Equation of state and Hugoniot locus for porous materials:: P-α model revisited

Foams, porous solids and granular materials have a characteristic Hugoniot locus that for weak shocks is concave in the (particle velocity, shock velocity)-plane. An equation of state (EOS) that has this property can be constructed implicitly from a Helmholtz free energy of the form Psi(V, T, phi) = Psi(s)(V, T) + B(phi) where the equilibrium volume fraction phi(eq) is determined by minimizing Psi, i.e., the condition partial derivative(phi)Psi = 0. For many cases, a Hayes EOS for the pure solid Psi(s) (V, T) is adequate. This provides a thermodynamically consistent framework for the P-alpha model. For this form of EOS the volume fraction has a similar effect to an endothermic reaction in that the partial Hugoniot loci with fixed phi are shifted to the left in the (V,P)-plane with increasing phi. The equilibrium volume fraction can then be chosen to match the concavity of the principal Hugoniot locus. An example is presented for the polymer estane. A small porosity of only 1.4 percent is required to match the experimental concavity in the Hugoniot data. This type of EOS can also be used to obtain the so-called "universal" Hugoniot for liquids.