The Rankine-Hugoniot equations for the shock jump conditions can be regarded as defining a relation between any pair of the variables ρ(density), P (pressure), e (energy), u_p (particle velocity) and U (shock velocity).

In many dynamic experiments making measurements of u_p and U it has been found that for most solids and many liquids over a wide range of pressure there is an empirical linear relationship between these two variables:

It is then convenient to establish a Mie-Gruneisen form of the equation of state based on the shock Hugoniot:

where it is assumed that Γ ρ = Γ₀ ρ₀ = constant and

Note that for s>1 this formulation gives a limiting value of the compression as the pressure tends to infinity. The denominator of the first equation above becomes zero and the pressure therefore becomes infinite for

1– (s-1)µ= 0

giving a maximum density of ρ = s ρ₀ (s-1). However, long before this regime is approached, the assumption of constant Γ ρ is probably not valid. Furthermore, the assumption of linear variation between the shock velocity U and the particle velocity u_p does not hold for too large a compression.

Γ is known as the Gruneisen coefficient and is often approximated to Γ ~2s-1 in the literature.

The Shock EOS linear model lets you optionally include a quadratic shock velocity, particle velocity relation of the form:

The input parameter, S₂, can be set to a non-zero value to better fit highly non-linear U_s - u_p material data.

Data for this equation of state can be found in various references and many of the materials in the explicit material library.

Custom results variables available for this model: