Hellenica World

# .

The Mie-Gruneisen equation of state is a relation between the pressure and the volume of a solid at a given temperature. It is often used to determine the pressure in a shock-compressed solid. Several variations of the Mie-Gruneisen equation of state are in use.

Expressions for the Mie-Gruneisen equation of state

A temperature-corrected version that is used in computational mechanics has the form[1] (see also [2], p.61)

$$p = \frac{\rho_0 C_0^2 (\eta -1) \left[\eta - \frac{\Gamma_0}{2}(\eta-1)\right]} {\left[\eta - S_{\alpha}(\eta-1)\right]^2} + \Gamma_0 E;\quad \eta := \cfrac{\rho}{\rho_0}$$

where $$C_0$$ is the bulk speed of sound,\rho_0 is the initial density, $$\rho$$ is the current density, $$\Gamma_0$$ is the Gruneisen's gamma at the reference state, $$S_{\alpha} = dU_s/dU_p$$ is a linear Hugoniot slope coefficient, $$U_s$$ is the shock wave velocity, $$U_p$$ is the particle velocity, and E is the internal energy per unit reference specific volume.

A rough estimate of the change in internal energy can be computed using

$$E = \frac{1}{V_0} \int C_v dT \approx \frac{C_v (T-T_0)}{V_0} where \( V_0 = 1/\rho_0$$ is the reference specific volume at temperature $$T = T_0$$, and $$C_v$$ is the specific heat at constant volume. In many simulations, it is assumed that $$C_p$$ and $$C_v$$ are equal.

Parameters for various materials

material $$C_0$$ (m/s) $$S_{\alpha}$$ $$\Gamma_0$$ ($$(T < T_1)$$) $$\Gamma_0$$ ($$T >= T_1$$) T_1 (K)
Copper 3933 [3] 1.5 [3] 1.99 [4] 2.12 [4] 700