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
*K*_{g} 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

_sat = _g(1 - c_2).
The new symbols used here are
*K*_{sat} for bulk modulus of solid containing pores saturated with melt,
for shear modulus of solid containing pores saturated
with melt, and *c _{1}* and

d v_p = 12d (K_sat+43_sat) - 12 c_1K_g+c_243_gK_g + 43_g , and

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

10/25/1999