Engineering Rock Mechanics

34 Stress

where
a:t+W = A B
axx + OX,’
and so on.
However, when II stress tensors have been specified by the magnitudes
and directions of their three principal stresses, qi, a2il qi, for i = 1 to n,
it is not correct to average the principal stress values and average their
directions. For example, if one stress state has the maximum principal
stress acting due north with a value of 5 MPa and a second stress state
has the maximum principal stress acting due west with a value of^10
MPa, the average stress state is a stress state with the maximum
principal stress acting northwest with a value of 7.5 ma.
Understanding how to average stress fields is important for data
reduction in stress measurement programmes. The components for each
tensor must be first specified relative to three reference axes, x, y and
z, and then the components averaged, as in the case above which
represents averaging for n = 2. Once this averaging has been done, the
principal stresses for the mean stress tensor can be calculated.

43.8 What are the first, second and third stress invariants?

A3.8 When the stress tensor is expressed with reference to sets of axes

oriented in different directions, the components of the tensor change.

However, certain functions of the components do not change. These are

known as the stress invariants, expressed as I,, Z2, 13. (Recall that in

A3.3, the mathematical definition of a tensor included ’.. .invariant with

respect to... permissible co-ordinate transformations.. .’). The three

invariants are:

The expression for the first invariant, 11, indicates that for a given

stress state, whatever the orientation of the x, y and z axes, i.e. whatever

the orientation of the reference cube shown in A3.1, the values of the

three normal stresses will add up to the same value Zl.

When the principal stresses have to be calculated from the components

of the stress tensor, a cubic equation is used for finding the three values,

al, a2, and a3. This equation is

or

a^3 - 1la2 + 1.0 - l3 = o

Because the values of the principal stresses must be independent of

the choice of axes, the coefficients in the equation above, i.e. ZI, 12, and

Z3, must be invariant with respect to the orientation of the axes.