Basic Continuum Kinematics - Continuum Mechanics of Anisotropic Materials - page 38

dx 2 /dt

dx 2

dx 1 /dt

dx 1

dx 3

dx 3 /dt

Fig. 2.6 Illustration for the geometric interpretation of the trace of the rate-of-deformation tensor
D as the instantaneous time rate of change of volume. The material time rate of change of an
element of volume in the spatial coordinate system, dv ¼ dx 1 dx 2 dx 3 , is shown to be d v _ ¼ tr Ddv

¼r v or tr D has the geometric interpretation as the instantaneous time rate of

change of material volume

udv , thus the ∇

v or tr D have the geometric interpretation as the instantaneous time
rate of change of material volume. Another way of viewing this result is to say that
the divergence of the velocity field is the time rate of change of a material volume
relative to how large it is at the instant ( 2.35 ).
The off-diagonal components of the rate-of-deformation tensor, for example
D 12 , represent rates of shearing. D 12 is equal to one-half the time rate of decrease
in an originally right angle between the filaments d x (1) and d x (2), Fig. 2.7 . To see
this, note that the dot product of material filaments d x (1) and d x (2) axes may
be written as

Thus the ∇

dxð 1 Þdxð 2 Þ¼jdxð 1 Þjjdxð 2 Þj cos y 12 ;

where y 12 is the angle between the two filaments. In the calculation of the material

time derivative of the dot product above, dx ð 1 Þ dx ð 2 Þ , we will employ the formula

dv i ¼ L ij dx j that follows from ( 2.31 ). The material time derivative of both

sides of the equation above is then computed;

d x i ¼ _

D
Dt ðdxð 1 Þdxð 2 ÞÞ ¼ d xð 1 Þdxð 2 Þþdxð 1 Þd xð 2 Þ
¼

L ij dx j ð 1 Þ dx 2 ð 2 Þþ dx i ð 1 Þ L ij dx j ð 2 Þ¼ 2 D ij dx i ð 1 Þ dx j ð 2 Þ

d x d x

¼j ð 1 Þjj dx ð 2 Þj cos y 12 þj dx ð 1 Þjj ð 2 Þj cos y 12

y 12 j dx ð Þjj dx ð Þj y 12

1 2 sin

and, since we are interested in the instant that y 12 ¼ p /2, it follows that