A note on a work inequality in linear viscoelasticity
In recent years various writers have studied the development and application of work or quasi-thermodynamic inequalities for common idealizations of material behavior. For time-dependent materials, a postulate of material stability was introduced by Drucker [1] as an extension of similar work for time-independent materials [2]. This postulate defines a class of materials for which the solutions to standard boundary value problems are unique. In linear viscoelasticity, Breuer and Onat [3] have used similar ideas to determine the restrictions which must be placed on the relaxation function in order to guarantee uniqueness in the same sense. Breuer and Onat assumed that the work required to deform a linear viscoelastic material from the virgin state is non-negative. This statement, together with the linearity, can be used for a uniqueness proof. In this note we consider an isotropic linear viscoelastic solid whose mechanical behavior can be represented by a generalization of a model composed of an infinite number of interconnected springs and dashpots. We assume that in the time interval (—0°, 0) the solid is in the unstressed and unstrained virgin state. We consider the specific work required to deform the solid, and we wish to show that the specific work can be bounded from below by a non-negative expression which depends only on the strain at time t and time t, and not on the history. Let d and e denote the components of the stress and infinitesimal strain tensors. Let s and e denote the stress and strain deviators and let p, v denote hydrostatic tension and isotropic volume strain. If ,/,(t) and J2{t) are the creep functions of the material in pure shear and dilatation respectively, the constitutive equations may be written as