A robust and efficient radial return algorithm based on incremental energy minimization for the 3D Souza-Auricchio model for shape memory alloys

The present paper focuses on the numerical simulation of quasi-static problems involving shape memory alloy (SMA) structures or components. Phenomenological constitutive models formulated within the continuum thermodynamics with internal variable framework describe phase transformation in a SMA by introducing a suitable set of internal variables, which may be constrained to satisfy a set of inequalities. The numerical treatment of such constraints, together with the presence of non-smooth functions and/or complementary conditions in the model formulation, is not an easy task and strongly influences the numerical convergence, algorithm robustness, and computational times. The aim of this paper is to propose a novel state-update procedure for the three-dimensional phenomenological model known as the Souza-Auricchio model. The proposed radial return algorithm, relying on an incremental energy minimization approach, allows for an easy implementation of model equations and internal constraints and avoids the use of regularization parameters for the treatment of non-smooth functions. Several numerical simulations assess the noticeable efficiency, robustness, and performance of the proposed approach, while comparisons with a classical algorithm proposed in the literature show the reduced computational times.