40 Stress
3.10 Concluding remarks
We emphasize again that stress is a tensor with six independent
components. When a force, F, is resolved through an angle 8, the resulting
components are F cos 8 and F sin 8. However, when a stress component,
0, contributes to the normal and shear stresses on a plane inclined at an
angle 8 to the direction in which the stress component acts, the resulting
components are CT cos’ 8 and o sin’ 8. It is crucial to note, as we showed in
Fig. 3.7, that by suitably orientating the reference cube it is possible to
eliminate all shear stresses. Conversely, it is not possible to determine an
orientation for the complementary circumstance where all the normal
stresses reduce to zero. An elegant method of directly indicating this result,
that the normal stresses cannot be reduced to zero, is that the first stress
invariant (a property of the second-order tensor),
a,, + oyy + o,, = q + 02 + 03 = a constant,
cannot be made equal to zero whatever the orientation of the cube-
because it is a constant. The exception is when the constant is zero, i.e. a
state of pure shear, for example, with normal stresses of 3, -1 and -2 MPa,
so that the first stress invariant is 3 - 1 - 2 = 0.
The material that has been presented in this chapter, and that which
follows in Chapter 4, is sufficient for a basic understanding of the nature
of the state of stress. However, an Appendix on stress analysis has been
included. The way in which the stress is taken into account in rock
mechanics and rock engineering is described in succeeding chapters.