What Can We Learn from Hugoniot Temperature as a Function of Shock Velocity?

Abstract:

Shock-wave experiments traditionally rely on impact techniques, whereby measured shock velocity (U_S) can be related to material velocity (u_p), determined from the impact velocity (= 2u_p for a symmetric impact), and resulting in the empirically observed linear U_S–u_p equation of state: U_S = c₀ + s u_p. Modern experiments relying on laser-driven compression have the advantage of reaching higher pressures than laboratory impact experiments, but u_p is typically not determined; instead, Hugoniot temperature (T_H) and shock velocity are more readily measured.

Assuming a linear U_S–u_p equation of state and that the Grüneisen parameter has the volume dependence g(V) = g₀ (V/V₀), measurements of the Hugoniot temperature as a function of shock velocity provide constraints on the specific heat along the Hugoniot

C_VH(U_S) = V₀ f(U_S)[c₀ g₀ T_H – s U_S dT_H/dU_S]^–1

 where the Walsh-Christian (1955) function f(U_S) = – (U_S – c₀)² U_S/(V₀ s c₀) = T_H dS_H/dV_H gives the entropy change along the Hugoniot (subscripts 0 and H indicate zero-pressure and Hugoniot states, respectively). In this sense, T_H(U_S) measurements are similar to calorimetry experiments. If specific heat and Grüneisen parameter are determined independently (e.g., from wave-velocity measurements and experiments on porous samples), the T_H(U_S) analog to the linear U_S–u_p equation of state is

 T_H(U_S) = {T₀ exp(g₀ /s) – ò[V₀ c₀ f(x)/(s x C_V)] exp[c₀ g₀ /(s x)] dx} exp[– c₀ g₀ /(s U_S)]

where the integration is from x = c₀ to x = U_S.

In addition, experiments can be considered with: 1) different initial volume, as in a porous sample; 2) different initial internal energy, as in a sample heated at constant volume; and 3) different initial volume and internal energy, as in a sample initially heated at ambient pressure. From these four initial states, we get four different Hugoniot curves, and can also consider the effect of phase transition latent heat. Temperature as a function of shock velocity may thus be benefit the analysis of melting and other phase transitions with small volume change and finite latent heat.