Mie–Gruneisen equation of state

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 C₀ is the bulk speed of sound,ρ₀ is the initial density, ρ is the current density, Γ₀ is the Gruneisen's gamma at the reference state, S_α = 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 V₀ = 1 / ρ₀ is the reference specific volume at temperature T = T₀, 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	C0 (m/s)	Sα	Γ0 (T < T1)	Γ0 (T > = T1)	T_1 (K)
Copper	3933 [3]	1.5 [3]	1.99 [4]	2.12 [4]	700

See also

References

1. ^ Zocher, M.A.; Maudlin, P.J. (2000), "An evaluation of several hardening models using Taylor cylinder impact data", Conference: COMPUTATIONAL METHODS IN APPLIED SCIENCES AND ENGINEERING, BARCELONA (ES), 09/11/2000--09/14/2000, http://www.osti.gov/energycitations/product.biblio.jsp?osti_id=764004, retrieved 2009-05-12 
2. ^ Wilkins, M.L. (1999), Computer simulation of dynamic phenomena, http://books.google.co.uk/books?hl=en, retrieved 2009-05-12 
3. ^ ^{a b} Mitchell, A.C.; Nellis, W.J. (1981), "Shock compression of aluminum, copper, and tantalum", Journal of Applied Physics 52: 3363, http://link.aip.org/link/?JAPIAU/52/3363/1, retrieved 2009-05-12 
4. ^ ^{a b} MacDonald, R.A.; MacDonald, W.M. (1981), "Thermodynamic properties of fcc metals at high temperatures", Physical Review B 24 (4): 1715–1724, doi:10.1103/PhysRevB.24.1715