Convergent spectral approximations for the thermomechanical processes in shape memory alloys (Q1818011)

In this article, discrete spectral approximations to the nonlinear partial differential equations arising from the conservation laws that model the dynamics of thermomechanical martensitic transformations (solid-solid phase transitions) in one-dimensional shape memory alloys with non-convex Landau-Ginzburg potentials are developed. By using the theories of analytic semigroups and interpolation spaces and a generalization of Gronwall's lemma for singular kernels, the convergence of the approximations is shown to hold not only in the state-space norm but also in the stronger \(\|\cdot \|_\delta\) -norm. The numerical experiments performed using this scheme show that under different initial conditions and distributed external actions the developed model is able to produce solutions whose qualitative behavior is found to be in close agreement with laboratory experiments performed on shape memory alloys under similar conditions. From a practical point of view it would be very important to find the values of the vector parameter \(q\) that ``best fit'' experimental data for a given alloy. This is called the parameter identification problem about which no results are yet known. In this regard the scheme presented here provides a friendly mathematical framework for attacking this problem.