CONTINUUM MECHANICS for ENGINEERS

Additionally, with regard to the material and spatial descriptions we may
interpret Eq 4.4-1 in either the material form
or the spatial form
u(X,t) = x(X,t) – X (4.4-3a)
u(x,t) = x – X(x,t) (4.4-3b)
In the first of this pair of equations we are describing the displacement that
will occur to the particle that starts at X, and in the second equation we
present the displacement that the particle now at x has undergone. Recalling
that since the material coordinates relate to positions in the reference configuration
only, and hence are independent of time, we may take the time
rate of change of displacement as an alternative definition for velocity. Thus,
du
d x−X dx
=
dt dt dt
(4.4-4)
Example 4.4-1
Obtain the displacement field for the motion of Example 4.3-1 in both material
and spatial descriptions.
Solution
From the motion equations of Example 4.3-1, namely,
x 1 = X 1e t + X 3 (e t – 1)
x 2 = X 2 + X 3 (e t – e –t )
x 3 = X 3
we may compute the displacement field in material form directly as
u 1 = x 1 – X 1 = (X 1 + X 3)(e t – 1)
u 2 = x 2 – X 2 = X 3(e t – e –t )
u 3 = x 3 – X 3 = 0
and by using the inverse equations from Example 4.3-1, namely,
X 1 = x 1e –t + x 3(e –t – 1)
X 2 = x 2 + x 3(e –t – e t )
X 3 = x 3
( ) = =
v