Limits on Transformation Strains for Non-Negative Dissipation

Abstract

We consider the maximum value of the magnitude of transformation strain for an Eshelby inclusion set by the requirement of non-negative dissipation. The general formulation for a linear elastic solid shows that the dissipation associated with a strain transformation can be calculated as an integral over the transformed inclusion. Closed-form expressions are given for the maximum transformation strain magnitude in an isotropic linear elastic solid for both cylindrical and spherical inclusions that have undergone transformations corresponding to either a pure volume (or area) change or a pure shear. Most results presented are for transformations in an infinite solid and presume uniform material properties. Examples of the effect of a finite boundary and of differing material properties inside and outside the transformed inclusion are also given. The analytical results indicate that non-negative dissipation typically limits the transformation strain to being a constant of order unity times the critical stress at transformation divided by a relevant elastic modulus.