Engineering Rock Mechanics

36 Stress

Figure 3.5 Illustration of the development of two shear stresses on each face of an
infinitesimal cube.

3.6 The symmetry of the stress matrix

From the text so far, we know that there are nine separate stress

components at a point. We also assume that the body is in equilibrium and

therefore there should be an equilibrium of forces and moments at all

points throughout the body. Thus, after listing the nine components in the

matrix above, we should inspect the equilibrium of forces at a point in terms

of these stress components.

In Fig. 3.6, we show the four stress components acting on the edges of

a small square (which is a cross-section through a cube of edge length AI)

at any given location and in any plane of given orientation in the body,

We now define a local Cartesian system of axes, perpendicular and parallel

to the edges of the square. Clearly, the forces associated with the normal

stress components, CY,, and ow, are in equilibrium; however, for there to

be a resultant moment of zero, then the two shear stress components have

to be equal in magnitude. This is demonstrated by taking moments about

the centre of the square:

(A1/2) X (AZ)’zXy - (AU2) x (AZ)2zyx = 0.

Thus, by considering moment equilibrium around the x, y and z axes, we

find that

%y - - zy,, zyz = zzy, z,, = zzx.

A1

I

• TYX
  tuyy

-mXX

XY

Figure 3.6 Consideration of the rotational equiIibrium about the z-axis of a small

cubic element at any position in a body.