On Rigidity for the Four-Well Problem Arising in the Cubic-to-Trigonal Phase Transformation

TL;DR

This paper classifies all stress-free solutions in the linearized elasticity model for the cubic-to-trigonal phase transformation, revealing that only simple laminates and crossing-twin structures are possible, and demonstrating their rigidity.

Contribution

It provides a complete classification of stress-free solutions for this phase transformation, showing their rigidity and limiting structures to laminates and crossing-twins.

Findings

Only simple laminates and crossing-twin structures are stress-free.

All solutions to the transformation are rigid.

The classification relies on compatibility and nonlinear strain conditions.

Abstract

We classify all exactly stress-free solutions to the cubic-to-trigonal phase transformation within the geometrically linearized theory of elasticity, showing that only simple laminates and crossing-twin structures can occur. In particular, we prove that although this transformation is closely related to the cubic-to-orthorhombic phase transformation, all its solutions are rigid. The argument relies on a combination of the Saint-Venant compatibility conditions together with the underlying nonlinear relations and non-convexity conditions satisfied by the strain components.