Continuum Mech. Thermodyn. (2011) 23:491–507 DOI 10.1007/s00161-011-0190-0 ORIGINAL ARTICLE Tomáš Roubíˇ cek Approximation in multiscale modelling of microstructure evolution in shape-memory alloys Received: 7 April 2010 / Accepted: 13 April 2011 / Published online: 26 May 2011 © Springer-Verlag 2011 Abstract Various models of microstructure in deformation gradient and its evolution arising in martensitic mechanically-induced isothermal phase transformation are surveyed and scrutinized, focusing on over-bridging of various scales of the problem and its numerical approximation. In particular, numerically efficient model of a relaxed problem is shown to be approximated by conventional but computationally less efficient model based on standard partial differential inequalities. Keywords Smart materials · Rate-independent processes · Young measures Mathematics Subject Classification (2000) 35K85 · 49S05 · 65Z05 · 74N15 1 Introduction, principles of SMAs Shape-memory alloys (=SMAs) are representatives of so-called smart materials which enjoy important appli- cations especially in engineering and human medicine. SMAs exhibit specific, hysteretic stress/strain response and a so-called shape-memory effect. The mechanism behind it is quite simple: atoms tend to be arranged in several crystallographic configurations having different symmetry groups: higher symmetrical one (referred to as the austenite phase, typically cubic) has higher heat capacity while lower symmetrical one (called the martensite phase, typically tetragonal, orthorhombic, monoclinic or a rhomboedric R-phase) has lower heat capacity and may exist, by symmetry, in several variants. We refer to [5, 11, 43, 49] for a thorough survey. Coexistence of various phases or phase variants and their (usually) fast reaction on (usually) slowly evolving external loading typically lead to complicated microstructure (cf. Fig. 1 (left) below) with very complex evo- lution behaviour, which gives an ultimate time/spatial multiscale character to the problem whose modelling thus becomes extremely difficult. Confining to isothermal models based on continuum mechanics, there are several kinds of models depend- ing on how the microstructure is described: here, we focus on a ‘microscopical’, or PDE-type model, based on conventional partial differential equations or inequalities in terms of deformation with possibly some order parameter, and on a ‘mesoscopical’ model expressed in terms of displacements combined with special gradient Young measures to reflect better a multiscale character of the problem, cf. [4, 31, 32] or also [55, Chap.6]. Other models may involve further internal variables like volume fractions; for a survey, see [57]. Communicated by Prof. Klaus Hackl. T. Roubíˇ cek (B ) Mathematical Institute, Charles University, Sokolovská 83, 186 75 Prague 8, Czech Republic E-mail: tomas.roubicek@mff.cuni.cz T. Roubíˇ cek Institute of Thermomechanics of the ASCR, Dolejškova 5, 182 00 Prague 8, Czech Republic