A study of the stress state associated with twin nucleation and propagation in anisotropic materials

Abstract

Abstract In this work we are concerned with the study of the stress state associated with the activation of twinning in elastically anisotropic materials, in an attempt to achieve a better understanding of the contribution that twinning makes to plastic deformation and to texture development during forming operations. The assumption usually made is that twinning is activated by a critical resolved shear stress on the twinning plane and in the twinning direction, although the experimental evidence suggests that twinning depends in a more complex way upon the other stress components present in the medium. Here we use a continuum approach to model the twin lamella as a flat inclusion of elliptic section embedded in an elastically anisotropic medium being acted upon by externally applied stresses. We calculate the Gibbs free energy of the system as the sum of an elastic term associated with size and inhomogeneity effects, and a surface energy term, associated with the twin-matrix interface. The minimization of the total energy with respect to the dimensions of the lamella is relevant both to twin nucleation and to twin propagation; in the former case, it gives the condition for when twin nucleation is energetically favourable with respect to a homogeneously strained matrix. In the latter case, it gives the condition of instability of an existing twin embryo, much the same as the Griffith's criterion does for crack extension. We derive explicit results for the twinning systems active in silicon-iron, calcite and hexagonal zirconium, titanium and zinc. We find that, although the critical condition for twin activation depends on all six independent stress components, the resolved shear dominates and the influence of the other stress components is about two orders of magnitude weaker. We conclude that any substantial dependence of twinning upon stress components other than the resolved shear must come from the dependence of the twin-boundary energy upon stress. Possible atomistic mechanisms which may be influenced by a general stress field are discussed qualitatively.