### Shock EOS Linear

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), up (particle velocity) and U (shock velocity).

In many dynamic experiments making measurements of up 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 up 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 Us - up material data.

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

Note:  This equation of state can only be applied to solid bodies.

The Poisson's ratio is assumed to be zero when calculating effective strain.

A specific heat capacity should be defined with this property to allow the calculation of temperature.

Table 126:  Input Data

NameSymbolUnitsNotes
Gruneisen coefficientΓNone
Parameter C1C1Velocity
Parameter S1S1None