54 In situ stress
N N
I A. Scalar meanE wI B. Tensor mean
s S
Tensor uI u2 u, trend u, u2 uI trend
1 18 10 000" A. 18 10 045"
2 18 10 090" B. 14 14 -Figure 4.10 Demonstration of stress tensor averaging. (a) Principal stresses to be
averaged. (b) Incorrect (A) and correct (B) methods of averaging.
In Fig. 4.11 we also illustrate the addition of two two-dimensional tensors
via the Mohr's circle representation of stress. This figure is particularly
interesting because it provides a further intuitive understanding of the
tensor concept-as being composed of one scalar component and one
vector component. Thus, when adding two tensors one adds the
hydrostatic components as scalars along the normal stress axis and the
deviatoric components as vectors in z-(T space. This representation can also
be extrapolated to any number of tensors.4.5 The representative elemental volume for
stress
Later on, and especially in Chapter 9 on permeability, we will be
discussing the concept of the representative elemental volume (REV).
When tests are conducted on rock there is a spread in the measured values.
This spread will occur through natural inhomogeneity of the material but,
more importantly in this context, the values will depend on how the
pre-existing discontinuities have affected the measured values. The
REV is the volume, for any given body, at which the size of the sample
tested contains a sufficient number of the inhomogeneities for the
'average' value to be reasonably consistent with repeated testing. This
concept is illustrated in Fig. 4.12 where the variability versus volume is
generically illustrated.
As shown, with low specimen volumes, the absence or presence of
discontinuities is highly variable but, as the specimen volume is increased,
the sample of discontinuities becomes more and more statistically repre-
sentative, until the REV is reached. This concept applies to all rock
properties and conditions which are affected by discontinuities, and is
especially pertinent (and paradoxical) for stress measurements. The