Mie–Grüneisen equation of state

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.

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

In that case p0 and e0 are independent of temperature and the values of these quantities can be estimated from the Hugoniot equations.

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] A temperature-corrected version that is used in computational mechanics has the form[6][7]: 61 where

is a linear Hugoniot slope coefficient,

is the internal energy per unit reference volume.

An alternative form is A rough estimate of the internal energy can be computed using where

is the specific heat capacity at constant volume.

are the pressure and internal energy at a reference state.

The Hugoniot equations for the conservation of mass, momentum, and energy are 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.

The momentum equation can then be written (for the principal Hugoniot where pH0 is zero) as Similarly, from the energy equation we have Solving for eH, we have With these expressions for pH and EH, the Grüneisen model on the Hugoniot becomes If we assume that Γ/V = Γ0/V0 and note that

, we get 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 where Ei[z] is the exponential integral.

The expression for p0 is For commonly encountered compression problems, an approximation to the exact solution is a power series solution of the form and Substitution into the Grüneisen model gives us the Mie–Grüneisen equation of state If we assume that the internal energy e0 = 0 when V = V0 (χ = 0) we have A = 0.

The Mie–Grüneisen equation of state can then be written as where E is the internal energy per unit reference volume.

If we take the first-order term and substitute it into equation (2), we can solve for C to get Then we get the following expression for p: This is the commonly used first-order Mie–Grüneisen equation of state.