Engineering Rock Mechanics

The representative elemental volume for stress 55

UH = (a, + u3y2

OD = (ul - u3)/2

1 & Vector iatoric addition component of

uD~ 282

1% aD1 28

U

I Scalar addition of

hydrostatic component

Figure 4.11 Adding two stress tensors using Mohr's circle representation. (a) The
hydrostatic and deviatoric components of the stress tensor. (b) Representation of
the hydrostatic and deviatoric components in Mohr's circle.

paradox occurs because stress is defined as a property at a point, i.e. the

property of a sample with zero volume. The value at zero volume is plotted

on the vertical axis in the top diagram of Fig. 4.12. It is immediately

apparent that we should expect wide variations in measured in situ stress

values because of the capricious effect of discontinuities at small volume.

It should be recalled in this context that the strain gauges mentioned in

Sections 4.3.3 and 4.3.4 measure strains only over distances of about 5 mm.

The paradox arises because we are generally attempting to estimate the in

situ stress which is being applied to a volume greater than the REV, but

stress is a property at a point. Certainly, this super-REV stress is the one

which we would require for input as a boundary condition to a numerical

analysis of an engineering structure. In the design though, it could well be

a local maximum in the stress field (the local sub-REV stress) acting on a

small volume of rock which is critical for the stability of the structure as a

whole.

There are many ramifications of the diagram in Fig. 4.12. The variability

of the stress state with sampled volume has strong implications for stress

measurement strategies, data reduction and presentation. It immediately

suggests the idea of measuring stresses on the super-REV scale through a

method such as 'tunnel undercoring' using very long extensometers for

strain measurement (Windsor, 1985). Also, the figure suggests that

numerical stress analyses of fractured rock should consist of continuum

methods for large volumes of rock and discontinuum methods for sub-REV

volumes. Moreover, the existence of discontinuities, together with their