-- was sufficient to describe
the behavior of isotropic systems. This analysis was good for
representing the behavior at very low crack densities. In order to arrive at
higher crack densities, these authors made use of an older effective
medium theory sometimes called ``self-consistent'', and sometimes
more accurately described as ``asymmetric self-consistent.''
This approach had the drawback
that it overpredicted the effect of cracks on reducing elastic compliance,
and therefore gave a relatively low value 
at which the cracked medium would fail. But it is known that failure
does not usually occur at such small crack densities, so these
overall predictions are often criticized on this basis. [See Henyey
and Pomphrey, 1982; Zimmerman, 1991.] Hudson (1980;
1996) used a different method, the so-called ``method of smoothing''
first introduced in the mathematics literature, for the crack problem.
Keeping density corrections just to first order in the Hudson approach
gives an improvement over the previously mentioned scheme. Hudson
also introduced a second order correction, but Sayers and Kachanov
(1991) point out that this approach then violates rigorous Hashin-Shtrikman
upper bounds on the moduli for this problem. They recommend instead using
a differential scheme [see Zimmerman (1991) for an excellent review of the DS],
because the DS tracks Hudson's first order model at low concentration
of cracks, but never violates the HS bounds at high concentrations.