The Cahn-Hilliard equation with elasticity – finite element approximation and qualitative studies.

*(English)*Zbl 0972.35164Summary: We consider the Cahn-Hilliard equation – a fourth-order, nonlinear parabolic diffusion equation describing phase separation of a binary alloy which is quenched below a critical temperature. The occurrence of two phases is due to a nonconvex double well free energy. The evolution initially leads to a very fine microstructure of regions with different phases which tend to become coarser at later times.

The resulting phases might have different elastic properties caused by a different lattice spacing. This effect is not reflected by the standard Cahn-Hilliard model. Here, we discuss an approach which contains anisotropic elastic stresses by coupling the expanded diffusion equation with a corresponding quasistationary linear elasticity problem for the displacements on the microstructure.

Convergence and a discrete energy decay property are stated for a finite element discretization. An appropriate timestep scheme based on the strongly A-stable \(\Theta\)-scheme and a spatial grid adaptation by refining and coarsening improve the algorithms efficiency significantly. Various numerical simulations outline different qualitative effects of the generalized model. Finally, a surprising stabilizing effect of the anisotropic elasticity is observed in the limit case of a vanishing fourth-order term, originally representing interfacial energy.

The resulting phases might have different elastic properties caused by a different lattice spacing. This effect is not reflected by the standard Cahn-Hilliard model. Here, we discuss an approach which contains anisotropic elastic stresses by coupling the expanded diffusion equation with a corresponding quasistationary linear elasticity problem for the displacements on the microstructure.

Convergence and a discrete energy decay property are stated for a finite element discretization. An appropriate timestep scheme based on the strongly A-stable \(\Theta\)-scheme and a spatial grid adaptation by refining and coarsening improve the algorithms efficiency significantly. Various numerical simulations outline different qualitative effects of the generalized model. Finally, a surprising stabilizing effect of the anisotropic elasticity is observed in the limit case of a vanishing fourth-order term, originally representing interfacial energy.

##### MSC:

35Q72 | Other PDE from mechanics (MSC2000) |

74N15 | Analysis of microstructure in solids |

74S05 | Finite element methods applied to problems in solid mechanics |