Formulation of Constitutive Equations

Abstract

The theme for this chapter is contained in a quote from Truesdell and Noll’s volume on the Non-Linear Field Theories of Mechanics: “The general physical laws in themselves do not suffice to determine the deformation or motion of. an object. subject to given loading. Before a determinate problem can be formulated, it is usually necessary to specify the material of which the…object…is made. In the program of continuum mechanics, such specification is stated by constitutive equations, which relate the stress tensor and the heat-flux vector to the motion. For example, the classical theory of elasticity rests upon the assumption that the stress tensor at a point depends linearly on the changes of length and mutual angle suffered by elements at that point, reckoned from their configurations in a state where the external and internal forces vanish, while the classical theory of viscosity is based on the assumption that the stress tensor depends linearly on the instantaneous rates of change of length and mutual angle. These statements cannot be universal laws of nature, since they contradict one another. Rather, they are definitions of ideal materials. The former expresses in words the constitutive equation that defines a linearly and infinitesimally elastic material; the latter, a linearly viscous fluid. Each is consistent, at least to within certain restrictions, with the general principles of continuum mechanics, but in no way a consequence of them. There is no reason a priori why either should ever be physically valid, but it is an empirical fact, established by more than a century of test and comparison, that each does indeed represent much of the mechanical behavior of many natural substances of the most various origin, distribution, touch, color, sound, taste, smell, and molecular constitution. Neither represents all the attributes, or suffices even to predict all the mechanical behavior, of any one natural material. No natural. object. is perfectly elastic, or perfectly fluid, any more than any is perfectly rigid or perfectly incompressible. These trite observations do not lessen the worth of the two particular constitutive equations just mentioned. That worth is twofold: First, each represents in ideal form an aspect, and a different one, of the mechanical behavior of nearly all natural materials, and, second, each does predict with considerable though sometimes not sufficient accuracy the observed response of many different natural materials in certain restricted situations.”