Basic Continuum Kinematics - Continuum Mechanics of Anisotropic Materials - page 39

dx 2 /dt

dx 1 /dt

dx 2

θ 12

dx 1

Fig. 2.7 An illustration for the geometric interpretation of the rate of shearing strain component
D 12 of the rate-of-deformation tensor D . The heavy black lines represent two material filaments of
infinitesimal length that are instantaneously perpendicular. The thin black lines represent the same
two material filaments in the next instant. The instantaneous time rate of change of the angle
between the two material filaments, the rate at which the two filaments are coming together or
separating is y 12 . The geometric interpretation of D 12 is that it is one half the instantaneous time of
decrease in an originally right angle between dx 1 and dx 2 , D 12 ¼y 12 = 2

Þ¼y 12 dx 2 D ij dx i ð 1 Þ dx j ð 2 j ð 1 Þ j j dx ð 2 Þ j:

Finally, if we take dxð 1 Þ¼ dx 1 e 1 and dxð 2 Þ¼ dx 2 e 2 it may be concluded that

y 12
2 ;

D 12 ¼

confirming that D 12 is equal to one-half the material time rate of decrease in an
originally right angle between dx 1 and dx 2 . The geometric interpretations of D 13
and D 23 are similar. These geometric interpretations of the components of D as the
instantaneous time rate of change of filaments, angles, and volume are the rationale
for calling D the rate of deformation tensor.

Example 2.2.1
Calculate the velocity and acceleration in the material representation of the motion
( 2.12 ) of Example 2.1.1, then determine the spatial representation. Verify that the
acceleration computed in the spatial representation is the same as the acceleration
computed in the material representation. Calculate the tensor of velocity gradients
L , the rate of deformation tensor D , and the spin tensor W for this motion.

Solution : The velocity and acceleration for this motion are given by ( 2.24 )as

_

x 1 ¼ X I þ X II þ 3 ; x 2 ¼ _ X I þ X II þ 2 ; x 3 ¼ _ 0 ; € x 1 ¼ € x 2 ¼ € x 3 ¼ 0 :

In order to find the spatial representation for this motion we must invert the
system of equations ( 2.12 ) representing the motion, thus