Thermal Control of a Rate-independent Model for Permanent Inelastic Effects in Shape Memory Materials

We address the thermal control of the quasi-static evolution of a polycrystalline shape memory alloy specimen. The thermomechanical evolution of the body is described by means of an extension of the phenomenological Souza-Auricchio model [6, 7, 8, 57] accounting also for permanent inelastic effects [9, 11, 27]. By assuming to be able to control the temperature of the body in time we determine the corresponding quasi-static evolution in the energetic sense. In a similar way as in [28], using results by Rindler [49, 50] we prove the existence of optimal controls for a suitably large class of cost functionals. Shape-memory alloys (SMA) are examples of active materials, showing a remarkable thermo-mechanical behaviour: at suitably high temperatures they are able to completely recover comparably large strains during the loading-unloading cycles (this is the so-called superelastic effect) while at lower temperatures permanent deformations appear when unloading, but the material can be forced to recover its original shape by means of a thermal cycle (this is the so-called shape memory effect). This characteristic macroscopic behaviour is the result of a solid-solid phase transition at the metallic lattice level between a highly symmetric crystallographic phase, called austenite, which is dominant at high temperatures, and less symmetric phases, called martensites, which are energetically favorable at lower temperatures or high stresses. This amazing thermo-mechanical behaviour of the SMA is at the basis of a great variety of innovative applications, going from biomedicine to different branches of engineering. Indeed the engineering literature of SMA models is large and the SMA behaviour has been investigated at all scales and by means of a great number of models (the interested reader can find a lot of references in this respect for instance in our previous paper [27]). Correspondingly, the mathematical treatment of SMA behaviour is comparably less developed. Some results in this sense refer to the original formulation or modifications of the Frémond model, [33] or in the Falk or Falk-Konopka models, [31, 32], see for instance, with no claim of completeness, [1, 2, 19, 22, 35, 38, 48, 62]; we also refer to more recent results concerning phase transitions in shape memory 1991 Mathematics Subject Classification. 74C05;49J20.