CONTINUUM MECHANICS for ENGINEERS

4 Kinematics of Deformation and Motion
4.1 Particles, Configurations, Deformation, and Motion
In continuum mechanics we consider material bodies in the form of solids,
liquids, and gases. Let us begin by describing the model we use to represent
such bodies. For this purpose we define a material body B as the set of
elements X, called particles or material points, which can be put into a oneto-one
correspondence with the points of a regular region of physical space.
Note that whereas a particle of classical mechanics has an assigned mass, a
continuum particle is essentially a material point for which a density is
defined.
The specification of the position of all of the particles of B with respect to
a fixed origin at some instant of time is said to define the configuration of the
body at that instant. Mathematically, this is expressed by the mapping
x = κ(X) (4.1-1)
in which the vector function κ assigns the position x relative to some origin
of each particle X of the body. Assume that this mapping is uniquely invertible
and differentiable as many times as required; in general, two or three
times will suffice. The inverse is written
X = κ –1 (x) (4.1-2)
and identifies the particle X located at position x.
A change in configuration is the result of a displacement of the body. For
example, a rigid-body displacement is one consisting of a simultaneous translation
and rotation which produces a new configuration but causes no
changes in the size or shape of the body, only changes in its position and/or
orientation. On the other hand, an arbitrary displacement will usually
include both a rigid-body displacement and a deformation which results in a
change in size, or shape, or possibly both.