On the non-uniqueness of solution in surface elasticity theory

Abstract

Recent researches have shown that the surface energy of some real elastic solids is unlikely to be positive semi-definite. Motivated by this, the present work studies counter-examples for uniqueness of the solution of surface elasticity theory in simple three-dimensional spherically symmetric and two-dimensional axi-symmetrical deformations. Simple sufficient conditions are derived for non-uniqueness of the solution in terms of bulk elastic modulus, surface elastic modulus and geometrical parameters of the elastic body. Unlike the non-uniqueness conditions for classical linear elasticity, which are given in terms of elastic constants alone, the geometrical dimensions of the elastic body play a key role in the non-uniqueness of the solution of surface elasticity theory. Roughly speaking, the solution of surface elasticity models can be non-unique if the smallest characteristic dimension of the body (such as thickness of a thin body, or diameter of a small body or a small hole in an elastic body) is below a certain critical value so that the magnitude of negative surface energy can compete with the positive bulk strain energy of the elastic body. For example, using the available data for the negative surface modulus suggested in the literature, our results predict that the solution could be non-unique for softer materials (such as rubbers) when the smallest characteristic dimension of the elastic body is below typically 1 µm. This result could predict self-buckling of softer elastic thin films in the presence of significant surface stresses.