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

Abstract

SUMMARYPrevious 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 ow rule and the inversion of the algorithmic tangent tensor. A uniÿed thermomechanical constitutive model, which does not take into account reorientation of martensitic variants but uniÿes 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.

Modeling of the thermomechanical behavior of porous shape memory alloys

“…The numerical return-mapping algorithm developed by Qidwai and Lagoudas (2000a) is used to describe the dense SMA matrix constitutive behavior in the unit cell finite element method. The algorithm is implemented in a user supplied material subroutine (UMAT) which is compiled with the commercial FEM package ABAQUS (HKS, 1997).…”

“…The modeling of dense SMA matrix is carried out using an existing rateindependent constitutive model Qidwai and Lagoudas, 2000a). The SMA constitutive model for isothermal conditions is briefly summarized in the appendix, while its details may be found in the cited references.…”

Hierarchical micromechanical modeling of the viscoelastic behavior coupled to damage in SMC and SMC-hybrid composites

“…Such formalism can be treated computationally using an incremental numerical scheme equivalent with those described in plasticity and viscoplasticity (Simo and Hughes, 1998;Qidwai and Lagoudas, 2000;Achour et al, 2015;Chatzigeorgiou et al, 2018). The advantage of these incremental methods is that they allow the identification of appropriate tangent modulus at each calculation increment, which is very useful in micromechanics schemes to estimate the overall response of a composite.…”

“…The convex cutting plane (CCP) approach is a special form of the more general class of return mapping algorithms (Simo and Hughes, 1998;Qidwai and Lagoudas, 2000) and it is an implicit scheme in terms of time integration. Two important operators are introduced in this algorithm:…”

Phenomenological multi-mechanisms constitutive modelling for thermoplastic polymers, implicit implementation and experimental validation

“…The comparison between these two methods (Qidwai and Lagoudas, 2000) pointed out that, the "convex cutting plane", in addition to its simplicity, involves less tensorial operations and computational cost than the "closest point projection" while both provide satisfactory results. For this reason, the proposed implementation is based on the "convex cutting plane" method.…”