Understanding and predicting damage and failure at grain boundaries in BCC Ta

Design of materials with tailored microstructures that can optimally perform and survive under dynamic loading (impact, shock, high strain rates, etc.) requires a fundamental understanding of the role of microstructure to nucleate defects and resist damage (voids) nucleation to initiate failure (spallation).^1–3 Microstructural features play a key role in determining the mechanisms for plastic deformation and hence control damage (voids) nucleation, growth, and failure modes in metals.^4–6 In polycrystalline and nanocrystalline metals, such microstructural features typically include preexisting heterogeneities such as grain boundaries (GBs), triple junctions, as well as defect structures (dislocations and deformation twins, etc.) generated under shock loading.^1,7 As a result, a significant body of study has been devoted to understanding how the above microstructural features, especially GBs, affect the overall dynamic strength, which is crucial for guiding the design of high-strength materials.⁸ It has been shown that the local structure of a GB determines the dislocation nucleation and adsorption behavior at the GB, and, in particular, GB structural units can effectively alter the activation of dislocation slip.^9–13 However, not all GBs contribute to plasticity and failure in a similar manner. GB properties such as the relative misorientation between the two grains, namely, misorientation angle significantly affects the ability of GBs to plastically deform and nucleate voids.¹⁴ Conventionally, the misorientation is represented by the inverse density of coincident sites, Σ value, explained in more detail in Sec. II. Wayne et al. showed that GBs within a certain range of misorientation angles serve as preferential void nucleation sites in Cu.¹⁵ Similarly, Escobedo et al. revealed that in polycrystalline Cu voids tend to nucleate preferentially at high misorientation angle GBs, whereas special GBs such as low-angle GBs and Σ3 GBs are more resistant to void nucleation.¹⁶ Another study on tungsten GBs showed Σ3 and Σ9 GBs to be more resistant to failure than other high coincident site lattice (CSL) GBs.¹⁷ These studies point to the deterministic manner in which voids nucleate in metals, in that certain GBs tend to be stronger than others, which brings up the question of which GBs are stronger, or, more resistant to void nucleation, and why.

Studying the deformation and failure behavior of BCC metals has gained substantial momentum in recent years, although it is still a challenging problem due to their complicated deformation response, as compared to their FCC counterparts.^18–20 As a model BCC metal, Ta is a promising candidate for defense-related applications such as armors and shape charges, due to its high energy density, stability, and strength.^21,22 The deformation behavior of Ta is characterized by an intricate interplay between dislocation slip and deformation twinning.²³ Although screw dislocation-based plasticity shows an overwhelming presence in static and quasistatic loading conditions, deformation twinning emerges as an active and even dominant deformation mode under high strain rate loading, such as shock.^19,24–27 The propensity for twinning in Ta is significantly affected by the specific loading conditions such as strain rate and pressure as well as microstructural features such as grain size and GB types.^28,29 For example, Lu et al. reported a critical threshold pressure of ∼40 GPa in single-crystal Ta, above which the dominant deformation modes transitioned from dislocation slip to twinning.²⁴ In nanocrystalline Ta, this threshold is found to be much higher (∼150 GPa).²⁵ Under plate-impact, deformation twinning prefers to nucleate at high-angle GBs and this propensity is also affected by the specific texture in polycrystalline Ta.³⁰ This suggests that the tendency of Ta GBs to nucleate voids can be significantly affected by GB properties such as misorientation angles. Weaver et al. studied the spall behavior of Ta bi-crystals and found that low-angle GBs tended to nucleate fewer voids as compared to high-angle ones.^31,32 Although the above studies suggest that the characteristics of GBs can critically determine the overall deformation and void nucleation behavior, there is a lack of systematic data that allows for the establishment of the links between GB characteristics and the failure behavior in Ta.

Atomistic modeling methods, such as molecular dynamics (MD) simulations, can offer valuable time-resolved data that can complement experimental studies. MD simulations are well-suited to study the role of grain orientation, grain size, strain rate, and temperature on the deformation behavior of single and nanocrystalline systems.^33–39 For example, it has been shown that in both nanocrystalline Cu and Ta, GBs oriented normal to the loading direction are more prone to failure due to the higher resolved normal stress, which affects plasticity and eventually failure.^40,41 Tramontina et al. studied the shock-induced plasticity of Ta [001] single-crystal and observed a mix of dislocation and deformation twinning at ∼30 GPa and predominantly deformation twinning above 70 GPa.⁴² Ravelo et al. studied the deformation behavior of Ta [111], [110], and [001] single-crystals and found twinning to be dominant in the [110] direction, whereas the other two directions were characterized by dislocation nucleation and multiplication.⁴³ Transition from dislocation slip to twinning has been shown to occur for Ta [001] under high strain deformation.⁴⁴ It is further shown that such transition of deformation modes could substantially lower the damage resistance of the microstructure. For example, Hahn et al. studied the spall behavior of single-crystal Ta and found an inverse correlation between twin volume fraction and the void nucleation stress due to the propensity of voids to form at twin-twin intersections.⁴⁵ This weakening effect is in contrast with that observed in nanocrystalline Cu, where the presence of deformation twins results in higher spall strengths.⁴⁶ Another study on nanocrystalline Mo suggested the prevalence of deformation twinning under tensile loading and its significant role in initiating damage nucleation and failure at twin-GB interaction.^47–49 In addition, MD simulations have also revealed the propensity for voids to nucleate at grain interiors in polycrystalline Ta. Such behavior is facilitated by the multiple GB-slip and GB-twin intersections in Ta due to the combination of a large number of slip systems and profuse twinning in the grain interior.³⁰ However, the role played by GBs in these processes is still not clear.

Spall failure, as a weak-link process, is dependent upon variables including microstructure features, strain rate and temperature, etc. Microstructural features discussed above, such as grain orientation, GB characteristics, and grain sizes, significantly affect the dislocation slip and deformation twinning behavior, which in turn determine the nucleation and growth kinetics of voids. The generation of the above defects, in addition, increases the local temperature, which furthermore modifies the kinetics for further defects nucleation and motion. Therefore, the above variables are not independent and oftentimes conspire in dictating the complicated spall failure process of a given microstructure. With MD simulations, fine control of these variables, such as the initial GB characteristics, makes it possible to study their effects. Although the strain rates that can be modeled with MD simulations (>10⁸s⁻¹) are typically much higher than that observed in experiments, they could provide valuable insights in assisting the interpretation of experimental results. It has been shown that the predicted spall strength values in MD simulations extrapolate quite well to the lower strain rate regime in experimental studies.³⁷ 

The foregoing considerations lead to three questions, which to our knowledge have not been systematically examined with MD simulations in BCC materials,

1. How does GB structure as quantified by its properties affect the deformation behavior?

2. How does the coupled existence of preexisting GBs and newly-generated defects affect the void nucleation behavior, and more specifically, where and under what stresses will voids nucleate (spall strength)?

3. Is there any correlation between these factors and the spall strength?

The goal of this work is to address these outstanding questions using Ta bi-crystals as a model system. Bi-crystals are chosen as a model system because they are conducive to a systematic study that allows one to alter different features of GB structures and hence investigate their effects on spall strength. This paper is organized in the following manner: the computational details and analysis methods are presented in Sec. II. The role of GB static properties such as GB plasticity and GB local structure on the deformation behavior and spall strengths of Ta bi-crystals, and a comparison of the results for Ta bi-crystals with that of the single-crystals are undertaken in Sec. III.

To determine the correlation between GB structure and spall strength, a statistically relevant data set of 74 symmetric tilt GBs along the [110] tilt axis is selected for this study. Due to limited computational resources, it is necessary to use a standard framework, coincident site lattice (CSL) theory to describe GBs in terms of the degree of fit between the two grains forming the GB.⁵⁰ CSL covers the whole range of GBs with rational normal planes, and although not all GBs are precisely CSL GBs, they can be approximated arbitrarily well by a CSL GB. For a GB with normal plane (hkl), Σ value is equal to h² + k² + l² if h² + k² + l² is odd or (h² + k² + l²)/2 if h² + k² + l² is even.⁵¹ Although there is no direct correlation between Σ values and the excess energy of GBs, GBs with small Σ values (such as Σ3, Σ5, Σ9, and Σ11 GBs) cover the majority of GBs with low excess energies.⁵² Additionally, these GBs show a relatively simple repetitive pattern (typically referred to as GB structural units) that forms the basis for larger Σ value GBs and, therefore, are ideal prototype GBs to investigate.⁵³ The set of GBs in this study covers a wide range of Σ values from 3 to 163.

The GBs are generated using the γ-surface approach: two half-crystals are joined at the GB plane, then one half-crystal is translated with respect to the other half-crystal along the GB plane, wherein any overlapping atoms are deleted, and a structural minimization is performed.^54–56 Different amounts of translations and overlap criteria yield different GB structures, namely, GBs with the same misorientation but different local atomic arrangements/structures. In experimental studies, the GB structure usually changes locally along the GB itself while the misorientation angle remains the same. This approach to alter only the GB structure allows us to measure an average response of the GB, similar to what is measured experimentally. In addition, previous works have shown that plasticity in metals is dependent on the local structure of the GB. Hence, modification in the deformation mechanisms would expectedly alter the void nucleation behavior as well.^38,57 Therefore, it is necessary to sample different GB structures to understand the collective deformation response of GBs that comprise a nanocrystalline material.

Figure 1 shows the variation of GB energy with misorientation angles for the chosen set of GBs. The GBs are chosen to cover the entire range of GB misorientation angle, energy, excess volume, as well as different local structures. The GB energy $(γGB)$ and excess volume $(Vex)$ are defined as the excess energy and volume per unit area for a periodic bi-crystal as compared to that of the bulk system with the same number of atoms,

where $EGB$ and $VGB$ are the total energy and volume of the Ta bi-crystal, N is the number of atoms in the Ta bi-crystal, $Ecoh$ and $Vatom$ are the cohesive energy and equilibrium volume of a Ta atom, and A is the unit area in the GB plane. These are calculated preshock.

To illustrate the choice of GBs in this work, static properties of a few representative GBs are listed in Table I, and the corresponding GB structures are shown in Fig. 2. These GBs have very different static properties as well as local structures. Moreover, for a given misorientation angle, two different GB structures are chosen, in order to investigate the role of local structures. The bi-crystals are created with a dimension of ∼40 × ∼40 × ∼150 nm³, with a total of 15– 20 × 10⁶ atoms. As shown in Fig. 3, the GB is positioned at 2/3 of the Z dimension, resulting in grain 1 with a Z dimension (100 nm) that is two times that of grain 2 (50 nm). The as-created bi-crystals are equilibrated using Nosé-Hoover isobaric-isothermal ensemble (NPT)^58,59 at zero pressure and 300 K for 50 ps. 1D shock loading is then introduced via the flyer-plate target method³⁷ as shown in Fig. 3. The left half of grain 1 is chosen as the flyer and the rest of the sample as the target. The particle velocity U_p is set at 750 m/s, and the flyer and target are given a velocity along the Z direction (normal to the GB plane) of 4/3U_p and −2/3U_p, respectively, causing them to impact against each other. 750 m/s is chosen since it is above the elastic-plastic transition for all the orientations considered here. This setup allows the rarefaction waves from the flyer and target to interact at the GB plane. Thus, spallation is introduced at the GB plane by design, allowing for direct evaluation of the spall strength of the GB. Periodic boundary conditions are applied in X and Y directions, and the shock (Z) direction is kept free. The Ta2 Embedded Atom Method (EAM) potential by Ravelo et al.⁶⁰ is used to model the interatomic interactions in the system. The open source software “LAMMPS” is used to carry out the MD simulations, with a time step of 2 fs.⁶¹ Atomic configurations are visualized in OVITO.⁶² 

Spall strength of the GBs is evaluated as the maximum stress in the shock direction 1 ps before the nucleation of voids at the GB, referred to as void nucleation stress here. To identify voids, the simulation cell is divided into a 3-dimensional grid of cubic cells, and empty cells (that contains no atoms) are identified. A void is defined as a cluster of two or more continuous (interconnected) empty cells.^63,64 Due to the presence of screw dislocation-based plasticity and deformation twinning in the system that can lead to void nucleation in the bulk region as well, we focus on a “GB region,” defined as a 5 nm region surrounding the GB and detect first void nucleation event in this region only. This is illustrated schematically in Fig. 4(a). Void nucleation stress is obtained by averaging over the longitudinal (Z) stresses of all the atoms within a 5 nm region surrounding the void (void region). In the case of multiple void nucleation events, void nucleation stress is calculated for each void separately, and the lowest value is reported as the void nucleation stress. Figure 4(b) shows the evolution of longitudinal stress, $σz$⁠, at the time of first void nucleation event $(tvoid)$⁠, 1 and 2 ps before as well as 1 and 2 ps after that. The local stress near the GB continues to decrease even at 1 ps following the void nucleation event due to a delay in void-induced stress relaxation. Therefore, the void nucleation stress reported here is typically 1–2 GPa lower than maximum supported spall stress, as noted in Ref. 45. It is also noted here that since the data from the simulation are outputted every 0.1 ps, the resulting accuracy of the void nucleation stress that depends on the rate of stress increase/relaxation is about ±0.3 GPa. Figure 4(c) shows the evolution of particle velocity, U_p, at the same time period. The strain rate, $ε˙$⁠, calculated from the spatial gradient of U_p is 1.5 ± 0.5 × 10¹⁰s⁻¹. Not surprisingly, for different GBs, the calculated strain rates are quite close due to similar sample dimensions. Therefore, the observed variation in the void nucleation stress arises not from strain rate effects but rather from different misorientations as well as the local structure of the GBs, similar to that observed in Ta single-crystal.⁴⁵ Our results thus contradict the conventional view that, under high strain rates, local structural differences are not as important due to the strong phonon-drag effects.^65,66

To quantify the amount of plastic deformation in the system, a quantity called total excess energy is evaluated as the extra amount of potential energy per atom in the void region 1 ps before the first void nucleation event. The plastic component of the total excess energy is calculated by subtracting the elastic contribution to the excess energy. The elastic contribution is approximated by the elastic strain energy at the corresponding stress value for single-crystals in the corresponding orientation.

The Dislocation Extraction Algorithm (DXA) is used to identify line dislocations in the microstructure.⁶⁷ For the BCC lattice, DXA identifies dislocations with Burgers vectors a₀/2⟨111⟩, a₀⟨100⟩, and a₀⟨110⟩, among which a₀/2⟨111⟩ is predominant. For identifying twins, Euler angles representing the local orientation of each atom are first calculated and compared to the initial (reference) Euler angles. There are three components of the proper Euler angles for a given orientation: phi, psi, and theta. These three components represent the sequential rotation from a general basis orientation to attain a given orientation. In the case of twinning, the original lattice undergoes a rotation, resulting in a different orientation in the twinned lattice, yet retaining the original crystal structure. Therefore, atoms with significantly changed Euler angles as compared to the initial values and with similar Euler angles as compared to their neighbors are identified as “twinned” atoms. Twin volume fraction is calculated as the total volume fraction (Voronoi volume) of the twinned atoms. In identifying twinned atoms, only the atoms inside the twinned region are considered, whereas atoms belonging to dislocations or twin boundaries are excluded. The presence of GBs could potentially affect the deformation behavior (screw dislocation-based plasticity and deformation twinning) near the GBs, whereas the deformation behavior of the bulk could be used as the single-crystal reference state for comparison. Therefore, it is important to characterize dislocation density and twin volume fraction for the entire system (total dislocation density, total twin volume fraction), as well as for the void region at the GB (local dislocation density, local twin volume fraction). This allows for the comparison of the total and local values, in order to evaluate how GBs affect the deformation behavior in their vicinity.

This study investigates the correlations between static properties of the GBs, including energy, excess volume, and misorientation angles with void nucleation stress. Figure 5 shows the variation of void nucleation stress with GB energy and excess volume. The wide scatter in the data indicates that no direct correlation exists between void nucleation stress and these two properties. Therefore, these properties are not sufficient to predict the spall resistance of the GBs. This is not surprising since GB energy and excess volume are average properties of the boundary and thus not representative of the local boundary structure and properties (repeat units, GB dislocations) that can have a greater impact on the ability of the GB to plastically deform and resist spall. This is also in line with previous studies that showed no direct correlation for Cu⁶⁸ and Ta GBs.⁶⁹ 

Figure 6 shows the variation of GB energy with misorientation angles, with the points colored based on void nucleation stress. Principal grain orientations, including [110], [111], [112], and [001], are marked in the plots by arrows based on their misorientation angle with respect to the [110] tilt axis. The GB normal direction tends toward [110] as the misorientation angle tends toward 0° and toward [001] as the misorientation angle tends toward 180°. The [111] and [112] orientations lie at misorientation angles of 70.53° and 109.47°, respectively. It can be observed that GBs with the normal direction close to [110] (misorientation angle less than 20°), [111] (misorientation angle 70.53°), [112] (misorientation angle between 109.47°) have a higher void nucleation stress, whereas GBs with the normal direction close to [001] (misorientation angle close to 180°) have a low void nucleation stress. The above results are in line with the experimental work on Ta bi-crystal that shows much higher void nucleation stress for low-angle GBs as compared to high-angle ones.³¹ The presence of these distinct subgroups with collectively high or low void nucleation stress indicates a significant role of misorientation angles, although no direct trend is observed here.

Excess energy is used as a first-order approximation to determine the amount of plastic deformation in the system right before void nucleation. Previous work has shown that there exists a direct correlation between the ability of the GB to plastically deform and the resulting void nucleation stress in Cu.⁷⁰ To see whether such correlations exist for Ta, total and plastic excess energy are calculated and plotted as a function of void nucleation stress in Fig. 7, with the points colored based on the misorientation angle. A direct correlation is observed between the void nucleation stress and the total, as well as plastic excess energy. A high total excess energy generally leads to a higher void nucleation stress. Moreover, it is also evident from the bifurcation of the plot that high-angle (red points) GBs follow a different trend as compared to medium-angle (green points) and low-angle (blue points) GBs. Specifically, at the same void nucleation stress, high-angle GBs show much lower total excess energy. The above bifurcation indicates possibly different deformation modes for GBs with different misorientation angles.

To understand the bifurcation observed in Fig. 7, four representative GBs with a void nucleation stress of ∼17 GPa are selected for additional detailed analysis: Σ163(1 1 18), Σ33(1 1 8), Σ41(4 4 3), and Σ33(5 5 4). Among these GBs, Σ163(1 1 18) and Σ33(1 1 8) GBs are high-angle GBs with low total excess energy (∼0.30 eV/atom), whereas Σ41(4 4 3) and Σ33(5 5 4) GBs are medium-angle GBs with high total excess energy (∼0.37 eV/atom). DXA and twinning analysis are performed on these GBs, as shown in Fig. 8. As can be seen from Figs. 8(a) and 8(b), Σ163(1 1 18) and Σ33(1 1 8) GBs show a substantial amount of twinning and a relatively sparse network of line dislocations in the microstructure. On the other hand, Σ41(4 4 3) [Fig. 8(c)] and Σ33(5 5 4) [Fig. 8(d)] GBs show little twinning and a dense network of line dislocations in the microstructure. Therefore, the bifurcation observed in Fig. 7 is due to the transition of the dominant deformation mechanism from twinning [high-angle GBs indicated by red points in Fig. 7(a)] to slip [medium- and low-angle GBs indicated by green and blue points in Fig. 7(a)].

Figure 9 shows the variation of void nucleation stress with total excess energy, with the points colored based on local twin volume fraction. Clearly, the high-angle GBs, as observed in Fig. 7(a), show higher twin volume fraction, which demonstrates quantitatively the transition of deformation mechanisms as seen in Fig. 8. Therefore, the observed transition in deformation modes is directly related to the misorientation angle, suggesting this is an important parameter in determining the ease of twinning in Ta bi-crystals. To understand this role clearly, Fig. 10 shows the variation of total twin volume fraction [Fig. 10(a)] and local twin volume fraction [Fig. 10(b)] with misorientation angles, with the points colored based on void nucleation stress. Interestingly, the total twin volume fraction increases with the misorientation angle. Therefore, the above result suggests that the overall ability of the bulk region to twin in the bi-crystal increases with the misorientation angle. Given that the twinning in the bulk region should be mostly affected by grain orientation only, the above trend should also comparable to our previous work in single-crystal Ta of the same orientation. For the local twin volume fraction, although this trend still holds true, many more local peaks are observed. This is not surprising since twinning near the local GB region is affected by both the grain orientation and the local GB structure, resulting in a more complicated variation.

The observed deformation twins are mostly lath-shaped, consisting of many one or two-layer steps, as shown in Fig. 11(a). Such steps are typically observed in polycrystalline Ta and are believed to be responsible for the self-thickening growth mechanism of deformation twins through dislocation reactions at these steps.⁷¹ Moreover, in all the high-angle GBs considered in this work, activation of multiple twin systems is observed. These twin systems can readily intersect and cross each other, as has been shown for Ta bi-crystals.³³ Figure 11(b) shows such interaction of two twin bands, which leads to the thinning of twin band 1 and the deviation of the original twinning plane in twin band 2.

Similarly, the dislocation densities are also evaluated for all the Ta bi-crystals and plotted as a function of misorientation angles in Fig. 12. The trends observed here for the total dislocation density are opposite to that of the total twin volume fraction. The variation of local dislocation density shows a more complicated pattern, although it is important to note here the clustering of low-angle GBs with high dislocation density and void nucleation stress on the left and high-angle GBs with low dislocation density and void nucleation stress on the right [Fig. 12(b)]. Therefore, the above dislocation and twinning analysis quantitatively demonstrates the transition from dislocation slip based plasticity to deformation twin based plasticity as a function of misorientation angles. Additionally, the different trends observed in the variation of the local twin volume fraction (local dislocation density) as compared to the total twin volume fraction (total dislocation density) suggest that local boundary structure also leads to significant variations in the plastic deformation behavior especially near the GBs.

Additionally, our results suggest that the specific location for void nucleation is also affected by the observed deformation modes. For low- and medium-angle GBs with dislocation slip dominated plasticity, as shown in Figs. 13(a) and 13(b), voids are observed to nucleate at the GBs. For high-angle GBs with deformation twinning-dominated plasticity, as shown in Figs. 13(c) and 13(d), voids are observed to initiate at the twin-GB intersection, as well as at the twin-matrix intersection, due to the high stress concentration. Therefore, the presence of deformation twins provides ample void nucleation in these sites,⁷² leading to lower void nucleation stresses for high-angle GBs as observed above.

It is important to note that the above variations in the deformation modes for GBs with different grain orientations are in line with previous results on BCC single-crystals that suggest twinning-dominated plasticity for the [001] orientation and dislocation slip dominated plasticity for [110], [111], and [112] orientations.^73,74 However, as pointed out in previous work, such competition between deformation twinning and dislocation slip cannot be explained by Schmid factor⁴⁵ and is also highly dependent on shock pressure. Table II lists the Schmid factors for four representative GBs. For low-angle Σ73(6 6 1) and high-angle Σ129(1 1 16) GB, although the Schmid factors for both slip and twin are very similar, the deformation mode is dominated by dislocation slip and deformation twinning, respectively. Therefore, more complicated models that incorporate the nonplanar nature of the dislocation core in BCC metals are required to explain the observed trends in this work.^9,18,75,76

The role of GB local structure is examined by quantifying the variability of the void nucleation stress with respect to the GB structure. This variability is quantified as the standard deviation in the void nucleation stress as a function of GB structure at the same misorientation angle. Figure 14 shows the variation of the average void nucleation stress for different GB structures as a function of misorientation angles, with the standard deviation indicated by the error bar. Most misorientation angles show a small standard deviation in the void nucleation stress, whereas for some misorientation angles the standard deviation is very high, exceeding 3 GPa in some cases. This suggests that there exists a complex relationship between local GB structure and misorientation angle and further indicates the importance of taking into account the local structures when examining the spall resistance of the GBs.

It is important to put the results of Ta bi-crystals in perspective by comparing it with previous results on Ta single-crystals. Table III lists the calculated void nucleation stress and local twin volume fraction for Ta single-crystal along four principal directions: [110], [111], [112], and [001] reported by Hahn et al..⁴⁵ For comparison, the results in four Ta bi-crystals: Σ99(7 7 1), Σ3(1 1 1), Σ3(1 1 2), and Σ163(1 1 18) are also listed. These are the GBs with a normal direction same as the single-crystal direction ([111], [112]) or close to the single-crystal direction ([110] and [001]) when there are no corresponding GBs along the same direction. The overall trend in the void nucleation stress for Ta bi-crystals is in line with the single-crystals, with much higher and similar values for the [110], [111], and [112] directions, and a much lower value for the [001] direction, due to higher twin volume fraction. It is of interest to note that the void nucleation stress observed for Ta bi-crystals in this work is generally higher than that reported for the corresponding single-crystals at the same piston velocity,⁴⁵ although one would expect the opposite due to the presence of GBs. This is likely owing to the smaller dimension of bi-crystals in this work (150 nm) as compared to that used in Ref. 45 (1 μm), resulting in a higher strain rate (∼1.5 × 10¹⁰s⁻¹) than that reported in Ref. 45 (∼2 × 10⁹s⁻¹). Thus, strain rate effects play an important role in determining the void nucleation stress.

For both single- and bi-crystals, there exists a negligible local twin volume fraction for the simulation cells along the [110] and [111] directions. In contrast, along the [112] direction, a fair amount of twin is observed in the bi-crystal (0.049), whereas there is none in single-crystal. Figure 15(a) shows the snapshots of the Σ3(1 1 2) bi-crystal and the twinned region, where the clustering of twins near the GB highlights the role GBs play in inducing twinning. However, along the [001] direction, the observed local twin volume fraction for the bi-crystal (0.165) is lower than that of the corresponding single-crystal (0.242). This is most likely due to the presence of GBs that can also hamper the growth and propagation of the twin bands nucleated in the bulk, as can be seen in Fig. 15(b). Therefore, comparison with Ta single-crystal illustrates the dual role that GBs play in controlling twinning in Ta bi-crystals: on the one hand, GBs can induce twin nucleation for low- and medium-angle GBs, and on the other hand, GBs can hinder twin growth for high-angle GBs.

Figure 16(a) reveals one such case where deformation twins are pinned at the GB. However, a closer examination of the results indicates that GBs do not always block the propagation of deformation twins. In some cases, as shown in Fig. 16(b), deformation twins can transmit across the GB. This transmission is enabled by the geometric alignment of the $(112)$ twin plane for the left grain and the $(112¯)$ twin plane for the right grain due to the symmetric nature of the GB.

MD simulations are performed to study the deformation behavior and spall strength of a set of 74 Ta bi-crystals, in order to understand the role of GB structure in dynamic failure of materials. The spall strengths, as represented by void nucleation stress in this work, are found to be highly dependent on the misorientation angle: GBs with the normal direction close to [110], [111], and [112] have a higher spall strength in comparison to GBs with the normal direction close to [001]. No direct correlation is found between average properties of the GB including energy, excess volume, and the spall strength, which emphasizes the importance of considering local structure and properties of the GB in predicting its resistance to dynamic failure. Local GB structure can significantly alter the spall strength of the bi-crystal, and this effect is highly dependent on the misorientation angle as well. Our results reveal a transition of dominant deformation modes from screw dislocation-based plasticity to deformation twinning, as the misorientation angle increases for the [110] tilt boundaries. The profuse twinning for high-angle GBs provides additional void nucleation sites at twin-GB intersections and twin boundaries itself and results in a lower spall strength. However, as is typical for BCC metals, the observed variation in spall strengths does not follow Schmid's law. Comparison with Ta single-crystals indicates that GBs can play a dual role in the determining twinning of bi-crystals, serving to initiate twinning for low- and medium-angle GBs and inhibit twin growth for high-angle GBs. This work highlights the complex role played by GBs in dynamic failure and demonstrates a correlation between GB structure, the specific plasticity mechanism, and the resulting spall strength. However, additional work is required to understand why specific misorientations are sensitive to minor changes in the structure while others are not, as well as to investigate the effect of the orientation of the tilt axis on these results.

MD simulations utilized resources provided by the LANL Institutional Computing Program. The work was performed as a part of LDRD project (No. LDRD-2017033DR). This work was supported by the U.S. Department of Energy through the Los Alamos National Laboratory. Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of U.S. Department of Energy (Contract No. 89233218CNA000001). J. Chen and A. M. Dongare also acknowledge the funding by National Science Foundation (NSF) CMMI (Grant No. 1454547).