Numerical implementation of a shape memory alloy thermomechanical constitutive model using return mapping algorithms

Previous studies by the authors and their co-workers show that the structur e of equations representing shape Memory Alloy (SMA) constitutive behaviour can be very similar to those of rate-independent plasticity models. For ex ample, the Boyd-Lagoudas polynomial hardening model has a stress-elastic st rain constitutive relation that includes the transformation strain as an in ternal state variable, a transformation function determining the onset of p hase transformation, and an evolution equation for the transformation strai n. Such a structure allows techniques used in rate-independent elastoplasti c behaviour to be directly applicable to SMAs. In this paper, a comprehensi ve study on the numerical implementation of SMA thermomechanical constituti ve response using return mapping (elastic predictor-transformation correcto r) algorithms is presented. The closest point projection return mapping alg orithm which is an implicit scheme is given special attention together with the convex cutting plane return mapping algorithm, an explicit scheme alre ady presented in an earlier work. The closest point algorithm involves rela tively large number of tensorial operations than the cutting plane algorith m besides the evaluation of the gradient of the transformation tensor in th e flow rule and the inversion of the algorithmic tangent tensor. A unified thermomechanical constitutive model, which does not take into account reori entation of martensitic variants but unifies several of the existing SMA co nstitutive models, is used for implementation. Remarks on numerical accurac y of both algorithms are given, and it is concluded that both algorithms ar e applicable for this class of SMA constitutive models and preference can o nly be given based on the computational cost. Copyright (C) 2000 John Wiley & Sons, Ltd.