It is not hard to show that the bulk and shear moduli can both be assumed to be decreasing functions of the volume fraction of partial melt. When there is no melt, the solid material constants are Kg for the purely solid (or grain) bulk modulus and for the purely solid shear modulus. As solid transforms into melt, the melt volume fraction is and the remaining solid volume fraction is . General relations for the changing elastic constants for small to modest values of are
K_sat = K_g(1 - c_1), and
_sat = _g(1 - c_2). The new symbols used here are Ksat for bulk modulus of solid containing pores saturated with melt, for shear modulus of solid containing pores saturated with melt, and c1 and c2 are nonnegative, dimensionless parameters. (If we were to do perturbation theory for small around the solid limit, then these parameters would be constant, independent of .But, we will instead use a more rigorous approach based on Gassmann's equation (Gassmann, 1951) and arrive at exact results for c1 that incorporate dependence and are therefore valid for a much wider range of values than would be possible using perturbation theory.) If in addition we make the assumption which is commonly made about these systems [see Williams and Garnero (1996)] that the melt density is approximately the same as that of the solid material, then we have the additional formula for changes in density
d v_p = 12d (K_sat+43_sat) - 12 c_1K_g+c_243_gK_g + 43_g , and
d v_s = 12d _sat - 12 c_2.
To simplify the expression in (dlnvp) further, we can make use of the well-known approximation that
v_pv_s 2. (We relax this strong assumption later in the paper.) Substituting (vp) and (vs) into (vpovervs) shows that
K_g 83_g, which when substituted into (dlnvp) shows that
d v_p - 16(2 c_1 + c_2).