Preprint no. 139
Mathematical Institute of the Academy of Sciences of the Czech Republic

On the hysteresis in martensitic transformations

Miroslav Silhavy

Mathematical Institute, Acad. Sci.of the Czech Republic, 115 67 PRAHA 1, Zitna 25, Czech Republic, e-mail: silhavy@math.cas.cz

Summary:

The paper proposes an explanation of the hysteresis in shape memory alloys using energy minimization in nonlinear elasticity and the entropy criterion. The stored energy has two wells describing the two phases. At elongations from some interval, the minimum energy is realized on two-phase mixtures. If the loaded phases are incompatible, that minimum energy is a {\it concave\/} function, thus giving a phase equilibrium curve of negative slope. The quasistatic evolution during loading experiments is realized in the class of mechanically but not thermodynamically equilibrated mixtures. This family contains states of elongation and force covering the whole area of the hysteresis loop. The evolution must satisfy the entropy criterion for moving phase interfaces which implies that in the region above the phase equilibrium line only processes with nondecreasing amount of the second phase are possible while below the situation is opposite. This picture provides all the elements necessary for the explanation of the hysteresis, including the internal hysteresis loops.

Keywords: hysteresis, pseudoelasticity, martensitic transformations, rank 1 convexity 

Mathematics Subject Classification 2000: 74N10, 74B20

Full version: M. Silhavy, 2003, On the hysteresis in martensitic transformations. In: Rational Continua, Classical and New. A collection of papers dedicated to Gianfranco Capriz. P. Podio--Guidugli, M. Brocato eds. Pp. 151--168. Springer. Berlin.

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