Energy scaling laws for geometrically linear elasticity models for microstructures in shape memory alloys^*

¹ Institut für Angewandte Mathematik, Universität Bonn, 53115 Bonn, Germany.
² Fakultät für Mathematik, Universität Wien, 1090 Wien, Austria.
³ Institut für Mathematik, Humboldt-Universität zu Berlin, 10117 Berlin, Germany.

^** Corresponding author: barbara.zwicknagl@math.hu-berlin.de

Received: 27 May 2019
Accepted: 16 April 2020

Abstract

We consider a singularly-perturbed two-well problem in the context of planar geometrically linear elasticity to model a rectangular martensitic nucleus in an austenitic matrix. We derive the scaling regimes for the minimal energy in terms of the problem parameters, which represent the shape of the nucleus, the quotient of the elastic moduli of the two phases, the surface energy constant, and the volume fraction of the two martensitic variants. We identify several different scaling regimes, which are distinguished either by the exponents in the parameters, or by logarithmic corrections, for which we have matching upper and lower bounds.