Mechanisms of compressive deformation and failure of porous bulk metallic glasses

The major kinematic idea of the constitutive theory is the multiplicative decomposition of deformation gradient into elastic and plastic parts.

$$\begin{eqnarray}&&{\boldsymbol{F}}={{\boldsymbol{F}}}^{{\boldsymbol{e}}}{{\boldsymbol{F}}}^{{\boldsymbol{p}}},\end{eqnarray} \tag{ 1 }$$

where, ${\boldsymbol{F}}$ is the deformation gradient. The spatial velocity gradient is

$$\begin{eqnarray}&&{\boldsymbol{l}}=\dot{{\boldsymbol{F}}}{{\boldsymbol{F}}}^{-1}={{\boldsymbol{l}}}^{{\boldsymbol{e}}}+{{\boldsymbol{l}}}^{{\boldsymbol{p}}},\end{eqnarray} \tag{ 2 }$$

where, ${{\boldsymbol{l}}}^{{\boldsymbol{e}}}=\dot{{{\boldsymbol{F}}}^{e}}\mathop{{{F}^{e}}^{-1}}\limits^{}$, ${{\boldsymbol{l}}}^{{\boldsymbol{p}}}={{\boldsymbol{F}}}^{{\boldsymbol{e}}}{{\boldsymbol{L}}}^{{\boldsymbol{p}}}{{{\boldsymbol{F}}}^{{\boldsymbol{e}}}}^{-1}$, ${{\boldsymbol{L}}}^{{\boldsymbol{p}}}={\dot{{\boldsymbol{F}}}}^{{\boldsymbol{p}}}{{{\boldsymbol{F}}}^{{\boldsymbol{p}}}}^{-1}$.

The plastic volumetric strain is defined by

$$\begin{eqnarray}&&\eta =\mathrm{ln}{J}^{p},\end{eqnarray} \tag{ 3 }$$

where, ${J}^{p}=\det \ {{\boldsymbol{F}}}^{{\boldsymbol{p}}}$.

The free energy function is

$$\begin{eqnarray}&&\psi =\hat{\psi }({{\boldsymbol{F}}}^{{\boldsymbol{e}}},\eta )\end{eqnarray} \tag{ 4 }$$

which also satisfies the relation

$$\begin{eqnarray}&&\psi =\tilde{\psi }({{\boldsymbol{b}}}^{{\boldsymbol{e}}},\eta ),\end{eqnarray} \tag{ 5 }$$

where, ${{\boldsymbol{b}}}^{{\boldsymbol{e}}}={{\boldsymbol{F}}}^{{\boldsymbol{e}}}{{{\boldsymbol{F}}}^{{\boldsymbol{e}}}}^{{\boldsymbol{T}}}$, the elastic left Cauchy–Green deformation tensor, in order to meet the requirements of isotropy and objectivity. Since ${{\boldsymbol{b}}}^{{\boldsymbol{e}}}$ can be expressed in terms of elastic principal stretches ${\lambda }_{A}^{e}$, A ∈ $\{1,2,3\}$,

$$\begin{eqnarray}&&\psi =\bar{\psi }({\lambda }_{1}^{e},{\lambda }_{2}^{e},{\lambda }_{3}^{e},\eta ).\end{eqnarray} \tag{ 6 }$$

Dissipation per unit reference volume is given in terms of Kirchhoff stress ${\boldsymbol{\tau }}$ and rate of deformation tensor '${\boldsymbol{d}}$' [28] as

$$\begin{eqnarray}&&D={\boldsymbol{\tau }}\,:{\boldsymbol{d}}-\dot{\psi }\geqslant 0.\end{eqnarray} \tag{ 7 }$$

The above equation can be simplified [28] to get the constitutive relation after applying the Coleman–Noll argument [28]

$$\begin{eqnarray}&&{\boldsymbol{\tau }}=2\displaystyle \frac{\partial \hat{\psi }}{\partial {{\boldsymbol{b}}}^{{\boldsymbol{e}}}}{{\boldsymbol{b}}}^{{\boldsymbol{e}}}.\end{eqnarray} \tag{ 8 }$$

Then, the expression for dissipation can be rewritten as

$$\begin{eqnarray}&&D={\boldsymbol{\tau }}\,:{{\boldsymbol{d}}}^{{\boldsymbol{p}}}+c\dot{\eta },\end{eqnarray} \tag{ 9 }$$

where, ${{\boldsymbol{d}}}^{{\boldsymbol{p}}}=\mathrm{symm}({{\boldsymbol{l}}}^{{\boldsymbol{p}}})$ and $c=-\tfrac{\partial \hat{\psi }}{\partial \eta }$, is the 'stress-like' internal variable called cohesion.

The spectral decomposition of ${\boldsymbol{\tau }}$ can be expressed as

$$\begin{eqnarray}&&{\boldsymbol{\tau }}=\displaystyle \sum _{A=1}^{3}{\hat{\tau }}_{A}{{\boldsymbol{n}}}_{{\boldsymbol{A}}}\otimes {{\boldsymbol{n}}}_{{\boldsymbol{A}}},\end{eqnarray} \tag{ 10 }$$

where, ${\hat{\tau }}_{A}$, A ∈ $\{1,2,3\}$ are the principal Kirchhoff stresses and ${{\boldsymbol{n}}}_{A}$ are the eigenvectors of ${\boldsymbol{\tau }}$. From equations (6) and (8),

$$\begin{eqnarray}&&{\hat{\tau }}_{A}={\lambda }_{A}^{e}\displaystyle \frac{\partial \bar{\psi }}{\partial {\lambda }_{A}^{e}}=\displaystyle \frac{\partial \bar{\psi }}{\partial {\hat{\varepsilon }}_{A}^{e}},\end{eqnarray} \tag{ 11 }$$

where, ${\hat{\varepsilon }}_{A}^{e}$ for A ∈ $\{1,2,3\}$ are the principal logarithmic elastic stretches given by

$$\begin{eqnarray}&&{\hat{\varepsilon }}_{A}^{e}=\mathrm{ln}({\lambda }_{A}^{e}).\end{eqnarray} \tag{ 12 }$$

The relationship between $\hat{{\tau }_{A}}$ and $\hat{{\varepsilon }_{A}^{e}}$ is taken to be governed by the isotropic Hooke's Law [29].

The flow rule corresponding to the plastic part is given as

$$\begin{eqnarray}&&{{\boldsymbol{l}}}^{{\boldsymbol{p}}}=\displaystyle \sum _{\alpha =1}^{6}{\dot{v}}^{(\alpha )}\{{{\boldsymbol{s}}}^{(\alpha )}\otimes {{\boldsymbol{m}}}^{(\alpha )}+\beta {{\boldsymbol{m}}}^{(\alpha )}\otimes {{\boldsymbol{m}}}^{(\alpha )}\},\end{eqnarray} \tag{ 13 }$$

where, β is the dilatancy function, ${{\boldsymbol{s}}}^{(\alpha )}$ and ${{\boldsymbol{m}}}^{(\alpha )}$ are the slip direction and the unit slip plane normal vectors, respectively, defined with respect to the principal directions of stress, and ${\dot{v}}^{(\alpha )}$ is the plastic shearing rate for the αth slip system.

The shearing rate is defined by the viscoplastic law as

$$\begin{eqnarray}&&{\dot{v}}^{(\alpha )}=\dot{{v}_{0}}{\left\{\displaystyle \frac{{\tau }^{(\alpha )}}{c+\mu {\sigma }^{(\alpha )}}\right\}}^{1/m}\geqslant 0,\end{eqnarray} \tag{ 14 }$$

where, $\dot{{v}_{0}}$ is a reference plastic shear strain rate. ${\tau }^{(\alpha )}$ and ${\sigma }^{(\alpha )}$ are shear and compressive normal tractions on the αth slip system and c is cohesion, defined as the yield strength in pure shear. μ is a material parameter defined as internal friction and when ${\tau }^{(\alpha )}=c+\mu {\sigma }^{(\alpha )}$, ${\dot{v}}^{(\alpha )}=\dot{{v}_{0}}$ for $m\gt 0;$ m being the strain rate sensitivity parameter. The dilatancy function β and cohesion c, are assumed to be exponential functions of plastic volumetric strain η as

$$\begin{eqnarray}&&\beta =\displaystyle \frac{{g}_{0}}{(e-1)}\left\{{e}^{\left(1-\tfrac{\eta }{{\eta }_{{\rm{cv}}}}\right)}-1\right\},\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&c={c}_{{\rm{cv}}}+\displaystyle \frac{b}{(e-1)}\left\{{e}^{\left(1-\tfrac{\eta }{{\eta }_{{\rm{cv}}}}\right)}-1\right\}.\end{eqnarray} \tag{ 16 }$$

As the volumetric strain increases from η to a saturation value ${\eta }_{{\rm{cv}}}$, β varies smoothly from g₀ to 0 and c from c₀ to c_cv, where initial cohesion ${c}_{0}={c}_{{\rm{cv}}}+b$.

The above constitutive model is numerically implemented in a thoroughly tested and benchmarked user-subroutine UMAT in a commercial finite element software ABAQUS [27] using a fully implicit backward-Euler update algorithm.

The complete stress–strain curve till final failure is simulated by adopting the damage model proposed by Anand and Su [26], described as follows.

The equivalent plastic shear strain rate $\tilde{{\gamma }^{p}}$ is defined as

$$\begin{eqnarray}&&\tilde{{\gamma }^{p}}\mathop{=}\limits^{\mathrm{def}}| {{\bf{D}}}_{0}^{p}| ,\end{eqnarray} \tag{ 17 }$$

where, ${{\bf{D}}}_{0}^{p}$ is the deviatoric part of plastic part of ${{\boldsymbol{L}}}^{p}$. Thus, equivalent plastic shear strain ${\gamma }^{p}$ is defined as

$$\begin{eqnarray}&&{\gamma }^{p}(t)\mathop{=}\limits^{\mathrm{def}}{\int }_{0}^{t}\tilde{{\gamma }^{p}}(\chi ){\rm{d}}\chi .\end{eqnarray} \tag{ 18 }$$

A damage variable d is defined as

$$\begin{eqnarray}&&d=\displaystyle \frac{{\gamma }^{p}-{\gamma }_{c}^{p}}{{\gamma }_{f}^{p}-{\gamma }_{c}^{p}}\end{eqnarray} \tag{ 19 }$$

such that d = 0 for ${\gamma }^{p}$ less than a critical value of plastic shear strain ${\gamma }_{c}^{p}$ and d = 1 as ${\gamma }^{p}$ evolves to failure value ${\gamma }_{f}^{p}$. After damage initiation, the cohesion c evolves linearly with the damage variable d and decays to zero as d increases to unity. An integration point is removed from calculation when the damage variable d at that point exceeds a critical value of 0.97. When all the integration points in an element are removed from calculation, the element is deleted from analysis. This signifies development of a crack in the section.

The above damage model along with the constitutive model presented earlier are again numerically implemented in ABAQUS/Explicit by writing another user material subroutine VUMAT. It should be noted that the above damage model represents the mechanism of ductile fracture of BMGs involving development of ductile shear cracks inside shear bands. All the simulations reported in the present work are carried out using ABAQUS/Explicit.

Thamburaja [30] proposed a non-local, finite deformation constitutive model for metallic glasses which includes mechanisms such as non-local interaction stress and free volume diffusion. Through finite element simulations, he showed that metallic glasses exhibit delayed shear localization or homogeneous deformation in sub-micron or nano-sized samples. Indeed, in the experiments of Jang and Greer [31] and Jang et al [32], the various size dependent phenomena in metallic glasses such as increase in yield strength with decrease in sample size are observed at length scales less than or equal to 1 μm. However, the present work attempts to capture the uniaxial compression behavior of porous BMGs containing pores having diameters ranging from 9 to 35 μm (see Wada et al [14, 21]). Thus, at such bigger length scales, mechanisms such as non-local interaction stress and free volume diffusion will have only negligible effect on the overall deformation behavior of the samples. Hence, in the present work, a local constitutive model proposed by Anand and Su [26] has been employed which is appropriate for the bigger length scales over which the present simulations have been carried out.

Uniaxial compression loading of low porosity (f = 1.9%, 3.6%, 5.7% and 20%) BMGs is simulated by the finite element method. The stress–strain curves obtained from the simulations in terms of macroscopic nominal stress versus macroscopic nominal strain are plotted in figure 2(a) for all the four low porosity cases. The stress–strain curves reported in Wada et al [14, 21] for the same porosities are also plotted in figure 2(a) for comparison purposes. The experimental stress–strain curve for the f = 5.7% porosity case was used to calibrate the material properties used in the present work. As expected, the simulated and experimental stress–strain curves for 5.7% porosity match very well with each other in terms of elastic modulus, yield strength, post yield response and failure strain. The simulated stress–strain curve for 3.6% and 20% porosity is also found to agree quite well with the corresponding experimental stress–strain curve (see figure 2(a)). This gives confidence in the calibrated material properties. However, the experimental stress–strain curve for 1.9% porosity case shows very little failure strain as compared to other low porosity BMGs. This is different from the trend of failure strain exhibited by 3.6% and 5.7% porosity cases. This discrepancy might have been caused by any defects present in the single sample of that porosity that was tested in [14]. Hence, the simulated stress–strain curve for 1.9% porosity case over-predicts the failure strain while the elastic modulus and yield strength are accurately captured.

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According to the present damage model, after damage initiation, the cohesion decreases linearly with the damage variable to zero causing strong softening in the response. While this softening due to damage may promote strain localization, it should be mentioned that in the constitutive model of Anand and Su [26], the plastic dilatation causes strain softening even before damage initiation. Furthermore, the use of Mohr–Coulomb type yield criterion and non-associated flow rule promotes strain localization inside shear bands irrespective of the damage model. The points delineating damage initiation and final fracture are indicated in figure 2(a). The softening in the damage model begins only after the damage initiation point marked on the curves.

The void volume fraction evolution of low porosity BMGs is also studied. Figure 2(b) shows the evolution of void volume fraction with respect to nominal strain obtained from the unit cell simulation study of low porosity BMGs. Points A and B in figure 2(b) indicate the void volume fraction at yield and failure points of the unit cell, respectively. At point A, the void volume fraction for the three simulations has reduced by 2% and at point B, the reduction in void volume fraction is by 45%. Consistency in the values of void volume fraction reduction at yielding and failure for all the three porosity cases, provides confidence in the simulation result for 1.9% porosity case.

The current model used follows a rate dependent viscoplastic law. Mesh size effects encountered in rate independent solids are not observed in rate dependent solids, as the boundary value problems remain well posed. It has been shown that, rate dependence introduces an artificial length scale into the boundary value problem irrespective of whether the constitutive model contains length scale parameter [40]. A mesh sensitivity study has been carried out for porous BMGs having f = 5.7%, through unit cell simulations. A coarse mesh model having 71 928 elements and fine mesh model having 114 000 elements are developed. The stress–strain curve obtained from the uniaxial compression of these unit cells match very well till damage initiation. The qualitative behavior post damage initiation is also found to be similar. This reveals that the overall behavior of the unit cell is independent of the mesh.

In the case of high porosity BMGs or 'BMG foams', the initial porosities are greater than 20%. Experiments of Wada and co-workers [21] show that the yield strength of BMG foams is much less than that of low porosity BMGs but they are more malleable and exhibit larger strain hardening. Figure 3(a) shows the simulated compressive stress–strain curves for BMG foams having porosities 41% and 60%. The experimental compressive stress–strain curves from Wada et al [21] for BMG foams having the same porosities are also presented in figure 3(a) for comparison purposes. It can be observed from this figure that the simulated stress–strain curves match reasonably accurately with the experimental stress–strain curves. In particular, the simulated and experimental curves match very well with each other till a nominal strain of 20%. Beyond 20% nominal strain, the simulated stress–strain curves capture the essential features of the experimental stress–strain curves such as apparent strain hardening very well, qualitatively.

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It should be noted that in the experiments of Wada et al [21], the high porosity BMG foams do not fail until a very large compressive strain. This is because the thin walls separating the cells in the BMG foam become plastic and there is hardly any formation of a dominant shear band. The damage model implemented in the present work is meant to simulate ductile shear cracking inside a dominant shear band. Thus, the present damage model is not applicable to BMG foams. Hence, in the simulations of uniaxial compression of BMG foams, the damage model is not employed. The mechanism of deformation of BMG foams will be discussed in detail in the next section.

Figure 3(b) shows the evolution of void volume fraction of BMG foams having porosity 41% and 60%. The void volume fraction evolution for BMG foams having initial porosities 41% and 60% are significantly different from that of low porosity BMGs. It is seen from figure 3(b) that the current void volume fraction first increases steadily till point B before it begins to drop smoothly. Point A indicates the void volume fraction at yield point. The percentage increase in peak void volume fraction rises with inital porosity, i.e., while void volume fraction at point B, for 41% inital porosity case rises by 4.45%, that for 60% initial porosity case rises by 11.11%. It should be noted that the current void volume actually decreases with applied nominal strain but the current unit cell volume decreases at a faster rate than that of the void volume. Thus, the net effect is slight increase in the void volume fraction up to point B. The void volume fraction beyond point B decreases with applied nominal strain.

The material parameter values used to simulate porous ${\mathrm{Pd}}_{42.5}{\mathrm{Cu}}_{30}{\mathrm{Ni}}_{7.5}{{\rm{P}}}_{20}$ BMG in the present work are obtained from literature. The influence of key material parameters on the mechanical response of porous BMGs in general is under investigation and will be reported in our forthcoming publication. However, a preliminary study of the effect of varying Mohr–Coulomb friction parameter μ on the compressive response of porous BMGs has been conducted. This study suggests that decrease in μ lowers the stress–strain curve while the hardening and strain to failure remain more or less unaffected.

In the work of Wei and co-workers [41], it was found that Poisson's ratio plays an important role in plastic deformation of thin plates. Thus, it was expected that in the present context of porous BMGs, especially in high porosity BMG foams, Poisson's ratio might play a role in the plastic deformation when the thin pore walls are subjected to bending. In order to investigate this issue, a preliminary study of effect of Poisson's ratio on the response of porous BMGs was conducted. In this study, uniaxial compression simulations were conducted using three different values (0.36, 0.41 and 0.43) of Poisson's ratio for unit cells having low (f = 1.9%) and high (f = 60%) porosities. It was found that Poisson's ratio does not influence the nominal stress–strain curves for both low and high porosity BMGs.

4.2.1. Suitability of the damage model for high porosity BMG foams

Experiments have shown that the geometry and orientation of pore also play a significant role in influencing the mechanical behavior of porous BMGs [21]. As observed in experiments, BMGs with ellipsoidal pores having porosity 3% show higher plastic strain to failure than BMGs with spherical pores of 3.6% porosity without compromising the strength [21]. It is also seen that the stress–strain curves of BMGs with ellipsoidal pores depend on the orientation of the major axis of the ellipsoidal pore, vis-a-vis the loading direction. Unit cells containing ellipsoidal pores with initial porosities 3% and 11% are analyzed for two different orientations (N-sample and P-sample) and compared with the experimental results [21]. In the case of N-sample, the direction of loading is perpendicular to the major axis and for P-sample, the direction of loading is parallel to the major axis of the ellipsoidal pore. Figures 4(a)–(d) show the comparison of the simulation results with experiments in terms of stress–strain curves. It can be noticed from figures 4(a) and (c), pertaining to the N-sample having 3% and 11% porosity, that the simulated stress–strain curves agree with the experimental stress–strain curves qualitatively well. In particular, the elastic modulus, yield point and failure strain for 3% porosity case is well predicted by the simulations. The post yield hardening behavior seen in the experimental stress–strain curves of both 3% and 11% porosity cases is also captured in the simulations. Thus, with respect to apparent hardening, the N-samples behave similar to that of BMG foams even though the porosity of N-samples is lesser.

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The elastic modulus and yield point are accurately captured in the simulated stress–strain curves for the P-sample for both 3% and 11% initial porosities (see figures 4(b) and (d)). However, the P-samples fail soon after reaching the yield point in experiments while the simulated stress–strain curves do not show failure for nominal strain as large as 20%. The reason for this discrepancy lies in the operative failure mechanism in the P-samples, which is different from that incorporated in the present simulations. This point will be further discussed in the next section.

In this subsection, the mechanism of deformation and failure of BMGs with spherical pores is discussed. The present three-dimensional finite element simulations enable one to do this. For this purpose, the behavior of representative low porosity (5.7%) and high porosity (41%) BMGs are contrasted by examining their simulated stress–strain curves and associated contour plots showing the evolution of plastic strain within the simulation model.

Figure 5 shows the nominal compressive stress–strain curves for BMGs with initial porosities f = 5.7% (with damage) and 41% (with and without damage). In view of section 4.2.1, the response of high porosity BMG foam without damage will only be discussed in the following. It can be seen from figure 5, the high porosity BMG foam exhibits malleability with plastic strains as high as 60%, while the low porosity BMG fails at about 20% strain. The stress–strain curves in figure 5 are marked with salient points A–F. These points are associated with various events shown in figures 6 and 7 within the simulation models of their unit cells that are responsible for the respective mechanical behavior. Figures 6 and 7 display the contour plots of maximum principal logarithmic plastic strain $\mathrm{ln}{\lambda }_{1}^{p}$ keyed to points A–F on the stress–strain curves in figure 5 for 5.7% and 41% porosity BMGs, respectively.

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For the simulation model with initial porosity 5.7%, the deformation histories are shown on 1/4th of the unit cell in figure 6. From point A to B in figure 5, the material is loaded elastically and no plastic strain is accumulated in the model, as seen from figures 6(a) and 7(a), for low and high porosity BMGs respectively. On loading the material beyond point B, it starts yielding, which is marked by the accumulation of plastic strains at the horizontal diametral plane of the pore (see figures 6(b) and 7(b)). It should be noted that the 'thick islands' marked in figure 6 of the unit cell remain elastic throughout the entire deformation history for low porosity BMGs. As the load is increased up to point C, the plastic strain spreads along the four corners of the low porosity unit cell and thus develops into inclined 'primary shear bands' (see figure 6(c)). These inclined shear bands join the adjacent unit cells and form a criss-cross network of shear bands bridging the spherical pores in the material. This feature is similar to that observed in the scanning electron microscopy (SEM) image of a section across a porous BMG in figure 8(b). In the case of high porosity simulation model, the plastic strains spread along the 'thin ligaments' connecting the pores as shown in figure 7(c).

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Further loading from point C to D leads to the intensification of plastic strain in the primary shear bands and thin ligaments for low and high porosity BMGs, respectively. Till this point, it is observed that the pore shape continues to be spherical in both the cases. At point D, a change in slope of the stress–strain response is observed. This is due to the nucleation of 'secondary shear bands' from the primary shear band at the pore for low porosity BMGs (see figure 6(d)) and intensification of plastic strains in the thin ligaments for high porosity BMGs (see figure 7(d)). For the low porosity BMG case, the figure 6(d) represents a stage between points D and E in the stress–strain curve, in order to get a better visualization of the secondary shear band. A note on the evolution of void volume fraction of BMG foams at this stage is pertinent here. It is observed that the current void volume fraction (see figure 3(b)) increases by 4.45% till point D, where the plastic strains increase in the thin ligaments. On loading beyond point D, the current void volume fraction begins to drop smoothly. As the loading is increased up to point E, localization of plastic strain in the secondary shear bands occurs for low porosity BMGs, while the plastic strain intensifies in the thin ligaments in high porosity BMGs. It is also noted that the pore shape is no longer spherical beyond point D.

After point E, the primary and secondary shear bands merge for low porosity BMGs as seen in figure 6(e). The change in slope from CD to DE in the stress–strain curve in figure 5 is about 22% for low porosity unit cell, which is much less compared to that for high porosity unit cell (∼82%). The low porosity unit cell does not take further load and begins to fail from the center of the void along the shear band (see figure 6(f)). Thus, the mechanism of failure of low porosity BMG involves formation of ductile shear cracks inside inclined dominant shear bands originating at the diameter of the spherical pore. For high porosity BMG foam, owing to the large bending plasticity of the thin ligaments, the material does not fail on loading beyond point E. But, the pore begins to collapse on itself leading to densification of the material (see figure 7(e)). This is indicated by a dramatic change in the slope at point E, for high porosity BMG foam in figure 5. Further loading of high porosity BMG leads to the yielding of the thick islands as seen in figure 7(f). Thus, the mechanism of deformation of high porosity BMG foams involves plastification of thin ligaments connecting the adjacent pores, collapse of the pores on themselves leading to self-contact and densification of the material. This causes the rapid apparent strain hardening observed in the stress–strain curves of porous BMG foams.

Comparing the 41% porosity unit cell simulation model results with the corresponding experiment, the events starting with the intensification of plastic strains in the thin ligaments denoted by points D and E in figure 5 (41% porosity) occur at strains of about 0.21 and 0.48, as compared to 0.12 and 0.26 in the experiment [21]. The early occurrence of events in the experiment as compared to the simulation may be due to non-regular arrangement of pores in actual material as compared to the uniform distribution of pores assumed in the present simulations.

Figure 8 compares the failed sample of porosity 5.7% obtained from the simulation with the experimental SEM image of the failed sample having the same porosity, reported in literature [14]. Figure 8(a) shows the contour plot of maximum principal logarithmic plastic strain ($\mathrm{ln}{\lambda }_{1}^{p}$) at failure corresponding to unit cell with porosity 5.7%. It is observed that the material starts to fail from the void diameter, which agrees with the SEM image of the failed sample [14], shown in figure 8(b).

In the present work, the effect of pore distribution has not been studied because of the use of unit cell approach. This is a limitation of the present work. Future works can study this effect by making use of a representative volume element containing a distribution of pores and PBCs.

Figure 9 shows the simulated stress–strain curves of N- and P-samples containing ellipsoidal pores and having a porosity of 11%. The salient events are marked by points A–D on the curves. The mechanism of deformation of both N- and P-samples vary significantly as seen from the failed samples in experiments reported in [21]. While the P-sample fails immediately after reaching the elastic limit, the N-sample of the same porosity shows substantial amount of plastic strain to failure. Figures 10(a)–(c) shows the contour plot of maximum principal logarithmic plastic strain $\mathrm{ln}{\lambda }_{1}^{p}$ for porous BMG with ellipsoidal pores having initial porosity 11%. These figures display the evolution of plastic strain in the simulation cell from the onset of yielding to final failure for the N-sample. On examining the simulated stress–strain curve of the N-sample, it is seen that as the material is loaded up to elastic limit (point B), stress concentration occurs at the corner of the major diameter. This leads to accumulation of plastic strain at the major axis, which spreads along its diameter as shown in figure 10(a). At point B, it can also be seen from figure 10(a) that the plastic strain spreads from the major axis diameter to the corner of the unit cell. Further loading from point B to C, intensifies the plastic strain at these regions, forming a primary shear band and thus improving the malleability of the material. This is shown in figure 10(b). However, there are 'thick islands' which remain elastic at this point. When loaded beyond point C, the pore tends to collapse on itself, similar to the behavior described earlier for BMG foams, leading to densification of the material. This event is marked by a slope change at point C. The pore begins to collapse on itself starting from the corner of the major diameter and gradually moving towards the center of the ellipsoid (see figure 10(c)). Beyond point D, the material begins to fail at the central plane of the pore.

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Referring to the uniaxial compressive stress–strain curve in figure 4(c) pertaining to the N-sample, a discrepancy in the yield point is observed. The simulation for 11% porosity unit cell simulation model predicts a yield strength about 300 MPa higher than that of the experiment. Also, while failure in the experiment occurs at a strain of 0.26, simulation shows that failure occurs at a strain of 0.34. These discrepancies can be attributed to the fact that the pore distribution is assumed uniform in the present unit cell analysis, while in reality, they are randomly arranged within the matrix material. This leads to a change in the actual interaction between the pores.

Figure 11 shows the comparison of contour plots of $\mathrm{ln}{\lambda }_{1}^{p}$ for the (a) 3% and (b) 11% porosity N-sample unit cell simulation models with that of the (c) experimental sample [21] at a nominal strain of 0.04. Both 3% and 11% samples show shear bands nucleating from the major axis and connecting the corners of the unit cell. Shear bands also connect the corner of minor axis diameter to the adjacent corners of the unit cell. These features are also observed in the experiments shown in figure 11(c).

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The present damage model is unable to capture the failure point of the P-samples in the simulations. This is because the failure mechanisms of P-samples and other unit cell simulation models differ significantly. This is explained with reference to figure 12. Figures 12(a) and (b) show the contour plots of $\mathrm{ln}{\lambda }_{1}^{p}$ for P-sample porous BMGs containing ellipsoidal pores of porosities 3% and 11%, respectively, at yield strain which corresponds to the failure point in the experiments. Figure 12(c) shows the SEM image of P-sample porous BMG with ellipsoidal pore after fracture, reproduced from the work of Wada et al [21]. It can be seen from figures 12(a) and (b) that the plastic strains are distributed homogeneously within the simulation model. Unlike the case of N-sample where shear bands emanating from the pore are observed (see figure 11(c)), no shear bands are observed around the pore in the P-sample (see figure 12(c)) and it fails immediately after the elastic limit in the experiments. Thus, in the case of P-sample, the ellipsoidal pore does not lead to nucleation and proliferation of shear bands in the simulation model of the unit cell. A limitation of the present work is that the failure point in the experimental stress–strain curves of the P-sample porous BMGs is not captured in the current simulations. This is because the P-samples seem to fail at yield point perhaps due to formation of a single dominant shear band in the whole specimen which cannot be predicted from unit cell approach. More experiments on P-samples are required to understand their failure.

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