Gas inclusion: $K_f << \pi\alpha\gamma_m$

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 $\gamma_m$ in P. We are going to make an approximation (only for analytical calculations) that $\gamma^* \simeq K^*[3(1-2\nu_m)/4(1-\nu_m^2)]$, so the effect of variations in Poisson's ratio away from $\nu_m$ 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 $\alpha (1-2\nu)$.But, we show in Appendix B that $\nu \to \nu_c \simeq \pi\alpha/18$,so only the dependence on $\alpha$ 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 $2 (\nu_c-\nu_m)$ and this term is of the order of 20% of b, and usually much less, for all the cases considered here.