Equation of state from weak shocks in solids

The Rankine-Hugoniot jump conditions for the increases across a shock of the normal stress, normal strain, and internal energy are not valid for weak shocks in solids. Correct jump equations for a solid can be obtained by integrating the equations for conservation of mass, momentum, and energy along the Rayleigh line through the shock process; these jump equations then depend on the details of the shock profile. Further, because a uniaxially compressed solid supports a nonzero shear stress, the locus of thermodynamic states reached behind planar shocks, which we call the anisotropic Hugoniot, requires for its description two stress variables and two strain variables. In the present paper the thermodynamic description of the anisotropic Hugoniot is given, and for the example of $6061-T6$ Al the shock-profile jump equations are derived, the weak-shock equation of state is computed, and the pressure on the principal adiabat is found to differ from the results of Rankine-Hugoniot theory by several percent in the range 0-100 kbar.