Finite element formulation of a phase field model based on the concept of generalized stresses

A finite element formulation of a phase field model for alloys is proposed within the general framework of continuum thermodynamics in conjunction with the concept of generalized stresses as proposed by Gurtin [1]. Using the principles of the thermodynamics of irreversible processes, balance and constitutive equations are clearly separated in the formulation. Also, boundary conditions for the concentration and order parameter and their dual quantities are clearly stated. The theory is shown to be well-suited for a finite element formulation of the initial boundary value problem. The set of coupled evolution equations, which are the phase field equation and the balance of mass, is solved using an implicit finite element method for space discretization and a finite difference method for time discretization. For an illustrative purpose, the model is used to investigate the growth of an oxide layer at the surface of a pure zirconium slab. Calculations in 1D show a good agreement with an analytical solution for the growth kinetics. Then, 2D calculations of the same process have been undertaken to investigate morphological stability of the oxide layer in order to show the ability of the finite element method to handle arbitrary conditions on complex boundaries.