Domain patterns and hysteresis in phase-transforming solids:
Analysis and numerical simulations of a sharp interface dissipative model via phase-field approximation

Antonio DeSimone; Martin Kružík; Antonio DeSimone; Martin Kružík

doi:10.3934/nhm.2013.8.481

Networks and Heterogeneous Media

2013, Volume 8, Issue 2: 481-499. doi: 10.3934/nhm.2013.8.481

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Domain patterns and hysteresis in phase-transforming solids: Analysis and numerical simulations of a sharp interface dissipative model via phase-field approximation

Antonio DeSimone ^{1
,},
Martin Kružík ^{2
,}

1.
SISSA, International School of Advanced Studies, Via Bonomea 265, 34136 Trieste
2.
Institute of Information Theory and Automation of the ASCR, Pod vodárenskou věží 4, 182 08 Prague

Received: 01 October 2012 Revised: 01 March 2013
Primary: 74N30, 49J45, 74D10; Secondary: 74S05.

We propose a sharp-interface model which describes rate-independent hysteresis in phase-transforming solids (such as shape memory alloys) by resolving explicitly domain patterns and their dissipative evolution. We show that the governing Gibbs' energy functional is the $\Gamma$-limit of a family of regularized Gibbs' energies obtained through a phase-field approximation. This leads to the convergence of the solution of the quasistatic evolution problem associated with the regularized energy to the one corresponding to the sharp interface model. Based on this convergence result, we propose a numerical scheme which allows us to simulate mechanical experiments for both spatially homogeneous and heterogeneous samples. We use the latter to assess the role that impurities and defects may have in determining the response exhibited by real samples. In particular, our numerical results indicate that small heterogeneities are essential in order to obtain spatially localized nucleation of a new martensitic variant from a pre-existing one in stress-controlled experiments.
- Hysteresis,
- shape memory alloys,
- multi-well energies,
- rate-independent processes,
- phase-field models,
- sharp-interface limits
Citation: Antonio DeSimone, Martin Kružík. Domain patterns and hysteresis in phase-transforming solids:Analysis and numerical simulations of a sharp interface dissipative model via phase-field approximation[J]. Networks and Heterogeneous Media, 2013, 8(2): 481-499. doi: 10.3934/nhm.2013.8.481

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Abstract

We propose a sharp-interface model which describes rate-independent hysteresis in phase-transforming solids (such as shape memory alloys) by resolving explicitly domain patterns and their dissipative evolution. We show that the governing Gibbs' energy functional is the $\Gamma$-limit of a family of regularized Gibbs' energies obtained through a phase-field approximation. This leads to the convergence of the solution of the quasistatic evolution problem associated with the regularized energy to the one corresponding to the sharp interface model. Based on this convergence result, we propose a numerical scheme which allows us to simulate mechanical experiments for both spatially homogeneous and heterogeneous samples. We use the latter to assess the role that impurities and defects may have in determining the response exhibited by real samples. In particular, our numerical results indicate that small heterogeneities are essential in order to obtain spatially localized nucleation of a new martensitic variant from a pre-existing one in stress-controlled experiments.

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