Thermodynamic relations for shock waves in materials with a linear relation between shock-wave and particle velocities

Abstract

The Rankine-Hugoniot relations for shock waves and the empirical linear relation between the shock-wave and particle velocities define an incomplete thermodynamic description of the states along the Hugoniot curve. This incomplete description defines the following along the Hugoniot: (1) internal energy and pressure as functions of specific volume, (2) the ratio of enthalpy to internal energy, (3) the ratio of the changes in enthalpy and internal energy across a shock wave, and (4) the relation between the Grüneisen coefficient and the effective isentropic exponent. We use the Dugdale-MacDonald relation for the Grüneisen coefficient at low pressure, an assumed constant value for the specific heat at constant volume, and reasonable physical assumptions for extremely strong shock waves together with the incomplete thermodynamic state description to define the following along the Hugoniot: (5) the Grüneisen coefficient, (6) the effective isentropic exponent, (7) the ratio of specific heats, and (8) thermal and elastic components of pressure, temperature, and entropy. We present representative numerical values of these parameters as functions of reduced volumetric compression. We show how the solutions for these parameters define tangent planes to the surfaces of the incomplete E,P, and V and P,V, and T equations of state at each point along the Hugoniot curve.