Weakly nonlinear theory of secondary rippling instability in surfaces of stressed solids

Abstract

Numerical simulations of the surface morphological evolution of uniaxially stressed elastic crystalline solids have demonstrated that in addition to Asaro-Tiller/Grinfeld (surface cracking) instabilities, long-wavelength perturbations from the planar surface morphology can trigger a tip-splitting instability that causes formation of a pattern of secondary ripples, which cannot be explained by linear stability theory. In this study, we develop a weakly nonlinear stability theory, which can explain the occurrence of such secondary rippling instabilities and predict the number of secondary ripples that form on the surface as a function of perturbation wavelength. The theory shows that this type of surface pattern formation arises entirely due to the competition between surface energy and elastic strain energy, regardless of surface diffusional anisotropy or the action of externally applied fields. The origin of secondary rippling is explained through nonlinear terms included in the analysis which generate sub-harmonic ripples in the surface morphology with wave numbers that are multiples of the original surface perturbation wave number. Based on the weakly nonlinear theory, we have developed simple analytical expressions that predict the critical wavelength for the onset of secondary rippling, the increase in the number of secondary ripples with increasing perturbation wavelength, and how the onset of the secondary rippling instability and the rippled surface pattern are affected by surface diffusional anisotropy and the action of an applied electric field. The conclusions of the theory are validated by systematic comparisons with results of self-consistent dynamical simulations of surface morphological evolution.