Constitutive Equations

All foregoing axioms, laws, equations are supposed to be valid for all material bodies and are thus labelled universal. In contrast, material theory aims at describing the individual behaviour of different materials. This is done by so-called material laws, constitutive equations or material models. Before going into details of the differences in material behaviour however, we want to find common features, something like the general form of material laws. For this, we first assume certain basic principles of material theory, which are partly based on our experience and partly on plausibility. Further on, we want to develop criteria to classify the universe of material behaviour to obtain more concrete models for certain classes of materials. Finally we will not be able to continue without experiments. But even for the design of appropriate experiments, theory is expected to give guidance. Any material characterisation is meaningful only in the context of a constitutive model.

The Principles of Material Theory

It is generally assumed that there is a deterministic relation between stresses in a body and the motion of the body, In order to put this into a functional form, we have to decide which quantities to use as independent variables and which as dependent variables. It has become more or less common practice to consider the stresses as dependent variables, and motions as independent ones. The whole history of motion may affect the present stresses, but of course, we do not allow the future motion of a body to have any effect on them.