II II II II II I - Waste Isolation Pilot Plant - U.S. Department of Energy

TRI-6L4S-2-O
Figure 2.1.1. Original, deformed, and intermediate configurations of a body.
Both W and fZ are antisymmetric and represent a rate of rotation (or angular velocity) about some axes. In
general, Q #W. The difference arises when the last term of Equation (2.1.5) is not symmetric. The symmetric part
of u U-l “K the unrotated deformation rate tensor d as defined below (note that both U and WI are symmetric).
d=#UU-l+U-%)=RTDR . (2. 1.7)
There are two possible cases which can cause rotation of a material line element: rigid body rotation and shear.
Because total shear vanishes along the axes of principal stretch, the rotation of these axes defines the total rigid body
rotation of a material point.
It is a simple exercise in vector analysis to show that Equation (2.1.6) represents the rate of rigid body rotation
at a material point as shown by Dienes (1979). It is equally simple to show that W represents the rate of rotation of
the principal axes of the rate of deformation D. Since D and W have no sense of the history of deformation, they are
not sufficient to define the rate of rotation in a finite deformation context.
Line elements where the rate of shear vanishes rotate solely due to rigid body rotations. These line elements are
along the principal axes of U. We will apply a similar observation below as we derive Dienes’ (1979) expression
for calculating !2:
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