82 Strain
Note that, because we now have complete isotropy, the subscripts can
be dispensed with, the shear modulus G is implicit and furthermore the
factor 1/E which is common to all terms can be brought outside the matrix.
Finally, we have
1/E1 -V -v 0 0 0
1 -v 0 0 0
1 0 0 0
2(1+v) 0 0
2(1+v) 0
symmetric 2(1+ vComplete anisotropy was characterized via the elastic compliance matrix
through 21 independent constants. By considering the architecture of the
full matrix and making all 'cross-coupled terms zero, we obtained the
orthotropic case with nine independent constants. This was further
reduced in the case of transverse isotropy to five constants, utilizing the
relation between shear modulus and Young's modulus and Poisson's ratio.
The ultimate reduction (also using the shear modulus relation) resulted in
two elastic constants for the case of a perfectly isotropic material. One is
reminded of the quotation given by Jacques Grillo in his book Form,
Function and Design, that "In anything at all, perfection is finally attained
not when there is no longer anything to add, but when there is no longer
anything to take away".
5.6 Implications for in situ stress
There are many ramifications of the elastic compliance matrix and the
possible reductions which we have presented in Section 5.5. One
particularly important corollary with reference to Chapter 4 on in situ stress
relates to the ratio of horizontal to vertical stress, as calculated by
the 'switched on gravity' analysis presented in Section 4.6.2. Recall the
ratiowhich meant that the horizontal stress could never exceed the vertical
stress. Implicit in the derivation of this relation is the fact that the rock was
assumed to be isotropic. We can generate similar relations for varying
degrees of anisotropy, in particular for transverse isotropy and
orthotropy. Using the matrices presented in Section 5.5, and for the case
where axis 3 is vertical and plane 12 is horizontal, these areV'
1-vfor tranverse isotropy oH = -or