On the modeling and computations of nonlinear thermodynamics in SMA wires

We present a mathematical model describing the thermodynamic behavior of shape memory alloy wires, as well as a computational technique to solve the resulting system of partial differential equations. The model consists of continuum-level conservation equations based on a new construction of the Helmholtz free energy potential. This construction allows specification of material constants for different shape memory wires. The computational technique introduces a viscosity-based continuation method, which allows the model to handle dynamic applications where the temporally local behavior of solutions is desired. Computational experiments document that this combination of modeling and solution techniques appropriately predicts the thermally- and stress-induced martensitic phase transitions, as well as the hysteretic behavior and production of latent heat associated with such materials.