*3.2. Cohesive Strength of Hydrogen-Free GBs*

The computational tensile tests were first performed for a perfect crystal of Ni loaded in crystallographic directions corresponding to the orientations of the loading axes in GB models (perpendicular to the GB planes), i.e., 111, 210, and 311 for Σ3, Σ5, and Σ11, respectively. We started with the uniaxial deformation keeping the transverse lattice parameters constant (see Table 1) and computed all the normal stresses as functions of the axial strain 1. The results are displayed in Figure 5, where the transverse stresses *σ*2, *σ*<sup>3</sup> are plotted as functions of the axial stress *σ*<sup>1</sup> up to *σ*max.

**Figure 5.** The relationship between the transverse stresses *σ*2, *σ*<sup>3</sup> and the axial stress *σ*<sup>1</sup> computed for the uniaxial deformation of bulk fcc Ni in the 111, 210, and 311 crystallographic directions.

One can note that omitting the Poisson contraction during the uniaxial deformation induces tensile transverse stresses that are superimposed to *σ*1, thus making the stress state triaxial. Values of *σ*<sup>2</sup> and *σ*<sup>3</sup> generally differ. They depend not only on the particular orientation of the *x*-axis but also on anisotropy of the perpendicular ({111}, {210}, and {311}) atomic planes. Therefore, we respected this anisotropy by keeping the same ratios *σ*3/*σ*<sup>2</sup> also in the cases of triaxial loading (see Table 1) applied to supercells simulating perfect crystals as well as the crystals with GBs. The greatest ratio *σ*3/*σ*<sup>2</sup> = 1.4 was obtained for the [210] direction, i.e., the same ratio was used in the triaxial tensile tests of supercells with the Σ5 GB. The ratio obtained for the [111] deformation is approximately equal to 1, thus, this ratio was also used for the triaxial tensile tests of Σ3 GB. The response of Ni crystal elongated in the {311} direction is somewhat complicated. For smaller strains (and the *σ*<sup>1</sup> values slightly above 15 GPa), the ratio is close to 1.0, but for greater strains (and the stresses close to the *σ*max value) it decreases to *σ*3/*σ*<sup>2</sup> = 0.8. The latter value was selected for the triaxial loading of Σ11 GB since the ratio close to the *σ*max value is of a higher relevance.

The tensile tests for all the loading conditions listed in Table 1 were then applied to the optimized GBs and the computed data are summarized in Figure 6. The left panel displays results of the triaxial loading in terms of the cohesive strength value *σ*max as a function of the applied transverse stress *σ*<sup>2</sup> for each GB model and the crystallographic direction. The right panel of Figure 6 shows the results received for special loading cases: uniaxial deformation, isotropic loading, and isotropic deformation.

**Figure 6.** Computed values of the cohesive strength *σ*max for the bulk crystal (symbols connected by dashed lines), clean GB (open symbols, solid lines), and H-charged GB (solid symbols, solid lines). The left panel shows *σ*max as a function of one of the transverse stresses (*σ*2) during triaxial loading. The right panel displays the *σ*max values computed for other specific loading types. Note that the data for *σ*<sup>2</sup> = 0 correspond to results of uniaxial loading.

The results shown in the left panel of Figure 6 reveal that *σ*max for Σ5 and Σ11 GBs linearly increases with increasing transverse stresses (as already reported for a majority of perfect cubic crystals and loading directions [28]), while it remains constant for the Σ3 GB and the bulk Ni crystal loaded in the [111] direction. Such an insensitivity to the superimposed transverse stresses is also indicated by the value of *σ*max achieved via uniaxial deformation that is practically equal to *σ*max from uniaxial loading (i.e., the value in the left panel of Figure 6 for *σ*<sup>2</sup> = 0).

One can also see that *σ*max values of the bulk crystal loaded in the crystallographic directions corresponding to orientations of grains in individual GBs (plotted in Figure 6 by symbols connected by dashed lines) mostly follow the values computed for the related GBs. In the cases of Σ3(111) and Σ11(311) GBs, values for the bulk and clean GB are almost equal (naturally, with slightly higher *σ*max values for the bulk crystal). This means that the presence of Σ3 and Σ11 GBs practically does not reduce the crystal strength. More remarkable reduction of *σ*max is caused by the presence of Σ5(210) GB.

The right panel in Figure 6 shows that the isotropic (hydrostatic) loading as well as the isotropic deformation yield lower *σ*max values than the uniaxial deformation. Let us note that, in the case of bulk crystals, both the isotropic loading and the isotropic deformation must lead to the same *σ*max value, regardless of the orientation of the loading axis. The corresponding data points (each obtained using a differently oriented supercell or deformation model) displayed in Figure 6 confirm reliability of our computational procedures by their negligible differences. Interestingly, almost the same values were also obtained for the supercell with the clean Σ3 GB, thus indicating that its effect on crystal strength can be considered negligible. On the other hand, presence of the other two GBs (with higher

energy *γ*GB) significantly reduce *σ*max under isotropic tensile loading. Therefore, one can presume that the cohesive strength of general GBs exhibiting greater GB energy will be even more reduced, particularly for highly triaxial stress states.
