View source for Mie–Gruneisen equation of state

The [[Gustav Mie|Mie]]–[[Eduard Grüneisen|Grüneisen]] [[equation of state]] is a relation between the [[pressure]] and the [[volume]] of a solid at a given temperature.<ref name=roberts>Roberts, J. K., & Miller, A. R. (1954). Heat and thermodynamics (Vol. 4). Interscience Publishers.</ref><ref name=bursh>Burshtein, A. I. (2008). Introduction to thermodynamics and kinetic theory of matter. Wiley-VCH.</ref>  It is used to determine the pressure in a [[Shock (mechanics)|shock]]-compressed solid.  The Mie–Grüneisen relation is a special form of the [[Grüneisen parameter|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
:<math>\Gamma = V \left(\frac{dp}{de}\right)_V</math>
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
:<math>
   p - p_0 = \frac{\Gamma}{V} (e - e_0)
 </math>
where ''p''<sub>0</sub> and ''e''<sub>0</sub> 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''<sub>0</sub> and ''e''<sub>0</sub> are independent of temperature and the values of these quantities can be estimated from the [[Rankine–Hugoniot conditions|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.<ref name=mie>Mie, G. (1903) "Zur kinetischen Theorie der einatomigen Körper." Annalen der Physik 316.8, p. 657-697.</ref>  In 1912 Eduard Grüneisen extended Mie's model to temperatures below the [[Debye model|Debye temperature]] at which quantum effects become important.<ref name=grun>Grüneisen, E. (1912). Theorie des festen Zustandes einatomiger Elemente. Annalen der Physik, 344(12), 257-306.</ref>  Grüneisen's form of the equations is more convenient and has become the usual starting point for deriving Mie–Grüneisen equations of state.<ref name=lemons>Lemons, D. S., & Lund, C. M. (1999). Thermodynamics of high temperature, Mie–Gruneisen solids. American Journal of Physics, 67, 1105.</ref>

== Expressions for the Mie–Grüneisen equation of state ==
A temperature-corrected version that is used in computational mechanics has the form<ref>{{Citation
 | title = An evaluation of several hardening models using Taylor cylinder impact data
 | url = http://www.osti.gov/energycitations/product.biblio.jsp?osti_id=764004
 | year = 2000
 | journal = Conference: COMPUTATIONAL METHODS IN APPLIED SCIENCES AND ENGINEERING, BARCELONA (ES), 09/11/2000--09/14/2000
 | last1 = Zocher	 | first1 =  M.A.
 | last2 =  Maudlin	 | first2 =  P.J.
 | accessdate = 2009-05-12	}}</ref>
(see also,<ref name=Wilkins1999>{{Citation
 | title =  Computer simulation of dynamic phenomena
 | url = https://books.google.com/books?hl=en
 | year = 1999
 | author = Wilkins, M.L.
 | accessdate = 2009-05-12
}}</ref> p.&nbsp;61)

:<math> 
   p = \frac{\rho_0 C_0^2 \chi
              \left[1 - \frac{\Gamma_0}{2}\,\chi\right]}
             {\left(1 - s\chi\right)^2} + \Gamma_0 E;\quad
   \chi := 1-\cfrac{\rho_0}{\rho}
  </math>
where <math>C_0</math> is the bulk speed of sound, <math>\rho_0</math> is the initial density, <math>\rho</math> is the current density,  <math>\Gamma_0</math> is Grüneisen's gamma at the reference state,  <math>s = dU_s/dU_p</math> is a linear Hugoniot slope coefficient,  <math>U_s</math> is the shock wave velocity, <math>U_p</math> is the particle velocity, and  <math>E</math> is the internal energy per unit reference volume.  An alternative form is
:<math> 
   p = \frac{\rho_0 C_0^2 (\eta -1)
              \left[\eta - \frac{\Gamma_0}{2}(\eta-1)\right]}
             {\left[\eta - s(\eta-1)\right]^2} + \Gamma_0 E;\quad
   \eta := \cfrac{\rho}{\rho_0} \,.
  </math>
A rough estimate of the internal energy can be computed using
:<math>
    E = \frac{1}{V_0} \int C_v dT \approx \frac{C_v (T-T_0)}{V_0} = \rho_0 c_v (T-T_0)
  </math>
where <math>V_0</math> is the reference volume at temperature  <math>T = T_0</math>, <math>C_v</math> is the [[heat capacity]] and <math>c_v</math> is the specific heat capacity at constant volume.  In many  simulations, it is assumed that <math>C_p</math> and <math>C_v</math> are equal.

=== Parameters for various materials ===
{| class="wikitable sortable" cellpadding="4" cellspacing="0" border="1" style="border-collapse: collapse"
|- bgcolor="#cccccc"
 !material!!<math>\rho_0</math> (kg/m<sup>3</sup>) !! <math>c_v</math> (J/kg-K) !! <math>C_0</math> (m/s)!! <math>s</math>!! <math>\Gamma_0</math> (<math>T < T_1</math>)!!<math>\Gamma_0</math> (<math>T >= T_1</math>)!! <math>T_1</math> (K)
|-
| [[Copper]]
| 8960
| 390
| 3933 <ref name=Mitchell>{{Citation
 | title = Shock compression of aluminum, copper, and tantalum
 | url = http://link.aip.org/link/?JAPIAU/52/3363/1
 | year = 1981
 | journal = Journal of Applied Physics
 | pages = 3363
 | volume = 52
 | last1 = Mitchell	 | first1 =  A.C.
 | last2 =  Nellis	 | first2 =  W.J.
 | accessdate = 2009-05-12
 | doi = 10.1063/1.329160
 | issue = 5	|bibcode = 1981JAP....52.3363M }}</ref>
| 1.5 <ref name=Mitchell/>          
| 1.99 <ref name=MacDonald>{{Citation
 | title = Thermodynamic properties of fcc metals at high temperatures
 | doi = 10.1103/PhysRevB.24.1715
 | year = 1981
 | journal = Physical Review B
 | pages = 1715–1724
 | volume = 24
 | issue = 4
 | last1 = MacDonald	 | first1 =  R.A.
 | last2 =  MacDonald	 | first2 =  W.M.	|bibcode = 1981PhRvB..24.1715M }}</ref>  
| 2.12 <ref name=MacDonald />
|700
|}

== Derivation of the equation of state ==
From Grüneisen's model we have
:<math>
   (1) \qquad p - p_0 = \frac{\Gamma}{V} (e - e_0)
 </math>
where ''p''<sub>0</sub> and ''e''<sub>0</sub> are the pressure and internal energy at a reference state.  The [[Rankine–Hugoniot conditions|Hugoniot equations]] for the conservation of mass, momentum, and energy  are
:<math>
   \rho_0 U_s = \rho (U_s - U_p) ~~, \quad p_H - p_{H0} = \rho_0 U_s U_p \quad \text{and} \quad
   p_H U_p = \rho_0 U_s \left(\frac{U_p^2}{2} + E_H - E_{H0}\right)
 </math>
where ''ρ''<sub>0</sub> is the reference density, ''ρ'' is the density due to shock compression, ''p''<sub>H</sub> is the pressure on the Hugoniot, ''E''<sub>H</sub> is the internal energy '''per unit mass''' on the Hugoniot, ''U''<sub>s</sub> is the shock velocity, and ''U''<sub>p</sub> is the particle velocity.  From the conservation of mass, we have
:<math>
   \frac{U_p}{U_s} = 1 - \frac{\rho_0}{\rho} = 1 - \frac{V}{V_0} =: \chi \,.
 </math>
Where we defined  <math>V = 1/\rho </math>, the specific volume (volume per unit mass).
:For many materials ''U''<sub>s</sub> and ''U''<sub>p</sub> are linearly related, i.e., ''U''<sub>s</sub> = ''C''<sub>0</sub> + ''s'' ''U''<sub>p</sub> where ''C''<sub>0</sub> and ''s'' depend on the material.  In that case, we have
:<math>
   U_s = C_0 + s\chi U_s \quad \text{or} \quad U_s = \frac{C_0}{1 - s\chi} \,.
 </math>
The momentum equation can then be written (for the principal Hugoniot where ''p''<sub>H0</sub> is zero) as
:<math>
   p_H = \rho_0 \chi U_s^2 = \frac{\rho_0 C_0^2 \chi}{(1 - s\chi)^2} \,.
 </math>
Similarly, from the energy equation we have
:<math>
  p_H \chi U_s = \tfrac{1}{2} \rho \chi^2 U_s^3 + \rho_0 U_s E_H = \tfrac{1}{2} p_H \chi U_s + \rho_0 U_s E_H \,.
 </math>
Solving for ''e''<sub>H</sub>, we have
:<math>
   E_H = \tfrac{1}{2} \frac{p_H \chi}{\rho_0} = \tfrac{1}{2} p_H (V_0 - V)
 </math>
With these expressions for ''p''<sub>H</sub> and ''E''<sub>H</sub>, the Grüneisen model on the Hugoniot becomes
:<math>
   p_H - p_0 = \frac{\Gamma}{V} \left(\frac{p_H \chi V_0}{2} - e_0\right) \quad \text{or} \quad
   \frac{\rho_0 C_0^2 \chi}{(1 - s\chi)^2}\left(1 - \frac{\chi}{2}\,\frac{\Gamma}{V}\,V_0\right) - p_0 = -\frac{\Gamma}{V} e_0 \,.
 </math>
If we assume that ''Γ''/''V'' = ''Γ''<sub>0</sub>/''V''<sub>0</sub> and note that <math>p_0 = -d e_0/d V</math>, we get
:<math>
  (2) \qquad \frac{\rho C_0^2 \chi}{(1 - s\chi)^2}\left(1 - \frac{\Gamma_0\chi}{2}\right) + \frac{d e_0}{d V} + \frac{\Gamma_0}{V_0} e_0 = 0\,.
 </math>
The above ordinary differential equation can be solved for ''e''<sub>0</sub> with the initial condition ''e''<sub>0</sub> = 0 when ''V = V''<sub>0</sub> (χ = 0).  The exact solution is 
:<math>
  \begin{align}
   e_0 =  \frac{\rho C_0^2 V_0}{2 s^4} \Biggl[&\exp(\Gamma_0\chi) (\tfrac{\Gamma_0}{s} - 3 ) s^2 - 
        \frac{ [\tfrac{\Gamma_0}{s} - (3 - s\chi)]s^2}{1 - s\chi} + \\
       & \exp\left[-\tfrac{\Gamma_0}{s} (1-s\chi)\right] (\Gamma_0^2 - 4 \Gamma_0 s + 2 s^2) (\text{Ei}[\tfrac{\Gamma_0}{s} (1-s\chi )] - \text{Ei}[\tfrac{\Gamma_0}{s}]) 
\Biggr]
  \end{align}
 </math>
where ''Ei[z]'' is the [[exponential integral]].  The expression for ''p''<sub>0</sub> is
:<math>
  \begin{align}
  p_0 = -\frac{de_0}{dV} = \frac{\rho C_0^2}{2s^4(1-\chi)} \Biggl[& \frac{s}{(1 - s\chi)^2} \Bigl (- \Gamma_0^2(1 - \chi)(1 -s\chi) 
   + \Gamma_0 [s \{4 (\chi-1) \chi s-2 \chi+3\}-1] \\
    & - \exp(\Gamma_0\chi)[\Gamma_0(\chi-1) -1](1-s\chi)^2(\Gamma_0-3s) + s [3-\chi s \{(\chi-2) s+4\}]\Bigr) \\
    & - \exp\left[-\tfrac{\Gamma_0}{s} (1-s\chi)\right][\Gamma_0(\chi-1) - 1](\Gamma_0^2 - 4 \Gamma_0 s + 2 s^2)(\text{Ei}[\tfrac{\Gamma_0}{s} (1-s\chi )] - \text{Ei}[\tfrac{\Gamma_0}{s}]) \Biggr] \,.
   \end{align}
 </math>
[[File:MieGruneisen0.svg|thumb|300px|right|Plots of ''e''<sub>0</sub> and ''p''<sub>0</sub> 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
:<math>
   e_0(V) = A + B \chi(V) + C \chi^2(V) + D \chi^3(V) + \dots
 </math>
and
:<math>
  p_0(V) = -\frac{de_0}{dV} = -\frac{de_0}{d\chi}\,\frac{d\chi}{dV} = \frac{1}{V_0}\,(B + 2C\chi + 3D\chi^2 + \dots) \,.
 </math>
Substitution into the Grüneisen model gives us the Mie–Grüneisen equation of state
:<math>
   p = \frac{1}{V_0}\,(B + 2C\chi + 3D\chi^2 + \dots) + \frac{\Gamma_0}{V_0} \left[e - (A + B \chi + C \chi^2 + D \chi^3 + \dots ) \right] \,.
 </math>
If we assume that the internal energy ''e''<sub>0</sub> = 0 when ''V = V''<sub>0</sub> (χ = 0) we have ''A'' = 0.  Similarly, if we assume ''p''<sub>0</sub> = 0 when ''V = V''<sub>0</sub> we have ''B'' = 0.  The Mie–Grüneisen equation of state can then be written as
:<math>
   p = \frac{1}{V_0}\left[2C\chi \left(1-\tfrac{\Gamma_0}{2}\chi\right) + 3D\chi^2\left(1 -\tfrac{\Gamma_0}{3}\chi\right) + \dots\right] + \Gamma_0 E
 </math>
where ''E'' is the internal energy per unit reference volume.  Several forms of this equation of state are possible.  
[[File:MieGruneisenP.svg|thumb|400px|right|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
:<math>
   C = \frac{\rho_0 C_0^2 V_0}{2(1-s\chi)^2} \,.
 </math>

Then we get the following expression for ''p'' :
:<math>
   p = \frac{\rho_0 C_0^2 \chi}{(1-s\chi)^2} \left(1-\tfrac{\Gamma_0}{2}\chi\right) + \Gamma_0 E \,.
 </math>
This is the commonly used first-order Mie–Grüneisen equation of state {{Citation needed|date=July 2019|reason=If it is commonly used there should be a citation.}}.

==See also==
* [[Impact (mechanics)]]
* [[Shock wave]]
* [[Shock (mechanics)]]
* [[Shock tube]]
* [[Hydrostatic shock]]
* [[ALEGRA]]
* [[Viscoplasticity]]

== References ==
<references />

{{DEFAULTSORT:Mie-Gruneisen equation of state}}
[[Category:Continuum mechanics]]
[[Category:Solid mechanics]]
[[Category:Equations of state]]