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Energy-minimizing Strains in Martensitic Microstructures


Shape memory alloys exhibit a solid/solid phase transformation between different crystallographic structures, known as austenite (stable at high temperature) and martensite (stable at low temperature). That phase transformation is triggered both by thermal and mechanical loading. In terms of crystallographic structure, the austenite has a higher symmetry than the martensite, which leads one to distinguish several symmetry-related martensitic variants. To each martensitic variant is attached a transformation strain that describes the deformation between the crystallographic structures of the austenite and the martensite. The number of martensitic variants as well as the corresponding transformation strains depend on the alloy considered, through the structure of the austenite and martensite lattices. Some common examples include the cubic to tetragonal transformation (MnCu, MnNi), the cubic to orthorombic transformation (CuAlNi) and the cubic to monoclinic transformations (NiTi), corresponding respectively to 3, 6 and 12 martensitic variants. This talk is concerned with the theoretical prediction of the set of strains that minimize the effective (or macroscopic) energy. Those strains, classically refered to as recoverable strains, play a central role in the shape memory effect. The macroscopic energy is defined as the quasiconvexification (or relaxation) of a multi-well energy function that models the behaviour of the material at a microscopic level. The relaxation procedure essentially consists in finding the austenite/martensite microstructures which minimize the total energy. The work presented aims at complementing existing results on that problem, essentially through the use of bounds on the set of energy-minimizing strains. Upper bounds are obtained using distinctive properties of Young measures. Lower bounds are constructed using lamination techniques. Both the geometrically nonlinear setting (finite strains) and the geometrically linear setting (infinitesimal strains) are covered, the latter being less accurate but significantly more tractable. In the geometrically nonlinear setting, analytical expressions of both lower and upper bounds are derived for a general three-well problem that encompasses the cubic to tetragonal transformation as a special case. In the geometrically linear setting, the twelve-well problems corresponding to cubic to monoclinic transformations are studied in detail.
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hal-00832134 , version 1 (10-06-2013)


  • HAL Id : hal-00832134 , version 1


Michaël Peigney. Energy-minimizing Strains in Martensitic Microstructures. Bristol Fluids and Materials Seminar, University of Bristol, Mar 2013, Bristol, United Kingdom. ⟨hal-00832134⟩
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