Engineering Rock Mechanics

60 Strain and the theory of elasticity

Table 5.2 Number of elastic constants required to characterize different forms of rock
mass symmetry

General anisotropic rock

Orthotropic rock

(three axes of symmetry, e.g. similar to a

rock mass with three orthogonal fracture

sets)

Transversely isotropic rock

(one axis of symmetry, e.g. similar to a

rock mass with distinct laminations or

with one main fracture set)

Perfectly isotropic rock

21 elastic constants:

all the independent Si, in the S matrix.

Because the matrix is symmetrical, there

are 21 rather than 36 constants.

9 elastic constants:

as in the matrix above - 3 Young’s

moduli, 3 Poisson‘s ratios and 3 shear

moduli

5 elastic constants:

2 Young’s moduli, 2 Poisson’s ratios, and

1 shear modulus (see 45.4)

2 elastic constants:

1 Young’s modulus, 1 Poisson’s ratio

In this way, by considering the architecture of the general elastic

compliance matrix, S, the number of different elastic constants required

to characterize rock masses with different basic forms of symmetry can

be established. This relates to the anisotropy of the rock mass, i.e. to

what extent it has different properties in different directions. The results

are in Table 5.2.

It is important to realize that most rock mechanics analyses have

been conducted assuming that the rock mass is completely isotropic, i.e.

assuming that the elastic moduli are the same in all directions, and that

the rock mass can therefore be characterized by two parameters: a single

value of Young’s modulus, E, and a single value of Poisson’s ratio, u. A

separate value for the shear modulus, G, is not required in the isotropic

case because then G is a function of E and v, see Q and A5.4.

In some cases, isotropy may be a useful simplifymg engineering

assumption; in other cases, especially where there is one dominant

low-stiffness fracture set, the isotropic assumption is not appropriate.

5.2 Questions and answers: strain and the theory of

elasticity

45.1 What is the meaning of the first stress invariant and the first

strain invariant?

A5.1 The first stress invariant, ZI, is the sum of the leading diagonal

terms of the stress matrix illustrated in Table 5.1: ZI = a,, + a,, + azz. It

can be considered as proportional to the mean normal stress, and hence

the mean pressure applied at a point.

The first strain invariant is the sum of the leading diagonal terms of

the strain matrix illustrated in Table 5.1: i.e. cxx + E,,, + szz. Normal strain

in one dimension is a measure of change in length, and so this invariant

is a measure of the volumetric change, the contraction or dilatation.