Iform = 0

Block Format Keyword The material law is based on a diffusive interface technique.

To get sharper interfaces between submaterial zones, refer to /ALE/MUSCL.

LAW51 is based on equilibrium between each material present inside the element. Radioss computes and outputs a relative pressure

Δ P

. At each cycle:(1)

Δ P = Δ P_{1} = Δ P_{2} = Δ P_{3}

Total pressure can be calculated with external pressure:(2)

P = Δ P + P_{ext}

Where,

P: Positive for a compression and negative for traction.

Hydrostatic stresses are computed from Polynomial EOS:(3)

- σ_{m} = Δ P = C_{0} + C_{1} μ + C_{2}^{'} μ^{2} + C_{3}^{'} μ^{3} + (C_{4} + C_{5} μ) E (μ)

(4)

d E_{int} = - (Δ P + P_{e x t}) d V

Where, $E = E_{int} / V_{0}, C_{2}^{'} = C_{2} δ_{μ \geq 0} a n d C_{3}^{'} = C_{3} δ_{μ \geq 0}$ mean that the EOS is linear for an expansion and cubic for a compression.

Deviatoric stresses are computed with shear modulus:(5)

σ_{d e v} = G ε

Format

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

#Global Parameters

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
`P`_ext		$ν$		$ν_{v o l}$

#Material1 Parameters

(1)	(3)	(5)	(7)	(9)
$α_{0}^{m a t_ 1}$	$ρ_{0}^{m a t_ 1}$	$E_{0}^{m a t_ 1}$	$Δ P_{\min}^{m a t_1}$	$C_{0}^{m a t_ 1}$
$C_{1}^{m a t_ 1}$	$C_{2}^{m a t_ 1}$	$C_{3}^{m a t_ 1}$	$C_{4}^{m a t_ 1}$	$C_{5}^{m a t_ 1}$
$G_{1}^{m a t_ 1}$

#Material2 Parameters

(1)	(3)	(5)	(7)	(9)
$α_{0}^{m a t_ 2}$	$ρ_{0}^{m a t_ 2}$	$E_{0}^{m a t_ 2}$	$Δ P_{\min}^{m a t_2}$	$C_{0}^{m a t_ 2}$
$C_{1}^{m a t_ 2}$	$C_{2}^{m a t_ 2}$	$C_{3}^{m a t_ 2}$	$C_{4}^{m a t_ 2}$	$C_{5}^{m a t_ 2}$
$G_{1}^{m a t_ 2}$

#Material3 Parameters

(1)	(3)	(5)	(7)	(9)
$α_{0}^{m a t_ 3}$	$ρ_{0}^{m a t_ 3}$	$E_{0}^{m a t_ 3}$	$Δ P_{\min}^{m a t_3}$	$C_{0}^{m a t_ 3}$
$C_{1}^{m a t_ 3}$	$C_{2}^{m a t_ 3}$	$C_{3}^{m a t_ 3}$	$C_{4}^{m a t_ 3}$	$C_{5}^{m a t_ 3}$
$G_{1}^{m a t_ 3}$

Definitions

Field	Contents	SI Unit Example
`mat_ID`	Material identifier. (Integer, maximum 10 digits)
`unit_ID`	Unit Identifier. (Interger, maximum 10 digits)
`mat_title`	Material title. (Character, maximum 100 characters)
`I`_form	Formulation flag. (Integer)
`P`_ext	External pressure. 2 Default = 0 (Real)	$[Pa]$
$ν$	Kinematic viscosity shear $ν = μ / ρ$ . 3 Default = 0 (Real)	$[\frac{m^{2}}{s}]$
$ν_{v o l}$	Kinematic viscosity (volumetric), $ν_{v o l} = \frac{3 λ + 2 μ}{ρ}$ which corresponds to Stokes Hypothesis. 3 Default = 0 (Real)	$[\frac{m^{2}}{s}]$
$α_{0}^{m a t_ i}$	Initial volumetric fraction. 4 (Real)
$ρ_{0}^{m a t_ i}$	Initial density. (Real)	$[\frac{kg}{m^{2}}]$
$E_{0}^{m a t_ i}$	Initial energy per unit volume. (Real)	$[\frac{J}{m^{3}}]$
$Δ P_{\min}^{m a t_i}$	Hydrodynamic cavitation pressure. 5 If fluid material ( $G_{1}^{m a t_ i} = 0$ ), then default = $- P_{e x t}$ . If solid material ( $G_{1}^{m a t_ i} \neq 0$ ), then default = -1e30. (Real)	$[Pa]$
$C_{0}^{m a t_i}$	Initial pressure. (Real)	$[Pa]$
$C_{1}^{m a t_i}$	Hydrodynamic coefficient. (Real)	$[Pa]$
$C_{2}^{m a t_i}$	Hydrodynamic coefficient. (Real)	$[Pa]$
$C_{3}^{m a t_i}$	Hydrodynamic coefficient. (Real)	$[Pa]$
$C_{4}^{m a t_i}$	Hydrodynamic coefficient. (Real)
$C_{5}^{m a t_i}$	Hydrodynamic coefficient. (Real)
$G_{1}^{m a t_i}$	Elasticity shear modulus. = 0 (Default) Fluid material (Real)	$[Pa]$

Example

/MAT/LAW51/1
99.99% Water + 0.01% Air-MULTIMAT: AIR+WATER,units{kg,m,s,Pa}
#(output is total pressure:Pext=0)
#--------------------------------------------------------------------------------------------------#
#                    Material Law No 51. MULTI-MATERIAL SOLID LIQUID GAS  ALE-CFD-SPH               
#--------------------------------------------------------------------------------------------------#
#     Blank format

#    IFORM
         0
#---Global parameters------------------------------------------------------------------------------#
#              P_EXT                  NU               LAMDA
                   0                   0                   0
#---Material#1:AIR(PerfectGas)---------------------------------------------------------------------#
#            ALPHA_1             RHO_0_1               E_0_1             P_MIN_1              C_0_1
              0.0001                 1.2             2.5E+05                   0                  0
#              C_1_1               C_2_1               C_3_1               C_4_1              C_5_1
                   0                   0                   0                 0.4                0.4
#                G_1
                   0
#---Material#2:WATER(Linear_Incompressible)--------------------------------------------------------#
#            ALPHA_2             RHO_0_2               E_0_2             P_MIN_2               C_0_2
              0.9999              1000.0                   0                   0                   0
#              C_1_2               C_2_2               C_3_2               C_4_2               C_5_2
             2.25E+9                   0                   0                   0                   0
#                G_2
                   0
#---Material#3:not defined-------------------------------------------------------------------------#
#            ALPHA_3             RHO_0_3               E_0_3             P_MIN_3               C_0_3
                 0.0                   0                   0                   0                   0
#              C_1_3               C_2_3               C_3_3               C_4_3               C_5_3
                   0                   0                   0                   0                   0
#                G_3
                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW51/1
99.99% Water + 0.01% Air-MULTIMAT: AIR+WATER,units{kg,m,s,Pa}
#(output is relative pressure to Pext=1E+5Pa)
#--------------------------------------------------------------------------------------------------#
#                    Material Law No 51. MULTI-MATERIAL SOLID LIQUID GAS -ALE-CFD-SPH               
#--------------------------------------------------------------------------------------------------#
#     Blank format

#    IFORM
         0
#---Global parameters------------------------------------------------------------------------------#
#              P_EXT                  NU               LAMDA
                1E+5                   0                   0
#---Material#1:AIR(PerfectGas)---------------------------------------------------------------------#
#            ALPHA_1             RHO_0_1               E_0_1             P_MIN_1               C_0_1
              0.0001                 1.2             2.5E+05                   0               -1E+5
#              C_1_1               C_2_1               C_3_1               C_4_1               C_5_1
                   0                   0                   0                 0.4                 0.4
#                G_1
                   0
#---Material#2:WATER(Linear_Incompressible)--------------------------------------------------------#
#            ALPHA_2             RHO_0_2               E_0_2             P_MIN_2               C_0_2
              0.9999              1000.0                   0                   0                   0
#              C_1_2               C_2_2               C_3_2               C_4_2               C_5_2
             2.25E+9                   0                   0                   0                   0
#                G_2
                   0
#---Material#3:not defined-------------------------------------------------------------------------#
#            ALPHA_3             RHO_0_3               E_0_3             P_MIN_3               C_0_3
                 0.0                   0                   0                   0                   0
#              C_1_3               C_2_3               C_3_3               C_4_3               C_5_3
                   0                   0                   0                   0                   0
#                G_3
                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

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
Radioss computes and outputs a relative pressure $Δ P$ .(6)
$Δ P = \max {Δ P \min, C_{0} + C_{1} μ + C_{2}^{'} μ^{2} + C_{3}^{'} μ^{3} + (C_{4} + C_{5} μ) E (μ)}$

However, total pressure is essential for energy integration ( $d E_{int} = - P d V$ ). It can be computed with the external pressure flag P_ext.

$P = Δ P + P_{e x t}$ leads to $d E_{int} = - (P_{e x t} + Δ P) d V$ .

This means that if P_ext = 0, the computed pressure $Δ P$ is also the total pressure: $Δ P = P$
Kinematic viscosities are global and is not specific to each material. It allows computing viscous stress tensor:(7)
$τ = μ [(\nabla \otimes V) +^{t} (\nabla \otimes V)] + λ (\nabla V) I$

Where,

$ν = μ / ρ$

Kinematic viscosity in shear

$ν_{v o l} = \frac{3 (λ + \frac{2 μ}{3})}{ρ}$

Kinematic volumetric viscosity
Volumetric fractions enable the sharing of elementary volume within the three different materials.
For each material $α_{0}^{m a t_ i}$ must be defined between 0 and 1.

Sum of initial volumetric fractions $\sum_{i = 1}^{3} α_{0}^{m a t _ i}$ must be equal to 1.

For automatic initial fraction of the volume, refer to the /INIVOL card.
$Δ P_{\min}^{m a t_i}$ flag is the minimum value for the computed pressure $Δ P$ . It means that total pressure is also bounded to:(8)
$P_{\min}^{m a t_i} = Δ P_{\min}^{m a t _ i} + P_{e x t}$

For fluid materials and detonation products, $P_{\min}^{m a t_i}$ must remain positive to avoid any tensile strength so $Δ P_{\min}^{m a t_i}$ must be set to $- P_{e x t}$ .

For solid materials, default value $Δ P_{\min}^{m a t_i}$ = 1e-30 is suitable but may be modified.
Material tracking is possible through animation files:
/ANIM/BRIC/VFRAC (All material volumetric fractions)