# /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 |
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mat_title |
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Blank | |||||||||

I_{form} |

## Formulation Types

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 |

`I`

_{form}=12) enables to define any formulation of type: 0, 1, 10, or 11 but with a simplified input.

Outlet formulation (`I`_{form}=3) is obsolete since 2018.0 version.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 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

C_{0} |
C_{1} |
C_{2} |
C_{3} |
C_{4} |
C_{5} |
E_{0} |
P_{ext} |
P_{min} |
||

Perfect gas (Example 43) | $P\left(\mu ,E\right)$ | $(\gamma -1)$ | $(\gamma -1)$ | $\frac{{P}_{0}}{\gamma -1}$ | ||||||

$\mathrm{\text{\Delta}}P\left(\mu ,E\right)$ | -P_{0} |
$(\gamma -1)$ | $(\gamma -1)$ | $\frac{{P}_{0}}{\gamma -1}$ | P_{0} |
|||||

Water (Linear EOS) | $P\left(\mu ,E\right)$ | P_{0} |
$\rho {c}^{2}$ | ${10}^{-30}$ | ||||||

$\mathrm{\text{\Delta}}P\left(\mu ,E\right)$ | $\rho {c}^{2}$ | P_{0} |
-P_{0} |
|||||||

Elastic Solid (Linear EOS) | $P\left(\mu ,E\right)$ | P_{0} |
$\frac{E}{3\left(1-2\nu \right)}$ | |||||||

$\mathrm{\text{\Delta}}P\left(\mu ,E\right)$ | $\frac{E}{3\left(1-2\nu \right)}$ | P_{0} |
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Mie-Gruneisen $\text{\Gamma}$ constant |
$\mathrm{\text{\Delta}}P\left(\mu ,E\right)$ | K_{1} |
${K}_{2}-\frac{\text{\Gamma}}{2}{K}_{1}$ | ${K}_{3}-\frac{\text{\Gamma}}{2}{K}_{2}$ | $\text{\Gamma}$ | $\text{\Gamma}$ | E_{0} |
P_{0} |
||

Mie-Gruneisen $\text{\Gamma}$ linear $\Gamma ={\Gamma}_{0}-a\left(\frac{\mu}{1+\mu}\right)$ |
$\mathrm{\text{\Delta}}P\left(\mu ,E\right)$ | K_{1} |
${K}_{2}-\frac{{\text{\Gamma}}_{0}}{2}{K}_{1}$ | ${K}_{3}-\frac{{\text{\Gamma}}_{0}}{2}{K}_{2}+a{K}_{1}$ | ${\text{\Gamma}}_{0}$ | ${\text{\Gamma}}_{0}-a$ | E_{0} |
P_{0} |

- $P\left(\mu ,E\right)$
- Total pressure and total energy formulation
- $\mathrm{\text{\Delta}}P\left(\mu ,E\right)$
- Relative pressure and total energy formulation
- $P\left(\mu ,\text{\Delta}E\right)$
- Total pressure and relative energy formulation
- $\mathrm{\text{\Delta}}P\left(\mu ,\mathrm{\text{\Delta}}E\right)$
- Relative pressure and relative energy formulation
`P`_{0}- Initial total pressure
`E`_{0}- Initial total energy
- $\gamma $
- Perfect gas constant
`E`- Young's modulus
- $\nu $
- Poisson coefficient
- $\text{\Gamma}$
- Gruneisen's gamma
- $a$
- Coefficient for first order volume correction to the Gruneisen gamma ${\text{\Gamma}}_{0}$
`c`- Speed of sound
- ${\rho}_{0}$
- Initial density
`S`- Linear Hugoniot slope coefficient

## Comments

- Numerical diffusion can be improved using the second order method for volume fraction convection, /ALE/MUSCL. The previous /UPWIND used to limit diffusion is now obsolete.
- Time step for ALE material laws can be tune with Engine card /DT/ALE; by default, scale factor on time step is 0.5
- This law can emulate /MAT/LAW37 (BIPHAS) (liquid and gas mixture) with less diffusion. It can also replace /MAT/LAW20 (BIMAT) in 2D analysis since /MAT/LAW51 is compatible with QUAD elements.
- /MAT/LAW51 (MULTIMAT) is based on the equilibrium between
each material present inside the element. Radioss
computes and outputs a relative pressure
$\text{\Delta}\mathrm{P}$
. At each cycle:
$\text{\Delta}P=\text{\Delta}{P}_{1}=\text{\Delta}{P}_{2}=\text{\Delta}{P}_{3}=\text{\Delta}{P}_{4}$
User can deduce total pressure using output value $\text{\Delta}\mathrm{P}$ and input parameter ${P}_{ext}$ :
(5) $$P=\text{\Delta}P+{P}_{ext}$$ - Tetra 4 elements can be used for this law, but BRICK elements are currently highly recommended for better numerical solution in ALE.