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Sayers and Kachanov (1991) present a useful method for
decomposing the elastic potential of a cracked system into parts due
to the (assumed) homogeneous and isotropic elastic background
material, and those due to the presence of cracks up to moderate densities.
The fundamental idea is that the elastic potential function is composed
of just nine terms, representing all combinations of stress invariants of
such a system. These invariants depend on the stress tensor and the crack density tensor . In particular, the tensor
is defined in three dimensions by
| |
(37) |

where *V* is the averaging volume, is the unit normal of
penny-shaped crack *c* having radius *a*_{c}. We use the notation
, where *T* is the transpose, to express the
outer product ()of two vectors; this notation is consistent with that
commonly used to express the singular value decomposition of an
arbitrary matrix in terms of its singular vectors. Another common, and
entirely equivalent, form of notation for the same quantity that is
often used in the mechanics literature is the dyadic
form .
The elastic potential then takes the form

| |
(38) |

where is the trace operation, and the dot notation
indicates a contraction over one set of indices.
(Note that the significance of crack-influence parameters , , and
have been changed from the definitions made by Kachanov (1980),
Kachanov and Sevostianov (2005), and Sayers and Kachanov (1991), so that here
is the coefficient of a contribution second order in , third order in , and fourth order in .)
The coefficients pertinent to the isotropic background elastic medium
are given by and ,where *E*_{0} is Young's modulus, and is Poisson's ratio.
Now, to illustrate the meaning of (38), we will
reduce this to component form in two cases. For the cases of interest, we
can assume
the crack density tensor itself reduces to the form

| |
(39) |

where , for *i* = 1,2,3, correspond to spatial directions
*x*, *y*, *z*, respectively. Furthermore, is the scalar crack density
defined in the main text.

*Horizontal cracks, *

If all the cracks in the system have the same axis of symmetry (which
we will take to be the *z*-axis), then and
(38) reduces to the following expression:

| |
(40) |

where the repeated index *j* is summed. At low crack densities
, we see that
only the terms proportional to and are important in
the crack-influence decomposition. As the crack density increases,
the terms proportional to , , and start to
contribute.
Then, at the highest crack densities considered, all seven of these
coefficients can come into play. Although we may imagine for
example that is actually a function of crack density ,it is clear from the form of (40) that such corrections
would be indistinguishable from corrections due to .So, for our present purposes, we do not need to consider any
coefficients except and at low crack
densities,
and we also do not need to consider any coefficients except
through , when we want to fit quadratic
corrections for the moderate crack density results.
Typical values of of interest in many applications are around
. So as long as the 's for higher order corrections
are of approximately the same order of magnitude as those for
and , we see that neglect of terms
like is entirely appropriate.

Now it is also easy to see how (40) gives rise
to the low density result (4).

*Vertical cracks, *

It is also straightforward now to repeat the previous exercise by
considering other types of crack density distribution. An interesting
case is the one with all vertical cracks, having their crack normals
in the *xy*-plane. Then, . A special case of
this type is when the crack normals are completely randomly distributed so
that . Then, we get simplified formulas for
all the terms in the elastic potential analogous to the previous
example. The results are:

| |
(41) |

where again the repeated index *j* is summed.
The basic conclusions reached in the previous example clearly apply
again. For small to moderate crack densities , we do not
need to consider dependence of crack-influence parameters
or ,as such dependence cannot be distinguished from the low order
contributions from and , respectively.
Similarly, comes into play whenever and
are important, while and can
presumably be neglected in many low to moderate crack density applications.

T

ABLE 2.Examples of Sayers and Kachanov (1991)
crack-influence parameters

and

when crack density

for penny-shaped cracks. Four choices of effective medium theory are
considered: NI (non-inteacting), DS, (differential scheme), CPA
(coherent potential approximation), and SC (the Budiansky and
O'Connell self-consistent scheme). Note that crack density is defined
here as

, where

*N*/

*V* is number density of cracks,
and

is the area of the circular crack face.

** Next:** APPENDIX C. DISCUSSION OF
** Up:** Berryman and Grechka: Random
** Previous:** Self-consistent Estimates for Hexagonal
Stanford Exploration Project

1/16/2007