Analysis of a thermo-mechanical model for shape memory alloys

This note addresses the mathematical study of a nonlinear model arising in the description of the macroscopic thermo-mechanical behavior of shape memory materials and previously introduced in Peyroux et al. 1996. In particular, we discuss the model derivation and investigate a system of PDE\u27s coupled with a vectorial variational inequality. We develop the analysis in both the dissipative and the non-dissipative case, providing indeed a quantitative asymptotic connection between the two regimes. Moreover, we prove the global in time well-posedness for suitable initial and boundary value problems. As a by-product of the well-posedness analysis, we address a variable time-step discretization procedure, proving indeed its convergence and providing some a priori error bounds. Finally, we deal with the asymptotic behavior of the system for large times and establish the convergence of the trajectories to the solution of a suitable stationary problem