Engineering Rock Mechanics

228 Rock mechanics interactions and rock engineering systems (RES!

14.3 Interaction matrices in rock mechanics

In Fig. 14.5, we show the conceptual link between stress and strain. In
Section 5.5, it was recalled that stress and strain are second-order tensors,
and that each component of the strain tensor can be linearly related to the
six components of the stress tensor, through an elastic compliance matrix
with 36 terms, of which 21 are independent. This is a result of the
application of linear elasticity. In the 2 x 2 matrix of Fig. 14.5(a), we show
this same link, except that we have now introduced the concept of path
dependency, i.e. how do we compute the strains from a knowledge of the
stresses, or vice versa? The top right-hand box of this matrix illustrates the
term Sll, or 1/E, of the elastic compliance matrix. Alternatively, we can
calculate the stresses from the strains as illustrated in the bottom left-hand
box. Note that the stress-strain curves are drawn with the independent
variable on the horizontal axis, in accordance with the conventions of
scientific presentation. Thus, this 2 x 2 matrix is not symmetrical in the sense
that the content of the off-diagonal terms are not equal, but is symmetric
in terms of functionality: that is, one can travel around the matrix with no
change occurring in the stress or strain state being represented.
However, the 2 x 2 matrix in Fig. 14.5@) represents constitutive behaviour
which includes rock failure-and is therefore beyond simple linear elastic-
ity-with the result that the behaviour of the rock now critically depends on
whether stress or strain is the independent variable. In some cases,
standardized methods for rock testing will specify values for the rate of stress
increase and, as illustrated in the top right-hand box of Fig. 14.5(b), this will
result in uncontrolled failure because the stress cannot be increased beyond
the compressive strength. Conversely, if the strain rate is specified then
strain becomes the independent variable and the complete stress-strain
curve is obtainable, as explained in Section 6.3. In every sense, therefore, this
matrix is asymmetric: the off-diagonal stress-strain behaviour is different;
and one cannot cycle repetitively through the matrix at all stress levels.

Independent

variable

stress

/ n

• Controlled 12
  failure

Strain

Uncontrolled

pfre

Stress

Strain

Figure 14.5 Stress-strain relations illustrated conceptually for elastic and inelastic

conditions utilizing 2 x 2 interaction matrices.