Nonlinear evolution equation for the stress-driven morphological instability

Yang Xiang, Weinan E

Research output: Contribution to journalArticle

39 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)9414-9422
Number of pages9
JournalJournal of Applied Physics
Volume91
Issue number11
DOIs
StatePublished - Jun 1 2002

All Science Journal Classification (ASJC) codes

  • Physics and Astronomy(all)

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