Four Linear Continuum Theories - Continuum Mechanics of Anisotropic Materials - page 126

Chapter 6
Four Linear Continuum Theories

Four linear theories are considered in this chapter. Each has a distinctive and
interesting history. Each one of the theories was originally formulated between
1820 and 1860. Representative of the theme of this chapter are the opening lines of
the Historical Introduction in A. E. H. Love's Theory of Elasticity (original edition,
1892): “ The Mathematical Theory of Elasticity is occupied with an attempt to
reduce to calculation the state of strain, or relative displacement, within a solid ..
object.. which is subject to the action of an equilibrating system of forces, or is in a
state of slight internal relative motion, and with endeavours to obtain results which
shall be practically important in applications to architecture, engineering, and all
other useful arts in which the material of construction is solid .”

6.1 Formation of Continuum Theories

Four linear continuum theories are developed in this chapter. These are the theories
of fluid flow through rigid porous media, of elastic solids, of viscous fluids, and of
viscoelastic materials. There are certain features that are common in the develop-
ment of each of these theories: they all involve at least one conservation principle
and one constitutive equation and for each, it is necessary to specify boundary or
initial conditions to properly formulate boundary value problems. Some of the
continuum theories involve more than one conservation principle, more than one
constitutive equation, and some kinematics relations. Thus this chapter draws
heavily upon the material in the previous chapters and serves to integrate the
kinematics, the conservation principles, and the constitutive equations into theories
that may be applied to physical situations to explain physical phenomena. This is
generally accomplished by the solution of partial differential equations in the
context of specific theories.
The differential equations that are formulated from these linear theories are
usually the familiar,

fairly well-understood differential equations, and they