The Mie–Grüneisen equation of state is an equation of state that relates the pressure and 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.
History
, 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.
Expressions for the Mie–Grüneisen equation of state
A temperature-corrected version that is used in computational mechanics has the form where is the bulk speed of sound, is the initial density, is the current density, is Grüneisen's gamma at the reference state, is a linear Hugoniot slope coefficient, is the shockwave velocity, is the particle velocity, and 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 reference volume at temperature , is the heat capacity and is the specific heat capacity at constant volume. In many simulations, it is assumed that and are equal.
Parameters for various materials
Derivation of the equation of state
From Grüneisen's model we have where p0 and e0 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. From the conservation of mass, we have Where we defined , the specific volume. The momentum equation can then be written 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 conditione0 = 0 when V = V0. The exact solution is where Ei 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 we have A = 0. Similarly, if we assume p0 = 0 when V = V0 we have B = 0. The Mie–Grüneisen equation of state can then be written as where E is the internal energy per unit reference volume. Several forms of this equation of state are possible. If we take the first-order term and substitute it into equation, 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.
