Numerical Aspects of Modelling Thermo- MechanicalWave Propagation With Phase Transformations

Due to their unique thermo-mechanical properties, the range of existing and potential applications of shape memory alloys (SMA) continues to grow. Many of these applications take advantage of the dynamic response of SMA under dynamic loading conditions. From the engineering design point of view, this requires a better understanding of the effect of phase transformations and thermo-mechanical coupling on wave propagation in the material. In achieving this goal, a fundamental task is to analyse the dynamics of first order phase transformations induced by shock loadings.

In this contribution, a mathematical model and its numerical discretization are constructed to analyse the wave propagation in shape memory alloy rods. The first order martensitic transformations and the associated effects of thermo-mechanical coupling are accounted for by employing the modified Ginzburg-Landau-Devonshire. The Landau-type free energy function characterizes different phases, while a Ginzburg term is introduced to account for the internal friction during phase transformations. The effect of the Ginzburg term on wave propagation patterns is analysed under shock loadings implemented via stress boundary conditions. For practical numerical simulations of SMA samples, the constructed model of coupled nonlinear system of PDEs is reduced to a system of differential-algebraic equations, where the Chebyshev collocation method is employed for the spatial discretization, while the backward differentiation is used for the integration in time.

A series of numerical experiments is carried out on copper-based SMA samples. Propagation of stress waves induced by shock loadings is analysed for different initial temperature. It is demonstrated that the patterns of wave propagation is complicated at low temperatures by phase transformations, while more regular patterns are observed for high temperature distributions. These observations are in agreement with experiments. Finally, the influence of viscosity effects on the overall thermo-mechanical behaviour of rods is analysed numerically by evaluating the contribution of the Ginzburg term responsible for the internal friction during phase transformations.