Positive definiteness of the elasticity tensor of a residually stressed material

Abstract

The purpose of this paper is to investigate the positive definiteness of the elasticity tensors associated with linear hyperelastic materials that support a residual stress. The motivation is that positive definiteness of the elasticity tensor is a central issue in establishing uniqueness of solution for boundary value problems. The uniqueness theorems of classical elasticity rely on the physically motivated condition that the work done to deform a body out of the reference configuration should be positive, with the associated result that the classical elasticity tensor is positive definite. For residually stressed bodies, it is not clear whether the condition that the elasticity tensor be positive definite is a reasonable one. It seems physically well motivated that the work done on a residually stressed body to deform it with small strains out of its unloaded configuration should be positive. But it also seems unlikely that the work will be positive pointwise: the residual stress is not uniform, and therefore must take both compressive and tensile values in the body; so it is plausible that, in any particular deformation, there may be points in the body at which the work may be negative if the deformation is such that it relieves the residual stress at the point. Even if the condition that the work be positive in a deformation is accepted, it does not translate directly into the condition that the elasticity tensor be positive definite because, instead of one, there are several elasticity tensors that are commonly used in the description of residually stressed materials, and each of these elasticity tensors is a function of the residual stress. In this paper, work and energy arguments are used to derive conditions on the elasticity tensors that assure that the total work to deform the body will be positive. The relationships of these derived conditions to the positive definiteness of the elasticity tensors are established. By means of an example, it is shown that in some cases the elasticity tensors associated with the first and second Piola-Kirchhoff stresses indeed fail to be positive definite.