Discussion of the linearized version of the Steigmann-Ogden surface model in plane deformation and its application to inclusion problems

Abstract

We re-examine two linearized versions of the Steigmann-Ogden model of a general elastic cylindrical surface under plane deformation. The first version appears quite commonly in the literature and is distinguished by the fact that the bending moment of the surface is defined by the nominal curvature of the deformed surface (which is independent of the change in the length scale of the surface during deformation). The second version is a modified version in which the bending moment is defined by the actual curvature of the deformed surface. In both versions, the tangential force of the surface is determined by its stretch during deformation. We demonstrate from an energetic point of view that the first version is self-consistent while the second is not in that it does not yield a unique positive-definite surface strain energy density. Accordingly, we propose a refined and self-consistent linearized version of the Steigmann-Ogden model in which the bending moment of a material surface remains dependent on the actual curvature of the deformed surface while the corresponding tangential force relies on the stretch of the surface, the initial curvature of the surface and the actual curvature of the deformed surface. We formulate, in the context of the complex variable formalism of elasticity, the boundary condition corresponding to this refined version for an elastic composite system with an arbitrary curved interface (including the case of an arbitrarily-shaped inclusion embedded within an elastic matrix) under plane deformation. Some additional formulae are also given for the analysis of the boundary value problem corresponding to an arbitrarily-shaped inclusion subjected to the proposed refined boundary condition.