/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
`I`_form

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 (I_form=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Outlet formulation (I_form=3) is obsolete since 2018.0 version.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
Material Hypothesis	Output	`C`₀	`C`₁	`C`₂	`C`₃	`C`₄	`C`₅	`E`₀	`P`_ext	`P`_min
Perfect gas (Example 43)	$P (μ, E)$ $P (μ, E)$					$(γ - 1)$ $(γ - 1)$	$(γ - 1)$ $(γ - 1)$	$\frac{P_{0}}{γ - 1}$ $\frac{P_{0}}{γ - 1}$
Perfect gas (Example 43)	$Δ P (μ, E)$ $Δ P (μ, E)$	`-P`₀				$(γ - 1)$ $(γ - 1)$	$(γ - 1)$ $(γ - 1)$	$\frac{P_{0}}{γ - 1}$ $\frac{P_{0}}{γ - 1}$	`P`₀
Water (Linear EOS)	$P (μ, E)$ $P (μ, E)$	`P`₀	$ρ c^{2}$ $ρ c^{2}$							$10^{- 30}$ $10^{- 30}$
Water (Linear EOS)	$Δ P (μ, E)$ $Δ P (μ, E)$		$ρ c^{2}$ $ρ c^{2}$						`P`₀	`-P`₀
Elastic Solid (Linear EOS)	$P (μ, E)$ $P (μ, E)$	`P`₀	$\frac{E}{3 (1 - 2 ν)}$ $\frac{E}{3 (1 - 2 ν)}$
Elastic Solid (Linear EOS)	$Δ P (μ, E)$ $Δ P (μ, E)$		$\frac{E}{3 (1 - 2 ν)}$ $\frac{E}{3 (1 - 2 ν)}$						`P`₀
Mie-Gruneisen $Γ$ $Γ$ constant	$Δ P (μ, E)$ $Δ P (μ, E)$		`K`₁	$K_{2} - \frac{Γ}{2} K_{1}$ $K_{2} - \frac{Γ}{2} K_{1}$	$K_{3} - \frac{Γ}{2} K_{2}$ $K_{3} - \frac{Γ}{2} K_{2}$	$Γ$ $Γ$	$Γ$ $Γ$	`E`₀	`P`₀
Mie-Gruneisen $Γ$ $Γ$ linear $Γ = Γ_{0} - a (\frac{μ}{1 + μ})$ $Γ = Γ_{0} - a (\frac{μ}{1 + μ})$	$Δ P (μ, E)$ $Δ P (μ, E)$		`K`₁	$K_{2} - \frac{Γ_{0}}{2} K_{1}$ $K_{2} - \frac{Γ_{0}}{2} K_{1}$	$K_{3} - \frac{Γ_{0}}{2} K_{2} + a K_{1}$ $K_{3} - \frac{Γ_{0}}{2} K_{2} + a K_{1}$	$Γ_{0}$ $Γ_{0}$	$Γ_{0} - a$ $Γ_{0} - a$	`E`₀	`P`₀

Where,(1)

K_{1} = ρ_{0} c^{2}

$K_{1} = ρ_{0} c^{2}$

(2)

K_{2} = ρ_{0} c^{2} (2 S - 1)

$K_{2} = ρ_{0} c^{2} (2 S - 1)$

(3)

K_{3} = ρ_{0} c^{2} (S - 1) (3 S - 1)

$K_{3} = ρ_{0} c^{2} (S - 1) (3 S - 1)$

Where,(4)

μ = \frac{ρ}{ρ_{0}} - 1

$μ = \frac{ρ}{ρ_{0}} - 1$

$P (μ, E)$ $P (μ, E)$: Total pressure and total energy formulation
$Δ P (μ, E)$ $Δ P (μ, E)$: Relative pressure and total energy formulation
$P (μ, Δ E)$ $P (μ, Δ E)$: Total pressure and relative energy formulation
$Δ P (μ, Δ E)$ $Δ P (μ, Δ E)$: Relative pressure and relative energy formulation
P₀: Initial total pressure
E₀: Initial total energy
$γ$ $γ$: Perfect gas constant
E: Young's modulus
$ν$ $ν$: Poisson coefficient
$Γ$ $Γ$: Gruneisen's gamma
$a$ $a$: Coefficient for first order volume correction to the Gruneisen gamma $Γ_{0}$ $Γ_{0}$
c: Speed of sound
$ρ_{0}$ $ρ_{0}$: 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 $Δ P$ $Δ P$ . At each cycle: $Δ P = Δ P_{1} = Δ P_{2} = Δ P_{3} = Δ P_{4}$ $Δ P = Δ P_{1} = Δ P_{2} = Δ P_{3} = Δ P_{4}$
User can deduce total pressure using output value $Δ P$ $Δ P$ and input parameter $P_{e x t}$ $P_{e x t}$ : (5)
$P = Δ P + P_{e x t}$ $P = Δ P + P_{e x t}$
Tetra 4 elements can be used for this law, but BRICK elements are currently highly recommended for better numerical solution in ALE.