Page 124 - Introduction to Continuum Mechanics

The Rate of Deformation Tensor 109

Now, from Eq. (3.12.4) and (3.13.1)

and by the definition of transpose of a tensor and the fact that W is an antisymmetric tensor
r
(i.e.,W = -W ), we have

Thus,

Therefore,

Equation (ii) then gives

With dx = dsn, Eq. (3.13.6a) can also be written:

Eq. (3.13.6b) states that for a material element in the direction of n, its rate of extension
(i.e., rate of change of length per unit length ) is given by D nn(no sum on n). The rate of
extension is also known as stretching. In particular
DH = rate of extension for an element which is in the ej direction
DII — rate of extension for an element which is in the 62 direction and
I>33 = rate of extension for an element which is in the 63 direction
We note that since \dt gives the infinitesimal displacement undergone by a particle during the
time interval dt, the interpretation just given can be inferred from those for the infinitesimal
strain components. Thus, we obviously will have the following results: [see also Prob. 3.45(b)]:

2 £>i2 = rate of decrease of angle (from —) of two elements in ej and 62 directions
^
JC
2 D|3 = rate of decrease of angle (from —) of two elements in ej and 63 directions and