Study of the cofactor conditions: conditions of supercompatibility between phases

The cofactor conditions, introduced in James and Zhang, are conditions of compatibility between phases in martensitic materials. They consist of three subconditions: i) the condition that the middle principal stretch of the transformation stretch tensor $\mathbf U$ is unity ($\lambda_2 = 1$), ii) the condition $\mathbf a \cdot \mathbf U\, \cof (\mathbf U^2 - \mathbf I)\mathbf n = 0$, where the vectors $\mathbf a$ and $\mathbf n$ are certain vectors arising in the specification of the twin system, and iii) the inequality ${\rm tr} \mathbf U^2 + \det \mathbf U^2 -(1/4) |\mathbf a|^2 |\mathbf n|^2\ge 2$. Together, these conditions are necessary and sufficient for the equations of the crystallographic theory of martensite to be satisfied for the given twin system but for any volume fraction f of the twins, $0 \le f \le 1$. This contrasts sharply with the generic solutions of the crystallographic theory which have at most two such volume fractions for a given twin system of the form $f^*$ and $1-f^*$. In this paper we simplify the form of the cofactor conditions, we give their specific forms for various symmetries and twin types, we clarify the extent to which the satisfaction of the cofactor conditions for one twin system implies its satisfaction for other twin systems. In particular, we prove that the satisfaction of the cofactor conditions for either Type I or Type II twins implies that there are solutions of the crystallographic theory using these twins that have no elastic transition layer. We show that the latter further implies macroscopically curved, transition-layer-free austenite/martensite interfaces for Type I twins, and planar transition-layer-free interfaces for Type II twins which nevertheless permit significant flexibility of the martensite. We identify some real material systems nearly satisfying the cofactor conditions.