Since the right hand sides of (MTvsKT) and (KT) are exactly the same if the host medium h is taken to be the same, note that
(^*_MT-_h)v_i ^hi (^*_KT-_h)^h*, showing that these two explicit approximations are very closely related but not generally equal. To achieve equality between the two approximations (), it is both necessary and sufficient that
^h* = v_i ^hi -- which is certainly not satisfied in general, but is true for spherical inclusions. To check satisfaction of (equalitycondition), I find, using the Ps and Qs for spheres from Table 1, that (equalitycondition) implies
K_h+43_hK^*+43_h = v_i K_h+43_hK_i+43_h must be satisfied by the bulk modulus, and
_h+_h^*+_h = v_i _h+_h_i+_h by the shear modulus. When the common factors depending only on host properties in the numerators are divided out of these two equations, the remaining formulas are exactly the results of the MT and KT theories for spherical inclusions. Since when the host and inclusion properties are the same regardless of the assumed host particle shape, the results (equalitycondition), (KTandMTforspheres_K), and (KTandMTforspheres_G) are also valid for both MT and KT whenever all the inclusions except the host are sphere shaped. In general, equation (equalitycondition) is just an approximation to (anotherid) that is not always valid.