Page 136 - Introduction to Continuum Mechanics

Kinematics of a Continuum 121

From Eq. (iii) we have at t = 0, F = I, and d\ = dX.
At t = 1, for all elements

3.19 Local Rigid Body Displacements

In Section 3.6, we discussed the case where the entire body undergoes rigid body displace-
ments from the configuration at a reference time t 0 to that at a particular time t. For a body
in a general motion, however, it is possible that the body as a whole undergoes deformations
while some (infinitesimally) small volumes of material inside the body undergo rigid body
displacements. For example, for the motion given in the last example, at t — 1 and X\ = 0,

T
It is easily to verify that the above F is a rotation tensor R (i.e., FF = I and det F = +1).
Thus, all infinitesimal material volumes with material coordinates (O^^s) undergo a rigid
body displacement from the reference position to the position at t =1.

3.20 Finite Deformation
Deformations at a material point X of a body are characterized by changes of distances
between any pair of material points within the small neighborhood of X. Since, through
motion, a material element dX becomes dx. = FrfX, whatever deformation there may be at X,
is embodied in the deformation gradient F. We have already seen that if F is a proper
orthogonal tensor, then there is no deformation at X. In the following, we first consider the
case where the deformation gradient F is a symmetric tensor before going to more general
cases.

We shall use the notation U for a deformation gradient F that is symmetric. Thus, for a
symmetric deformation gradient, we write

In this case, the material within a small neighborhood of X is said to be in a state of pure
stretch deformation (from the reference configuration). Of course, Eq. (3.20.1) includes the
special case where the motion is homogeneous, i.e., x = UX, (U = constant tensor) in which
case the entire body is in a state of pure stretch.