1.8 Constitutive Relations

Although the physical laws presented so far are of general validity, very little can be said concerning the behavior of any substance, as can be ascertained by noting that at this point there are many more unknown quantities than there are equations. The missing relations are those that describe the connection between how a material is made up, or constituted, and the relation between stress and the geometric and thermodynamic variables. Hooke s law and the state equations of an ideal gas are two familiar examples of constitutive equations. Whereas in a few cases constitutive relations can be derived from statistical mechanics considerations using special mathematical models for the molecular structure, the usual procedure is to decide, based on experiments, which quantities must go into the constitutive equation and then formulate from these a set of equations that agree with fundamental ideas, such as invariance with respect to the observer and the like.

There is much inventiveness in proceeding along the path of determining constitutive equations, and much attention has been given to the subject in past decades. The mental process of generating a description of a particular fluid involves a continuous interchange between theory and practice. Once a constitutive model is proposed, mathematical predictions can be made that then, it is hoped, can be compared with the experiment. Such a procedure can show a model to be wrong, but it cannot guarantee that it will always be correct, since many models can predict the same velocity...