Finite Element Method for Conserved Phase Fields: Stress-Mediated Diffusional Phase Transformation

Phase-field models with conserved phase-field variables result in a 4th order evolution partial differential equation (PDE). When coupled with the usual 2nd order thermo-mechanics equations, such problems require special treatment. In the past, the finite element method (FEM) has been successfully applied to non-conserved phase fields, governed by a 2nd order PDE. For higher order equations, the convergence of the standard Galerkin FEM requires that the interpolation functions belong to a higher continuity class.We consider the Cahn-Hilliard phase-field model for diffusion-controlled solid state phase transformation in binary alloys, coupled with elasticity of the solid phases. A Galerkin finite element formulation is developed, with mixed-order interpolation: C0 interpolation functions for displacements, and C1 interpolation functions for the phase-field variable.To demonstrate convergence of the mixed interpolation scheme, we first study a one-dimensional problem - nucleation and growth of the intermediate phase in a thin-film diffusion couple with elasticity effects. Then, we study the effects of completeness of C1 interpolation on parabolic problems in two space dimensions by considering the growth of the intermediate phase in a binary system. Quadratic convergence, expected for conforming elements, is achieved for both one- and two-dimensional systems