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Materials from mathematics. (English) Zbl 1409.74035

This review paper covers solid-solid structural transformations that manifest at the macroscopic scale as hysteresis. From a mathematical point of view, the necessary ingredients at the microscale are the symmetry properties of the phases and some non-convexity assumptions on the free-energy function (Section 3). The Cauchy-Born rule is the cornerstone of the theory, and in order to account for the symmetry of the lattice, a minimal set of assumptions leads to an energy well structure. Generic experimental results show that finite microstructures accompany the phase transformation: Section 4 discusses the situation when along a sequence of piecewise homogeneous deformations supported by rank-one connected deformation gradients in the energy-wells, the infimum of the energy functional is not attained. Finite-size microstructures observed in experiments and the nonattainment result draw attention to additional missing microscopic physics. Two possible issues are mentioned in Section 5: energy of the interfaces between some variants of the martensite, and the elastic energy in the transition austenite/martensite layer. Section 6 introduces a degeneracy related to crystallographic theory, the cofactor condition that enriches significantly the class of microstructures. As this condition can also be interpreted in terms of relaxed energy, this angle opens a path toward a general theory of supercompatibility. The last section illustrates the reversible transforming materials as a potential candidate for heat-to-electricity conversion.

MSC:

74N30 Problems involving hysteresis in solids
74N05 Crystals in solids
74N20 Dynamics of phase boundaries in solids
74G65 Energy minimization in equilibrium problems in solid mechanics
82B26 Phase transitions (general) in equilibrium statistical mechanics
74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids

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