under bi-directional loading. To our knowledge, explicit definition of DIPA(B) has so far been attempted only in two studies In this background, the present work selects the following admissible expressions of DIPA(B) in line with the related literature DIPA(B)=max·(DIx,DIy)+ExorEyEx+Ey·min(DIx,DIy)DIPA(B)=max·(DIx,DIy)+DIxorDIyDIx+DIy·min(DIx,DIy)in which DIx and DIy respectively refer to DIPA computed independently using peak deformations (Δx, Δy) and hysteretic energy (Ex and Ey) respectively in x and y directions, when structure is subjected to bi-directional shaking. λ, a constant parameter, has been taken as 0.5 are fundamental to the present work, it may be useful to judge these formulations physically. Since these are applied to combine the responses that account for interaction, responses in two principal axes are correlated (even when the record components are uncorrelated). However, the degree of cross-correlation is highly uncertain and generally unknown. It seems that the different equations above aim to model this non-zero cross-correlation in different empirical approaches, wherein is a special case that assumes zero cross-correlation. Eq. models the vectorial combination of responses in two orthogonal directions to compute Park-Ang index for bi-directional loading. In the strict sense, this requires the estimation of ultimate deformation under monotonic loading (Δu) and yield strength (Fy) in the direction of the resultant maximum displacement obtained from dynamic analysis. This may be complex especially to study the progression of damage over time which is beyond the scope of the current investigation. Thus for simplicity, corresponding values estimated from pushover analysis are taken in the present work and hence minor modification has been made in Eq. . This may be reasonable especially for pier with square section.In order to examine the performance of the admissible expressions above, we have computed damage indices for simple bridge pier due to a suite of carefully selected ground motions. In the following sub-sections, we briefly summarize details of modeling of the pier, selection of ground motions and analysis procedures before evaluating DIPA(B).A simple pier (relevant details are shown in Typical reinforced concrete (RC) section consists of unconfined concrete, confined concrete and reinforcing steel. Uni-axial constitutive behavior of confined and unconfined RC element (c and d respectively) is simulated after the improved model of Martinez-Rueda and Elnashai Reinforcing steel is modeled using Menegotto-Pinto steel model e). Values of the parameters involved in the idealization of the pier (in . System so modeled reasonably represents interaction between bending and axial force, but could not account for ‘cyclic flexure-shear interaction’, which is a current subject of wide research Characteristics of seismic excitations are known to depend on several factors such as magnitude (Mw), distance (r), rupture process, travel path from source to site and local site conditions. In the vicinity of an active fault, pulses in the form of ‘directivity’ may be potentially damaging In this background, we choose a total of fifty-six (56) ground motions with a range of geophysical parameters, viz., magnitude-distance-soil triads in order of increasing Mw, is representative of near-fault (NF) motions. Almost all NF motions are selected from the catalogue of NIST GCR 11-917-15 presents the inventory of far-fault (FF) motions. Mw (and r) for the NF and FF motions ranges from 5.8 to 7.9 (0.5–13.6 km) and 5.8 to 7.3 (17.6–50.2 km) respectively. Selected motions, downloaded from the strong motion database of Pacific Earthquake Engineering Research (PEER) Centre Selected motions are suitably scaled to achieve a similar response reduction factor (R taken herein as 4). For each component of a record, this scale factor is decided observing the elastic seismic demand of structure under each original record component in relation to the pier capacity (estimated from capacity curve constructed by pushover analysis shown in f). Scale factors of two components of a record so computed are compared and the average factor is applied to both the components. Thus the relative amplitude of the components remains unaffected due to this scaling Response of structure has been evaluated per standard response history analysis conducted in the time domain using Hilber-Hughes-Taylor integration scheme. While Hilber-Hughes-Taylor parameters α, γ and β are chosen respectively as −0.1, 0.6 and 0.3025, time step of integration is taken to be sufficiently small to ensure convergence. In nonlinear dynamic analysis, a major portion of dissipation of energy introduced through seismic action is implicitly modeled though fiber formulation. Thus a relatively small damping may be warranted in fiber formulation. Considering the inherent uncertainties and diverging opinions, 2% of initial stiffness proportional damping is assumed Independently applying two components of motion, responses due to two components of a selected pair are computed. Peak displacements in two displacement histories are compared and the greater one is taken as displacement under uni-directional excitation (Δ(U)). Comparing the area of force-displacement histories, greater one is similarly taken as E(U) (hysteretic energy under uni-directional shaking). Since Δ(U) corresponds to a specific instant and E(U) is related to the entire history of motion, it is intuitive that the component of motion that yields Δ(U) may or may not result in E(U). In the present investigation, it has been observed that, for about 70% of cases, Δ(U) and E(U) are caused by the same component. For bi-directional analysis, computed displacement histories in two principal directions are combined (as square root of sum of squares, i.e., SRSS) through the entire history (not independent peaks as they may not occur together) and the maximum value of displacement (overall peak) is taken as the drift under bi-directional excitation (Δ(B)). In each principal axis of pier, we have also independently noted the peak displacement, viz., Δx and Δy (directional peak) for bi-directional shaking. It is, therefore, apparent that Δ(B) may not be equal to Δx2+Δy2 since directional peak displacements (i.e., Δx, Δy) may not occur concurrently. Under two-component shaking, area of the force-displacement histories in two principal axes, viz., Ex and Ey are summed to compute E(B).Directional and overall peak as well as cumulative responses under bi-directional shaking so computed are next utilized to estimate Park-Ang damage under bi-directional shaking.Conducting pushover analysis of the bridge pier so modeled, yield strength (Fy |
= 1066 kN as in b) of pier is estimated. DIPA(B) has then been computed for each record in all alternative approaches (Eqs. for near-fault and far-fault motions respectively. It appears that the alternative formulations are similar to one another to quantitatively assess damage (values of COV is low: 0.01–0.15 and 0.02–0.14 respectively for NF and FF records).It is, however, essential to examine how well DIPA(B)∗ is co-related with the physical damage of structure. Damage thresholds at component level that ‘could be directly incorporated in the bridge design codes’ have been outlined in a recent work corresponding to the important damage states described therein (in terms of physical damage appearance (PDA) and degree of damage (DOD)). Relevant limits of Park-Ang parameter, as interpreted from the observations of damage occurred during real earthquake . These limits of Park-Ang indices are also consistent with the recommendations of other studies f, it may be noted that extensive crushing of concrete (in terms of strain in core concrete = 0.006) occurs at a drift of around 3.64%. Appreciating that the corresponding strain limit may be rather judgmental, we choose ultimate displacement (Δu) as 3.8% (intermediate of 3.64% and 4% specified in also lists the perceived DOD, viz., moderate (M), severe (S) and collapse (C) independently interpreted in terms of the computed DIPA(B)∗, component strains as well as Δ(B). The comparative evaluation of three damage parameter transpires a fair agreement in predicted damage states in terms of all three parameters for all NF and FF motions. DIPA(B)∗ so evolved may, therefore, be used as an adequate damage index under bi-directional seismic excitation.Before assessing damage amplification due to bi-directional loading relative to the uni-directional counterpart using this Park-Ang model, consequences of the normalization error inherent in Park-Ang model may be briefly examined.It may be noted that the Park-Ang damage model, despite its extensive use, suffers from normalization problem and, consequently, does not converge at its upper and lower limits (i.e., DIPA |
> 0 for system in elastic range as well as DIPA |
> 1 for structures loaded monotonically to collapse). In order to redress this fundamental deficiency in Park-Ang model, a modified definition, proposed by Jiang et al. Δyield represents yield deformation of pier. For pier under bi-directional shaking, the present work makes a reappraisal by extending this modified damage parameter for bi-directional loading (DIPA-J(B)). From the response history analyses of the pier due to bi-directional shaking, DIx and DIy as per Eq. are evaluated and substituted them in Eqs. as under, DIPA-J(B) has also been estimated in six parallel alternatives.DIPA-J(B)=(1-β)ΔxorΔyΔu+βE(B)Fy(Δu-Δyield)DIPA-J(B)=(1-β)Δ(B)Δu+βE(B)Fy(Δu-Δyield)Values of DIPA-J(B) so estimated per six extended alternatives have been averaged. Mean values of the modified damage index (DIPA-J(B)∗) so estimated have been appended in respectively for each NF and FF motion. A comparison of the companion values of DIPA(B)∗ and DIPA-J(B)∗ clearly shows that DIPA-J(B)∗ may, in relation to DIPA(B)∗, increase/decrease or may even alter negligibly. Corrections introduced in Eq. appear to reduce the first term, while the second term increases and the order of such changes may be dependent on the values of β. At a relatively low β (=0.042), the changes are not, therefore, significant. This justifies the trend upon accounting for the normalization problem. In the backdrop of the nominal impact of this correction when β = 0.042, we may infer that DIPA-J(B)∗ may, by and large, be consistent with other damage indicators. However, the implication of normalization may be sensitive to the values of β.In order to examine the damage amplification due to bi-directional shaking, we have plotted NDIPA∗ (=DIPA(B)∗/DIPA(U)) and NDIPA-J∗ (=DIPA-J(B)∗/DIPA-J(U)) for all selected earthquakes (). From the values of Δ(U) and E(U), we have calculated DIPA(U) and DIPA(B) by Eqs. respectively. Δ(B)/Δ(U) and E(B)/E(U) are also superimposed therein for introspection (to be read with reference to right-hand axis). Normalized damage parameter, plotted discretely against each motion, has been joined to conveniently observe the trend. It may be noted that the records are chosen in ascending order of Mw that strongly influences spectral shape and hence may be linked to damage that the bi-directional shaking may amplify the damageability even to the tune of around 35%–55% respectively for FF and NF motions. Magnification of hysteretic energy dissipation (∼2 times) expectedly exceeds that for the peak displacement (∼25%). Since Park-Ang index combines peak and cumulative parameters, magnification in damage measured through NDIPA∗ or NDIPA-J∗ appears to lie in between Δ(B)/Δ(U) and E(B)/E(U). The implication is that the estimates of amplification in damage due to bi-directional shaking may be sensitive to selected damage index. We may thus infer that any strategy of predicting damage under bi-directional shaking by efficiently exploiting the companion damage due to uni-directional shaking may be dependent on selection of damage index.The previous study thus reveals that damage of bridge pier, measured through a conceptually sound index, may substantially amplify due to bi-directional excitation. This, coupled with the implications of incidence angle reviewed already, inspires the authors to investigate how amplification in damage due to bi-directional shaking changes with incidence angle.In order to examine the implications of incidence angle, a brief scrutiny of the variation of important ground motion characteristics ax(ψ)(t)ay(ψ)(t)=cosψsinψ-sinψcosψax(t)ay(t)in which ax(t), ay(t) are horizontal components of original record and axψ(t), ayψ(t) are components of the record when rotated anti-clockwise by an angle ψ. We use ‘orientation’ (denoted by ψ) to refer to the issues related to ground motion alone, while ‘incidence angle’ (denoted by θ) is used to refer to response of pier due to such rotated motions. illustrates the correspondence between orientation and incidence angle. Choosing the alignment of recording sensors as reference, records are rotated herein by ψ to apply the motions along the principal axes of the structure. Defining incidence angle as the angle between the principal axis of structure with the chosen reference axis (alignment of recording sensors), ψ = θ in general for our purpose.Of particular interest are the parameters that may be independent of structural properties and hence may be applicable for every structure. Peak deformation is generally dependent on amplitude and frequency content, while hysteretic energy dissipation also depends on duration of motion In parallel, we are impelled to choose characteristics intensity (Ic) of motion. This pure ground motion parameter condenses the amplitude, frequency as well as duration of motion. It may also be recalled that the damage estimated by Park-Ang parameter has been shown to be well correlated to IcThus rotating motion components to multiple orientations, these component properties (Le and Ic) individually (for uni-directional shaking) and collectively (combined by SRSS for bi-directional shaking) have been studied. We rotate ground motions by 360° with an interval of 15° in the anti-clockwise sense and properties of each component at each orientation are computed. We define normalized component intensity (NCI) as the ratio of the intensity of stronger to weaker component and hence is always greater than or equal to unity. Representative results presented in Component characteristics appear to interchange over 90° and repeat over 180° as may also be expected from the transformation matrix in Eq. Ground motion intensity is sensitive to orientations component-wise and relatively stable pair-wise.Orientation corresponding to the peak intensity of one component is closely associated with the minimum of the other. Thus NCI assumes a peak in this orientation and is denoted as ψmax(Le or Ic).There exists certain orientations (intersection points) where intensities for two orthogonal components approach to be proximate. Thus NCI may approach to minimum in this orientation and is denoted as ψmin(Le or Ic).As mentioned earlier, two components are separately applied and the maximum response is considered in case of uni-directional analysis while, for bi-directional shaking, both the components are together applied. For such definitions, in view of the observations (a) and (b) above, we have computed response for different incidence angle in the range of 0°–90° only. This is adequate for solitary pier with symmetric shape of cross-section (such as square and circular) which is practically desirable In view of the promise of the orientations ψmax(Le), ψmin(Le) and ψmax(Ic), ψmin(Ic) discussed in the previous sub-section, we have calculated Δ(U) and E(U) as well as Δ(B) and E(B) respectively for each motion corresponding to these four incidence angles. presents the variation of normalized drift (Δ(U) and Δ(B) normalized by pier height, H) and normalized energy dissipation (E(U) and E(B) normalized by the product of Fy and Δy) in the sample form. It should be noted that the responses correspond to the selected orientations characterized by maximum or minimum values of NCI and the actual angle (depending on arbitrary orientation of sensor) is of no interest. It appears from that both Δ(B) and E(B) remain relatively stable over orientations and hence damage under bi-directional shaking covering all admissible incidence angles appears invariant with respect to incidence angles. However, damages due to corresponding uni-directional excitations are generally sensitive to orientations. This is expected since the cross-section of the pier is square and ground motion intensity has been observed sensitive component-wise, but insensitive pair-wise to orientations. In sum, we note that the difference in response (peak and cumulative) between uni-directional and bi-directional excitation may be at variance with incidence angle. that uni-directional analysis yields a better estimate of peak displacement and hysteretic energy dissipation in the orientations of ψmax(Le) and /ψmax(Ic). Further, it appears that the difference between E(B) and E(U) generally exceeds than that between Δ(B) and Δ(U) (as before). Recalling the discussion on Le and Ic in the previous sub-section, upon closer inspection, orientations corresponding to ψmax(Le) and ψmax(Ic), as marked in , may be chosen as the estimated most preferred orientation (where parameters contributing to damage caused by uni-directional shaking is most proximate to bi-directional counterpart) respectively for peak displacement and hysteretic energy. On the contrary, the estimated least preferred orientations (where parameters contributing to damage caused by uni-directional shaking is least proximate to bi-directional counterpart) for peak displacement and hysteretic energy may be correlated to the orientations defined by ψmin(Le) and ψmin(Ic) respectively. discussed above may be useful to estimate MCD under two-component shaking (measured through DIPA(B)∗ and DIPA-J(B)∗) for all admissible incidence angles from uni-directional analysis. We have, therefore, computed DIPA(B)∗ (and DIPA-J(B)∗) as well as DIPA(U) (and DIPA-J(U)) for four identified orientations. For generic evolution, two values of β equal to 0.05 and 0.5 have been selected that may respectively assign lesser and greater weightage to cumulative response. It appears that the MCD under bi-directional shaking is insensitive to incidence angle. From the representative results presented in , orientations defined by ψmax(Le or Ic) and ψmin (Le or Ic) appear potential for estimating MCD from uni-directional analysis. To be more definitive, values of NDIPA (=DIPA(B)∗/DIPA(U)) and NDIPA-J (=DIPA-J(B)∗/DIPA-J(U)) computed in the identified orientations for each motion are plotted in . Since the values of NDIPA and NDIPA-J are very similar (although DIPA and DIPA-J may differ), we have plotted them together against two different axes (left-hand/right hand) in , together with summary of the statistics, transpires that:Magnification in damage under bi-directional shaking estimated by DIPA and DIPA-J appears similar, albeit the actual value may change, especially when β is large.DIPA(U) approaches to be the most proximate to DIPA(B)∗ (for NF motion: average difference of ∼20% for β = 0.05 and ∼25% for β = 0.5; for FF motion: average difference of ∼10% for β = 0.05 and ∼20% for β = 0.5) in the orientations of ψmax(Le) and/ ψmax(Ic). This observation holds good for NDIPA-J as well. The implication is that the orientations corresponding to ψmax(Le) and ψmax(Ic) may be considered as the estimated most preferred orientations for all values of β.NDIPA and NDIPA-J may increase (for NF motion: average difference of ∼37% for β = 0.05 and ∼42% for β = 0.5; for FF motion: average difference of ∼20% for β = 0.05 and ∼30% for β = 0.5) in the orientations of ψmin(Le) and/ ψmin(Ic). This suggests that the estimated least preferred orientations may be correlated to ψmin(Le) and ψmin(Ic) for all values of β.Thus, Selecting DIPA (or DIPA-J) as appropriate damage index, some latitude may exist to consider [ψmax(Le), ψmax(Ic)] and [ψmin(Le), ψmin(Ic)] as estimated most and least preferred orientations respectively. Low values of COV (0.07–0.14) also suggest the stability of this inference. From the quantitative analysis, it is evident that the damage computed per uni-directional analysis in the estimated most preferred orientations, however, requires further improvement. To this end, we examine the efficacy of the codified combination rule in the next section.Response under bi-directional shaking may be estimated by combining corresponding responses due to simpler uni-directional analysis. Combination rules, reviewed elsewhere DIPA(c)=max·[(1.0×DIPA(U),x+γ×DIPA(U),y)and(1.0×DIPA(U),y+γ×DIPA(U),x)]DIPA-J(c)=max·[(1.0×DIPA-J(U),x+γ×DIPA-J(U),y)and(1.0×DIPA-J(U),y+γ×DIPA-J(U),x)]where DIPA(U),x and DIPA(U),y are Park-Ang index; DIPA-J(U),x and DIPA-J(U),y represent Park-Ang index modified by Jiang when two components of motion are separately applied along two principal axes of the pier. In the present work, γ equals to 0.3 has been chosen (i.e., 30% rule as specified in NDIPA(C) (i.e., DIPA(B)∗/DIPA(c)) and NDIPA-J(C) (i.e., DIPA-J(B)∗/DIPA-J(c)) have been calculated for each motion for two selected values of β (0.05 and 0.5). This normalized parameter corresponding to all four orientations of interest have been presented in for NF and FF motions respectively. Difference in damage estimated by combination rules with respect to that under bi-directional shaking has been summarized below:For NF motion, in ψmax(Le): 3% for β = 0.05 and 7% for β = 0.5; ψmax(Ic): 7% for β = 0.05 and 10% for β = 0.5.For FF motion, in ψmax(Le): −6% for β = 0.05 and −1% for β = 0.5; ψmax(Ic): −6% for β = 0.05 and 1% for β = 0.5.For NF motion, in ψmin(Le): 11% for β = 0.05 and 15% for β = 0.5; ψmin(Ic): 9% for β = 0.05 and 10% for β = 0.5.For FF motion, in ψmin(Le): −4% for β = 0.05 and 6% for β = 0.5; ψmin(Ic): −4% for β = 0.05 and 5% for β = 0.5.A close introspection of the summary above elucidates that the damage under uni-directional shaking combined by 30% rule in the orientation of ψmax(Le) or ψmax(Ic) offers very good estimates of MCD due to bi-directional shaking. Clearly, ψmax(Le) may be favored especially for NF motion. In parallel, it may be noted that the orientation defined by ψmin(Le) is more (ψmin(Ic)) efficiently co-related to the incidence angle that results in maximum error in damage prediction using combination rule. We thus recommend to choose ψmax(Le) and ψmin(Le) as the estimated most preferred orientation and estimated least preferred orientation respectively to apply 30% combination rule.Further, for NF motion, it appears that the uncertainty in incidence angle may lead to underestimating damage even to the order of ∼15%. This may, however, be reduced to ∼7% only by employing combination rule (30%) in the orientation of ψmax(Le). In contrast, damage estimated per 30% combination rule may be less-sensitive to the incidence angle (error limited to ±6%) for FF motion. We, therefore, recommend 30% combination rule to assess damage under bi-directional shaking in the orientation of ψmax(Le). This offers confidence to employ simplified uni-directional analysis for estimating ‘true’ damage of bridge pier dispelling the ‘threat’ of incidence angle, too. This conclusion is, however, specific to the damage index chosen and may not be generalized.It may be noted that the axial load levels on piers are typically low respectively. Ground motions are carefully selected such that a broad range of Mw may be covered. Damage under bi-directional and uni-directional shaking has been evaluated for these selected motions. DIPA(U) and DIPA(B)∗ computed with and without axial force have been furnished in . Damage parameters with axial force are denoted as DIPA(U),N or DIPA(B),N respectively. Axial load may increase overall lateral load capacity of pier, while fracture/buckling of reinforcement may, for a few cases, occur earlier. Thus with some irregularity, damage appears to decrease when axial load is included under both uni-directional and bi-directional shaking. However, amplification in damage due to bi-directional shaking over uni-directional companion, parameter of major significance in the present work, may not be significantly affected due to axial load at least in the range of practical interest. A similar trend has been noticed in terms of DIPA-J(U), DIPA-J(B)∗ and DIPA-J(U, N), DIPA-J-N(B)∗ and has not been included for brevity.Yet, with the intent of more explicit verification of the existence of preferred orientation for bridge pier subjected to axial force, we re-compute damage of pier under uni-directional as well as bi-directional shaking. In this part of investigation, we have arbitrarily selected a subset of records employed in . Conducting nonlinear response history analysis under uni-directional and bi-directional shaking for several incidence angles, we have computed NDIPA,N (=DIPA(B),N/DIPA(U),N) in the range of 0–90°. Variation of NDIPA,N at each incidence angle, (to be read from left-hand axis) along with the NCI (in terms of both Le and Ic) in the corresponding orientation (to be read from right-hand axis) has been plotted in . Incidence angles yielding maximum (θmax) and minimum (θmin) amplification in damage as well as the orientations relevant to ψmax(Le), ψmin(Le), ψmax(Ic) and ψmin(Ic) |
are marked therein. Clearly, NDIPA, N tends to attain its maximum and minimum as the NCI approaches to its minimum and maximum respectively. Thus, the correspondence of [ψmax(Le), ψmax(Ic)] and [ψmin(Le), ψmin(Ic)] respectively with the most preferred orientation and least preferred orientation, as already identified, appears to have been confirmed even when bridge pier is subjected to axial load in the practical range of interest.It may be contended that, across incidence angles, relative stability in response under bi-directional shaking may not exist for pier with other section geometry such as rectangular (other than square or circular which is generally preferred because of omnidirectional strength and stiffness). For such system with difference in strength and stiffness along two principal axes, one axis (weak) appears to govern the response ) has been conducted for a more definitive conclusion.To this end, owing to the difference in properties along two principal axes, intuitive extension of the uni-directional and bi-directional analyses, defined hitherto, may be useful. In uni-directional analysis, each component of a selected pair of horizontal motion is separately applied along 1–1 and 2–2 axis. Thus four distinct cases, viz., Case U-1(x): x component along 1–1, Case U-1(y): y component along 1–1, Case U-2(x): x component along 2–2 and Case U-2(y): y component along 2–2 have appeared for a pair of motion at a specified orientation. During bi-directional shaking, both the components are simultaneously applied along two principal axes of the pier. However, owing to the difference in geometry, two sets of bi-directional analyses are conducted. This yields two cases for bi-directional shaking (viz., Case B-1: x component along 1–1 and y component along 2–2 and Case B-2: x component along 2–2 and y component along 1–1).For uni-directional shaking, we compute DIPA(U) for Case U-1(x) and Case U-1(y) independently and the maximum is defined as damage under uni-directional shaking along 1–1 axis (DIPA(U),1-1). DIPA(U),2-2 is similarly calculated from the results of Case U-2(x) and Case U-2(y). During bi-directional shaking, DIPA(B) has been calculated by Eq. (one alternative is chosen for simplicity) separately for Case B-1 and Case B-2 and the greater one, representing the worst possible damage, has been taken as damage under bi-directional shaking (MDIPA(B)). Normalized damage parameters redefined as NDIPA,1-1 (=MDIPA(B)/DIPA(U),1-1) and NDIPA,2-2 (=MDIPA(B)/DIPA(U),2-2), for seven arbitrarily selected NF and FF motions, have been summarized in . Values of NDIPA,2-2 are substantially greater than the corresponding NDIPA,1-1. This implies that the selection of weak axis (along 2–2) in uni-directional analysis may itself reduce the difference between damage due to bi-directional and uni-directional shaking to a significant extent. This prediction is further improved when such uni-directional analysis (in weak direction) is conducted for incidence angle defined by the most preferred orientation (). This statistically stable (COV is low) trend, in turn, appears to explain why the current investigation has primarily dealt with the pier having omnidirectional stiffness and strength.To portray ‘true’ damage of bridge pier, nonlinear response history analysis under bi-directional excitation representative of the seismic activity of a region may be required. A simple yet rational damage index, applicable for bi-directional loading, has first been evolved herein. Considering challenges of bi-directional analysis, simple combination rules are often used that involves uni-directional analyses. In addition, damage estimates are known to be strongly affected by the orientation/incidence angle of recorded components of accelerogram. The present work thus intends to remove the limitations of uni-directional analysis and incidence angle by way of identifying suitable orientation/ incidence angle. MCD of bridge pier may be closely predicted by combining damage due to uni-directional shaking conducted for appropriate incidence angles so recognized. The following broad conclusions emerge.Conceptually sound and extensively used damage parameter, viz., Park-Ang index (accounting for existing deficiency of normalization) has been verified, for the first time, under bi-directional seismic excitation. Park-Ang damage so computed correlates well with the physical damage of pier and is in excellent agreement with the damage at component level. This is also in compliance with the drift limits outlined in modern codes. Park-Ang index evolved herein may thus be used to quantify damage under bi-directional seismic loading.Observed that the magnification due to bi-directional excitation relative to uni-directional shaking may be consistently greater for cumulative response (∼2 times hysteretic energy dissipation) relative to the noncumulative response (∼25% for peak displacement). Clearly, amplification in damage measured through Park-Ang index (as high as around 35%–55% respectively for FF and NF motions) appears to lie in between noncumulative and cumulative quantities. This implies that the prediction of damage under bi-directional shaking from the results of uni-directional analyses may be dependent on damage parameter of interest. However, problem of normalization inherent in Park-Ang model appears to little influence amplification in damage.It is evident that pure ground motion parameters (Le and Ic), with change of orientations, are at variance component-wise and stable pair-wise. Thus normalized component intensity (NCI) has been defined as the ratio of intensity of the stronger to weaker component. On the one hand, it has been noted that the orientation corresponding to the maximum intensity of one component is closely associated with the minimum of the other, while on the other, there exists certain orientations where intensities of two orthogonal components approach to be proximate. In the former NCI approaches to the peak (denoted as ψmax(Le or Ic)), while NCI may assume its minimum (denoted as ψmin(Le or Ic)) to the latter.Representative case studies reveal that the response under uni-directional shaking is sensitive to incidence angle, while that under bi-directional shaking is relatively stable. For pier with omnidirectional stiffness/ strength, this trend is in agreement with the previous conclusion. From physically-motivated arguments and close inspection of results, we choose ψmax(Le) and ψmax(Ic) as estimated most preferred orientation for peak response and estimated most preferred orientation for cumulative response respectively. Orientations defined by ψmin(Le) and ψmin(Ic) respectively showing a converse trend in response relative to those in ψmax(Le) and ψmax(Ic) appear to confirm the existence of these orientations of special significance.Further, the orientations that correspond to the most or least margin in estimating maximum credible damage (per Park-Ang index) under bi-directional shaking relative to uni-directional counterpart appear important. Choosing Park-Ang index evolved as appropriate damage index, it appears that some latitude may exist considering [ψmax(Le), ψmax(Ic)] and [ψmin(Le), ψmin(Ic)] as estimated most and least preferred orientations respectively. Since Park-Ang parameter is dependent on peak and energy dissipation together, this trend may be intuitive. It is inspiring to note that the selection of appropriate incidence angle so defined may, on average, reduce the error in damage prediction by around 17% and 10% respectively for NF and FF motions.Amplification is damage due to bi-directional shaking appears to be marginally influenced by (a) correction for normalization error inherent in Park-Ang model and (b) axial force within the range of practical interest. Limited yet representative case studies further confirm the correspondence of [ψmax(Le), ψmax(Ic)] and [ψmin(Le), ψmin(Ic)] respectively with the most preferred orientation and least preferred orientation even when bridge pier is subjected to axial load.Difference in estimating damage, measured through Park-Ang index, may be reduced appreciably by conducting uni-directional analysis in the estimated most preferred orientations defined by ψmax(Le) or ψmax(Ic). Damage estimated under uni-directional shaking in this preferred orientation may further by combined per 30% rule, especially for NF motion. This may, however, marginally overestimate damage at far-fault. Damage statistics reveal that the orientation defined by ψmax(Le) may be more promising to apply combination rule. To evaluate MCD, we thus recommend to:Rotate ground motions in the estimated most preferred orientation defined by ψmax(Le)Apply both the components along the principal axes of pier separately and calculate damage for each componentCombine damage indices so estimated per 30% combination rule.This conclusion may, however, be specific to the damage index chosen.By appropriate selection of weak direction (for pier with rectangular cross-section), amplification in damage due to bi-directional shaking relative to uni-directional counterpart appears to be substantially reduced in itself. The difference sharply decays when damage due to uni-directional shaking is estimated employing record components rotated to the estimated most preferred orientations defined by ψmax(Le) or ψmax(Ic). This indirectly suggests that establishing a framework to estimate damage under bi-directional shaking employing uni-directional analysis may be of primary interest for pier with omnidirectional section-geometry (as dealt herein primarily).It may be interesting to estimate the Park-Ang index for bi-directional loading over time to study the progress of damage. However, this requires the estimation of ultimate deformation under monotonic loading (Δu) and yield strength (Fy) at every instant in the direction of the resultant maximum displacement obtained from dynamic analysis. This physically motivated approach should be considered in future with a more rigorous structure idealization Simulation of Burden Distribution and Charging in an Ironmaking Blast FurnaceThe changing of solid particulate matter in the ironmaking blast furnace is studied in the paper using modeling and experiments in small scale. A small-scale charging rig is used to study the layer formation when coke and pellets are charged using a bell-less charging equipment. The modeling involved the use of a simplified geometric model as well as discrete element modeling (DEM). In DEM, both spherical and non-spherical (clumped particles) models were used to describe the coke, while for pellets a spherical model was used. The results from the different models are compared to the experimental findings, and conclusions concerning the appropriateness of the models are drawn. Finally, a challenging phenomenon, i.e., the coke push effect, where lighter coke on the burden surface is shifted by charging of the heavier pellets is discussed and some simulation results for the full-scale problem are presented.A study on the influence of oil film lubrication to the strength of engine connecting rod componentsDynamic lubrication analysis of connecting rod is a very complex problem. Some factors have great effect on lubrication, such as clearance, oil viscosity, oil supplying hole and surface roughness. This work establishes the EHD characteristic of bearing and bush to analyze the peak oil film pressure, min oil film thickness and node force with consideration of deformation and cavitation; then create connecting rod components to calculate the stress distribution with consideration of film lubrication or not and finally contract the results of two conditions. The results show that the stress distribution of two conditions is different in individual positions, but the stress calculation method considering the oil film lubrication is more consistent with the actual working conditions.Connecting rod is an important component of engine which transforms the reciprocating piston motion in the cylinder into rotary motion of shaft, as well as converting the power from piston to shaft. In such operating condition, the connecting rod experiences complex dynamic loads as stretching, compressing and bending during the service. Therefore it is considered to be a critical component.The majority of damages have been reported to take place at some parts of connecting rod, such as rounded filet of small connecting rod end, rounded filet of big connecting rod end and the connecting rod bolt. As shown in , the failure of connecting rod occurs and the crack was found at rounded filet of big connecting rod end. A number of efforts have been made to calculate the strength of connecting rod. The traditional way to calculate the strength of connecting rod simplified the 3D model into 2D stress problem. The dimensionless thickness of rounded filet and ribs are determined by the area equivalent principle Over the past decades, a considerable number of theoretical and experimental studies have been made on bearing performance. The effects of oil viscosity, bearing deformation and cavitation were taken into consideration gradually. He zhenpeng calculated the minimum oil film, maximum oil pressure and friction loss of connecting rod bearing with consideration of oil hole location, oil supply pressure and surface roughness, but do not calculate the connecting rod stress distribution.the relative velocity of the contact surfaceviscosity of the lubricant in the film regiona parameter, in full film area: a=P-PcPref-Pc, in cavitation area: a=ρ-ρcρcthe bearing deformation caused by oil pressureThe oil film can be divided into full film area and cavitation area. The flow in the gap can be described by Reynolds equation ∂∂θρch312μ∂Fα∂θ+R2∂∂yρhc312μ∂Fα∂y=UR2Pref−Pc∂ρc1+1−Fαh∂θ.where ρc is the lubricant density, μ is the lubricant viscosity in the film region, U is the relative velocity of the contact surface, Pc is the air pressure and Pref is the reference pressure.Since in full film area: a |
= (p |
− |
pc)/(pref |
− |
pc), Eq. ∂∂θρch312μ∂α∂θ+R2∂∂yρch312μ∂a∂y=UR2pref−pc∂ρch∂θ.In cavitation area: a |
= (ρ |
− |
ρc)/ρc, Eq. Real surface are not ideal smooth and the film thickness is generally given by:where c is the radial clearance and δ(θ) is the bearing deformation caused by oil pressure. During the calculation, we neglected the roughness of the contact surface. The nominal film thickness no longer represents the mean film gap and in fact it can even become negative for large contact fraction The bearing can be unwrapped onto a local cylinder coordinate system (R, |
θ, |
Y), as shown in . In this coordinate system, R corresponds to the radial coordinate, θ corresponds to the circumferential coordinate and Y corresponds to the axial coordinate. Due to symmetry with respect to the plane y |
= 0, the computational domain can be divided into half. Periodical boundary conditions apply to the surface at θ |
= 0 and 2π. The boundary conditions can be written as 1. at the side end of the bearing: y=±L2:P |
= 0;2. the Reynolds cavitation boundary condition with p equals to the vapor pressure:3. the JFO boundary condition at the film rebuild side:The mathematical model presented by Eqs. is solved numerically using an iterative scheme. The big-end bearing and small-end bush are both meshed with rectangle, the density of the finite element is 80 ∗ 120. The mapping schematic is shown in 112μ3hij2∗hi+1,j−hi−1,j2Δθ∗αi+1,j−αi−1,j2Δθ+hi,j3αi+1,j−2αi,j+αi−1,jΔθ2+R212μ3hi,j2hi,j+1−hi,j−12Δy∗αi,j+1−αi,j−12Δy+hi,j3αi,j+1−2αi,j+αi,j−1Δy2=UR2pref−pc∗hi+1,j−hi−1,j2Δθ.Mathematical solution of the Reynolds equation may predict a negative value for the pressure in diverging parts of the film. In practice, cavitation area is found due to dissolved gasses that appear when the pressure drops below the air pressure. During the calculation, set the parameter P first, then by the parameter α to judge whether the oil is in full film area or in cavitation area. When the parameter P meets Eq. , the loop ends. The numerical scheme is shown in tol=∑i=1m∑j=1npi,jk+1−pi,jk∑i=1m∑j=1npi,jk+1≤10−4Based on the analysis described in the present paper, a computer program was developed to study the performance of connecting rod bearing and bush. lists the essential parameters of connecting rod, bearing and bush.presents the distribution of maximum oil pressure and minimum oil thickness changing with angle for connecting rod big end bearing and connecting rod small end bush. (a) shows that MOFT of bearing in one cycle is 3.49 μm, which appears at 84°CA, meeting the requirement of 1.5 μm for cars (b) shows that the MOFT of bush in one cycle is 1.8 μm, which appears at 274°CA, also meeting the requirement of 1.5 μm for cars. The maximum oil film pressure in one cycle is 216.42 MPa, which appears at 9°CA. The MOFT and MOFP in one cycle appear at different crank angles.According to the pressure, calculate the node force of bearing and bush by Eq. Fr=−∫−w/2w/2∫0πP∗R∗cosθ∗dθdyradialcoordinateFa=∫−w/2w/2∫0πP∗R∗sinθ∗dθdyaxialcoordinatewhere w is the width of bearing and bush, P is the oil film pressure, R is the radial of bearing and bush. presents the distribution of node force changing with crank angle for bearing and bush. (a) shows that because of gas force in cylinder, the maximum force of bearing in radial coordinate is 954.65 kN, which appears at 9°CA. The minimum force of bearing in radial coordinate is − 199.47 kN, which appears at 360°CA. (b) shows that the maximum node force in axial coordinate is 79.12 kN, which appears at 434°CA. And at 645°CA the minimum node force in axial coordinate is − 76.25 kN. (c) shows that the maximum force of bush in radial coordinate is 77.12 kN, which appears at 360°CA. And the minimum force of bush in radial coordinate is − 1064.4 kN. (d) shows that the maximum node force of bush in axial coordinate is 8.8 kN, which appears at 18°CA. The minimum node force of bush in axial coordinate is − 7.15 kN. Compared with the force in radial coordinate, the force in axial coordinate is small relatively. If the calculation of connecting rod strength is not strict, the node force in axial coordinate can be neglected.Forces acting on connecting rod of diesel engine are periodic and complex since the connecting rod is used to transform the piston reciprocating motion into crankshaft rotary motion as shown in . The force connecting rod suffered can be described as:where F is the gas force and ω is the rotation speed. Referring to Eq., it can be seen that the connecting rod is suffered from tensile load and compressive load.The displacement of piston designated as xdue to the connecting rod with the length of L is given by:The assumption that the connecting rod with the length of L can be considered as two masses, one at big connecting rod end (M2) and the second at piston (M1), the inertia load of connecting rod can be describe as:Fr=M1*r*ω2∗cosα+rLcos2αreciprocatinginertialoadFL=M2*r*ω2rotationinertiaload., it is concluded that during service the connecting rod is subjected to time-varying tensile and compressive cosinusoidal force. Such complex forces acting at the end of connecting rod may become critical, especially that the peak load is achieved. These dynamic loads have a potency to cause fatigue failure.The traditional way to calculate the connecting rod stress distribution was proposed to set the load calculated by Eqs. on connecting rod directly without consideration of oil film lubrication. This work built the connecting rod components and meshed it by the software ABAQUS as shown in . The dimension of the model was built according to the failed connecting rod, as shown in . In order to consider the function of oil film, the element density of bearing and bush is 80 ∗ 120, which equals to the number of elements in numerical calculation. Constitutive properties for all elements including Young modulus and Poisson's ratio were assumed to be 210 GPa and 0.45 respectively.In order to calculate the stress distribution more accurately, the preload of bolt was considered, as well as the interference of bearing and bush. The values of preload and interference are listed in . The node forces taken from bearing numerical calculation are applied to bush and bearing. The load that connecting rod suffered was shown in , the compressive load reached maximum at 9°CA. Imposing the node force taken from numerical calculation on bearing and bush, we got the stress distribution of maximum explosion condition. The tensile load reached maximum at 360°CA. Imposing the node force of this condition at bearing and bush, we also got the stress distribution of maximum reciprocating inertia force condition. The Von Misses distribution obtained from FEA is shown in (a) and (b) presents the connecting rod stress distribution in maximum compressive load condition and maximum tensile load condition. As shown in (a), the maximum stress is around 400 MPa, which occurs at the filet region near the big connecting rod. The stress concentration region is consistent with the failure position as shown in (b) presents that in maximum reciprocating inertia force condition the stress is relatively lower. Of note is that the maximum operating stress is less than the yield stress of the connecting rod material, i.e. 900 MPa, fatigue failure occurs under dynamic loads due to high stress concentration.As comparison, the compressive load and tensile load were directly imposed on the connecting rod. The parameters of the connecting rod were listed in and these loads were calculated by Eqs. . The Von Misses distribution obtained from FEA is shown in (a) and (b)presents the connecting rod stress distribution of in maximum compressive load condition and maximum tensile load condition without consideration of oil film. The results are almost similar with the condition considering the oil film lubrication.Based on the analysis describe in this paper, contract the stress distribution of two condition, as shown in show that the maximum stress of connecting rod is close, but its distribution has some individual differences. Region B refers to rounded filet of the big connecting rod end and region D refers to rounded filet of the small connecting rod end, these positions are easily turn to be the location of crack origin by dynamic load. With consideration of oil film lubrication, the stress is lower at these two regions.In maximum compressive load condition which considering the influence of oil film lubrication, the maximum stress is 375.8 MPa in region B and 360.6 MPa in region D. However without consideration of lubrication, the maximum stress reached 400.7 MPa in region B and 367.4 MPa in region D. Comparing the result of two calculation methods, the deviation was 6.63% in region B and 1.89% in region D.In maximum tensile load condition with consideration of oil film lubrication, the maximum stress is 45.3 MPa in region B and 154.6 MPa in region D. However without consideration of lubrication, the maximum stress reached 57.3 MPa in region B and 171.3 MPa in region D. In region D the deviation was 10.8%. In region B the deviation reached 26.5%.The deviation of two condition in region B and region D was caused by the difference of imposing method. When considering the oil film lubrication, the gas force was turned into node force of bearing and bush in radial coordinate and axial coordinate by numerical solution. However without consideration of oil film lubrication, the gas force was imposed on connecting rod directly. The maximum operation stress of two conditions is less than the yield stress of the connecting rod materials, fatigue failure can occur under dynamic loads due to high stress concentration and material defects.The connecting rod stress calculation was introduced with consideration of oil film lubrication or not. And the node force of bearing and bush was calculated by Reynolds equation. From the results presented in this paper, the following conclusions can be drawn:The stress in rounded filet of connecting rod was higher relatively. The fatigue failure can occur under dynamic loads due to high stress concentration and material defects. In connecting rod design, the rounded filet should be smooth.With consideration of oil film lubrication or not, the stress distribution of two conditions are different in individual positions. When considering the lubrication, the stress in round filet is relatively lower, and it is much more close to actual operation condition.To prevent fatigue failure of connecting rod, the rounded filet should be smooth. Also using longer filet radius can reduce stress concentration.Application of stochastic finite element method in estimation of elastic constants for NCF compositesThe main aim of this paper is to evaluate the effect of micro-structure variations on elastic constants of NCF (non-crimp fabric) composites. To reach this aim, a three-dimensional unit cell model is developed by the finite element code ABAQUS and mean value of elastic constants in NCF composites are determined by analytical and unit cell based methods with various boundary conditions. In addition, first- and second-order stochastic finite element formulations are derived and variances of elastic constants are computed with respect to the dispersion in length and width of the voids. Results show up to 9.1% scattering in the elastic constants.The microstructure of NCF (non-crimp fabric) composites shows the imperfections due to the stitch thread (). The fibers are straight and aligned with principal material direction of the ply except around the stitch insertion locations that lead to the generation of in-plane fiber distorted regions. In addition, fiberless regions are also generated around the stitches in the form of a crack-shaped void. Sometimes the crack-shaped voids are connected and merged together to form a continuous channel. These regions will turn into the resinfull regions or resin voids when the fabric is subjected to manufacturing processes. During the manufacturing process, the fibers may also nest into the adjacent plies resin voids that leads to the out-of-plane fiber misalignments. However, in most cases, both in-plane and out-of-plane fiber misalignments are negligible. compute the in-plane and out-of-plane elastic constants of NCF composites. calculate the elastic constants of stitched NCF composites. predict in-plane strength and stress–strain characteristics of stitched NCF composites with different stitching and loading configurations using finite element unit cell model. address the variations of microstructural parameters. However, the effect of variations in microstructural parameters has not been considered. The main aim of this paper is to investigate the effects of these variations on the elastic constants of NCF composites. To this end, due to the variations of void length, void width, stitch pitch and stitch spacing, a stochastic procedure is used. The mean value and variance of elastic constants are calculated and effect of variation in microstructural parameters are taken into account using first- and second-order stochastic finite element method. A three-dimensional unit cell model considering NCF heterogeneities (resin voids) and three-dimensional structure (stitch thread) is developed. The mean value of the elastic constants is computed with both analytical and unit cell based methods. For obtaining the elastic constants and the first- and second-order variances, a user-defined MATLAB program is linked to a parametric ABAQUS model.A bidiagonal (−45°/45°) carbon NCF with total areal weight of 343 (g/m2) is used as sample in this paper. The pattern of stitching is warp and it is oriented along direction “2” (. Void length (L), void width (W), stitch pitch (P), and stitch spacing (S) are measured at more than 50 points (b). Among these parameters, the variation of the void length and the void width is significant and the remaining parameters are considered to be constant due to their small standard deviation. Different distributions such as normal, lognormal, and Weibull are fitted to the measured data and results show the best fitting with the normal distribution curve (For taking into account heterogeneities (resin voids) and three-dimensional structure (stitch thread), a three-dimensional linear finite element model based on the NCF unit cell is generated. The unit cell is a basic building block which can describe the NCF periodic structure. To save time and obtain more flexibility, we use a parametric model capable of considering various unit cell dimensions and architecture such as void length, void width, thickness, in-plane fiber orientation, stitch pitch, stitch spacing, and stitch thread cross section. The model is discretized by three-dimensional linear 8-node solid element and 25 elements are used in directions “1” and “2” (). We use two and four elements through the thickness. It is shown that the number of elements through the thickness does not have a noticeable effect on the results. Therefore, two elements are used in the through-thickness direction. We control the mesh generation process so that most elements have an aspect ratio less than 2 and around the voids tip less than 4. Based on the microstructure observations, the cross section of the stitch thread is circular through the thickness and elliptical on the layer surface, hence, it is modeled as an external reinforcement and represented by three-dimensional linear 2-node beam element that has circular and elliptical cross sections through the thickness and on the layer surface, respectively.In order to determine the elastic constants of in-plane fibers region, a micromechanical approach based on Composite Cylinder Assemblage model () is used. The model assumes a group of composite cylinders, each consisting of a circular fiber core and a concentric matrix shell. The size of fiber core and matrix shell is chosen such that the fiber volume fraction in each cylinder is equal to the fiber volume fraction in whole composite. In the context of NCF composites, comparing with other well-known models, a desirable feature of this model is the randomness of fiber placement that can handle imperfections of in-plane fibers around the resin voids due to the stitch thread.For simplification, the fiber volume fraction is assumed to be constant in the unit cellwhere VfNCF and Vf denote the fiber volume fraction after and before stitching, respectively. Avoid is the area of the crack-shaped void or channel that is WL/2 or WL, respectively. The elastic constants of in-plane fibers region areExx=EmVmNCF+EfVfNCF+4(νf−νm)2VmNCFVfNCF(VmNCF/Kf)+(VfNCF/Km)+(1/Gm)νxy=νxz=νmVmNCF+νfVfNCF+(νf−νm)((1/Km)−(1/Kf))VmNCFVfNCF(VmNCF/Kf)+(VfNCF/Km)+(1/Gm)Gyz=Gm1+1+βmVfNCFρ−(1+(3βm2(VmNCF)2/α(VfNCF)2+1))VfNCFα=βm−γβf1+γβf,ρ=γ+βmγ−1,βm=13−4νm,βf=13−4νf,γ=GfGmIn these equations, x–y–z is local coordinate system. E, G, ν, and K are the Young's modulus, shear modulus, Poisson's ratio, and bulk modulus of the in-plane fibers (index f) and matrix (index m), respectively, as listed in The NCF unit cell is a mixture of three-dimensional (due to stitch thread), orthotropic (due to in-plane fibers), and isotropic (due to resin voids) behaviors which interact in a complicated manner to form the unit cell's anisotropic behavior. To fully characterize the elastic behavior of the unit cell, the stress–strain relation is considered aswhere C is the matrix of the mean value of the effective elastic constants defined byC=C11C12C13C14C15C16C12C22C23C24C25C26C13C23C33C34C35C36C14C24C34C44C45C46C15C25C35C45C55C56C16C26C36C46C56C66where σ¯ij, ε¯ij and σij, ɛij are macroscopic and microscopic stresses and strains, respectively. The global coordinate system 1–2–3 is employed in this equation (In order to compute the effective elastic constants, both analytical and unit cell based methods are used. In the analytical method, the NCF lamina is treated as a unidirectional lamina and effective elastic constants can be obtained from C=T1−1CmT2 where Cm is the matrix of the mean value of the elastic constants in local coordinate system (x–y–z) and T1 and T2 are the transformation matrices defined by In the unit cell based method, a set of load cases are applied to the unit cell boundaries that allow the terms of the matrix C to be determined (). The load cases are either uniform strain or uniform stress. The uniform strain load case consists of the following six independent relations that are applied separately to Eq. L1:ux=ε¯11x,uy=0,uz=0L2ux=0,uy=ε¯22y,uz=0…L6ux=ε¯12y2,uy=ε¯12x2,uz=0The uniform stress boundary condition can be applied in a similar way. shows the mean value of the elastic constants obtained by analytical, uniform strain, and uniform stress boundary condition. The analytical method always predicts higher values than uniform strain and uniform stress boundary conditions. The maximum and minimum differences between analytical and uniform strain predictions are 4.7% (C26) and 2.7% (C13), respectively. The maximum and minimum differences between analytical and uniform stress predictions are 10.6% (C26) and 3.2% (C13), respectively. In the analytical method, the NCF layer is considered as a uni-directional layer and therefore the decreasing effect of the resin voids on the elastic constants is neglected. It can be the main reason for the overestimation of the analytical method. The elastic constants calculated by uniform strain and uniform stress boundary conditions show difference up to 6.9% (C55). It can be a result of the boundary effect. To reduce the boundary effect, 9-celled and 25-celled models are considered and the elastic constants are computed for the centroidal cell (). The differences between the values of Cij with uniform strain and uniform stress boundary conditions show that the 9-celled model is accurate enough and therefore it is used in our calculations.Knowing the zero terms of elastic constants (σ¯11σ¯22σ¯33σ¯23σ¯13σ¯12=C11C12C1300C16C12C22C2300C26C13C23C33000000C44000000C550C16C26000C66ε¯11ε¯22ε¯33ε¯23ε¯13ε¯12 and applying load cases L1 to L6, we obtain the matrix of the variance of Cij as[var(Cij)]=var(σ¯11)[E(ε¯11)]2var(σ¯22)[E(ε¯11)]2var(σ¯33)[E(ε¯11)]200var(σ¯12)[E(ε¯11)]2var(σ¯11)[E(ε¯22)]2var(σ¯22)[E(ε¯22)]2var(σ¯33)[E(ε¯22)]200var(σ¯12)[E(ε¯22)]2var(σ¯11)[E(ε¯33)]2var(σ¯22)[E(ε¯33)]2var(σ¯33)[E(ε¯33)]2000000var(σ¯23)[E(ε¯23)]2000000var(σ¯13)[E(ε¯13)]20var(σ¯11)[E(ε¯12)]2var(σ¯22)[E(ε¯12)]2000var(σ¯12)[E(ε¯12)]2By first order stochastic finite element method with uncertainty in length (L) and width (W) of the voids, the microscopic stress vector can be approximated asand index “e” denotes the element number. From the above relations, the macroscopic stress vector and its variance can be expressed as (σ¯=1V∑e=1neσ0eVe+aL∑e=1ne∂σe∂aLVe+aW∑e=1ne∂σe∂aWVevar(σ¯)=∂σ¯∂aL2var(L)[E(L)]2+∂σ¯∂aW2var(W)[E(W)]2The uncertainty in the void length and the void width, leads to the uncertainty of the void's boundary. Therefore we have∂σe∂aL=∑i=1nu∂σe∂αi∂αi∂aL+∂σe∂βi∂βi∂aL,∂σe∂aW=∑i=1nu∂σe∂αi∂αi∂aW+∂σe∂βi∂βi∂aW∂σe∂αi=De∂Be∂αiU0e+DeB0e∂Ue∂αi,∂σe∂βi=De∂Be∂βiU0e+DeB0e∂Ue∂βiand αi and βi are the uncertainties in the coordinates of the nodes on the void's boundary. In addition, De, Be, Ke, and K are element material properties, transformation, element stiffness, and global stiffness matrices, respectively, and Ue and U are the element and global displacement vectors.The terms ∂αi/∂aL, ∂βi/∂aL, ∂αi/∂aW, ∂βi/∂aW in Eq. , relate the uncertainty of the nodal coordinates to the uncertainty of the void length and the void width. By assuming a cosinusoidal path for the fibers around the stitches (), with some mathematical calculations, we can writeBy the second order stochastic finite element method, the microscopic stress vector can be approximated asσe=σ0e+∂σe∂aLaL+∂σe∂aWaW+12∂2σe∂aL2aL2+∂2σe∂aL∂aWaLaW+12∂2σe∂aW2aW2The macroscopic stress vector and its variance can be expressed as (σ¯=1V∑e=1neσ0eVe+aL∑e=1ne∂σe∂aLVe+aW∑e=1ne∂σe∂aWVe+aL22∑e=1ne∂2σe∂aL2Ve+aLaW∑e=1ne∂2σe∂aL∂aWVe+aW22∑e=1ne∂2σe∂aW2Vevar(σ¯)=∂σ¯∂aL2var(L)[E(L)]2+∂σ¯∂aW2var(W)[E(W)]2+12∂2σ¯∂aL22var(L)[E(L)]22+12∂2σ¯∂aW22var(W)[E(W)]22∂2σe∂aL2=∑i=1nu∑j=1nu∂2σe∂αi∂αj∂αi∂aL∂αj∂aL∂2σe∂aL∂aW=∑i=1nu∑j=1nu∂2σe∂αi∂βj∂αi∂aL∂βj∂aW∂2σe∂aW2=∑i=1nu∑j=1nu∂2σe∂βi∂βj∂βi∂aL∂βj∂aL∂2σe∂αi∂αj=De∂2Be∂αi∂αjU0e+De∂Be∂αi∂Ue∂αj+De∂Be∂αj∂Ue∂αi+DeB0e∂2Ue∂αi∂αj∂2σe∂αi∂βj=De∂2Be∂αi∂βjU0e+De∂Be∂αi∂Ue∂βj+De∂Be∂βj∂Ue∂αi+DeB0e∂2Ue∂αi∂βj∂2σe∂βi∂βj=De∂2Be∂βi∂βjU0e+De∂Be∂βi∂Ue∂βj+De∂Be∂βj∂Ue∂βi+DeB0e∂2Ue∂βi∂βj∂2U∂αi∂αj=−K0−1∂2K∂αi∂αjU0+∂K∂αi∂U∂αj+∂K∂αj∂U∂αi∂2U∂αi∂βj=−K0−1∂2K∂αi∂βjU0+∂K∂αi∂U∂βj+∂K∂βj∂U∂αi∂2U∂βi∂βj=−K0−1∂2K∂βi∂βjU0+∂K∂βi∂U∂βj+∂K∂βj∂U∂βi shows the procedure for determination of the variance of the elastic constants. Once the load case is applied to the finite element model, the elements stress and displacement vectors are computed by the solver and called into the MATLAB program for calculating macroscopic stress and its variance. The related variance of elastic constants is computed and the procedure continues by updating the load case. Results show that the matrix of the variance of elastic constants (Eq. ) is symmetric. The variance coefficient of the elastic constants according to the variance coefficient of the length and the width of the voids is shown in . Our observations show that the variance coefficient of the length and the width of the voids in different NCF samples varies from 5% to 15%. However, greater variance coefficients are addressed by . Therefore, for the sake of generality, the variance coefficient of the length and the width of the voids is allowed to change from 0% to 20%.With the first order of uncertainty, the sensitivity of each elastic constant to the variation in the length and to the variation in the width of the voids is almost equal. The most and least affected elastic constants to the length of the voids are C66 and C23. By enhancing the variance coefficient of the voids length by 20%, the variance coefficient of C66 increases to 0.9% and the variance coefficient of C23 takes the value of 0.5%. Dispersion in the width of the voids has the most and least effect on C16 and C13, respectively. By increasing the variance coefficient of the voids width to 20%, the variance coefficient of C16 reaches the value of 1.1% and the variance coefficient of C13 takes the value of 0.5%.With the second order of uncertainty, the variance coefficient of the elastic constants increases nonlinearly with respect to the variance coefficient of the length and the width of the voids. Therefore, the second order uncertainty predicts higher values of variance coefficients than the first order uncertainty. The most and least difference between first order and second order variance coefficients due to the variation in the voids length appears in C11 and C33, respectively. In addition, the most and least difference between first order and second order variance coefficients due to the variation in the voids width appears in C16 and C33, respectively. Unlike the first order uncertainty, it is found that the sensitivity of the elastic constants to the length of the voids and to the width of the voids is different. C11, C22, C26, and C55 are more sensitive to the variation in the length of the voids while C12, C16, C44, and C66 are more sensitive to the variation in the width of the voids. In addition, C13, C23, and C33 are marginally more sensitive to the width of the voids. The most and least sensitive elastic constants to the length of the voids are C11 and C33 with 8.1% and 0.6% variance coefficients, respectively. Moreover, the most and least sensitive elastic constants to the width of the voids are C16 and C33 with 9.1% and 0.6% variance coefficients, respectively.In this paper, a general procedure for computation of the mean value and variance of the elastic constants of NCF composites using stochastic finite element method is presented. The mean value of elastic constants is computed with analytical and unit cell based methods. In the analytical method, the NCF layer is considered as a unidirectional layer and therefore the effect of the resin voids and the stitch thread is ignored. In the unit cell based method, the resin voids and the stitch thread are considered and the mean value of the elastic constants is determined with both uniform strain and uniform stress boundary conditions. By comparing the elastic constants obtained by analytical and unit cell based methods, the following conclusions can be made:Resin voids expose their effect on in-plane elastic constants rather than out-of-plane elastic constants. Therefore, in-plane elastic constants must be computed from unit cell based methods.Due to the direction of the fibers, the resin voids are extended along 45° direction. Therefore, they show the most effect on the in-plane shear coupling elastic constants (C16, C26, C66).The difference between elastic constants predicted by uniform strain and uniform stress boundary conditions reduces by increasing the number of cells. Results show enough accuracy with the 9-celled model. Therefore, the 9-celled model can be used to compute the elastic constants with either uniform strain or uniform stress boundary condition.The difference between the out-of-plane elastic constants, predicted by analytical and unit cell based methods, is negligible. Therefore, the out-of-plane elastic constants can be determined from the analytical method.Stitch thread, due to its orientation along the NCF thickness, is expected to show its major effect on the out-of-plane elastic constants. However, according to the above conclusion, we can say that the effect of stitch thread on the elastic constants is negligible.By observing the variance of the elastic constants obtained by first- and second-order formulations, the following conclusions can be made:With the first order uncertainty, the sensitivity of the elastic constants to the voids length and to the voids width is almost equal, and the most effective elastic constants to the void parameters are in-plane elastic constants.The second order uncertainty predicts more significant variations in the elastic constants with respect to the dispersion in the void parameters and therefore it must be used for computation of the variance of elastic constants.Due to the existence of resin voids, the in-plane elastic constants are considerably sensitive to the variation in the void parameters.The most dispersed elastic constant with variation in the voids length is the coupling between normal stress and normal strain in direction “1” (C11).The most dispersed elastic constant with variation in the voids width is the coupling between normal stress in direction “1” and in-plane shear strain (C16).The out-of-plane elastic constants are less sensitive to the variation in the void parameters.C11, C22, C26, and C55 are more sensitive to the variation in the voids length while C12, C16, C44, and C66 are more sensitive to the variation in the voids width. In addition, the out-of-plane elastic constants (C13, C23, C33) are marginally more sensitive to the voids width.Impact of ultrafine-grained microstructure on the corrosion of aluminium alloy AA2024► Successful grain size reduction of an aircraft alloy. ► Transition from IGC to pitting. ► Newly generated grain boundaries are not prone to IGC. ► ECAP processing “desensitizes” grain boundaries.Ultrafine-grained microstructures produced by severe plastic deformation upon AA2024-T351 were investigated with respect to pitting and intergranular corrosion. Refined microstructures were achieved using multiple pass equal-channel angular pressing (ECAP). Polarization in 0.5 M NaCl indicates that anodic behavior is affected by deformation when compared with coarse-grained specimens. Susceptibility to intergranular corrosion (IGC) via potentiostatic experiments reveals differences in the corrosion morphology between specimens with varying degrees of processing. In the unrefined state, IGC primarily occurs along grain boundaries; whilst ECAP processing predominantly results in pitting corrosion owing to a complete rearrangement of grain boundaries, revealing a desensitization effect arising from processing.High-strength aluminum alloys are widely used in aerospace and transport applications. However, localised corrosion, including pitting, intergranular corrosion (IGC), exfoliation, and stress corrosion cracking can be a major cause of aluminium alloy failures in practice Of studies that consider grain size alteration in the context of Al or Al alloys, the majority of works suggest that as grain size decreases, corrosion rate also decreases A consolidated view of relevant conclusions indicates that overall electrochemical behavior is dominated by second phase particles and how ECAP changes particle size and distribution, with less corrosion originating from grain boundaries. However, Brunner et al. The material studied was AA2024-T351 (exact composition (wt.%): Cu 4.15, Mg 1.14, Mn 0.53, Fe 0.23, Si 0.12, Zn 0.13, Ti 0.03, balance Al), which was provided by EADS (France) as bulk plate. The ECAP processing was carried out at room temperature using a solid die having a channel containing a 90° bend, employing a backpressure of 69 MPa similar to previous work by Horita Before each experiment, specimens were ground with a microcut paper disc (600 grit), and subsequently polished using 3 μm and1 μm diamond spray suspension and a pure ethanol lubricant. After polishing, samples were sonicated in pure ethanol and dried under stream of warm air prior to masking.To help prevent the possibility of crevice corrosion during electrochemical measurements the epoxy specimen interface was painted with Turco Form Mask 527 HT® (Henkel) and cured for 12 h at room temperature prior to specimen mounting in epoxy. For each sample-electrolyte combination, the corrosion potential was allowed to stabilise for 450 s. Potentiodynamic experiments were carried out in quiescent 0.5 mol/l NaCl. All potentiodynamic measurements were initiated at 250 mV cathodic to the open circuit potential (OCP) and scanned upwards until a current density of 500 μA/cm2 was reached at which point scans were reversed. The measurements were carried out using a three electrode setup consisting of an Ag/AgCl (E |
= +207 mV SHE) reference electrode, which was equilibrated with 3 mol/l KCl solution and a platinum counter electrode. All electrodes were connected to a specially modified Jaissle IMP 83 PC T-BC potentiostat/galvanostat which has a current-detection limit of 10 fA.Susceptibility to intergranular corrosion and to other forms of localized corrosion depends strongly on the alloy microstructure. The typical elongated pancake-shaped grain structure of AA2024-T351 rolled plate can be seen in a. The apparent grain size and aspect ratio vary depending on the specimen orientation. After ECAP processing the average grain size (and shape) is refined. The one pass sample (Bc1) exhibits an elongated textured grain structure as visible in the optical image (left) as well as in the EBSD unique grain image (right) (b). The microstructures of the four pass specimen (Bc4) consist of grains that have been further refined in average size to less than 200 nm, which is evident from the EBSD unique grain image (c). In the microstructure, there are many wavy or poorly defined grain boundaries as well as many dislocations within the grains (the latter of which effects EBSD pattern quality and can be qualitatively assessed by contrast changes with a grain in the unique grain image). All these features indicate that the microstructures of the ECAP specimens are in relatively high energy and non-equilibrium state illustrates the evolution of the OCP of the complete set of samples in a quiescent 0.5 mol/l NaCl solution. For sake of clarity, only the representative data for the L-ST plane and both ECAP samples are plotted in the main figure – the complete dataset is shown in the inset. During the first hour the OCP of all samples increases towards more noble values accompanied by potential transients ( inset). Due to Mg dissolution on immersion or following dealloying or incongruent dissolution ). It is assumed that the entire Al surface experiences the total potential drop due to propagation and repassivation of a metastable pit After ∼10 h of immersion all bulk orientations and the one pass ECAP sample (Bc1) reach a stable potential of approximately −600 mV vs. Ag/AgCl. In contrast, the four pass ECAP specimen exhibits a stable potential of approximately −575 mV after similar times. This shift towards more noble potentials may be related to both surface copper enrichment a the as received state contains the needle shaped Al2CuMg, which are clearly defragmented to round shaped particles after four passes of pressing (b). One may conclude from these results an enrichment of matrix composition resulting in a change in macroscopically electrochemical behaviour. Reports on dissolution and fragmentation of precipitates are also given in Refs. Representative cyclic polarization experiments after 450 s of OCP stabilisation in 0.5 mol/l NaCl solution are shown in . As one may anticipate, the polarization curves for the three different orientations from the bulk plate are almost identical in shape and form with repassivation noted to occur at a potential of approximately −800 mV vs. Ag/AgCl. This is consistent with the notion that the bulk potentiodynamic response is largely dictated by the bulk composition of the alloy matrix and typically is less sensitive to intermetallics/microstructure , the cathodic current density on the forward scan appears to be unaffected by the ECAP process. This suggests that a possible defragmentation of cathodic sites affects the oxygen reduction reaction either in a positive or negative way. The corrosion potential (Ecorr) of the one pass ECAP sample was observed to be shifted cathodically relative to other specimens by about 40 mV. The anodic behavior is dominated by active dissolution as typified by a sharp increase in current density at potentials above Ecorr. Comparing only the different orientations of the bulk plate, the reverse scan differs somewhat in shape, however, repassivation occurs at approximately the same potential. The Ecorr of the Bc4 sample correlates closely with that of the plate material, however, small variations in anodic curve shape were observed. Typically, the hysteresis in the reverse scan is the smallest for the Bc4 specimen in all tests. The initiation sites for corrosion damage are still localized at intermetallics, resulting in selective particle dissolution or pitting in the matrix adjacent to the particle. The hysteresis loop for both UFG materials suggest differences in repassivation, and consequently in attack morphology To evaluate the susceptibility to intergranular corrosion, potentiostatic polarization experiments were conducted in 0.5 mol/l NaCl for 20 h. shows the current response obtained for all samples while potentiostatically holding samples at −580 mV vs. Ag/AgCl. The current densities for the L-ST and LT-ST directions increase rapidly with time. In contrast, the current density for the Bc1 sample increases slowly, whereas the Bc4 sample shows very low relative anodic activity at this potential. Corresponding to the section of the polarization curve (inset in ), the Bc4 sample shows a small plateau previous to −580 mV, suggesting a pseudo passive range. For this reason a second experiment with a potentiostatic hold 10 mV more anodic (−570 mV) was used to induce an anodic current. Despite an increase in potential, the current density observed from the ECAP four specimen remains at a low level compared to the other samples. This result may indicate different IGC susceptibility for the Bc4 material, and may be due to the differences induced from processing (since the same bulk chemistry exists with all samples tested, as they came from the same plate).The IGC behavior was further examined by metallographic cross sections of the corroded samples. Optical micrographs confirmed severe intergranular corrosion for all bulk samples, which is what is expected from long potentiostatic holds a). After 20 h, IGC penetrates to depths of approximately 1000 μm from the surface. In contrast, the ECAP samples show different behavior and much less penetration. The cross section in b depicts the typical intergranular attack observed for the Bc1 test specimen. As expected, the IGC occurs along the textured microstructure grain boundaries following one ECAP pass, however, the associated attack depth is significantly reduced compared to that observed from bulk material. In contrast, at a potential of −580 mV vs. Ag/AgCl the four pass sample exhibits no severe corrosion, especially IGC. For this reason, cross sections were taken from the sample polarized to −570 mV vs. Ag/AgCl (c) for observation. Large pits are visible and reach a depth of ∼200 μm. It should be mentioned that small isolated regions do exhibit intergranular corrosion but not in a significant or ordered capacity. This apparent but limited quantity IGC observed in ECAP specimens may be due to portions of the original AA2024 plate microstructures remaining locally intact (i.e. with no re-dissolving of precipitates) after the SPD process. Overall, the investigations herein reveal a substantial change in corrosion morphology after ECAP; transitioning from an initial IGC dominance to pitting with processing. Furthermore, the deformation process decreases corrosion penetration depths, in agreement with literature data It is generally accepted that intergranular corrosion in 2xxx alloys at open circuit conditions in chloride containing environments commences with the dissolution of S-phase particles (Al2CuMg) on grain boundaries which is promoted, not only by microgalvanic coupling a schematically illustrates the well-known mechanism of IGC in 2xxx series alloys, and includes a depiction of precipitate-free zones, which are synonymous with precipitate microstructures. The SEM micrograph for a bulk 2xxx sample (a) shows the “chain link” shaped attack morphology associated with S-phase dissolution from grain boundaries – similar behavior has been observed in 7xxx series alloys by SEM In contrast, the results obtained from the ECAP samples require a different explanation to explain the decreased susceptibility to IGC. We recall that all specimens processed by ECAP have the same nominal composition. However, the microstructure and local chemical composition, as well as the character and type of the grain boundaries are altered by ECAP processing. The schematic sketch in b attempts to explain the electrochemical behavior and the corrosion morphology of the ECAP samples. Due to the shearing process and large local strains associated with SPD, the initial grain boundaries, and therefore their unique arrangement of chemical composition, is destroyed. It can be assumed that the S-phase particles and the solute-depleted zone are “mixed up” in the matrix material during ECAP. At the very least it is reasonable to assume that significant second phase particle breakup and redistribution occurs To verify the mechanistic considerations, AES investigations were applied to validate the hypothesis as it relates to corrosion. Therefore, the potentiostatically corroded Bc4 and L-LT samples were carefully ground and subsequently polished. Areas of intergranular attack on both samples served as reference points for the chemical composition mapping of single grain boundaries. A compositional map of Cu near a grain boundary for the L-LT direction is given in a. On the left hand side the elemental map indicates distribution in relation to the grain boundary. The image on the right hand side shows a corresponding SEM image of the area of interest. The map clearly reveals increased copper concentrations in the vicinity of the grain boundary observed from the bulk material b shows complimentary SEM and AES results for an ECAPed sample. In contrast to a, increased concentrations of Cu associated with grain boundaries, which are much finer (), were not observed in the ECAP specimens. Additionally, qualitatively the size of Cu-rich second phases appears to decrease with ECAP – the general homogenization and redistribution of Cu throughout the matrix are in agreement with the model presented in b. The investigated GB exhibits no copper-enrichment verifying the suggested mechanistic hypothesis. Generally, a more homogeneous distribution of the copper is noticeable. This emphasizes that in the case of AA2024, it is more strictly the changes in micro-chemical distribution of alloying elements that dominate the corrosion mode over the grain size effect in grain refined specimens.The results presented herein indicate the influence of the ECAP processing on the corrosion response of AA2024-T351. In particular, the number of ECAP passes and hence the microstructural evolution (and compositional redistribution) plays a dominant role in corrosion and IGC response. Subtle differences in behavior are detectable in electrochemical polarization curves, whereby different grain orientations of the bulk material show comparatively identical behavior, while the ECAP samples differ vastly – in particular Bc4.IGC inducing experiments reveal a clear change in the attack morphology between bulk coarse grained and ECAPed specimens. The IGC susceptibility and mode changes as a function of the number of ECAP passes. This is believed to be associated with changes in microstructure and hence local chemical composition that ECAP induces. With increasing ECAP passes the dominant corrosion propagation mode transitions from IGC to pitting, resulting in a relative reduction in attack depth. Moreover, it should be noted that the SPD process acts to ‘desensitize’ grain boundaries through redistribution of Cu to other areas of the microstructure. The desensitization effect of the ECAPed samples is supported further by AES investigation, which showed little copper in the vicinity of grain boundaries. A correlation between grain size and a higher susceptibility to IGC was not found, as the newly generated grain boundaries are not prone to IGC.Effect of laser cleaning in Laser Assisted Joining of CFRP and PC sheetsThe present paper investigates the influence of laser cleaning of Carbon Fibre Reinforced Plastic (CFRP) surface when joining CFRP to Polycarbonate (PC) sheet by means of laser assisted Joining.Experimental tests were conducted to perform polycarbonate-CFRP (with epoxy matrix) joints. The laser cleaning treatment was carried out on CFRP laminate adopting a 30 W Q-switched Yb:YAG fibre laser. Laser assisted joining was performed adopting a continuous wave 200 W diode laser. Untreated samples were adopted as reference. Morphological analysis and single lap shear tests were conducted to characterize the joints. Infrared thermography (IRT) was carried out to determine the temperature distribution and variation during the joining process. ANalysis Of VAriance was applied to investigate the effect of the process parameters, (laser power, energy, and treatment) on the extension of the bonded area and the mechanical properties. The results show that laser pre-treatment enables a significant increase of the joint strength which is, under the optimal conditions, more than double than the reference samples: However, considering the apparent shear strength, the laser pre-treatment does not produce a significant advantage, as it mainly resulted in the enlargement of the bonded area.The growing demand of light-weight structures, especially in transport, results in wider employment of hybrid structures made of different materials. In this context, the employment of multi-material assemblies represents a possible solution; coupling materials with different physical and mechanical behaviours can offer different opportunities. For example, in the case of Carbon Fibre Reinforced plastics CFRP coupled with polycarbonate (PC), it is possible to obtain hybrid structures, characterized by high strength and toughness, with the presence, if necessary, of transparent areas, for inspection or aesthetical reasons. On the other hand, as PC shows good impact resistance, it can be adopted either as matrix for Carbon Fibre Reinforced Thermoplastics (CFRTP), as adopted in Ref. [], or as core between two thin plies of carbon cross textile fibre/epoxy (CFRP/PC/CFRP) [The main concern of multi-material assemblies is the adoption of a suitable joining process. Mechanical joining processes (such as riveting or bolting) [] are often adopted for this purpose. However, the stress concentration and the long working time, due to the drilling operations, represent great limitation to their application. In recent years, fast mechanical joining methods, which do not require pre-drilled holes, were developed, such as Self-Pierce Riveting (SPR) and Mechanical Clinching (MC). Owing to their simplicity, robustness, low cost, and high productivity, many studies were conducted to adapt these processes to hybrid metal–polymer [] and metal–Fibre Reinforced Plastics (FRP) [] joints. However, these processes still suffer some drawbacks; e.g. high stress concentration (due to the spot joints) and the requirement of access from both sides of the joint. In addition, when continuous fibres are adopted as reinforcement, delamination is present near the joint []. Finally, these processes can be employed only when the material placed on the punch side has high strength and shows certain plasticity.Besides mechanical joining processes, adhesive bonding [] are also widely used. Adhesive bonding enables good stress distribution, good fatigue life, corrosion resistance, and high strength-to-weight ratio as compared to mechanically fastened joints. In addition, it does not cause stress concentration or fibres interruption. Despite these advantages, adhesive bonding shows some drawbacks and special requirements, including: substrate preparation, long processing time (due to preparation of substrate and curing time), specialized workers and involves high environmental impact, due to emission of volatile organic compounds as well as the employment of solvents used for substrates preparation []. In addition, adhesive bonding introduces a third material (the adhesive) during the process [] and bonding thermoplastic matrix composites requires special attention given the low chemical affinity of this matrix with the structural adhesives.In order to overcome the aforementioned limitations, in last years, new joining processes, such as Friction Lap Welding (FLW), friction spot welding, and friction based stacking, were developed and studied. For example, FLW was adopted to joint metals to thermoplastics []. Friction Assisted Joining was employed to join thermoplastic polymers [] and reinforced thermoplastic polymers []. Moreover, it was also adopted to joint metals with FRP with long fibres []. On the other hand, these processes enable obtaining joints with low quality finishing and they require very stiff clamping systems, due to high loads involved during the processes.A valid alternative to the aforementioned joining processes are Laser Transmission Welding (LTW) and Laser-Assisted Joining (LAJ) [] to weld thermoplastics and to join hybrid metal-plastic structures, respectively. During LTW and LAJ processes, the laser beam heats and melts the plastic component and promotes the adhesion (both mechanical and chemical). The substrates interface can be heated either by transmission of the radiation through the plastic, or by means of conduction throughout a heated metal. LAJ process is characterized by several advantages as compared to adhesive bonding, e.g. fast joining (no curing time), localized heating, no vibration, low residual stresses [] and no employment of further components (i.e. the adhesive). During LTW and LAJ the parts need to be pressed together to reduce the gap and allow the adhesion []. However, fine calibration of the laser processing parameters (e.g. power, scanning speed, stand-off-distance, clamping pressure) is required. Indeed, previous studies indicate that these parameters influence the temperature distribution inside the bonded area [], the presence of contact between the materials, the polymer crystallinity [] and, hence, the bond quality and strength.Recently, LTW was adopted to weld fibre-reinforced thermoplastics [], like Glass Fibre Reinforced Polyamide (GFR-PA66) to PC, reaching a shear strength of 4 MPa []. To estimate the temperature distribution inside the bonded area and consequently to optimize the weld performance, LTW of composites requires accurate modelling of the interaction of the laser beam with semi-transparent composites []. However, when reinforced plastics are welded, high care should be paid to the fibres orientation. Indeed, carbon fibres have great thermal conductivity compared to the matrix material. Consequently, the heat is conducted along the fibres and the weld seam geometry is affected by the local orientation of the fibres [So far, Laser Assisted Joining processes have been employed to weld thermoplastics [], Polytetrafluoroethylene (PTFE)-titanium [] as well as to join fibre-reinforced thermoplastics (PC, PA66, PE) [], the LAJ of thermoplastics sheet and composite with thermosetting matrix was reported for the first time. In that work, the process was conducted without prior removing the layer of epoxy that covered the irradiated carbon fibres. Thus, the main mechanisms developed during the laser processing were determined: heating of the carbon fibres, vaporization of the covering layer of epoxy, heating the overlying thermoplastic material and adhesion between the fibres and the thermoplastic material. Different phenomena were involved in the process that limited the strength of the joints, including: production of fumes (of both the epoxy and polycarbonate materials since the high temperatures involved), thermal degradation of the polycarbonate with formation of entrapped fumes at the interface and bubbles, which reduced the effective adhesion area, degradation of the epoxy layer underlying the layer of exposed carbon fibres. These issues were found under all processing conditions. On the other hand, when low energies were adopted, poor epoxy was removed from the interface layer. This affected the joint strength as it reduced the adhesion area between the PC and the carbon fibres. Generally, in order to achieve the formation of the joint, the processing conditions should lead to the removal of the epoxy layer. To this end, a temperature of at least 400 °C should be reached. Such temperature is very close to that of degradation of the PC material (540 °C).The present study deals, for the first time, on the effect of laser pre-treatment on the joint strength of PC sheet to CFRP laminate, obtained by Laser Assisted Joining. To this end, laser joining at different process conditions was performed on autoclave cured CFRP laminate and PC sheets, with and without a laser cleaning pre-treatment of the CFRP surface. The laser pre-treatment was carried out adopting a 30 W Q-switched Yb:YAG fiber laser; while the joints were performed adopting a 200 W diode laser.After joints production, mechanical tests were carried out; ultimate tensile strength and apparent shear strength were measured. Moreover, fracture surfaces analysis was performed. Infrared thermography (IRT) was also performed in order to understand the interaction phenomena occurring during the laser joining. ANalysis Of VAriance was applied to study the influence of the process parameters (laser power, energy, and treatment), on the bonded area extension and the mechanical properties.Polycarbonate (PC) sheets, 2.0 mm in thickness, supplied by Bayer, were coupled to CFRP laminates with thickness of 1.5 mm. The CFRP laminate was manufactured by means of hot pressing for 2 h at 130 °C and 5 MPa, as suggested by the resin supplier. Plain weave (SK Chemicals, UGN200) carbon fibre prepegs (0/90°, 50% in the warp and weft directions, MRC Pyrofil, TR30S) and a thermosetting epoxy resin (bisphenol-A type epoxy + phenol novolac type epoxy) were used for the production of the CFRP laminate []. The thermal characteristics of the materials are summarized in the laser system characteristics are summarized. In order to treat the entire surface to be joined, more parallel lines, spaced of fixed quantities (called hatch distance, Hd), were performed, according to the procedures described in Refs. [. During the cleaning treatment the following parameters were adopted: hatch distance of 40 μm, beam speed of 2000 mm/s, average power of 30 W, pulse frequency of 30 kHz and pulse duration of 50 ns. Under these conditions the pulse power resulted of 20 kW and the pulse energy of 1 mJ. These values were selected on the basis of previous experiences [] and preliminary tests. Under these conditions, it was possible to remove the first matrix layer up to the fibres exposition, avoiding apparent damages, according to [ shows the CFRP laminate surface in both the analysed conditions: “As Received” (AR) and “Laser Cleaned” (LC).A diode source (DLR-200-AC by IPG) with maximum power of 200 W was used to perform Laser-Assisted Joining process. A collimator was adopted to obtain a laser beam with circular shape of 6 mm in diameter. In the laser system characteristics are summarized.CFRP samples, 40 mm × 80 mm in plane dimensions, were placed in a clamping system, mounted in a 3 + 1 CNC axis machine, with maximum speed of 200 mm/s. The PC sheets were placed on the CFRP samples with an overlap of 40 mm. The sheets were clamped with a pressure of about 1 MPa; such a value was selected on the bases of preliminary tests. The laser beam was moved at the centre of the overlapping length area. In a schematic of the Laser Assisted Joining setup is depicted. Note that the PC sheet was placed at the laser beam side (being almost transparent to the diode laser radiation []). The adopted setup enabled to produce joints similar to those suggested by ASTM the sample geometry, with the dimensions, is shown. In the figure, the central area namely “Bonded Region” represents the irradiated area.In order to investigate the effect of the surface treatment and the effect of the process parameters adopted in the LAJ process, a full factorial plan was developed and tested, according to the Design of Experiments (DOE) methodology. The levels of the experimental plan, based on preliminary tests and previous experiences [. The tests were conducted by varying: the CFRP surface condition (AR/LC), the laser power (P) and the linear energy (LE = P/Ss, were Ss is the scanning speed). It worth noting that the choice of the LE values was driven by the previous experience [], where the maximum strength was reached for LE = 4.5 J/mm.After the joining, single lap shear tests were carried out adopting a universal test machine, model 322.31 by MTS. The tests were conducted under quasi-static conditions (constant transverse speed of 0.5 mm/min). During the tests, the load and the displacement were recorded. Three replicates were performed for each testing condition; average and standard deviation of the Ultimate Tensile Force (UTF) and the apparent shear strength (τa) were determined. UTF was calculated as the maximum of the load measured during the test, while, τa was calculated as the ratio between the UTF and the joined area (which was measured after the test).After the mechanical tests, a digital microscope (KH-8700 by Hirox), equipped with a 2.11 mega-pixel CCD sensor and motorized x-y table, was used to measure the adhesion area and to study the joint morphology.Analysis of variance (ANOVA) was conducted to test the statistical significance of the process parameters on both the geometric and mechanical characteristics of the joints. The analysis was carried out at a 95% confidence level (α = 0.05). ANOVA assumes that the observations are normally and independently distributed with the same variance for each treatment or factor level. Before the analysis, these assumptions were successfully checked via analysis of residuals. Moreover, the main effects plot and the interaction plot were adopted to analyse the effect of the process parameters.In addition, to better understand the interaction phenomena occurring during the laser joining, the measure of the temperature reached on the laminate surface during the joining was carried out by Infrared thermography (IRT), adopting a IR-camera (model A655SC by FLIR), equipped with 24.5 mm lens.As the PC is not transparent to the mean IR radiation (the camera has a spectral range of 7.5–14.0 μm), the thermal analysis was performed on the CFRP laminate without the PC sheet. This enabled to determine the temperature of the CFRP surface (as the poor thermal conductivity of the PC). Because of the great transmittance of the PC at the wavelength of λ = 975 nm (almost 90% []), the test were carried out at an equivalent laser power (P*) lowered by 10% with respect to that used during the LAJ tests. During the tests, the IR camera was placed at an angle of 60° with respect to the plane of the CFRP laminate and a distance of 250 mm. During the thermal analysis, shielding gas (N2 at 2 bar) was adopted to avoid excessive formation of plasma. show the typical appearance of the PC surface after the single-lap shear tests for the as received (AR) and laser treated (LT) samples, respectively. For the AR samples (), the surfaces appearance is very similar to those reported in Ref. []. In particular, in the bonding area two different regions are visible: the first is placed at the centre of the irradiated area and a second one all around. The first region is characterized by the presence of fibre fragments coming from the CFRP laminates or by their imprint. In this area the CFRP is directly exposed to the laser radiation; the temperature reached by the fibres is so high to cause both the vaporization of the epoxy matrix and the PC softening. This produced an intimate contact between the softened polycarbonate and the heated fibres; thus, the bonding between the two materials is obtained.The second region is characterized by the yellow surrounding area; in this region, the achieved temperature is not enough to produce the epoxy matrix vaporization because this area lies outside that directly irradiated by the laser beam. Thus, the heating occurs only by conduction. Moreover, due to the absence of direct contact between the fibres and the PC (as the presence of the epoxy resin layer that covers the carbon fibres), the temperature reached by the PC is lower than the first area. This leads to insufficient material softening. In this region, the vapours, produced by epoxy degradation and vaporization in the central zone, tend to flow sideways from the irradiated area and to condense. In these regions, the formation of an intimate contact between the PC and the laminate is even more difficult.Similar remarks can be done in the case of laser cleaned surfaces (), even if the first area appears more regular along the scanning direction and an increase of the numbers of fibres attached to the PC is observable; this is due to the absence of epoxy matrix layer. Moreover, the yellowed area appears smaller in width and characterized by a comb shape. When the CFRP is cleaned from the superficial layer of epoxy, less vapour is produced during the laser joining as there is less amount of epoxy resin to vaporize; thus, initial cleaning has left voids between the weft and the warp from where the vapour can flow out.The ANOVA was performed on the measured joined area width, to study the effect of the process parameters (P, LE, Treatment) on the bonding regions dimension. The result of the analysis is reported in . On the basis of assumptions made (α = 0.05), all the process parameters are statistically significant (i.e the p-value is less than 0.05). Moreover, also the interaction between the linear energy and the surface treatment results significant. the main effects plot and the interaction plot (only significant interactions) for the width are reported, respectively. From , as expected, the width increases at the increasing of the linear energy and the average power, or when the laser pre-treatment is adopted on the laminate (LC). The significant interaction (LE*Treatment) (shown in ) reveals that on the untreated laminate (AR), the width remains almost constant at the LE changing. Conversely, when the laser cleaned laminate is adopted (LC), the increase of the width is more consistent (The ANOVA results on ultimate tensile force and apparent shear strength results are summarized in . The results indicate that, the linear energy, the surface treatment and the interaction between the linear energy and the surface treatment are statistically significant for the UTF (). On the other hand, the apparent shear strength is affected only by the linear energy ( the main effects plots for the ultimate tensile force (UTF) and the apparent shear strength (τa) are reported, respectively. In the plots, the significant parameters are highlighted using continuous lines. The figures show that, both UTF and τa decrease with increasing LE. In addition, the ultimate tensile force is much higher for treated (LC) as compared to as received (AR) condition. depicts the significant interaction (LE*Treatment) for the UTF. Similarly to what observed in the case of the width, the UTF of LC samples is more sensible to the linear energy changing as compared to the AR laminate ( depicts the IR image of a CFRP specimen during laser irradiation, under P* = 200 W and LE = 5 J/mm. The IR maps show the presence of vapours, which are due to the degradation and vaporization of the epoxy matrix, covering the specimen even though the adoption of a shielding gas. These vapours exert a shield effect. Actually, they absorb the infrared radiation and prevent the correct absorption of the laser radiation as well as temperature measurement issues. Consequently, the thermal analysis is able to provide only qualitative and comparative information about LAJ performed on AR and LC samples. The maximum temperature versus the process time is reported in for the different process conditions. In the figures, the curves are shifted for more clarity. The maximum temperature tends to increase with increasing LE; moreover, the temperature measured on LC samples was always higher than that observed on AR samples. Actually, when LAJ is performed on AR samples, part of the laser energy is spent to remove the epoxy resin on the laminate surface. This limited the temperature at the CFRP/PC interface. shows the acquired temperature in the transverse direction with respect to the joining one, just after the beam passing for LE = 5, P* = 100 W, and LE = 3, P* = 200 W. In the figure the dashed lines represent the melting (Tm) and the degradation (Td) temperatures of PC (230 °C and 540 °C, respectively). The figure clearly indicates for LC samples a slight enlargement of the areas where Tm and Td intersect the temperature profile. Similar results were obtained for the other combinations. However, these diagrams were not reported for sake of briefness.The influence of the process parameters on the width of the joint region can be explained considering that: the increase of LE involves an increase of the energy released on the same surface. This comes with an increase of the maximum temperature and, consequently, an enlargement of the area where the same temperature is reached (The use of higher power levels involved the adoption of higher scan speeds (at the same LE) and, consequently, lower interaction times. Under these conditions, the heat has less time to diffuse into the inner layers and tends to be restricted on the surface of the CFRP. This results in a larger joined area (i.e. width) that exceeds the threshold temperature required for joining. It is worth noting that, not all the heated area concurs in the joining. In the central part (in correspondence of the beam axis) PC degradation occurs while peripheral regions do not reach the PC melting temperature. These region have a marginal contribute to the strength of the joint.The adoption of the laser cleaning involves a direct contact between the carbon fibres and the PC. This leads to a more efficient heating of the PC and to a better gas evacuation (as confirmed by the analysis of ). Moreover, the part of the energy that was lost in the epoxy matrix degradation (i.e. vaporization) becomes available to heat the fibres. This also influences the joined area (i.e. the width) that is larger for LC samples.The LE*Treatment interaction can be explained as the following: in the case of AR laminate, the laser radiation vaporizes only the portion of epoxy matrix directly exposed to the radiation. Consequently, in the surrounding area, the bonding is prevented by the resin layer that, independently from the temperature achieved, covers the carbon fibres. On the contrary, in the case of LC laminate, the absence of the first matrix layer facilitates the joining between the PC and the carbon fibres, making the joint width more sensitive to the increase of the linear energy.However, the increase of the bonding area (i.e. width) does not necessarily involve an increase of UTF. Actually, UTF depends on several factors, including: the dimension of the adhesion area, the adhesion of the PC to the carbon fibres, the extension of the area where PC degradation occurs, as well as the laminate degradation.The increase of the maximum temperature, due to increase of LE or P, produces a larger joined area and width where the degradation temperature of the PC rises (as showed in ). Moreover, here, during the joining (i.e. the diode laser irradiation), the laminate undergoes a strong thermal stress. This over-heating may result in degradation/sublimation of the matrix placed under the exposed fibres, leading to formation of cavities, according to the observations reported in Refs. []. Thus, it is reasonable to suppose that a reduction in the load capability of the fibres, which detach from the underlying layers, may occur. This effect is higher when the laminate is cleaned (i.e. without the first matrix layer that covers the fibres), because of the highest temperature (). Thus, an increase of the linear energy or of the laser power, lead to the increase of both the apparent adhesion surface (i.e. width) and the degradation region, these aspects having adverse effects on the UTF. The increase in LE damages both the materials (laminate and PC) leading to a reduction of UTF. This reduction is partially compensated by the increase in the adhesion surface which occurs when the laminate is cleaned (LC). As a consequence, the laser cleaning treatment, promoting the increase in the adhesion area, results in an improvement in the UTF. Conversely, within the analysed processing window, considering the apparent shear strength, laser pre-treatment does not produce a significant advantage.From the process point of view, although the apparent shear strength does not increases when laser cleaning is adopted, from a practical point of view, the laser cleaning results in a reduction of the process time as the same joint resistance can be obtained by way of a single scanning line or highest scan speed.Laser Assisted Joining Process (LAJ) between CFRP and PC sheet was investigated under different process conditions with and without laser cleaning pre-treatment of CFRP. Single lap joint test was performed to determine the quality of the joints. From the results, for the adopted laser sources and under the experimental conditions adopted here, the main conclusions are the following:laser cleaning process enables removing the first epoxy matrix layer from the laminate. This has great benefits on the joints morphology since the complete absence of degraded epoxy within the joined region, the easy removal of vapours due to the polymers degradation, as well as the increase of the joined area;under the adopted conditions, the temperatures reached on the CFRP during the joining, may arise the PC degradation temperature and produce damages on the CFRP laminate. Thus, in the centre of the bonded region a damaged area can be produced. This reduces the overall joint efficiency;although the increase of the linear energy involves an increase of the apparent bonded area (i.e. the joint width), for both the types of joined samples (AR and LC), the ultimate tensile failure load and the apparent shear strength decrease with the increasing of the linear energy;the average power does not result to be a statistically significant neither for UTS, neither for the apparent shear strength;in the industrial applications, the use of laser cleaning results in a reduction of the process time as the same joint resistance can be obtained by way of a single scanning line or highest scan speed.Destructive and nondestructive evaluations of the effect of moisture absorption on the mechanical properties of polyester-based compositesIn this study the effect of moisture absorption on the mechanical properties of glass-reinforced polyester composites is evaluated using both destructive and nondestructive tests. The composite resins were produced with two different production processes, while the mechanical properties of the composite materials were measured using DMA destruction tests. According to the DMA tests, the dependency in terms of temperature for the real component of the complex elastic modulus (E′), the imaginary component of the complex elastic modulus (E″), as well as tan(δ) can be traced. For a more efficient use of the composite materials, the compliance tensor was obtained with nondestructive tests based on ultrasound. A method for the generation and reception of Lamb waves in plates of composite materials is described, based on using air-coupling, low-frequency, ultrasound transducers in a pitch–catch configuration. The results of the nondestructive measurements made in this study are in good agreement with those obtained when using the DMA destructive tests.The mechanical properties of a material determine its manufacturability, performance, and longevity. This means that a knowledge of the mechanical properties is essential for making good design decisions. Polymers are exceptionally complex materials – their mechanical properties depend on chemistry, processing, and the thermo-mechanical history, as well as on volume constraints. Thus, in order to gain useful information for making sound decisions when designing with polymer composites, mechanical property measurements should be made on a relevant sample in a relevant context The performance of GFRP pipes is critical in many engineering applications when they are subjected to a combination of high-temperature/high-humidity environments. The diffusion of water or an aqueous fluid into GFRP pipes may lead to changes in the thermo-physical, mechanical, and chemical characteristics. Many of these changes can result in a degradation of the material’s performance. In order to properly predict the service life of a GFRP pipe, we need to understand the mechanisms that govern these changes.Water absorption by the resin may cause both reversible and irreversible changes to the resin, including hydrolysis, plasticization, micro-cracking, and even the glass-transition temperature Samples of GFRP plates having as their reinforcement six sheets of ravings with 250 ± 50 gm−2 density and a matrix made from different types of unsaturated Orthophthalic polyester resins, made by Helios, Slovenia, were used in this study. The characteristics of the studied GFRP samples are presented in The effect of water absorption on the GFRP composites was investigated. Initially, DMA measurements on non-immersed samples and immersed samples were conducted on unconditioned samples. The obtained results were inconclusive. For this reason, the measurements were remade, this time on conditioned samples, i.e., with the water that might be in the initial samples being eliminated.The conditioning involved maintaining all the samples in a drying oven at 50 °C for 5 days. For each type of composite, one sample was weighed in order to record the initial mass. This sample was marked. The immersion was carried out in distilled water and the immersion periods were 3, 6, 9, 12, 20, 40 and 100 days. As a result of a mistake, the sample from the 7524 composite immersed for 100 days is missing from the results.After each period of immersion, two examples of each type of sample were removed, one of samples was used for the determination of the adsorbed water and the second one was used for the DMA measurements. The adsorbed water was determined from weighing samples after each period.where M0 is the mass of the non-immersed sample after the conditioning and Mt is the mass of the same sample immersed for time t. presents the dependency of Mt(%) on the immersion time for the three types of composites used in the study. denote linear Fickian behavior, where the moisture-related weight gain gradually attains equilibrium after a rapid initial take off.In order to determine the water-diffusion coefficient in a composite, the data from were converted so that the abscissa of the graphic in It is assumed that the studied composites absorb the water in accordance with the law of Fick, as shown in Shen et al. where M∞ represents the maximum quantity of water that diffuses into the composite and h is the thickness of the sample. The values for the diffusion coefficients are given in The measurements of the elastic modulus along the three directions were made with a Dynamic Mechanical Analyzer DMA 242C from Netzsch, Germany with a three-point bending fixture and using the analysis software Proteus v.4.8.5. The measurements were carried out at a frequency of 1 Hz.Polymers are often employed in products because of their ability to both store and damp energy. The complex modulus E∗ is a phase vector that incorporates both capacities:The real part (E′) of the complex modulus is called the storage modulus because it quantifies the material’s ability to store energy elastically. In materials with insignificant damping, the storage modulus is equivalent to the Young’s modulus. The imaginary part of the complex modulus (E″) is called the loss modulus, because it quantifies the material’s ability to dampen out the energy. The dimensionless loss factor tan(δ) is independent of the contact energy, because it is the ratio of the loss to the storage modulus:The mechanical performance of a composite material depends strongly on its glass-transition temperature, Tg. Above this temperature the mechanical properties decrease rapidly. Therefore, it is necessary to check the influence of the moisture on the glass-transition temperature of the GFRP specimen. The glass-transition temperature of the samples was determined with a DMA instrument. Both dry and immersed specimens were tested to determine the aging produced by water immersion. present the DMA results for dry and wet GFRP specimens. The storage modulus, the loss modulus, and the loss factor are taken as the ordinate, while the temperature is taken as the abscissa. Some observations were made. First, there are some changes with respect to the storage modulus, the loss modulus, and the loss factor, i.e., the peak value of the loss factor. At relatively low temperatures (0–40 °C) the polymer is stiff and has a high storage modulus E′ and a low modulus E″. The chains are frozen in fixed positions because insufficient energy for the translational and rotational motions of the polymer segments is available. As the temperature increases, the polymer obtains sufficient thermal energy to enable its chains to move more freely. At temperatures higher than 100 °C the storage modulus decreases to about 500 MPa. The increase in E′ after 3 days of immersion is due to the fact that the water acts as plasticizer, initially, and then, once with the increasing of the immersion time, E′ decreases (). In analogy with this process a transition occurs at temperatures between 60 and 100 °C, and the loss modulus E″ goes through a maximum. This region is referred to as the glass–rubber transition or glass transition Tg and can be qualitatively interpreted as the onset of large-scale conformational rearrangements of the polymer chain’s backbone. The temperature at which this peak maximum is observed is conventionally defined as Tg. shows the loss factor tan(δ) of both dry and wet specimens. The peak value of tan(δ) is considered as the glass-transition temperature. For each immersion time and each composite, the glass-transition temperature Tg and the activation energy for the glass transition Eg as the basis of the Arrhenius rule were determined. One can see clearly that moisture absorption by the specimen results in a change of Tg, which indicates a degradation process in the material (). It can also be seen that the water acts initially as a plasticizer.However, after 100 days of immersion in water, the glass-transition temperature is almost the same as it is for the non-immersed specimens ( also presents the glass-transition activation energy vs. the immersion time for all three composites.In order to measure the propagation speed of the ultrasound longitudinal and transversal waves we used a Pulser Receiver 5073 PR Panametrics NDT, USA, in impulse–echo mode, with the signal received being visualized on Digital Oscilloscope LeCroy WaveRunner 64Xi with a 600-MHz passband and a 10 GS/s sampling rate. A transducer G5KB–GE USA was used for the generation of the longitudinal waves with ZG-F gel Krautkramer, Germany for the coupling and MB4Y-GE, USA for the transversal waves using honeybee wax as the coupling. The transducers were placed on a delay block made from Plexiglas with a thickness of 20 mm.The principle scheme of the equipment is presented in a, and the physical realization is presented in The measurements for the phase velocity of the Lamb waves, modes A0 and S0 were made using air-coupling US transducers, pitch-catch configuration, NCG100S25–UltranGroup, USA, having a central frequency of 100 kHz. The critical incident angle was calculated according to Eq. , considering cair |
= 340 m/s and cs having the value obtained by a determination of the transversal waves-propagation speed. The value of θcritic was 10.2°.The incident angle, θ, is chosen so θ |
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