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

摘要

Previous studies by the authors and their co-workers show that the structure of equations representing shape memory Alloy (SMA) constitutive behaviour can be very similar to those of rate-independent plasticity models. For example, the Boyd-Lagoudas polynomial hardening model has a stress-elastic strain constitutive relation that includes the transformation strain as an internal state variable, a transformation function determining the onset of phase transformation, and an evolution equation for the transformation strain. Such a structure allows techniques used in rate-independent elastoplastic behaviour to be directly applicable to SMAs. In this paper, a comprehensive study on the numerical implementation of SMA thermomechanical constitutive response using return mapping (elastic predictor-transformation corrector) algorithms is presented. The closest point projection return mapping algorithm which is an implicit scheme is given special attention together with the convex cutting plane return mapping algorithm, an explicit scheme already presented in an earlier work. The closest point algorithm involves relatively large number of tensorial operations than the cutting plane algorithm besides the evaluation of the gradient of the transformation tensor in the flow rule and the inversion of the algorithmic tangent tensor. A unified thermomechanical constitutive model. which does not take into account reorientation of martensitic variants but unifies several of the existing SMA constitutive models. is used for implementation. Remarks on numerical accuracy of both algorithms are given, and it is concluded that both algorithms are applicable for this class of SMA constitutive models and preference can only be given based on the computational cost.