Some analytical results for K^*

We now assume the inclusion phase is a fluid so K_i = K_f and G_i = G_f = 0. The fluid can be either a liquid or a gas. We consider three cases: (1) liquid inclusion: $K_f \gt\gt \pi\alpha\gamma_m$,(2) gas inclusion: $K_f << \pi\alpha\gamma_m$,(3) general inclusion: $K_f \simeq \pi\alpha\gamma_m$.Case 1 corresponds to liquid inclusions, Case 2 to gas inclusions, and Case 3 to a circumstance in which crack aspect ratio is tuned to fluid modulus, or in which we do not want to limit ourselves to the assumptions of either of the previous two cases.