Finite element procedure and simulations for a multiphase phase field approach to martensitic phase transformations at large strains and with interfacial stresses

A detailed finite element procedure for a new phase field approach (Basak and Levitas, 2018) to temperature- and stress induced multivariant martensitic transformations at large strains and with interfacial stresses is developed. A system with austenite and N martensitic variants is considered. N+1 order parameters related to the transformation strains are used, one of which describes the austenite↔martensite transformation; the other N order parameters describe N martensitic variants. Evolution of the order parameters is governed by coupled Ginzburg-Landau and mechanics equations. Assuming a non-monolithic strategy for solving the governing equations by using Newton's iterative method, a weak formulation with emphasis on the derivation of the tangent modulus has been presented. Notably, the fourth order tangent modulus for the equilibrium equations has a contribution not only from the elastic stresses but also from the structural interfacial stresses, which appears here for the first time. A second order backward difference scheme is used to discretize the time derivative in the Ginzburg-Landau equations. An adaptive time stepping is considered. A finite element code has been developed within an open source package deal.II for a system with austenite and two martensitic variants and used to solve three problems: (i) simple shear deformation of a rectangular parallelepiped with evolution of austenite and single martensitic variant; (ii) twinning in martensite and the effect of sample size on the twinned microstructures; (iii) a rectangular block under nanoindentation. The results for the first two problems describe the well-known analytical solutions. Two kinematic models (KMs) for the transformation deformation gradient tensor are used and the corresponding results are compared: KM-I represents a linear transformation rule in the Bain tensors and KM-II is an exponential-logarithmic type of transformation rule in the Bain tensors. The algorithm can naturally be extended for the study of phase transformations in multiphase solids, solidification, diffusive phase transitions, interaction between phase transformations and plasticity and/or fracture, etc.