Mie-Grüneisen's equation of state

The Mie Grüneisen equation of state (engl. Also Mie-Gruneisen equation of state ), named after Gustav Mie and Eduard Grüneisen is a state equation of physics that a special for high density matter functional relationship between the density , the pressure and the absolute temperature represents. She will u. a. to calculate the speed of sound and shock waves at high ambient pressures, as well as for modeling of seismic studies of the Earth's interior uses. ${\displaystyle \rho }$ ${\displaystyle p}$ ${\displaystyle T}$

The special assumption made by Mie-Grüneisen relates to the temperature dependence, which may only occur in the form of a "scaled temperature" : ${\displaystyle t}$

${\displaystyle t(T,\rho )={\frac {T}{TD(\rho )}},}$

where the density- or volume- dependent "temperature parameter " generally represents the frequency spectrum of the lattice vibrations and usually contains several material parameters . ${\displaystyle TD(\rho )}$

Special form of the equation

A special form of the Mie-Grüneisen equation of state represents the measurement results of high pressure experiments on the basis of three material parameters in the temperature-independent part:

${\displaystyle p=p_{0}\cdot \left(1-\Gamma \cdot \eta \right)+{\frac {\rho _{0}\cdot C_{0}^{2}\cdot \eta }{\left(1-s\cdot \eta \right)^{2}}}\cdot \left(1-{\frac {\Gamma \cdot \eta }{2}}\right)+\Gamma \cdot \rho _{0}\cdot \left(e-e_{0}\right)}$

With

${\displaystyle \eta =1-{\frac {\rho _{0}}{\rho }}}$.

Here referred to

• ${\displaystyle \rho _{0}}$the density in the normal state
• ${\displaystyle C_{0}}$the speed of sound in the normal state
• ${\displaystyle \Gamma =\Gamma _{0}}$the dimensionless Green Iron coefficient in the normal state
• ${\displaystyle s}$the linear Hugoniot -Steigungskoeffizient (engl. linear Hugoniot slope coefficient ), a dimensionless material constant
• ${\displaystyle e-e_{0}}$the specific internal energy , which in the Mie-Grüneisen case may only depend on the scaled temperature (see above). ${\displaystyle t}$

Examples of parameters of the Mie-Grüneisen equation of state

Water : kg / m ³  ; m / s;  ; ${\displaystyle \rho _{0}=1000}$ ${\displaystyle C_{0}=1489}$ ${\displaystyle s=1{,}79}$ ${\displaystyle \Gamma =1{,}65}$

Steel : kg / m ³  ; m / s;  ; ${\displaystyle \rho _{0}=7850}$ ${\displaystyle C_{0}=4500}$ ${\displaystyle s=1{,}49}$ ${\displaystyle \Gamma =2{,}17}$

Copper : kg / m ³  ; m / s;  ; ${\displaystyle \rho _{0}=8930}$ ${\displaystyle C_{0}=3940}$ ${\displaystyle s=1{,}48}$ ${\displaystyle \Gamma =1{,}96}$

Relationship of the parameters with other thermodynamic state variables

The speed of sound with which small pressure and density fluctuations propagate in a medium is given by: with a reversible adiabatic change of state (i.e. with constant entropy ): ${\displaystyle S}$

${\displaystyle c_{S}={\sqrt {\left.{\frac {\partial p}{\partial \rho }}\right|_{S}}}={\sqrt {{\frac {p}{\rho }}\cdot \gamma }}}$

The speed of sound is a state variable.

The adiabatic exponent results from: ${\displaystyle \gamma }$

${\displaystyle \gamma =-{\frac {V}{p}}\cdot \left.{\frac {\partial p}{\partial V}}\right|_{S}}$

The Green Iron coefficient is defined by:

${\displaystyle \Gamma =-{\frac {V}{T}}\cdot \left.{\frac {\partial T}{\partial V}}\right|_{S}={\frac {\beta }{\kappa \cdot \rho \cdot c_{V}}}}$

where the Maxwell relation and the following terms were used: ${\displaystyle \left.{\frac {\partial S}{\partial V}}\right|_{T}=\left.{\frac {\partial p}{\partial T}}\right|_{V}}$

Thermal expansion :

${\displaystyle \beta ={\frac {1}{V}}\cdot \left.{\frac {\partial V}{\partial T}}\right|_{p}=-{\frac {1}{\rho }}\cdot \left.{\frac {\partial \rho }{\partial T}}\right|_{p}}$

Isothermal compressibility :

${\displaystyle \kappa =-{\frac {1}{V}}\cdot \left.{\frac {\partial V}{\partial p}}\right|_{T}}$

Isochore specific heat capacity :

${\displaystyle c_{V}={\frac {T}{\rho \cdot V}}\cdot \left.{\frac {\partial S}{\partial T}}\right|_{V}}$

literature

• Debye, P .: On the theory of specific heats. In: Annalen der Physik 39, 789–839 (1912)
• Grüneisen, E .: Theory of the solid state of monatomic elements. In: Annalen der Physik 39, 257–306 (1912)
• Mie, G .: Basics of a theory of matter. In: Annalen der Physik 2, 1-40 (1912)
• G. McQueen, SP Marsh, JWTaylor, JNFritz, WJCarter: "High Velocity Impact Phenomena", (1970), p. 230
• MAZocher et al .: An evaluation of several hardening models using Taylor cylinder impact data. Proc. European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS, Barcelona, ​​Spain
• WBHolzapfel: Equations of state for solids under strong compression. In: Journal of Crystallography. 216 (2000) pp. 473-488