Differential Equations and Their Solutions for Hugoniot Relation of Porous Materials along the Isobaric Path

Shock compression techniques can be applied to porous materials to reproduce thermodynamic states at high temperatures and high densities, and a lot of Hugoniot data were published in the past two decades. To investigate the intrinsic relationship between Hugoniot curves of dense metals and porous ones is of importance for development of more reasonable equation of state (EOS) models. In this paper, a set of differential equations, which relate the temperature and density of the shocked states along an isobaric path in p-V-T space to its initial densities of the porous material, is deduced from the traditional three-terms EOS and Grneisen EOS of solids. The differential formula of Wu-Jing EOS and the analytic expression of the Wu-Jing variable (Rp) are given. It is emphasized that Wu-Jing EOS is resulted not only from the contribution of crystal vibration as traditionally considered but also from most of the thermal electron effects. The new differential equations are applied to porous copper, and the effects of the thermal electrons to shock temperature, compression density, and value of Wu-Jing variable are discussed. It shows that the existence of the thermal electrons will reduce the increase rate of Rp with decrease of density.