Formulation of Constitutive Equations - Continuum Mechanics of Anisotropic Materials - page 106

Chapter 5
Formulation of Constitutive Equations

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

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 instanta-
neous 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 mechani-
cal 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.”

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