Shear deformation, ideal strength, and stacking fault formation of fcc metals: A density-functional study of Al and Cu

Abstract

Ab initio density-functional calculations have been used to study the response of two face-centered-cubic metals (Al and Cu) to shearing parallel to the close-packed (111) planes along two different directions, $[11\overline{2}]$ and $[\overline{1}10]$. Two different types of deformations---affine and alias---have been investigated. Under an affine shear deformation, all atoms are shifted parallel to the shearing direction by a distance proportional to their distance from the fixed basal plane. In the alias regime, only the top layer is displaced in the shearing direction. In both regimes, calculations have been performed with (pure shear) and without (simple shear) relaxation. For a pure alias shear, due to the interaction between the atoms, the displacement propagates through the sample; this is certainly the most realistic description of the shearing processes. In the pure alias regime, shear deformation, theoretical shear strength, and stacking fault formations may be described on a common footing. For small strains (in the elastic region), affine and alias shears lead to very similar results. Beyond the elastic limit, relaxation has a strong influence of the response on an applied shear strain. The elastic shear moduli are significantly larger for Cu than for Al, but a much higher shear strength is calculated for Al, although the shear strength is limited by the occurrence of a stacking fault instability before the stress maximum is reached. Under $⟨\overline{1}10⟩$ {111} shear the analysis of the atomistic deformation mechanism shows that in this case the formation of a stacking fault leads to a splitting of the $\frac{1}{2}[\overline{1}10]$ dislocation into two partial Shockley dislocations. Due to the repulsive interaction between the atoms in adjacent close-packed planes, the atoms in the top A layer move along $\frac{1}{6}[\overline{2}11]$ to a position directly above the B layer such that the stable intrinsic stacking fault configuration is the same for both slip systems. The analysis of the variation in the lattice parameters under strain reveals significant differences in the relaxation behavior of both metals: Al is very stiff, but Cu is rather soft along the $⟨112⟩$; in-plane relaxation is very strong for Cu but modest for Al. This much stronger relaxation explains that while the differences in the unstable stacking fault energies of both metals are only modest, the intrinsic stacking fault energies differ by as much as a factor of 4. A detailed comparison of the response to shear and tensile deformations has been performed. A phonon instability of the uniaxial tensile deformation along the [110] direction has been explained by the close connection with the shear system $⟨11\overline{2}⟩$ {111}.