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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 \( C_0 \) 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 |
See also
References
^ 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, retrieved 2009-05-12
^ Wilkins, M.L. (1999), Computer simulation of dynamic phenomena, retrieved 2009-05-12
^ a b Mitchell, A.C.; Nellis, W.J. (1981), "Shock compression of aluminum, copper, and tantalum", Journal of Applied Physics 52: 3363, retrieved 2009-05-12
^ 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
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