Perturbation analysis of an undulating free surface in a multi-layered structure

This paper presents an analysis of the effect of an undulating free surface on the stress state in a multi-layered structure with an arbitrary number of layers. We present a simple procedure to derive in terms of stress functions the first-order perturbation solution for a multi-layered structure subjected to residual stresses and to external edge loads. The solution is valid for undulation amplitudes that are small compared to undulation wavelength, and it reproduces results reported earlier for homogeneous materials and bi-layers. Using this perturbation solution, the stability of a stressed undulating surface with a given surface energy is evaluated. This analysis leads naturally to the definition of a critical undulation wavelength above which undulations are preferred energetically over a flat surface. The results obtained in the perturbation analysis can be applied to surface diffusion/etching problems in multi-layered structures and to a range of other engineering problems related to undulating surfaces and surface stability.