Nonlinear evolution equation for the stress-driven morphological instability

A flat surface of stressed solid is unstable to small perturbations. This morphological instability is called Asaro-Tiller-Grinfield instability. Nonlinear evolution of this instability will result in the formation of cusp singularities. This instability can be described by a continuum model with surface diffusion driven by a stress-dependent chemical potential. The stress and strain in the solid are coupled with surface morphology and an elasticity problem must be solved numerically. We derive a nonlinear approximation equation governing the evolution of this instability in which the stress-dependent chemical potential is expressed explicitly as a function of the surface morphology. Linear instability analysis using our equation shows the same results as the well-known Asaro-Tiller-Grinfeld instability result. Compared with the exact solution of the elasticity problem for cycloid surface obtained by Chiu and Gao, our nonlinear approximation has a much wider range of applicability than linear approximation. Numerical simulation using our nonlinear evolution equation shows that the surface evolves towards a cusplike morphology from small perturbations, which agrees very well with results obtained by solving the full elasticity problem.