An incremental energy minimization state update algorithm for 3D phenomenological internal-variable SMA constitutive models based on isotropic flow potentials

An incremental energy minimization approach for the solution of the constitutive equations of 3D phenomenological models for shape memory alloys (SMA) is presented. A robust algorithm for the solution of the resulting nonsmooth constrained minimization problem is devised, without introducing any regularization in the dissipation or chemical terms. The proposed algorithm is based on a thorough detection of the singularities relevant to the incremental energy formulation, in conjunction with a Newton-Raphson method equipped with a Wolfe line search dealing with regular solutions. The saturation constraint on the transformation strain is treated by means of an active set strategy, thus avoiding any need for a two-stage return-mapping algorithm. A parametrization of the saturation constraint manifold is introduced, thus reducing the problem dimensionality, with improved computational performance. Finally, an efficient algorithm for the computation of the dissipation function in terms of Haigh-Westergaard invariants is presented, allowing for a quite general choice of deviatoric transformation functions. Numerical results confirm the robustness and consistency of the proposed state update algorithm.