For this limit, the stiffness form and the compliance form
of the DEM equations are of equal difficulty to integrate, but a complication
arises due to the presence of shear modulus dependence in the
term in *P*. We are going to make an approximation
(only for analytical calculations) that
, so
the effect of variations in Poisson's ratio away from
for the matrix material is
assumed *not* to affect the results significantly (*i.e.*,
to first order)
over the range of integration. Without this assumption, the DEM
equations for bulk and shear are coupled and must be solved
simultaneously (and therefore numerically in most cases).

With this approximation, the equation to be integrated then becomes

(1-y)dK^*(y)dy = 1b[K_f-K^*(y)], where

b = 3(1-2_m)4(1-_m^2). The result of the integration is

K^* - K_f = (K_m-K_f)(1-)^1b.
This result seems to show a very strong dependence of *K ^{*}* on the aspect
ratio and Poisson's ratio through the product .But, we show in Appendix B that ,so only the dependence on is significant.

It seems that this decoupling approximation might have a large effect
for a dry system, but an exact decoupling can be achieved in this
case (see Appendix B). The result shows that the only significant
approximation we have made in (b) is one of order
and this term is of the order of 20% of *b*,
and usually much less, for all the cases considered here.

4/29/2001