Equilibrium for Multiphase Solids with Eulerian Interfaces
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Equilibrium for Multiphase Solids with Eulerian Interfaces Diego Grandi1 · Martin Kružík2,3 · Edoardo Mainini4 · Ulisse Stefanelli5,6,7
Received: 12 May 2020 / Accepted: 7 October 2020 / Published online: 23 November 2020 © Springer Nature B.V. 2020
Abstract We describe a general phase-field model for hyperelastic multiphase materials. The model features an elastic energy functional that depends on the phase-field variable and a surface energy term that depends in turn on the elastic deformation, as it measures interfaces in the deformed configuration. We prove existence of energy minimizing equilibrium states and -convergence of diffuse-interface approximations to the sharp-interface limit. Mathematics Subject Classification (2010) 74G25 · 49J45
B U. Stefanelli
[email protected] D. Grandi [email protected] M. Kružík [email protected] E. Mainini [email protected]
1
Dipartimento di Matematica e Informatica, Università degli Studi di Ferrara, Via Machiavelli 30, 44121 Ferrara, Italy
2
Institute of Information Theory and Automation, Czech Academy of Sciences, Pod vodárenskou veží 4, 182 08, Prague 8, Czech Republic
3
Faculty of Civil Engineering, Czech Technical University, Thákurova 7, 166 29, Prague 6, Czech Republic
4
Dipartimento di Ingegneria meccanica, energetica, gestionale e dei trasporti, Università degli studi di Genova, Via all’Opera Pia, 15, 16145 Genova, Italy
5
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
6
Vienna Research Platform on Accelerating Photoreaction Discovery, University of Vienna, Währingerstraße 17, 1090 Wien, Austria
7
Istituto di Matematica Applicata e Tecnologie Informatiche E. Magenes, via Ferrata 1, 27100 Pavia, Italy
410
D. Grandi et al.
Keywords Elasticity · Eulerian-Lagrangian description · Phase transition · Variational methods · Gamma-convergence
1 Introduction Mathematical models of multi-component (or multi-phase) materials have attracted the attention of researchers for decades. A prominent example of multi-phase materials is provided by shape memory alloys, i.e., intermetallic materials having a high-temperature phase called austenite and a low-temperature phase called martensite, existing in many symmetryrelated variants, see [6, 9]. Mathematical analysis of elastostatic problems of such materials is involved because of the lack of suitable convexity properties. In fact, these materials exhibit complicated microstructures which are reflected in faster and faster oscillations of minimizing sequences driving the elastic energy functional to its infimum. Consequently, no minimizer generically exists and various methods have been developed to cope with this difficulty. A possibility to overcome the nonexistence issue is to search for a lower semicontinuous envelope of the energy functional that describes macroscopic behavior of the specimen [13]. This provides us with a solvable minimization problem and ensures that every minimizer is reachable by a minimizing sequence of the original pro
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