A Peierls Criterion for Deformation Twinning at a Mode II Crack

Summary

The theoretical criterion for deformation twinning (DT) at a crack tip due to Tadmor and Hai [TH03] is reviewed. The criterion quantifies the competition between slip and DT at a crack tip in terms of the critical stress intensity factors necessary to nucleate a dislocation and a two-layer microtwin there. The analysis is based on Rice’s Peierls framework for dislocation emission from a crack tip [Ric92]. It is found that DT at a crack tip is controlled by a new material parameter named the “unstable twinning energy”, which plays an analogous role for twinning as Rice’s unstable stacking energy plays for slip. The derivation presented in this paper is for the simplest possible special case: a crack with a co-planar slip plane loaded in pure mode II shear along a crystallographic slip direction. The analysis is limited to a face-centered cubic (fcc) crystal. While this derivation contains all of the features in the full derivation in [TH03], the simpler geometry helps to clarify the underlying concepts. Some of the expressions derived for this special case are new.