Notes on the Theory of Constitutive Equations

Abstract

The object of these lectures is to present the manner in which constitutive equations in mechanics and other branches of continuum physics can be formulated in a systematic manner on the basis of clearly stated concepts of the type of physical behavior that it is intended to model. In order to do this, we have first to decide on the variables which it is appropriate to relate to each other, in view of the physical situations to which it is intended to apply the constitutive equations. Secondly, we have also to consider such questions as the smoothness of the relations between these variables, since different assumptions regarding continuity or differentiability of the constitutive relations can radically alter the physical behavior which they model.

Both of these matters can be conveniently illustrated by studying one-dimensional constitutive equations, without introducing the complexities which arise in the three-dimensional case. In this chapter, these aspects of the problem of formulating constitutive equations are discussed in the context of the mechanics of viscoelastic materials, in which a tensile force is assumed to depend on a tensile strain, or rate-of-strain, or on the history of these. It will be evident that similar situations apply in a wide variety of other physical contexts.