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. 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

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

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.

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

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

where ${\displaystyle C_{0}}$ is the bulk speed of sound,${\displaystyle \rho _{0}}$ is the initial density, ${\displaystyle \rho }$ is the current density, ${\displaystyle \Gamma _{0}}$ is Grüneisen's gamma at the reference state, ${\displaystyle s=dU_{s}/dU_{p}}$ is a linear Hugoniot slope coefficient, ${\displaystyle U_{s}}$ is the shock wave velocity, ${\displaystyle U_{p}}$ is the particle velocity, and ${\displaystyle E}$ is the internal energy per unit reference volume. An alternative form is

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