Phase field-elasticity analysis of austenite–martensite phase transformation at the nanoscale: Finite element modeling

Abstract

In the present work, a nonlinear finite element method is utilized to solve the coupled system of time-dependent phase field and elasticity equations for phase transformations (PTs) at the nanoscale in the Cartesian coordinate system. The Galerkin residual weighted method is used to derive the finite element equations. The alpha family and the explicit methods are used for time discretization. Since the local free energy includes a 3rd degree polynomial in terms of the phase order parameter the kinetics Ginzburg-Landau equation is a nonlinear function of the order parameter. Thus, the Newton-Raphson method is used to linearize the nonlinear equation. Linear triangle elements have been used in the self-developed FEM code. Stability and mesh and time step independence of the solutions have been discussed. The system of equations and the numerical procedure are verified using the existing analytical solutions. For the phase field equation, the isolated boundary condition is considered everywhere, imposing the constant surface energy over the simulation domain. Examples of austenite (A) to martensite (M) phase transformations in 2D for a single martensitic variant are presented including planar/nonplanar interface propagation, martensitic nucleus growth and reverse phase transformation under thermal and different mechanical loadings. The A-M interface velocity, width and energy have been obtained. The threshold stresses for the growth of a martensitic nucleus in an austenitic matrix under uniaxial and biaxial loadings and for reverse PTs are calculated. It is found that the numerical results are in a good agreement with the transformation work based criterion. The developed FEM code represents a proper and accurate tool to study the PTs including nucleation, growth and propagation of transformed phase, reverse PTs and equilibrium and stability conditions for PTs. A further development of the numerical procedure provides a powerful tool for the study of more complicated PTs-related phenomena in 2D and 3D.