Finite element approximation of a phase field model for surface diffusion of voids in a stressed solid

Abstract:

We consider a fully practical finite element approximation of the degenerate Cahn–Hilliard equation with elasticity: Find the conserved order parameter, $\theta (x,t)\in [-1,1]$, and the displacement field, $\underline {u}(x,t) \in \mathbb {R}^2$, such that \begin{align*} \gamma \textstyle {\frac {\partial \theta }{\partial t}} & = \nabla . (b(\theta ) \nabla [- \gamma \Delta \theta + \gamma ^{-1} \Psi ’(\theta ) + \textstyle \frac 12 c’(\theta ) \mathcal {C} \underline {\underline {\mathcal {E}}} (\underline {u}): \underline {\underline {\mathcal {E}}} (\underline {u})] ) , \nonumber \nabla . (c(\theta ) \mathcal {C} \underline {\underline {\mathcal {E}}} (\underline {u}) ) & = \underline {0} , \end{align*} subject to an initial condition $\theta ^0(\cdot ) \in [-1,1]$ on $\theta$ and boundary conditions on both equations. Here $\gamma \in {\mathbb R}_{>0}$ is the interfacial parameter, $\Psi$ is a non-smooth double well potential, $\underline {\underline {\mathcal {E}}}$ is the symmetric strain tensor, $\mathcal {C}$ is the possibly anisotropic elasticity tensor, $c(s):=c_0+\textstyle \frac 12 (1-c_0) (1+s)$ with $c_0(\gamma )\in {\mathbb R}_{>0}$ and $b(s):=1-s^2$ is the degenerate diffusional mobility. In addition to showing stability bounds for our approximation, we prove convergence, and hence existence of a solution to this nonlinear degenerate parabolic system in two space dimensions. Finally, some numerical experiments are presented.