The elastic compliance matrix 81Such a material could arise from the microstructure of the intact rock, or in
the case of rock masses when three mutually perpendicular sets of discon-
tinuities with different properties and/or frequencies are present. The double
subscripts applied to the Poisson’s ratios are required in order to differentiate
the effects in the two different axial directions in each case.The reader should
note that there are in fact six Poisson’s ratios: the symmetry of the matrix
ensures that there are three relations of the form v12/El = vZl/Ez.
We can reduce the elastic compliance matrix even further by considering
the case of transverse isotropy. This is manifested by a rock mass with a
laminated fabric or one set of parallel discontinuities. In the case when the
plane of isotropy is parallel to the plane containing Cartesian axes 1 and 2,
we can say thatEl = Ez = E and E3 = E‘
v1z = vZl = v and ~13 = ~23 = v’
G12 # GZ3 and GZ3 = GJ1 = G‘.
The associated elastic compliance matrix is then./E -v/E -VfIE’ 0 0 0
1/E -v’/E’ 0 0 0
1/E‘ 0 0 00 01/G‘ 02(1+ v)
Esymmetric 1/GNote that in the above matrix, the term 2(1 + v)/E has been substituted
for 1/G12 because in the plane of isotropy there is a relation between the shear
modulus and the Young’s modulus and Poisson’s ratio. It is vital,
however, to realize that this relation, i.e. G = E/2(1 + v), only applies for
isotropic conditions and so we cannot make a similar substitution for either
1/GZ3 or 1/G31, which are out of the plane of isotropy. Thus, the number of
independent elastic constants for a transversely isotropic material is not six
but five, i.e.
E E‘ G‘ v VI.The final reduction that can be made to the compliance matrix is to
assume complete isotropy, whereEl = E2 = E3 = E