Page 194 - Introduction to Continuum Mechanics

Symmetry of Stress Tensor- Principle of Moment of Momentum 179

Now, whether the elements is in static equilibrium or not, 2(^,4)3 is equal to zero because the
angular acceleration term is proportional to the moment of inertia which is given by (1/12)
2 2 an
(density) (AxiAK2A*3)[(A*i) +(Ax2) ] d is therefore a small quantity of higher order than
the right side of Eq. (ii). Thus,

Similarly, one can obtain

and

r
These equations state that the stress tensor is symmetric, i.e., T = T . Therefore, there are
only six independent stress components.

Example 4.4.1

The state of stress at a certain point is T = -pi, where/? is a scalar. Show that there is no
shearing stress on any plane containing this point.
Solution, The stress vector on any plane passing through the point with normal n is

Therefore, it is normal to the plane. This simple stress state is called a hydrostatic state of
stress,

Example 4.4.2
With reference to anxyz coordinate system, the matrix of a state of stress at a certain point
of a body is given by:

(a) Find the stress vector and the magnitude of the normal stress on a plane that passes through
the point and is parallel to the plane

(b)If