Mie–Gruneisen equation of state

History

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.

Expressions for the Mie-Grüneisen equation of state

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

p = ρ 0 C 0 2 χ [ 1 − Γ 0 2 χ ] ( 1 − s χ ) 2 + Γ 0 E ; χ := 1 − ρ 0 ρ

where C 0 is the bulk speed of sound, ρ 0 is the initial density, ρ is the current density, Γ 0 is Grüneisen's gamma at the reference state, s = d U s / d U p is a linear Hugoniot slope coefficient, U s is the shock wave velocity, U p is the particle velocity, and E is the internal energy per unit reference volume. An alternative form is

p = ρ 0 C 0 2 ( η − 1 ) [ η − Γ 0 2 ( η − 1 ) ] [ η − s ( η − 1 ) ] 2 + Γ 0 E ; η := ρ ρ 0 .

A rough estimate of the internal energy can be computed using

E = 1 V 0 ∫ C v d T ≈ C v ( T − T 0 ) V 0 = ρ 0 c v ( T − T 0 )

where V 0 is the reference volume at temperature T = T 0 , C v is the heat capacity and c v is the specific heat capacity at constant volume. In many simulations, it is assumed that C p and C v are equal.

Derivation of the equation of state

From Grüneisen's model we have

( 1 ) p − p 0 = Γ V ( e − e 0 )

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

ρ 0 U s = ρ ( U s − U p )     , p H − p H 0 = ρ 0 U s U p and p H U p = ρ 0 U s ( U p 2 2 + E H − E H 0 )

where ρ₀ is the reference density, ρ is the density due to shock compression, p_H is the pressure on the Hugoniot, E_H is the internal energy per unit mass on the Hugoniot, U_s is the shock velocity, and U_p is the particle velocity. From the conservation of mass, we have

U p U s = 1 − ρ 0 ρ = 1 − V V 0 =: χ .

Where we defined V = 1 / ρ , the specific volume (volume per unit mass).

For many materials U_s and U_p are linearly related, i.e., U_s = C₀ + s U_p where C₀ and s depend on the material. In that case, we have U s = C 0 + s χ U s or U s = C 0 1 − s χ .

The momentum equation can then be written (for the principal Hugoniot where p_H0 is zero) as

p H = ρ 0 χ U s 2 = ρ C 0 2 χ ( 1 − s χ ) 2 .

Similarly, from the energy equation we have

p H χ U s = 1 2 ρ χ 2 U s 3 + ρ 0 U s E H = 1 2 p H χ U s + ρ 0 U s E H .

Solving for e_H, we have

E H = 1 2 p H χ ρ 0 = 1 2 p H ( V 0 − V )

With these expressions for p_H and E_H, the Grüneisen model on the Hugoniot becomes

p H − p 0 = Γ V ( p H χ V 0 2 − e 0 ) or ρ C 0 2 χ ( 1 − s χ ) 2 ( 1 − χ 2 Γ V V 0 ) − p 0 = − Γ V e 0 .

If we assume that Γ/V = Γ₀/V₀ and note that p 0 = − d e 0 / d V , we get

( 2 ) ρ C 0 2 χ ( 1 − s χ ) 2 ( 1 − Γ 0 χ 2 ) + d e 0 d V + Γ 0 V 0 e 0 = 0 .

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

e 0 = ρ C 0 2 V 0 2 s 4 [ exp ⁡ ( Γ 0 χ ) ( Γ 0 s − 3 ) s 2 − [ Γ 0 s − ( 3 − s χ ) ] s 2 1 − s χ + exp ⁡ [ − Γ 0 s ( 1 − s χ ) ] ( Γ 0 2 − 4 Γ 0 s + 2 s 2 ) ( Ei [ Γ 0 s ( 1 − s χ ) ] − Ei [ Γ 0 s ] ) ]

where Ei[z] is the exponential integral. The expression for p₀ is

p 0 = − d e 0 d V = ρ C 0 2 2 s 4 ( 1 − χ ) [ s ( 1 − s χ ) 2 ( − Γ 0 2 ( 1 − χ ) ( 1 − s χ ) + Γ 0 [ s { 4 ( χ − 1 ) χ s − 2 χ + 3 } − 1 ] − exp ⁡ ( Γ 0 χ ) [ Γ 0 ( χ − 1 ) − 1 ] ( 1 − s χ ) 2 ( Γ 0 − 3 s ) + s [ 3 − χ s { ( χ − 2 ) s + 4 } ] ) − exp ⁡ [ − Γ 0 s ( 1 − s χ ) ] [ Γ 0 ( χ − 1 ) − 1 ] ( Γ 0 2 − 4 Γ 0 s + 2 s 2 ) ( Ei [ Γ 0 s ( 1 − s χ ) ] − Ei [ Γ 0 s ] ) ] .

For commonly encountered compression problems, an approximation to the exact solution is a power series solution of the form

e 0 ( V ) = A + B χ ( V ) + C χ 2 ( V ) + D χ 3 ( V ) + …

and

p 0 ( V ) = − d e 0 d V = − d e 0 d χ d χ d V = 1 V 0 ( B + 2 C χ + 3 D χ 2 + … ) .

Substitution into the Grüneisen model gives us the Mie-Grüneisen equation of state

p = 1 V 0 ( B + 2 C χ + 3 D χ 2 + … ) + Γ 0 V 0 [ e − ( A + B χ + C χ 2 + D χ 3 + … ) ] .

If we assume that the internal energy e₀ = 0 when V = V₀ (χ = 0) 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

p = 1 V 0 [ 2 C χ ( 1 − Γ 0 2 χ ) + 3 D χ 2 ( 1 − Γ 0 3 χ ) + … ] + Γ 0 E

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 (2), we can solve for C to get

C = ρ C 0 2 V 0 2 ( 1 − s χ ) 2 .

Then we get the following expression for p :

p = ρ C 0 2 χ ( 1 − s χ ) 2 ( 1 − Γ 0 2 χ ) + Γ 0 E .

This is the commonly used first-order Mie-Grüneisen equation of state.