Implicit implementation of a coupled transformation – plasticity crystal mechanics model for shape memory alloys that includes transformation rotations

A rate-dependent crystal-plasticity (CP) framework that captures the coupled phase transformation - plastic deformation behavior of shape memory alloys (SMAs) is presented. Here, different from previous models, the flow rule for martensitic phase transformation incorporates the entire deformation gradient for transformation, including the rotation. Predictions of transformation strain and variant selection of Nickel-Titanium (NiTi) using this model are directly compared with previous formulations that did not include the rotation. The results show that the rotation is essential to accurately calculate the single crystal and polycrystal micromechanics of variant selection and transformation strains of SMAs. The constitutive law formulation also includes current formulations for both slip and deformation twinning plasticity mechanisms, and the differences in transformation mechanisms are further shown to impact plasticity calculations through transformation-plasticity interactions. In addition to the advancement of the constitutive law, a computationally efficient implicit time integration scheme is given for numerical implementation and demonstrated using a user material subroutine (UMAT) in the commercial finite element code ABAQUS Standard. The proposed framework and the associated numerical protocols achieve stable solutions using strain increments on the order of $0.05$ mm/mm in simulating inelastic deformations and strain increments $0.01$ mm/mm in the elastic-inelastic transitions. Furthermore, the use of an analytic Jacobian results in stable convergence in fewer than 10 global Newton iterations while calculating solutions for elastic-inelastic transitions, making the computational benefits evident.