To give one example, we find that the corrections to the compliance
matrix *S*_{ij} due to the presence of an isotropic crack distribution
take the form:

(1) |

The real advantage of this approach can now be shown very simply using a couple of examples.

First, consider the situation in which all the cracks in the system
have the same vertical (*z*-)axis of symmetry. Then, the cracked/fractured
system is not isotropic, and we have the compliance correction matrix

(2) |

ranpc2vert
Example of a vertical cross-section
(Figure 1 xz-plane) through a medium having penny-shaped cracks with radius
r/L = 0.05, where L = 1.0 is the length on each side of a cube in 3D.
This image was produced by randomly placing 2000 crack centers in the
box of volume = L (so crack density ), and testing
to see if the center is within a
distance of the central square at ^{3}y = 0.5. If so, then
a random angle is chosen for the crack. If this crack orientation results in
an intersection with the plane y=0.5, the line of intersection is plotted
here. The resulting lines can have any length from 2r = 0.1 to
zero. The number of intersections found for this realization was 114,
whereas the expected value for any particular realization
is approximately . |

Next, consider the case when all cracks have their normals lying
randomly in parallel planes. Then, if the parallel planes are taken to
be horizontal, the cracks are all vertically aligned as in Figure 1.
So, I immediately find the anisotropic (*i.e.*, vertical
transverse isotropy or VTI) result that

(3) |

This same basic concept then works very well for any assumed symmetry
that we might like to model. There is no additional work to be done
once (*i*) the isotropic results are known (for some EMT)
and (*ii*) the layout of the two 's
in the correction matrix have been determined once and for
all for a given elastic symmetry resulting from a specific choice of
crack orientation distribution. Sayers and Kachanov (1991) give a
precise prescription for this. Although I make use of this
prescription here,
I will not show the details in order to avoid some of the
mathematical complications inherent in their tensorial expressions.

There are interesting and important questions of uniqueness related to
the inverse problem (*i.e.*, deducing the 's from seismic
wave observations) since more than one type of distribution can give
rise, for example, to vertical transverse isotropy (VTI). Then, the
question is whether quantities such as the Thomsen parameters of anisotropy
can help us to remove some of these possible ambiguities from the
interpretations of field measurements.

4/5/2006