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| 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–Gruneisen equation of state are in use.
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| The Grüneisen model can be expressed in the form
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| :<math>\Gamma = V \left(\frac{dp}{de}\right)_V</math>
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| 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
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| :<math>
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| p - p_0 = \frac{\Gamma}{V} (e - e_0)
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| </math>
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| 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.
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| | |
| == History ==
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| 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>
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| | |
| == Expressions for the Mie-Grüneisen equation of state ==
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| 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
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| | journal = Conference: COMPUTATIONAL METHODS IN APPLIED SCIENCES AND ENGINEERING, BARCELONA (ES), 09/11/2000--09/14/2000
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| | last1 = Zocher | first1 = M.A.
| |
| | last2 = Maudlin | first2 = P.J.
| |
| | accessdate = 2009-05-12 }}</ref>
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| (see also,<ref name=Wilkins1999>{{Citation
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| | title = Computer simulation of dynamic phenomena
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| | url = http://books.google.co.uk/books?hl=en
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| | year = 1999
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| | author = Wilkins, M.L.
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| | accessdate = 2009-05-12
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| }}</ref> p. 61)
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| | |
| :<math>
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| p = \frac{\rho_0 C_0^2 \chi
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| \left[1 - \frac{\Gamma_0}{2}\,\chi\right]}
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| {\left(1 - s\chi\right)^2} + \Gamma_0 E;\quad
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| \chi := 1-\cfrac{\rho_0}{\rho}
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| </math>
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| 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 Gruneisen'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
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| :<math>
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| p = \frac{\rho_0 C_0^2 (\eta -1)
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| \left[\eta - \frac{\Gamma_0}{2}(\eta-1)\right]}
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| {\left[\eta - s(\eta-1)\right]^2} + \Gamma_0 E;\quad
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| \eta := \cfrac{\rho}{\rho_0} \,.
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| </math>
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| A rough estimate of the internal energy can be computed using
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| :<math>
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| 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)
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| </math>
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| 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.
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| === Parameters for various materials ===
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| {| class="wikitable sortable" cellpadding="4" cellspacing="0" border="1" style="border-collapse: collapse"
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| |- bgcolor="#cccccc"
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| !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>)!! T_1 (K)
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| |-
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| | [[Copper]]
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| | 8960
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| | 390
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| | 3933 <ref name=Mitchell>{{Citation
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| | title = Shock compression of aluminum, copper, and tantalum
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| | url = http://link.aip.org/link/?JAPIAU/52/3363/1
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| | year = 1981
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| | journal = Journal of Applied Physics
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| | pages = 3363
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| | volume = 52
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| | last1 = Mitchell | first1 = A.C.
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| | last2 = Nellis | first2 = W.J.
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| | accessdate = 2009-05-12
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| | doi = 10.1063/1.329160
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| | issue = 5 |bibcode = 1981JAP....52.3363M }}</ref>
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| | 1.5 <ref name=Mitchell/>
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| | 1.99 <ref name=MacDonald>{{Citation
| |
| | title = Thermodynamic properties of fcc metals at high temperatures
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| | doi = 10.1103/PhysRevB.24.1715
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| | year = 1981
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| | journal = Physical Review B
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| | pages = 1715–1724
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| | volume = 24
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| | issue = 4
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| | last1 = MacDonald | first1 = R.A.
| |
| | last2 = MacDonald | first2 = W.M. |bibcode = 1981PhRvB..24.1715M }}</ref>
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| | 2.12 <ref name=MacDonald />
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| |700
| |
| |}
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| | |
| == Gruneisen constant for perfect crystals with pair interactions ==
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| | |
| The expression for Gruneisen constant of a perfect crystal with pair interactions in <math>d</math>-dimmensional space has the form:<ref name=Krivtsov_Kuzkin>{{Citation | |
| | title = Derivation of Equations of State for Ideal Crystals of Simple Structure
| |
| | doi = 10.3103/S002565441103006X
| |
| | year = 2011
| |
| | journal = Mechanics of Solids
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| | pages = 387–399
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| | volume = 46
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| | issue = 3
| |
| | last1 = Krivtsov | first1 = A.M.
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| | last2 = Kuzkin | first2 = V.A. }}</ref>
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| | |
| <math>
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| \Gamma_0 = -\frac{1}{2d}\frac{\Pi'''(a)a^2 + (d-1)\left[\Pi''(a)a - \Pi'(a)\right]}{\Pi''(a)a + (d-1)\Pi'(a)},
| |
| </math>
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| | |
| where <math>\Pi</math> is the interatomic potential, <math>a</math> is the equilibrium distance, <math>d</math> is the space dimensionality. Relations between the Gruneisen constant and parameters of Lennard-Jones, Morse, and Mie potentials are presented in the table below.
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|
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| {|class="wikitable"
| |
| |-
| |
| !Lattice
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| !Dimensionality
| |
| !Lennard-Jones potential
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| !Mie Potential
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| !Morse potential
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| |-
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| | Chain
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| ! <math> d=1 </math>
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| ! <math>10\frac{1}{2} </math>
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| ! <math>\frac{m+n+3}{2}</math>
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| ! <math>\frac{3\alpha a}{2}</math>
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| |-
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| | Triangual lattice
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| !<math>d=2 </math>
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| ! <math>5</math>
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| ! <math> \frac{m+n+2}{4}</math>
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| ! <math> \frac{3\alpha a - 1}{4}</math>
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| |-
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| | FCC, BCC
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| ! <math>d=3 </math>
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| ! <math>\frac{19}{6} </math>
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| ! <math>\frac{n+m+1}{6}</math>
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| ! <math>\frac{3\alpha a-2}{6}</math>
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| |-
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| | "Hyperlattice"
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| ! <math>d=\infty</math>
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| ! <math>-\frac{1}{2}</math>
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| ! <math>-\frac{1}{2}</math>
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| ! <math>-\frac{1}{2}</math>
| |
| |-
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| | General formula
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| ! <math>d</math>
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| ! <math>\frac{11}{d}-\frac{1}{2}</math>
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| ! <math>\frac{m+n+4}{2d}-\frac{1}{2}</math>
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| ! <math>\frac{3\alpha a + 1}{2d}-\frac{1}{2}</math>
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| |-
| |
| |}
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| | |
| The expression for Gruneisen constant of 1D chain with Mie potential exactly coincides with the results of McDonald and Roy<ref name=McDonald_Roy>{{Citation
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| | title = Vibrational Anharmonicity and Lattice Thermal Properties. II
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| | doi = 10.1103/PhysRev.97.673
| |
| | year = 1955
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| | journal = Phys. Rev.
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| | pages = 673–676
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| | volume = 97
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| | last1 = MacDonald | first1 = D. K. C.
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| | last2 = Roy | first2 = S.K. }}</ref>
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| | |
| == Derivation of the equation of state ==
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| From Grüneisen's model we have
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| :<math>
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| (1) \qquad p - p_0 = \frac{\Gamma}{V} (e - e_0)
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| </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
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| :<math>
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| \rho_0 U_s = \rho (U_s - U_p) ~~, \quad p_H - p_{H0} = \rho_0 U_s U_p \quad \text{and} \quad
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| p_H U_p = \rho_0 U_s \left(\frac{U_p^2}{2} + E_H - E_{H0}\right)
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| </math>
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| 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
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| :<math>
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| \frac{U_p}{U_s} = 1 - \frac{\rho_0}{\rho} = 1 - \frac{V}{V_0} =: \chi \,.
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| </math>
| |
| 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
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| :<math>
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| p_H = \rho_0 \chi U_s^2 = \frac{\rho C_0^2 \chi}{(1 - s\chi)^2} \,.
| |
| </math>
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| Similarly, from the energy equation we have
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| :<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
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| :<math>
| |
| E_H = \tfrac{1}{2} \frac{p_H \chi}{\rho_0} = \tfrac{1}{2} \frac{p_H \chi V_0}{\rho V} \quad \text{or} \quad
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| e_H = \tfrac{1}{2} p_H \chi V_0
| |
| </math>
| |
| where ''e''<sub>H</sub> is the total internal energy. With these expressions for ''p''<sub>H</sub> and ''e''<sub>H</sub>, the Grüneisen model on the Hugoniot becomes
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| :<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 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
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| :<math>
| |
| \begin{align}
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| 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} + \\
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| & \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}])
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| \Biggr]
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| \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] \\
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| & - \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>
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| [[File:MieGruneisen0.svg|thumb|300px|right|Plots of ''e''<sub>0</sub> and ''p''<sub>0</sub> for copper as a function of χ.]]
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| 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 C_0^2 V_0}{2(1-s\chi)^2} \,.
| |
| </math>
| |
| | |
| Then we get the following expression for ''p'' :
| |
| :<math>
| |
| p = \frac{\rho 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.
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| | |
| ==See also==
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| * [[Impact (mechanics)]]
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| * [[Shock wave]]
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| * [[Shock (mechanics)]]
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| * [[Shock tube]]
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| * [[Hydrostatic shock]]
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| * [[ALEGRA]]
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| * [[Viscoplasticity]]
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| | |
| == References ==
| |
| <references />
| |
| | |
| {{DEFAULTSORT:Mie-Gruneisen equation of state}}
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| [[Category:Continuum mechanics]]
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| [[Category:Solid mechanics]]
| |