Failure mode in first-principles computational tensile tests of grain boundaries: effects of a bulk-region size, dominant factors, and local-energy and local-stress analysis

Figures 2 and 3 show the energy–strain and stress–strain curves of the {0 0 1} $\Sigma$5 twist GB in Al by using the 70-atom and 50-atom supercells. The present results of the 50-atom cell were obtained via the VASP code [40] using the same PAW–GGA method with similar conditions, in contrast to the usage of the QMAS code for the 70-atom cell. Note that the results of the 50-atom cell are quite similar to our previous results of the same supercell using the QMAS code in [19]. We aim to show that our results of the dependency on the bulk-region size are general one, independent of used computational codes.

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For the 70-atom cell, the tensile stress normal to the interface reaches the maximum value, 9.785 GPa, at the strain of 26%, where the tensile stress parallel to the interface is 7.176 GPa in the stress–strain curve of figure 3. Then just after this strain, the spontaneous failure occurs with catastrophic energy release in the energy–strain curve in figure 2. At the maximum stress point, the accumulated strain energy as 13.285 eV already exceeds the final energy state of fractured surfaces of about 10.95 eV, and thus the failure occurs as an exothermic reaction just after the maximum stress point. This kind of failure is usually observed in FPCTTs of pure GBs in various materials.

For the 50-atom cell, the stress–strain curve in figure 3 is rather similar to that of the 70-atom cell, while each stress component is a little bit smaller and the maximum stress point is a little bit smaller as 24% than 26% for the 70-atom cell. The maximum tensile stress normal to the interface is 9.07 GPa, where the stress parallel to the interface is 6.81 GPa. The energy-strain curve for the 50-atom cell shows quite different features in figure 2. The accumulated strain energy at 24% or 26% does not yet reach the level of the final energy state of fractured surfaces. Thus no spontaneous failure occurs, and successive tensile straining, namely successive work has to be added as a slow endothermic procedure, until reaching the converged energy state of fractured surfaces in figure 2. The final energy state is the same as that of the 70-atom cell. After the maximum stress point in figure 3, the stresses are greatly decreased, while the decreases are rather gradual, compared to the 70-atom cell. The 50-atom cell is greatly softened and easily stretched in this regime. All these features of the 50-atom cell are quite similar to the results by QMAS in [19].

Here we name the failure mode of the 70-atom cell as an exothermic reaction with catastrophic energy release, Type A, and the failure mode of the 50-atom cell as an endothermic reaction without any energy release, Type B. The occurrence of the Type-B failure should be caused by the thin bulk-region size in the 50-atom cell. The accumulated strain energy in one bulk slab can be regarded to be used to create fracture surfaces for one GB in a repeated supercell normal to the interface. The accumulated strain energy should be nearly proportional to the bulk-region thickness for the same strain. Figure 4 shows the ratio of the energy increases of the two GB supercells in figure 2. Until the failure, the ratio is nearly 1.4, similarly to the ratio of the bulk slab thickness, 7:5. However, the ratio is a little larger than 1.4. This point is concerned with the result that the stress value of the 70-atom cell is a little larger than that of the 50-atom cell at each strain in figure 3. These are caused by a little larger averaged elastic modulus of the 70-atom cell than that of the 50-atom cell, due to a higher ratio of the bulk-like layers with a little higher local elastic modulus than the interface layer.

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In all the GB supercells, the failure starts when the applied stress exceeds the common maximum stress, namely the GB strength in figure 3. We call this as the stress condition. Note that the cell strain to satisfy the stress condition is almost the same in both the GB supercells, because of thin GB regions in the present case. However, the strain to satisfy the Griffith condition (energy condition), namely the condition that the accumulated strain energy exceeds the necessary fracture energy (work of separation), remarkably depends on the bulk-region size as shown in figures 2 and 4. The occurrence of Type A or Type B is determined by the relation between the strains to satisfy the energy and stress conditions, respectively. For a GB supercell with thin bulk regions, even after the strain to satisfy the stress condition, the Griffith condition is possibly not yet satisfied in the energy-strain curve, leading to the failure of Type B.

As another factor for the occurrence of Type B, the work of separation (required fracture energy) is quite high for the present configuration of the {0 0 1} $\Sigma$5 twist GB in Al. This corresponds to the final energy state in the energy-strain curve in figure 2, which is common in both the 50-atom and 70-atom cells. In this manuscript, the term, work of separation, means the necessary energy in creating the final fracture surfaces for the GB configuration during the FPCTT. This is not the required one in ideal cleavage into the most stable (0 0 1) flat surfaces, nor the peak energy value just before the failure of Type A in figure 2. As analyzed in [19], the failure process of the Al {0 0 1} $\Sigma$5 twist GB in the FPCTT is complicated. In the two interface layers facing each other, one atom per CSL cell in each interface layer is attracted and transferred toward the counter interface layer to each other, during the failure process, resulting in atomically rough (0 0 1) surfaces with quite high surface energies. This is caused by local strong reconstructed bonds at the interface, due to the covalent nature of Al–Al bonds. From the energy-strain curve in figure 2, a half of the work of separation (fracture energy) is 5.459 eV/CSL cell (0.134 eV $\mathring{\rm A}^{-2}$  =  2.154 J m$^{-2}$) as the final energy state of one surface against the initial state of a GB. On the other hand, the surface energy of the final relaxed fractured surface is 5.904 eV/CSL (0.145 eV $\mathring{\rm A}^{-2}$  =  2.329 J m$^{-2}$) against the bulk crystal, which is obtained by adding a half of the GB excess energy listed in table 1 in [26] to the above value. This final surface energy is twice as large as the energy of a simple flat (0 0 1) surface of Al as 0.94 J m$^{-2}$ in [41], for example. This is because of atomistic roughness of the fractured surface. In our recent FPCTT of the {2 2 1} $\Sigma$9 tilt GB in Al [20], a half of the work of separation for this tilt GB is 2.130 eV/CSL cell (0.087 eV $\mathring{\rm A}^{-2}$  =  1.401 J m$^{-2}$). Similarly, the surface energy of the final fractured (2 2 1) surface is 2.480 eV/CSL (0.101 eV $\mathring{\rm A}^{-2}$  =  1.631 J m$^{-2}$) against the bulk crystal by adding the value from [26]. This value is in the range of usual Al surface energies [20], and is much smaller than that of the {0 0 1} $\Sigma$5 twist GB. It can be said that the present configuration of the {0 0 1} $\Sigma$5 twist GB in Al requires quite large work of separation (fracture energy), due to the formation of atomically rough surfaces, compared to the {2 2 1} $\Sigma$9 tilt GB, while the maximum tensile stress, namely the strength of the $\Sigma$5 twist GB, about 9.8 GPa, is only a little larger than that of the {2 2 1} $\Sigma$9 tilt GB, about 8.5 GPa. These points should cause the easy occurrence of the Type-B failure for the Al {0 0 1} $\Sigma$5 twist GB.

Here we consider three factors in FPCTTs of GBs; the accumulated strain energy, the work of separation, and the GB strength. The relation among the three factors and their dependence on the bulk-region size should dominate the failure mode. We deal with a GB supercell consisting of alternate stacking of bulk slabs with a thickness of $L$, where we assume that the elastic modulus $Y$ is uniform in a supercell, namely constant in both the interface and bulk regions, and also constant during the tensile procedure, leading to a straight line as $Yx$ for the strain $x$ in the stress–strain curve. Of course, rigorously the elastic modulus is not uniform in the GB supercell as discussed in section 3.1, and such an elastic (linear) regime is only limited for small strains, while the present simple model can describe the essential features as will be shown. In the present model, the strain energy per interface is the integration of the force $SYx$ multiplied by $L{\rm d}x$ until the strain $x$, where $S$ is a cross section of the supercell. The resultant strain energy $E_{\rm st}$ is expressed as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{resstrene} E_{\rm st} = \frac{1}{2}SYLx^{2}. \nonumber \end{align} \tag{ 1 }$$

The work of separation, namely the required energy for creating fracture surfaces in a GB, is expressed by $W$, which is regarded to be already normalized by surface area $S$ here. This is the energy of the final state after the failure against the initial GB state in the energy-strain curve in figure 2, common in both Types A and B. The strain $x_{\rm G}$ to satisfy the Griffith condition, namely the condition that the strain energy exceeds the work of separation as $WS=E_{\rm st}$ is expressed as follows:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{strexceworksep} x_{\rm G} = \left(\frac{2W}{Y}\right)^{1/2}L^{-1/2}. \nonumber \end{align} \tag{ 2 }$$

For given values of $W$ and $Y$, the strain to satisfy the Griffith condition is inverse square-root proportional to the bulk-region size $L$ in equation (2). Here we also assume that the GB strength, namely the maximum sustained stress $\sigma_{\rm M}$ is common for the GB supercells of the same GB independent of a bulk-region size. This leads to the common critical strain $x_{\rm M}$ to generate $\sigma_{\rm M}$ within the elastic (linear) model, $\sigma_{\rm M}=Yx_{\rm M}$, independent of $L$ as follows:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{cristrelastic} x_{\rm M} = \frac{\sigma_{\rm M}}{Y}. \nonumber \end{align} \tag{ 3 }$$

In this way, $x_{\rm G}$ depends on $L$, while $x_{\rm M}$ does not.

Figure 5 shows the plots of equations (2) and (3) as a function of a bulk-region size $L$ for given $W$, $Y$ and $\sigma_{\rm M}$. The vertical axis means the strain $x$. The plot of equation (2) is a downward-convex curve, indicating that the necessary strain for the Griffith condition $x_{\rm G}$ becomes larger for smaller $L$, while the plot of equation (3) is a horizontal line, showing the constant critical strain $x_{\rm M}$ to generate the maximum stress $\sigma_{\rm M}$, independent of $L$. The cross point of the two plots leads to the critical bulk-region size $L_{\rm c}$ from the condition, $x_{\rm G}=x_{\rm M}$ as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{xgequxm} L_{\rm c} = \frac{2WY}{\sigma^{2}_{\rm M}}. \nonumber \end{align} \tag{ 4 }$$

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This divides the region of $L$ in the horizontal axis into the areas of Types A and B. A GB supercell with the bulk-region size of $L<L_{\rm c}$ has $x_{\rm M}<x_{\rm G}$, showing a failure of Type B, where the strain $x$ in the FPCTT reaches the stress condition before the Griffith condition, as the present 50-atom cell. On the other hand, a GB supercell with a bulk-region size of $L\geqslant L_{\rm c}$ has $x_{\rm G}\leqslant x_{\rm M}$, showing a failure of Type A, where the strain $x$ in the FPCTT reaches the Griffith condition before the stress condition, as the present 70-atom cell. In this way, the critical size $L_{\rm c}$ should exist between the 50-atom and 70-atom cells for the present twist GB in Al.

The diagram in figure 5 depends on the GB strength $\sigma_{\rm M}$ and the work of separation (fracture energy) $W$. The smaller $\sigma_{\rm M}$ leads to the downward shift of the line of equation (3), which moves the cross point $L_{\rm c}$ to the right, resulting in a wider region of $L$ for Type B. The lower $W$ leads to the downward shift of the curve of equation (2), which moves the cross point $L_{\rm c}$ to the left, resulting in a wider region of $L$ for Type A. All these are consistent with the form of equation (4).

Strictly speaking, the strain energy should be deviated from the simple elastic-energy descriptions as equations (1)–(3) for larger strains. For example, as shown in figure 3, the stress is lower than the linear value except for small $x$. Thus the shape of the curve of equation (2) in figure 5 should be changed to the upper direction for larger $x$ or smaller $L$, while the essential feature of a downward-convex curve of equation (2) should not be changed. And for the line of equation (3), the coefficient to determine $x_{\rm M}$ from $\sigma_{\rm M}$ may be changed for larger $x$. However, the above behavior of $L_{\rm c}$ in equation (4) does not seem to be seriously changed, at least qualitatively.

As discussed above, figure 5 shows the relation between the bulk-region size $L$ and the necessary strains for the stress and Griffith conditions, $x_{\rm M}$ and $x_{\rm G}$, deciding the failure mode for given $L$. Moreover, we can consider the failure-mode change for fixed $L$ by the shifts of the line and curve of equations (3) and (2) due to the changes of $\sigma_{\rm M}$ and $W$. This is possible for the impurity segregation in the same GB supercell. Indeed, failure-mode changes were observed in the previous FPCTTs of the Al {1 2 2} $\Sigma$9 tilt GB with segregation of various $sp$ elements [8–10]. In these works, the common GB supercell with fixed $L$ was dealt with, while different segregated species induced different failure modes, perhaps according to the change of $\sigma_{\rm M}$ and $W$ by each impurity species. In this subsection, the previous FPCTT results are reinterpreted from the present viewpoint of the shifts of the curve and line of equations (2) and (3) in figure 5.

From [8, 9], the supercell of the clean {1 2 2} $\Sigma$9 GB in Al shows usual Type-A failure with $x_{\rm M}$ larger than $x_{\rm G}$ for the given $L$ as expressed by black lines in figure 6. However, the same GB supercell with Ca segregation and that with Na segregation show Type-B failure in the FPCTTs in spite of the same $L$. In the energy–strain and stress–strain curves [8, 9] (private communication), the GB strength $\sigma_{\rm M}$ for the Al $\Sigma$9 GB with Ca is reduced by about 13% than the clean GB, while the work of separation $W$ is not changed. For the Al $\Sigma$9 GB with Na, $\sigma_{\rm M}$ is much lower by about 50% than the clean GB, and $W$ is also much reduced by about 60% (as about 40%). Figure 6 explains the occurrence of Type-B failure for the segregation of Ca and Na by the shifts of the lines of equations (2) and (3) for the fixed $L$. For the Ca-segregation, the curve of equation (2) is not changed from that of the clean GB as described in black, due to no change of $W$, while the line of equation (3) is shifted downward a little as described in blue, due to the decrease of $\sigma_{\rm M}$. For the clean GB (black line in figure 6), the bulk-region size $L$ should be located near the critical point $L_{\rm c}$ and in the right-hand side of it. Then for the Ca-segregation, the downward shift of the line of equation (3) moves $L_{\rm c}$ to the right of $L$ as indicated by $L_{\rm c}$ in blue, leading to the Type-B relation for $L$ as $x_{\rm M}$ smaller than $x_{\rm G}$ in figure 6. For the Na-segregation, both the line of equation (3) and the curve of equation (2) are greatly shifted downward as described in red, compared to the clean GB. Due to the linear and square-root dependences of equations (3) and (2) on $\sigma_{\rm M}$ and $W$, respectively, the shift of the line of equation (3) should be larger than that of equation (2), which also moves $L_{\rm c}$ to the right of $L$ as indicated by $L_{\rm c}$ in red, leading to the Type-B relation for $L$ as $x_{\rm M}$ smaller than $x_{\rm G}$ as shown in figure 6. In both the Ca-segregation and Na-segregation cases, the effect of the downward shift of the line of equation (3) is more significant, leading to Type-B failure.

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In [9], the stress–strain and energy–strain curves of the same Al GB with co-segregation of Na and Si show that $\sigma_{\rm M}$ is reduced by about 20% than the clean GB, and that $W$ is much smaller as about 35% (reduced by about 65%) than that of the clean GB, leading to Type-A failure similar to the clean GB, in contrast to Type-B failure for the Ca-segregation and Na-segregation cases. The reason is explained by green lines in figure 6. The line of equation (3) is a little lower than that of the Ca segregation expressed in blue, while the curve of equation (2) is also a little lower than that of the Na segregation expressed in red. The latter change is more significant, which moves $L_{\rm c}$ to the left as expressed in green, keeping the Type-A failure for the fixed $L$. In [10], S segregation in the same Al GB greatly reduces $W$, while the reduction of $\sigma_{\rm M}$ is not so significant, similarly to the case of co-segregation of Na and Si. Thus the Type-A failure of this case [10] can be understood from the viewpoint of figure 6.

The present analysis has clarified that the failure mode observed in a FPCTT of a GB seriously depends on the bulk-region size $L$. The critical bulk-region size $L_{\rm c}$ to decide the ranges of $L$ for the two types of failure modes is dominated by $\sigma_{\rm M}$, $W$ and $Y$. These three quantities should be associated with a configuration of a clean CSL GB as intrinsic quantities, and can be examined by a FPCTT for a given $L$. As examined above, the impurity segregation can change the quantities of $\sigma_{\rm M}$ and $W$, possibly leading to a different failure mode from that of a clean GB, even with the same $L$. As mentioned in section 3.2, the present argument is based on the analytical model of elastic energy and stress. The deviation from the elastic expression should lead to the change of the curve shape of equation (2) and some shift of the line of equation (3), while this does not seem to change the essential relation between the downward-convex curve and straight line in figures 5 and 6 as discussed above. On the other hand, the assumption of the uniformity of $Y$ within a GB supercell may cause some problems in the cases of small values of $L$ or impurity segregation cases, where the difference in $Y$ at GB regions may be enhanced. In the near future, it may be possible to incorporate such local changes of $Y$ into the above analytical model, while the total effect does not seem to be so serious. Of course, too thin bulk regions should not be used in a FPCTT, because the bulk-like electronic structure or bonding is not well reproduced in such thin bulk regions, which can be examined by using our local-energy and local-stress schemes [26].

Experimentally, failure initiates at some weakest GB in a polycrystalline sample, and the grain size in a sample does not necessarily affect the failure mode, differently from FPCTTs, because the strain energy accumulated in the whole polycrystalline body can be used for creating failure surfaces at the weakest GB, resulting in Type-A failure usually. This indicates a serious question on the physical reality or real occurrence of Type-B failure in usual experiments of polycrystals. In Type-B failure, after the maximum-stress point, extremely large bond stretching seems to occur in back-bond and bulk regions, so that the total strain energy in small bulk regions reaches the work of separation. Such extreme degrees of deformation in a bulk region should become more significant for a smaller bulk-region size. This phenomenon does not seem to occur in usual experiments, except for some special nanoscale systems such as contacting nanoparticles or nanorods [42]. In other words, the atomic and electronic behaviors in Type-B failure in a FPCTT, at least, those after the maximum-stress point, cannot be regarded as general phenomena, and cannot be used as ab initio data for meso- or macroscopic simulations of usual polycrystal samples [18]. For this purpose, it is desirable to use FPCTT results of Type-A failure, at least. We should be aware of the distinction of the failure mode in a FPCTT, dominated by the three quantities and the bulk-region size, as analyzed in the present study for the first time.

Here we discuss the applicability of the local-energy and local-stress schemes to extract the local deformation and failure behaviors from the FPCTT results of Type-A failure of a GB, as suitable forms to be combined with meso- or macroscopic simulations. Figure 7 shows the variations of local-energy and local-stress (diagonal sum) values of each symmetrically-independent atom in the 70-atom GB supercell during the FPCTT showing Type-A failure. As shown in figure 1, the atoms Nos. 17 and 20 belong to the interface layer, Nos. 11 and 12 belong to the second (back) layer from the interface, Nos. 6 and 10 belong to the third layer from the interface, and Nos. 1 and 5 belong to the central bulk layer. In figure 7, the atoms on the interface layer, Nos. 17 and 20, mainly show remarkable variations of local energies and local stresses. In the early stage of the cell strain, the local energy and local stress are quite high at the interface atom No. 20, and in the late stage after around 13%, the interface atom No. 17 shows rapid increases of the local energy and local stress, becoming larger than those of No. 20, while at the final stage, the local energy and stress at No. 20 become large again, associated with the transfer of this atom toward the counter interface layer. Figure 8 shows the variations of the sum or average of local energies and local stresses of each layer. In the energy variation, initially the interface and second layers show increases, while after 13% only the interface layer shows a remarkable increase, in contrast to a gradual decrease in the second layer. It is interesting that the mid layer in the bulk region shows a rather constant energy increase until the failure.

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About the initial interface configuration [19, 28] shown in figure 1(b), the atoms of Nos. 20 and 21 are located on the four-fold symmetry sites. We call these atoms Type I. The atom of No. 20 on the lower interface layer neighbors the four atoms of Nos. 22–25 on the upper interface layer, while the atom of No. 21 on the upper interface layer neighbors the four atoms of Nos. 16–19 on the lower interface layer. We call the atoms of Nos. 16–19 and Nos. 22–25 Type II. Each Type I atom on the interface layer has twelve neighbors; four Type II atoms on the counter interface layer, four on the same (0 0 1) layer and four as the back bonds. Each Type-II atom on the interface layer has two neighbor atoms on the counter interface layer, one Type-I atom and one nearest Type-II atom. Thus each Type-II atom has ten neighbors.

Figure 9 shows bond-length changes around Type-II (No. 17) and Type-I (No. 20) interface atoms during the FPCTT. In the early stage till around 13%, the back bonds as No. 17–No. 15 and No. 20–No. 14 bonds at the interlayer between the interface and second layers are mainly stretched, while in the late stage after around 13%, the interfacial bonds between Type-II atoms (Nos. 17–25 bond) and the back bonds of a Type-I atom (Nos. 20–14 bond) are greatly stretched and broken. These features are consistent with the variations of the local energy and local stress of each atom and layer in figures 7 and 8. In the early stage, the local energies and local stresses of Type-I interface atoms (No. 20) and the atoms on the second layer (Nos. 11 and 12) are mainly increased, while in the late stage, Type-II interface atoms (No. 17) show remarkable increases of local energies and local stresses, until breaking of the interfacial bonds between Type-II interface atoms. Just before the failure, Type-I interface atoms also show rapid increases of local energy and local stress, associated with breaking of the back bonds of Type-I interface atoms due to the transfer to the counter interface layer.

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From the present analysis, it is clear that the atoms on the interface and second layers dominate the failure behavior, which is similar to the 50-atom supercell in [19]. From the viewpoint of extracting energy-traction relation for continuum fracture mechanics such as a CZM or RGS model [21–24] as mentioned in section 1, we propose the usage of the variation curve of a local energy of each layer as $f_{n}(x)$, where $n$ means the layer index from the interface and $x$ means the strain. We can assume $f_{n}(x)=f_{4}(x)$ for $n\geqslant4$ as extension for wider regions on both sides of a GB. About the variable $x$, it may be better to use a quantity like a local strain or a local interlayer-distance change, instead of cell strain. If we obtain similar layer-energy functions to represent local-energy variations during the FPCTT for various CSL GBs via our local-energy scheme, we can embed such functions in each GB region with proper connection to macroscopic regions, which should open a new way of multi-scale simulations. In the practical application, it should be necessary to make further examination and consideration. The introduction of the response for shear deformation might be necessary. Of course, there may exist other effective ways of utilizing the local-energy and local-stress scheme in such multi-scale simulations. We would like to emphasize that our local-energy and local-stress scheme should be powerful tools to fully utilize rich information in FPCTTs or FPCSTs in meso or macroscopic mechanics simulations of practical polycrystalline materials.