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 = Γ_{0} ρ_{0} = 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 ρ_{0} (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, S_{2}, 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.

**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**

Name | Symbol | Units | Notes |
---|---|---|---|

Gruneisen coefficient | Γ | None | |

Parameter C1 | C_{1} | Velocity | |

Parameter S1 | S_{1} | None | |

Parameter Quadratic S2 | S_{2} | 1/Velocity |

Custom results variables available for this model:

Name | Description | Solids | Shells | Beams |
---|---|---|---|---|

PRESSURE | Pressure | Yes | No | No |

DENSITY | Density | Yes | No | No |

COMPRESSION | Compression | Yes | No | No |

VISC_PRESSURE | Viscous Pressure | Yes | No | No |

INT_ENERGY | Internal Energy | Yes | No | No |

TEMPERATURE | Temperature | Yes | No | No |