Coherent stress relaxation in a half space: Modulated layers, inclusions, steps, and a general solution

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Abstract
We first calculate analytically the elastic displacement and strain fields and the elastic energy of the system composed of a misfitting layer of finite thickness coherently deposited on a bulk semi-infinite substrate and covered by a capping layer, when the intrinsic stress-free strain in the intermediate layer is a sinusoidal modulation of dilatation along a direction parallel to the substrate surface. This is achieved by the direct application of Eshelby’s method and the determination of the appropriate stress function. We show that, combined with Fourier analysis, this basic calculation provides a new general analytical solution to the problem of the coherent relaxation of any misfitting dilatational inhomogeneity in a half space. More specifically, this method is here applied to the full analytical solution of the cases of a coherent misfitting parallelepipedic inclusion buried under a planar free surface, and of a coherent step at the interface between a substrate and a misfitting overlayer. Applications are briefly discussed.