Quasiconvexity at the boundary and the nucleation of austenite

TL;DR

This paper explains why austenite nucleates at corners in a shape-memory alloy by analyzing energy minimization and quasiconvexity conditions in a nonlinear elasticity model.

Contribution

It introduces a simplified nonlinear elasticity model demonstrating how quasiconvexity conditions influence nucleation sites in shape-memory alloys.

Findings

Nucleation points are at corners due to energy minimization.

Quasiconvexity conditions are satisfied in interior, faces, and edges.

Corners allow microstructures that lower energy, explaining nucleation locations.

Abstract

Motivated by experimental observations of H. Seiner et al., we study the nucleation of austenite in a single crystal of a CuAlNi shape-memory alloy stabilized as a single variant of martensite. In the experiments the nucleation process was induced by localized heating and it was observed that, regardless of where the localized heating was applied, the nucleation points were always located at one of the corners of the sample - a rectangular parallelepiped in the austenite. Using a simplified nonlinear elasticity model, we propose an explanation for the location of the nucleation points by showing that the martensite is a local minimizer of the energy with respect to localized variations in the interior, on faces and edges of the sample, but not at some corners, where a localized microstructure, involving austenite and a simple laminate of martensite, can lower the energy. The result for the interior, faces and edges is established by showing that the free-energy function satisfies a set of quasiconvexity conditions at the stabilized variant in the interior, faces and edges, respectively, provided the specimen is suitably cut.