14 1 Deformation and Fracture of Perfect Crystals 1.1.1.2 Ab initio Methods Nowadays,so-called ab initio (or first principle)approaches enable us to compute the crystal energy in a very accurate manner.This particularly holds for single crystals of pure elements or compounds,crystals with periodical arrangements of atoms of various kinds and also for local crystal defects. These methods utilize the density functional theory 37,38 in which the problem of many interacting electrons is transformed into a study of single electron motion in an effective potential.A brief description of principles of such methods is presented in Appendix A. Most probably,the first ab initio calculation of the uniaxial IS oiut was that of Esposito et al.39 for the copper crystal.However,these authors did not perform relaxations of atomic positions inside the loaded crystal in directions perpendicular to the loading axis(Poisson's type of expansion or contraction).Probably the first ab initio simulation of a tensile test that included the relaxation in perpendicular directions to the loading axis was performed by Price et al.[40]for TiC along the 001]axis.Later,oiut was calculated for [001]and [111]loading axes for a variety of cubic crystals by Sob et al.[41-43].Kitagawa and Ogata [44,45]studied the tensile IS of Al and AlN but also did not include the Poisson's contraction.Further calculations of oiut, performed for a-SiC,diamond,Si,Ge,Mo,Nb and SigN4,have already taken that effect into account by allowing a transversal relaxation of atoms [46-51. The values Tis,b of the shear IS were first calculated by Paxton et al.[32 for V,Cr,Nb,Mo,W,Al,Cu and Ir.The values Tis calculated according to the model of a uniform shear(see Figure 1.3)were later reported by Moriarty et al.52,53 for Mo and Ta.These calculations did not include any relaxations. Recently,the relaxed values of Tis were calculated for many crystals such as TiC,TiN,HfC,Mo,Nb,Si,Al,Cu and W by groups of Morris et al.,Kitamura et al.and Pokluda et al.[54-58].In these models,the interplanar distance was relaxed towards the minimum energy during deformation.More advanced models also enabled relaxations of the arrangement of atomic positions within the planes [48,59-see also Sections 1.1.2 and 1.1.3. Since 1997,ab initio calculations of oiht have been performed by Pokluda et al.[60-62]and other authors (e.g.,63)).Since spherical symmetry was maintained during deformation,the relaxation procedures were not necessar- ily applied in these models. In the majority of older studies on IS,the deformation process was assumed to proceed in a stable manner until the applied stress reached its maximum value.This means that the crystal failed in the same mode in which it was originally deformed from the very beginning.However,this assumption was disputed in many works [64-66.Under both tensile and compressive load- ings,the shear stresses in some slip systems can exceed their critical values (corresponding to the related Tis)well before the stress reaches its maximum. This was also observed in tensile tests on whiskers [30,33.Indeed,some of the whiskers evidently failed by shear across some favourable crystallographic14 1 Deformation and Fracture of Perfect Crystals 1.1.1.2 Ab initio Methods Nowadays, so-called ab initio (or first principle) approaches enable us to compute the crystal energy in a very accurate manner. This particularly holds for single crystals of pure elements or compounds, crystals with periodical arrangements of atoms of various kinds and also for local crystal defects. These methods utilize the density functional theory [37, 38] in which the problem of many interacting electrons is transformed into a study of single electron motion in an effective potential. A brief description of principles of such methods is presented in Appendix A. Most probably, the first ab initio calculation of the uniaxial IS σiut was that of Esposito et al. [39] for the copper crystal. However, these authors did not perform relaxations of atomic positions inside the loaded crystal in directions perpendicular to the loading axis (Poisson’s type of expansion or contraction). Probably the first ab initio simulation of a tensile test that included the relaxation in perpendicular directions to the loading axis was performed by Price et al. [40] for TiC along the [001] axis. Later, σiut was calculated for [001] and [111] loading axes for a variety of cubic crystals by Sob ˇ et al. [41–43]. Kitagawa and Ogata [44,45] studied the tensile IS of Al and AlN but also did not include the Poisson’s contraction. Further calculations of σiut, performed for α-SiC, diamond, Si, Ge, Mo, Nb and Si3N4, have already taken that effect into account by allowing a transversal relaxation of atoms [46–51]. The values τis,b of the shear IS were first calculated by Paxton et al. [32] for V, Cr, Nb, Mo, W, Al, Cu and Ir. The values τ ∗ is calculated according to the model of a uniform shear (see Figure 1.3) were later reported by Moriarty et al. [52,53] for Mo and Ta. These calculations did not include any relaxations. Recently, the relaxed values of τis were calculated for many crystals such as TiC, TiN, HfC, Mo, Nb, Si, Al, Cu and W by groups of Morris et al., Kitamura et al. and Pokluda et al. [54–58]. In these models, the interplanar distance was relaxed towards the minimum energy during deformation. More advanced models also enabled relaxations of the arrangement of atomic positions within the planes [48, 59] – see also Sections 1.1.2 and 1.1.3. Since 1997, ab initio calculations of σiht have been performed by Pokluda et al. [60–62] and other authors (e.g., [63]). Since spherical symmetry was maintained during deformation, the relaxation procedures were not necessar￾ily applied in these models. In the majority of older studies on IS, the deformation process was assumed to proceed in a stable manner until the applied stress reached its maximum value. This means that the crystal failed in the same mode in which it was originally deformed from the very beginning. However, this assumption was disputed in many works [64–66]. Under both tensile and compressive load￾ings, the shear stresses in some slip systems can exceed their critical values (corresponding to the related τis) well before the stress reaches its maximum. This was also observed in tensile tests on whiskers [30, 33]. Indeed, some of the whiskers evidently failed by shear across some favourable crystallographic