A thermomechanically coupled finite deformation constitutive model for shape memory alloys based on Hencky strain

This paper presents a new thermomechanically coupled constitutive model for polycrystalline shape memory alloys (SMAs) undergoing finite deformation. Three important characteristics of SMA behavior are considered in the development of the model, namely the effect of coexistence between austenite and two martensite variants, the variation of hysteresis size with temperature, and the smooth material response at initiation and completion of phase transformation. The formulation of the model is based on a multi-tier decomposition of the deformation kinematics comprising, a multiplicative decomposition of the deformation gradient into thermal, elastic and transformation parts, an additive decomposition of the Hencky strain into spherical and deviatoric parts, and an additive decomposition of the transformation stretching tensor into phase transformation and martensite reorientation parts. A thermodynamically consistent framework is developed, and a Helmholtz free energy function consisting of elastic, thermal, interaction and constraint components is introduced. Constitutive and heat equations are then derived from this energy in compliance with thermodynamic principles. Time-discrete formulations of the constitutive equations and a Hencky-strain return-mapping integration algorithm are presented. The algorithm is then implemented in Abaqus/Explicit by means of a user-defined material subroutine (VUMAT). Numerical results are validated against experimental data obtained under various thermomechanical loading conditions. The robustness and efficiency of the proposed framework are illustrated by simulating a SMA helical spring actuator.