Applications of fully conservative schemes in nonlinear thermoelasticity: modelling shape memory materials

In this paper, we consider a strongly coupled model of nonlinear thermoelasticity describing the dynamics of materials with shape memory. The model is not amenable to analytical treatments and the development, analysis, and applications of effective numerical approximations for this model is in the focus of the present paper. In particular, we discuss a recently proposed fully conservative difference scheme for the solution of the problem. We note that a standard energy inequality technique, applied to the analysis of convergence properties of the scheme, would lead to restrictive assumptions on the grid size and/or excessive smoothness assumptions on the unknown solution. We show how such assumptions can be removed to achieve unconditional convergence of the proposed scheme. Next, we apply the proposed scheme to the analysis of behaviour of a shape memory alloy rod. We demonstrate that the proposed approximation can describe a complete range of behaviour of the shape memory material, including quasiplastic, pseudoelastic, and almost elastic regimes. We discuss the influence of nonlinear effects in each of these regimes focusing on hysteresis effects.