Page 94 - Introduction to Continuum Mechanics

3

Kinematics of a Continuum

The branch of mechanics in which materials are treated as continuous is known as
continuum mechanics. Thus, in this theory, one speaks of an infinitesimal volume of material,
the totality of which forms a body. One also speaks of a particle in a continuum, meaning, in
fact an infinitesimal volume of material. This chapter is concerned with the kinematics of such
particles.

3.1 Description of Motions of a Continuum

In particle kinematics, the path line of a particle is described by a vector function of time,
i.e.,

e e z e tne
where r(f) = *(0 i+y(0 2+ (0 3 is position vector. In component form, the above equa-
tion reads:

If there are N particles, there are N pathlines, each is described by one of the equations:

That is, for the particle number 1, the path line is given by r^t), for the particle number 2, it
is given by r2(t), etc.
For a continuum, there are not only infinitely many particles, but within each and every
neighborhood of a particle there are infinitely many other particles. Therefore, it is not
possible to identify particles by assigning each of them a number in the same way as in the
kinematics of particles. However, it is possible to identify them by the positions they occupy
at some reference time t 0. For example, if a particle of a continuum was at the position (1,2,3)
at the reference time t 0, the set of coordinates( 1,2,3) can be used to identify this particle. In
general, therefore, if a particle of a continuum was at the position (Xi^^i) at tne reference
time t 0, the set of coordinate (^^2^3) can be used to identity this particle. Thus, in general,

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