# /MAT/LAW51 (MULTIMAT)

Block Format Keyword Up to four material laws can be defined: elasto-plastic solid, liquid, gas and detonation products. The material law is based on a diffusive interface technique to get sharper interfaces between submaterial zone (/ALE/MUSCL in Radioss Starter Input).

It is not recommended to use this law with Radioss single precision engine.

## Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
/MAT/LAW51/mat_ID/unit_ID
mat_title
Blank
Iform

## Formulation Types

Table 1. Material Law
Formulation Number of Sub-materials Plasticity Explosive
Iform = 0 3 -- --
Iform = 1 3 Johnson-Cook --
Iform = 10 4 Johnson-Cook Jones-Wilkins-Lee
Iform = 11 4 Johnson-Cook

Drucker-Prager

Jones-Wilkins-Lee
Iform = 12 5 Johnson-Cook

Drucker-Prager

Jones-Wilkins-Lee
General formulation (Iform=12) enables to define any formulation of type: 0, 1, 10, or 11 but with a simplified input.
Table 2. Elementary Boundary Conditions
Formulation Type
Iform = 2 INLET
Iform = 4 GAS INLET (state defined from stagnation point)
Iform = 5 LIQUID INLET (state defined from stagnation point)
Iform = 6 OUTLET (non-reflective)

Outlet formulation (Iform=3) is obsolete since 2018.0 version. It is replaced by new Non-Reflecting-Frontier (Iform = 6)

## Modeling Technique with Polynomial EOS

Material Hypothesis Output Modeling
C0 C1 C2 C3 C4 C5 E0 Pext Pmin
Perfect gas (Example 43) $P\left(\mu ,E\right)$         $\left(\gamma -1\right)$ $\left(\gamma -1\right)$ $\frac{{P}_{0}}{\gamma -1}$
$\mathrm{\text{Δ}}P\left(\mu ,E\right)$ -P0       $\left(\gamma -1\right)$ $\left(\gamma -1\right)$ $\frac{{P}_{0}}{\gamma -1}$ P0
Water (Linear EOS) $P\left(\mu ,E\right)$ P0 $\rho {c}^{2}$             ${10}^{-30}$
$\mathrm{\text{Δ}}P\left(\mu ,E\right)$   $\rho {c}^{2}$           P0 -P0
Elastic Solid (Linear EOS) $P\left(\mu ,E\right)$ P0 $\frac{E}{3\left(1-2\nu \right)}$
$\mathrm{\text{Δ}}P\left(\mu ,E\right)$   $\frac{E}{3\left(1-2\nu \right)}$           P0
Mie-Gruneisen

$\text{Γ}$ constant

$\mathrm{\text{Δ}}P\left(\mu ,E\right)$   K1 ${K}_{2}-\frac{\text{Γ}}{2}{K}_{1}$ ${K}_{3}-\frac{\text{Γ}}{2}{K}_{2}$ $\text{Γ}$ $\text{Γ}$ E0 P0
Mie-Gruneisen

$\text{Γ}$ linear

$\Gamma ={\Gamma }_{0}-a\left(\frac{\mu }{1+\mu }\right)$

$\mathrm{\text{Δ}}P\left(\mu ,E\right)$   K1 ${K}_{2}-\frac{{\text{Γ}}_{0}}{2}{K}_{1}$ ${K}_{3}-\frac{{\text{Γ}}_{0}}{2}{K}_{2}+a{K}_{1}$ ${\text{Γ}}_{0}$ ${\text{Γ}}_{0}-a$ E0 P0
Where,(1)
${K}_{1}={\rho }_{0}{c}^{2}$
(2)
${K}_{2}={\rho }_{0}{c}^{2}\left(2S-1\right)$
(3)
${K}_{3}={\rho }_{0}{c}^{2}\left(S-1\right)\left(3S-1\right)$
Where,(4)
$\mu =\frac{\rho }{{\rho }_{0}}-1$
$P\left(\mu ,E\right)$
Total pressure and total energy formulation
$\mathrm{\text{Δ}}P\left(\mu ,E\right)$
Relative pressure and total energy formulation
$P\left(\mu ,\text{Δ}E\right)$
Total pressure and relative energy formulation
$\mathrm{\text{Δ}}P\left(\mu ,\mathrm{\text{Δ}}E\right)$
Relative pressure and relative energy formulation
P0
Initial total pressure
E0
Initial total energy
$\gamma$
Perfect gas constant
E
Young's modulus
$\nu$
Poisson coefficient
$\text{Γ}$
Gruneisen's gamma
$a$
Coefficient for first order volume correction to the Gruneisen gamma ${\text{Γ}}_{0}$
c
Speed of sound
${\rho }_{0}$
Initial density
S
Linear Hugoniot slope coefficient

4. /MAT/LAW51 (MULTIMAT) is based on the equilibrium between each material present inside the element. Radioss computes and outputs a relative pressure $\text{Δ}\mathrm{P}$ . At each cycle: $\text{Δ}P=\text{Δ}{P}_{1}=\text{Δ}{P}_{2}=\text{Δ}{P}_{3}=\text{Δ}{P}_{4}$
User can deduce total pressure using output value $\text{Δ}\mathrm{P}$ and input parameter ${P}_{ext}$ : (5)
$P=\text{Δ}P+{P}_{ext}$