Compact zones of shear transformation in an anisotropic solid

Abstract

The possible configurations of transformed regions in a material that undergoes mechanical phase change are investigated. Anti-plane shear kinematics are assumed. A constitutive law is proposed for a cubic crystal which can sustain deformations with strain discontinuities, such as twinning. It is characterized by an anisotropic stored energy function which loses ellipticity and has multiple potential wells with corresponding stress-free states, connected by transformation strains which are simple shears. The problem of a bounded transformed high-strain zone of unknown shape in an infinite body is studied. The strains must take values within different phases, i.e. disjoint subsets of the domain of strong ellipticity. This condition imposes severe restrictions on the shape of the inclusion ; its boundary cannot be smooth and cannot have corners. Only sufficiently slender regions, terminating in cusps and oriented along special directions are possible. Qualitative geometrical features of twin lamellae are thus predicted. Integral representations of the deformation and estimates of the strains in such zones are obtained.