Upper and lower bounds on the set of recoverable strains and on effective energies in cubic-to-monoclinic martensitic phase transformations

¹ University of Würzburg, Institute for Mathematics, Emil-Fischer-Str. 40, 97074 Würzburg, Germany
² University of Bristol, School of Mathematics, University Walk, Bristol, BS8 1TW, UK
³ Ruhr-University Bochum, Institute for Mechanics, 44780 Bochum, Germany

^a e-mail: anja.schloemerkemper@mathematik.uni-wuerzburg.de

Abstract

A major open problem in the mathematical analysis of martensitic phase transformations is the derivation of explicit formulae for the set of recoverable strains and for the relaxed energy of the system. These are governed by the mathematical notion of quasiconvexity. Here we focus on bounds on these quasiconvex hulls and envelopes in the setting of geometrically-linear elasticity. Firstly, we will present mathematical results on triples of transformation strains. This yields further insight into the quasiconvex hull of the twelve transformation strains in cubic-to-monoclinic phase transformations. Secondly, we consider bounds on the energy of such materials based on the so-called energy of mixing thus obtaining a lamination upper bound on the quasiconvex envelope of the energy. Here we present a new algorithm that yields improved upper bounds and allows us to relate numerical results for the lamination upper bound on the energy with theoretical inner bounds on the quasiconvex hull of triples of transformation strains.