Questions and answers: strain and the theory of elasticity 65principal strains from the measured strains, and then determine the prin-
cipal stresses from the principal strains: the results will be identical to
those shown here, and can be used as a useful check of the calculations.45.6 Explain clearly why an isotropic rock has two independent
elastic constants, a transversely isotropic rock has five independent
constants and an orthotropic rock has nine independent constants
(compared to the general anisotropic case where there are twenty-
one independent elastic constants).A5.6 Adding further detail to Table 5.2 provides the answers.
Perfectly isotropic rock. If all properties are equal in all directions, all the
Young’s moduli and Poisson’s ratios in the different directions are equal.
Similarly, all the shear moduli in different directions are equal and can
be calculated from the single E and u values (A5.4). Thus, there are two
elastic Constants: one Young’s modulus, one Poisson’s ratio.
Transversely isotropic rock. Say, a rock mass with one main fracture set
so that the properties are the same in a plane but different perpendicular
to the plane. Five elastic constants: two Young’s moduli, two Poisson’s
ratios, and one shear modulus (not two, see A5.4).Orthotropic rock. Say a rock mass with three mutually perpendicular
fracture sets. Assuming there is no linkage between (a) normal and shear
components, and (b) different shear components (see the S matrix in
Section 5.1), this leaves nine elastic constants: three Young’s moduli, three
Poisson’s ratios and three shear moduli.
E9
General anisotropic rock. The 6 x 6 elastic compliance matrix is symmet-
rical, reducing the number of independent constants to 21. No further
reduction can be achieved. Thus, there are 22 elastic constants: all the S,,
in the matrix. ;i‘
45.7 Each of the following four rock masses is to be modelled us-
ing elasticity theory. State whether you think that an isotropic rock
assumption is justified, or whether one of the anisotropic assump-
tions would be more appropriate. Assume that the fractures have a
significant effect on the rock deformability.
A limestone with effectively three fracture sets, Le. the bedding
with mean strata thicknesses of 1 m, plus fracture set 1 (perpen-
dicular to the bedding) with two fractures/m, and fracture set 2
(perpendicular to both the bedding and fracture set 1) with five
fractureslm.
A welded volcanic tuff with five fracture sets.
Fracture set 1 : dip direction 089O; dip 50°; frequency 2.9/m.
Fracture set 2: dip direction 278O; dip 88O; frequency 1.3/m.
Fracture set 3: dip direction 224O; dip OSo; frequency 0.9/m.
Fracture set 4: dip direction 169O; dip 23O; frequency 2.1 /m.
Fracture set 5: dip direction 11 3O; dip 70°; frequency 0.7/m.