Mie–Grüneisen equation of state

The Mie–Grüneisen equation of state is an equation of state that relates the pressure and volume of a solid at a given temperature.Roberts, J. K., & Miller, A. R. (1954). Heat and thermodynamics (Vol. 4). Interscience Publishers.Burshtein, A. I. (2008). Introduction to thermodynamics and kinetic theory of matter. Wiley-VCH. It is used to determine the pressure in a shock-compressed solid. The Mie–Grüneisen relation is a special form of the Grüneisen model which describes the effect that changing the volume of a crystal lattice has on its vibrational properties. Several variations of the Mie–Grüneisen equation of state are in use.

The Grüneisen model can be expressed in the form
:$\Gamma = V \left(\frac\right)_V$
where is the volume, is the pressure, is the internal energy, and is the Grüneisen parameter which represents the thermal pressure from a set of vibrating atoms. If we assume that is independent of and , we can integrate Grüneisen's model to get
:$p - p_0 = \frac (e - e_0)$
where $p_0$ and $e_0$ are the pressure and internal energy at a reference state usually assumed to be the state at which the temperature is 0K. In that case ''p''₀ and ''e''₀ are independent of temperature and the values of these quantities can be estimated from the Hugoniot equations. The Mie–Grüneisen equation of state is a special form of the above equation.

History

Gustav Mie, in 1903, developed an intermolecular potential for deriving high-temperature equations of state of solids.Mie, G. (1903) "Zur kinetischen Theorie der einatomigen Körper." Annalen der Physik 316.8, p. 657-697. In 1912, Eduard Grüneisen extended Mie's model to temperatures below the Debye temperature at which quantum effects become important.Grüneisen, E. (1912). Theorie des festen Zustandes einatomiger Elemente. Annalen der Physik, 344(12), 257-306. Grüneisen's form of the equations is more convenient and has become the usual starting point for deriving Mie–Grüneisen equations of state.Lemons, D. S., & Lund, C. M. (1999). Thermodynamics of high temperature, Mie–Gruneisen solids. American Journal of Physics, 67, 1105.

Expressions for the Mie–Grüneisen equation of state

A temperature-corrected version that is used in computational mechanics has the form

:$p = \frac + \Gamma_0 E;\quad \chi := 1-\cfrac$
where $C_0$ is the bulk speed of sound, $\rho_0$ is the initial density, $\rho$ is the current density, $\Gamma_0$ is Grüneisen's gamma at the reference state, $s = dU_s/dU_p$ 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 volume. An alternative form is
:$p = \frac + \Gamma_0 E;\quad \eta := \cfrac \,.$
A rough estimate of the internal energy can be computed using
:$E = \frac \int C_v dT \approx \frac = \rho_0 c_v (T-T_0)$
where $V_0$ is the reference volume at temperature $T = T_0$, $C_v$ is the heat capacity and $c_v$ is the specific heat capacity at constant volume. In many simulations, it is assumed that $C_p$ and $C_v$ are equal.

Parameters for various materials

Derivation of the equation of state

From Grüneisen's model we have

where $p_0$ and $e_0$ are the pressure and internal energy at a reference state. The Hugoniot equations for the conservation of mass, momentum, and energy are
: \begin
\rho_0 U_s &= \rho (U_s - U_p) \,, \\ exp_H - p_ &= \rho_0 U_s U_p \,, \\ exp_H U_p &= \rho_0 U_s \left(\frac + E_H - E_\right)
\end
where ''ρ''₀ is the reference density, ''ρ'' is the density due to shock compression, ''p''_H is the pressure on the Hugoniot, ''E''_H is the internal energy per unit mass on the Hugoniot, ''U''_s is the shock velocity, and ''U''_p is the particle velocity. From the conservation of mass, we have
:$\frac = 1 - \frac = 1 - \frac =: \chi \,.$
Where we defined $V = 1/\rho$, the specific volume (volume per unit mass).
:For many materials ''U''_s and ''U''_p are linearly related, i.e., where ''C''₀ and ''s'' depend on the material. In that case, we have
:$U_s = C_0 + s\chi U_s \quad \text \quad U_s = \frac \,.$
The momentum equation can then be written (for the principal Hugoniot where ''p''_H0 is zero) as
:$p_H = \rho_0 \chi U_s^2 = \frac \,.$
Similarly, from the energy equation we have
:$p_H \chi U_s = \tfrac \rho \chi^2 U_s^3 + \rho_0 U_s E_H = \tfrac p_H \chi U_s + \rho_0 U_s E_H \,.$
Solving for ''e''_H, we have
:$E_H = \tfrac \frac = \tfrac p_H (V_0 - V)$
With these expressions for ''p''_H and ''E''_H, the Grüneisen model on the Hugoniot becomes
:$p_H - p_0 = \frac \left(\frac - e_0\right) \quad \text \quad \frac\left(1 - \frac\,\frac\,V_0\right) - p_0 = -\frac e_0 \,.$
If we assume that and note that $p_0 = -d e_0/d V$, we get

The above ordinary differential equation can be solved for ''e''₀ with the initial condition ''e''₀ = 0 when ''V'' = ''V''₀ (''χ'' = 0). The exact solution is
:
\begin
e_0 = \frac \Biggl &\exp(\Gamma_0\chi) \left(\tfrac - 3 \right) s^2 - \frac + \\
& \exp\left[-\tfrac (1-s\chi)\right\left(\Gamma_0^2 - 4 \Gamma_0 s + 2 s^2\right) \left(\text\left tfrac (1-s\chi )\right- \text\left tfrac\rightright)
\Biggr]
\end

where Ei[''z''] is the exponential integral. The expression for ''p''₀ is
:
\begin
p_0 = -\frac = \frac \Biggl[
& \frac \Bigl (- \Gamma_0^2(1 - \chi)(1 -s\chi)
+ \Gamma_0 \-1\\
& \qquad\qquad\quad - \exp(\Gamma_0\chi) Gamma_0(\chi-1) -11-s\chi)^2(\Gamma_0-3s) + s -\chi s \Bigr) \\
& - \exp\left \tfrac (1-s\chi)\right\left Gamma_0(\chi-1) - 1\right\left(\Gamma_0^2 - 4 \Gamma_0 s + 2 s^2\right) \left(\text\left tfrac (1-s\chi )\right- \text\left tfrac\rightright)
\Biggr] \,.
\end

For commonly encountered compression problems, an approximation to the exact solution is a power series solution of the form
:$e_0(V) = A + B \chi(V) + C \chi^2(V) + D \chi^3(V) + \cdots$
and
:$p_0(V) = -\frac = -\frac\,\frac = \frac\,(B + 2C\chi + 3D\chi^2 + \cdots) \,.$
Substitution into the Grüneisen model gives us the Mie–Grüneisen equation of state
:
p = \frac\,(B + 2C\chi + 3D\chi^2 + \cdots) + \frac \left - (A + B \chi + C \chi^2 + D \chi^3 + \cdots ) \right\,.

If we assume that the internal energy ''e''₀ = 0 when ''V = V''₀ () we have ''A'' = 0. Similarly, if we assume ''p''₀ = 0 when ''V = V''₀ we have ''B'' = 0. The Mie–Grüneisen equation of state can then be written as
:
p = \frac\left C\chi \left(1-\tfrac\chi\right) + 3D\chi^2\left(1 -\tfrac\chi\right) + \cdots\right+ \Gamma_0 E

where ''E'' is the internal energy per unit reference volume. Several forms of this equation of state are possible.

If we take the first-order term and substitute it into equation (), we can solve for ''C'' to get
:$C = \frac \,.$

Then we get the following expression for ''p'':
:$p = \frac \left(1-\tfrac\chi\right) + \Gamma_0 E \,.$
This is the commonly used first-order Mie–Grüneisen equation of state.

See also

* Impact (mechanics)
* Shock wave
* Shock (mechanics)
* Shock tube
* Hydrostatic shock
* Viscoplasticity

References

{{DEFAULTSORT:Mie-Gruneisen equation of state
Continuum mechanics
Solid mechanics
Equations of state