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Material Description and Spatial Description 83

Example 3.1.3
s
The position at time t, of a particle initially at (J^^^s), * given by the equations:

(a) Find the velocity at t-2 for the particle which was at (1,1,0) at the reference time.
(b) Find the velocity at t=2 for the particle which is at the position (1,1,0) at t = 2.
Solution, (a)

For the particle (Xi^^s) - (14,0), the velocity at t - 2 (and any time /) is

i.e.,

(b)To calculate the reference position (Xi^C^s) which was occupied by the particle which
=
is at (xi^2^3) (14,0) at /= 2 , we substitute the value of (*i,*2»*3) — (1,1,0) and t - 2 in
e
Eq. (i) and solve for (^1^2*^3), i- ->

3.2 Material Description and Spatial Description

When a continuum is in motion, its temperature 0, its velocity v, its stress tensor T (to be
defined in the next chapter) may change with time. We can describe these changes by:

I. Following the particles, i.e., we express 0, v, T as functions of the particles (identified by
the material coordinates, (^^2*^3)) an< * time t. In other words, we express