Mie–Gruneisen equation of state

The Mie–Grüneisen equation of state is a relation between the pressure and the volume of a solid at a given temperature.[1][2] 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

Γ=V(dpde)V

where V is the volume, p is the pressure, e is the internal energy, and Γ is the Grüneisen parameter which represents the thermal pressure from a set of vibrating atoms. If we assume that Γ is independent of p and e, we can integrate Grüneisen's model to get

p−p0=ΓV(e−e0)

where p0 and e0 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 p0 and e0 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.

## Contents

* 1 History
* 2 Expressions for the Mie–Grüneisen equation of state
* 2.1 Parameters for various materials
* 3 Derivation of the equation of state
* 4 See also
* 5 References

## History

Gustav Mie, in 1903, developed an intermolecular potential for deriving high-temperature equations of state of solids.[3] In 1912 Eduard Grüneisen extended Mie's model to temperatures below the Debye temperature at which quantum effects become important.[4] Grüneisen's form of the equations is more convenient and has become the usual starting point for deriving Mie–Grüneisen equations of state.[5]

## Expressions for the Mie–Grüneisen equation of state

A temperature-corrected version that is used in computational mechanics has the form[6] (see also,[7] p. 61)

p=ρ0C02χ[1−Γ02χ](1−sχ)2+Γ0E;χ:=1−ρ0ρ

where C0 is the bulk speed of sound, ρ0 is the initial density, ρ is the current density, Γ0 is Grüneisen's gamma at the reference state, s=dUs/dUp is a linear Hugoniot slope coefficient, Us is the shock wave velocity, Up is the particle velocity, and E is the internal energy per unit reference volume. An alternative form is

p=ρ0C02(η−1)[η−Γ02(η−1)][η−s(η−1)]2+Γ0E;η:=ρρ0.

A rough estimate of the internal energy can be computed using

E=1V0∫CvdT≈Cv(T−T0)V0=ρ0cv(T−T0)

where V0 is the reference volume at temperature T=T0, Cv is the heat capacity and cv is the specific heat capacity at constant volume. In many simulations, it is assumed that Cp and Cv are equal.

### Parameters for various materials

material | ρ0 (kg/m3) | cv (J/kg-K) | C0 (m/s) | s | Γ0 (T<T1) | Γ0 (T>=T1) | T1 (K)

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Copper | 8960  | 390  | 3933 [8] | 1.5 [8] | 1.99 [9] | 2.12 [9] | 700

## Derivation of the equation of state

From Grüneisen's model we have

(1)p−p0=ΓV(e−e0)

where p0 and e0 are the pressure and internal energy at a reference state. The Hugoniot equations for the conservation of mass, momentum, and energy are

ρ0Us=ρ(Us−Up),pH−pH0=ρ0UsUpandpHUp=ρ0Us(Up22+EH−EH0)

where ρ0 is the reference density, ρ is the density due to shock compression, pH is the pressure on the Hugoniot, EH is the internal energy per unit mass on the Hugoniot, Us is the shock velocity, and Up is the particle velocity. From the conservation of mass, we have

UpUs=1−ρ0ρ=1−VV0=:χ.

Where we defined V=1/ρ, the specific volume (volume per unit mass).

For many materials Us and Up are linearly related, i.e., Us = C0 \+ s Up where C0 and s depend on the material. In that case, we have
Us=C0+sχUsorUs=C01−sχ.

The momentum equation can then be written (for the principal Hugoniot where pH0 is zero) as

pH=ρ0χUs2=ρ0C02χ(1−sχ)2.

Similarly, from the energy equation we have

pHχUs=12ρχ2Us3+ρ0UsEH=12pHχUs+ρ0UsEH.

Solving for eH, we have

EH=12pHχρ0=12pH(V0−V)

With these expressions for pH and EH, the Grüneisen model on the Hugoniot becomes

pH−p0=ΓV(pHχV02−e0)orρ0C02χ(1−sχ)2(1−χ2ΓVV0)−p0=−ΓVe0.

If we assume that Γ/V = Γ0/V0 and note that p0=−de0/dV, we get

(2)ρC02χ(1−sχ)2(1−Γ0χ2)+de0dV+Γ0V0e0=0.

The above ordinary differential equation can be solved for e0 with the initial condition e0 = 0 when V = V0 (χ = 0). The exact solution is

e0=ρC02V02s4[exp⁡(Γ0χ)(Γ0s−3)s2−[Γ0s−(3−sχ)]s21−sχ+exp⁡[−Γ0s(1−sχ)](Γ02−4Γ0s+2s2)(Ei[Γ0s(1−sχ)]−Ei[Γ0s])]

where Ei[z] is the exponential integral. The expression for p0 is

p0=−de0dV=ρC022s4(1−χ)[s(1−sχ)2(−Γ02(1−χ)(1−sχ)+Γ0[s{4(χ−1)χs−2χ+3}−1]−exp⁡(Γ0χ)[Γ0(χ−1)−1](1−sχ)2(Γ0−3s)+s[3−χs{(χ−2)s+4}])−exp⁡[−Γ0s(1−sχ)][Γ0(χ−1)−1](Γ02−4Γ0s+2s2)(Ei[Γ0s(1−sχ)]−Ei[Γ0s])].
Plots of e0 and p0 for copper as a function of χ.

For commonly encountered compression problems, an approximation to the exact solution is a power series solution of the form

e0(V)=A+Bχ(V)+Cχ2(V)+Dχ3(V)+…

and

p0(V)=−de0dV=−de0dχdχdV=1V0(B+2Cχ+3Dχ2+…).

Substitution into the Grüneisen model gives us the Mie–Grüneisen equation of state

p=1V0(B+2Cχ+3Dχ2+…)+Γ0V0[e−(A+Bχ+Cχ2+Dχ3+…)].

If we assume that the internal energy e0 = 0 when V = V0 (χ = 0) we have A = 0. Similarly, if we assume p0 = 0 when V = V0 we have B = 0. The Mie–Grüneisen equation of state can then be written as

p=1V0[2Cχ(1−Γ02χ)+3Dχ2(1−Γ03χ)+…]+Γ0E

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

Comparison of exact and first-order Mie–Grüneisen equation of state for copper.

If we take the first-order term and substitute it into equation (2), we can solve for C to get

C=ρ0C02V02(1−sχ)2.

Then we get the following expression for p :

p=ρ0C02χ(1−sχ)2(1−Γ02χ)+Γ0E.

This is the commonly used first-order Mie–Grüneisen equation of state [citation needed].

## See also

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

## References

1. ↑ Roberts, J. K., & Miller, A. R. (1954). Heat and thermodynamics (Vol. 4). Interscience Publishers.
2. ↑ Burshtein, A. I. (2008). Introduction to thermodynamics and kinetic theory of matter. Wiley-VCH.
3. ↑ Mie, G. (1903) "Zur kinetischen Theorie der einatomigen Körper." Annalen der Physik 316.8, p. 657-697.
4. ↑ Grüneisen, E. (1912). Theorie des festen Zustandes einatomiger Elemente. Annalen der Physik, 344(12), 257-306.
5. ↑ Lemons, D. S., & Lund, C. M. (1999). Thermodynamics of high temperature, Mie–Gruneisen solids. American Journal of Physics, 67, 1105.
6. ↑ 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
7. ↑ Wilkins, M.L. (1999), Computer simulation of dynamic phenomena, https://books.google.com/books?hl=en, retrieved 2009-05-12
8. ↑ 8.0 8.1 Mitchell, A.C.; Nellis, W.J. (1981), "Shock compression of aluminum, copper, and tantalum", Journal of Applied Physics 52 (5): 3363, doi:10.1063/1.329160, Bibcode: 1981JAP....52.3363M, http://link.aip.org/link/?JAPIAU/52/3363/1, retrieved 2009-05-12
9. ↑ 9.0 9.1 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, Bibcode: 1981PhRvB..24.1715M

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