Analysis of a thermomechanical model of shape memory alloys, pp. 487-536
Authors: Aida Timofte and Vlad Timofte
Abstract: This chapter is devoted to the mathematical study of a thermomechanical model describing the macroscopic behavior of shape memory alloys (SMA). Various experiments (slow uniaxial traction-compression tests on cylindrical SMA samples) emphasized the temperature changes during hysteresis loops and relaxation and creep phenomena. They also established the absence of permanent deformations at the end of a loading-unloading cycle, the so-called pseudoelastic behavior of SMA. The macroscopic phenomena observed during the experiments can be predicted by using a classical (Gibbs) thermomechanical model founded on a free energy, which is convex with respect to the strain and to the martensitic volume fraction, and concave with respect to the temperature. The model takes into account the non-isothermal character of the phase transition, as well as the existence of the intrinsic dissipation. The first law of thermodynamics, the balance of momentum in its quasistatic form, together with the evolution equation for the internal variables, form a partial differential equation system, whose solutions fulfill the second principle of thermodynamics (the entropy inequality). In the circular cylindrical case with non-negligible radius, this system reduces to an ordinary differential system. In this case, we prove the uniqueness of solutions in a large class of spaces (abstract differential structures). Existence and regularity of solutions are then established in various functions spaces endowed with natural derivatives. At the end of the chapter, for a particular traction-compression test we present the exact solution, together with the associated hysteresis loop.