Page 193 - Introduction to Continuum Mechanics

178 Stress

T
We should also note that some authors use the convention t = T n so that t^ = 70«y. Under
that convention, for example, TI\ and T 2^ are tangential components of the stress vector on
the plane whose normal is 62 etc. These differences in meaning regarding the nondiagonal
elements of T disappear if the stress tensor T is symmetric,

4.4 Symmetry of Stress Tensor- Principle of Moment of Momentum

By the use of moment of momentum equation for a differential element, we shall now show
that the stress tensor is generally a symmetric tensor .

Fig. 4.3

Consider the free-body diagram of a differential parallelepiped isolated from a body as
shown in Fig. 4.3. Let us find the moment of all the forces about an axis passing through the
center points and parallel to the*3-axis:

In writing Eq. (i) we have assumed the absence of body moments.
Dropping the terms containing small quantities of higher order, we obtain

f See Prob. 4.27 for a case where the stress tensor is not symmetric