CONTINUUM MECHANICS for ENGINEERS

1.2 Continuum Mechanics
The analysis of the kinematic and mechanical behavior of materials modeled
on the continuum assumption is what we know as continuum mechanics.
There are two main themes into which the topics of continuum mechanics
are divided. In the first, emphasis is on the derivation of fundamental equations
which are valid for all continuous media. These equations are based
upon universal laws of physics such as the conservation of mass, the principles
of energy and momentum, etc. In the second, the focus of attention is
on the development of so-called constitutive equations characterizing the
behavior of specific idealized materials, the perfectly elastic solid and the
viscous fluid being the best known examples. These equations provide the
focal points around which studies in elasticity, plasticity, viscoelasticity, and
fluid mechanics proceed.
Mathematically, the fundamental equations of continuum mechanics mentioned
above may be developed in two separate but essentially equivalent
formulations. One, the integral or global form, derives from a consideration
of the basic principles being applied to a finite volume of the material. The
other, a differential or field approach, leads to equations resulting from the
basic principles being applied to a very small (infinitesimal) element of
volume. In practice, it is often useful and convenient to deduce the field
equations from their global counterparts.
As a result of the continuum assumption, field quantities such as density
and velocity which reflect the mechanical or kinematic properties of continuum
bodies are expressed mathematically as continuous functions, or at
worst as piecewise continuous functions, of the space and time variables.
Moreover, the derivatives of such functions, if they enter into the theory at
all, likewise will be continuous.
Inasmuch as this is an introductory textbook, we shall make two further
assumptions on the materials we discuss in addition to the principal one of
continuity. First, we require the materials to be homogeneous, that is, to have
identical properties at all locations. And second, that the materials be isotropic
with respect to certain mechanical properties, meaning that those properties
are the same in all directions at a given point. Later, we will relax this isotropy
restriction to discuss briefly anisotropic materials which have important
meaning in the study of composite materials.