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&PSgr;(V,T,&fgr;)=&PSgr;s(V,T)+B(&fgr;)where the equilibrium volume fraction&fgr;eqis determined by minimizing &PSgr;,i.e., the condition∂&fgr;&PSgr;=0.For many cases, a Hayes EOS for the pure solid&PSgr;s(V,T)is adequate. This provides a thermodynamically consistent framework for theP-&agr;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 &fgr; are shifted to the left in the(V,P)-plane with increasing &fgr;. 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. ©2000 American Institute of Physics.