Note on the deformation-induced change in the curvature of a material surface in plane deformations

Abstract

The deformation-induced change in the curvature of a material surface is a fundamental parameter used in establishing accurate boundary conditions on the surface of a solid when the incorporation of surface energies becomes necessary in the model of deformation. In the context of small strain plane deformations of an elastic solid, the literature basically employs various versions of the expression for the deformation-induced change in the curvature of a material surface. To the authors’ knowledge, no explanation has been offered to clarify either the relationship between each version or the rationale for the use of one as opposed to the other. In this note, we apply complex-variable techniques to derive the exact expression for the deformation-induced change in the curvature of a material surface subjected to the aforementioned assumptions of small strain plane deformations. Based on this exact expression, the origin of each of the above-mentioned versions of the curvature becomes quite clear. In particular, by using this exact expression the stress boundary condition in the complete Gurtin-Murdoch model is easily recovered and a modified stress boundary condition is suggested.