Derivation of the stress states for (111) (112) multiple slip and twinning

Abstract

The yield equation of a slip or twinning system is represented by a surface in five dimensional stress coordinates. Because of crystallographic symmetry, the yield surfaces of the twelve {111}〈112〉 twin systems can be arranged as faces of three separate tetrahedra in three-dimensional shear stress coordinates σ₂₃, σ₃₁, σ₁₂. The other two stress components then determine the size of the tetrahedra. For {111}〈112〉 (or {112}〈111〉) slip, three additional tetrahedra are needed for slip in the reverse direction. A general shape change requirement of five or more active twin (or slip) systems may be found as intersections of five or more faces among the tetrahedra with the requirement that the stresses at the intersections do not exceed the yield value for the nonactive systems. These intersections have been obtained systematically. It is verified that the lists of stress states previously reported by Hosford and Chin for {111}〈112〉 multiple slip and twinning are complete.