Nonlinear Elasticity Theory with Discontinuous Internal Variables

Abstract

In this paper, a modified theory of nonlinear elasticity in which the strain energy function depends on discontinuous internal variables is proposed. Specifically, the internal variables are allowed to be discontinuous across one or more surfaces. The objective is to model nonclassical phenomena in which two or more material phases are separated by a surface or surfaces of discontinuity. While in the present theory the internal variables may suffer discontinuities, the deformation itself is smooth, and this distinguishes the theory from that initiated by Ericksen, which involves discontinuities in the deformation gradient. The governing equilibrium equations and jump conditions are derived from a variational principle and then specialized to the case of an incompressible isotropic elastic solid with a single internal variable by application to the equilibrium of the radially symmetric deformation of a thick walled circular cylindrical tube under combined extension and inflation. The governing equations include an equation relating the deformation implicitly to the internal variables. By taking a suitable model for the dependence of the internal variable on the deformation, it is shown that a jump in the internal variable may occur across a circular cylindrical surface concentric with the cylinder. At a critical value of the internal radius, the jump surface is initiated at the inner boundary and then propagates through the material as inflation proceeds, and the two phases, separated by the jump surface, coexist in equilibrium. It is then shown that for the unloading process, the theory allows for the possibility of a residual strain remaining once the pressure is removed, and this aspect of the theory is illustrated by use of a simple material model.