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

In this second limit, the equation for G^* is especially simple, since

Q^*f = 15[1 + 8(1-_m)(5-_m)3(2-_m)] 1d   is a constant under our constant Poisson's ratio approximation. The DEM equation is then integrated to obtain

G^* = G_m(1-)^1d   which should be compared to (DEMKr21). Within the analytical approximation, we will use (DEMmur21) as our defining equation for G_dry, and note that we can then replace the volume fraction factor $1-\phi$ by

(1-) = (G_dryG_m)^d   whenever it is convenient to do so.

Our decoupling approximation for shear modulus in this case turns out to be somewhat better than the corresponding one for the bulk modulus. The result in Appendix B shows that the only significant approximation we have made in (ddef) is one of order $0.7 (\nu_c-\nu_m)$ and this term is of the order of 7% of d or less for all the cases considered here. The relative error is therefore about one third of that made in the case of the bulk modulus.