# 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 $\text{Δ}P$ . At each cycle:(1)
$\text{Δ}P=\text{Δ}{P}_{\text{ }1}=\text{Δ}{P}_{\text{ }2}=\text{Δ}{P}_{\text{ }3}$
Total pressure can be calculated with external pressure:(2)
$P=\text{Δ}P+{P}_{\mathrm{ext}}$
Where,
P
Positive for a compression and negative for traction.
Hydrostatic stresses are computed from Polynomial EOS:(3)
$-{\sigma }_{m}=\text{Δ}P={C}_{0}+{C}_{1}\mu +{C}_{2}^{\text{'}}{\mu }^{2}+{C}_{3}^{\text{'}}{\mu }^{3}+\left({C}_{4}+{C}_{5}\mu \right)E\left(\mu \right)$
(4)
$d{E}_{\mathrm{int}}=-\left(\text{Δ}P+{P}_{ext}\right)dV$

Where, $E={E}_{\mathrm{int}}/{V}_{0},\text{\hspace{0.17em}}{C}_{2}^{\text{'}}={C}_{2}{\delta }_{\mu \ge 0}\text{\hspace{0.17em}}and\text{\hspace{0.17em}}{C}_{3}^{\text{'}}={C}_{3}{\delta }_{\mu \ge 0}$ mean that the EOS is linear for an expansion and cubic for a compression.

Deviatoric stresses are computed with shear modulus:(5)
${\sigma }_{dev}=G\epsilon$

## Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
/MAT/LAW51/mat_ID/unit_ID
mat_title
Blank
Iform
#Global Parameters
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
Pext $\nu$ ${\nu }_{vol}$
#Material1 Parameters
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
${\alpha }_{0}^{mat\text{ }_\text{​}\text{ }1}$ ${\rho }_{0}^{mat\text{ }_\text{​}\text{ }1}$ ${\text{E}}_{0}^{\text{m}\text{a}\text{t}\text{ }_\text{​}\text{ }1}$ $\text{Δ}{P}_{\mathrm{min}}^{mat\text{ }_1\text{ }}$ ${C}_{0}^{mat\text{ }_\text{​}\text{ }1}$
${C}_{1}^{mat\text{ }_\text{​}\text{ }1}$ ${C}_{2}^{mat\text{ }_\text{​}\text{ }1}$ ${C}_{3}^{mat\text{ }_\text{​}\text{ }1}$ ${C}_{4}^{mat\text{ }_\text{​}\text{ }1}$ ${C}_{5}^{mat\text{ }_\text{​}\text{ }1}$
${G}_{1}^{mat\text{ }_\text{​}\text{ }1}$
#Material2 Parameters
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
${\alpha }_{0}^{mat\text{ }_\text{​}\text{ }2}$ ${\rho }_{0}^{mat\text{ }_\text{​}\text{ }2}$ ${E}_{0}^{mat\text{ }_\text{​}\text{ }2}$ $\text{Δ}{P}_{\mathrm{min}}^{mat\text{ }_2\text{ }}$ ${C}_{0}^{mat\text{ }_\text{​}\text{ }2}$
${C}_{1}^{mat\text{ }_\text{​}\text{ }2}$ ${C}_{2}^{mat\text{ }_\text{​}\text{ }2}$ ${C}_{3}^{mat\text{ }_\text{​}\text{ }2}$ ${C}_{4}^{mat\text{ }_\text{​}\text{ }2}$ ${C}_{5}^{mat\text{ }_\text{​}\text{ }2}$
${G}_{1}^{mat\text{ }_\text{​}\text{ }2}$
#Material3 Parameters
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
${\alpha }_{0}^{mat\text{ }_\text{​}\text{ }3}$ ${\rho }_{0}^{mat\text{ }_\text{​}\text{ }3}$ ${E}_{0}^{mat\text{ }_\text{​}\text{ }3}$ $\text{Δ}{P}_{\mathrm{min}}^{mat\text{ }_3\text{ }}$ ${C}_{0}^{mat\text{ }_\text{​}\text{ }3}$
${C}_{1}^{mat\text{ }_\text{​}\text{ }3}$ ${C}_{2}^{mat\text{ }_\text{​}\text{ }3}$ ${C}_{3}^{mat\text{ }_\text{​}\text{ }3}$ ${C}_{4}^{mat\text{ }_\text{​}\text{ }3}$ ${C}_{5}^{mat\text{ }_\text{​}\text{ }3}$
${G}_{1}^{mat\text{ }_\text{​}\text{ }3}$

## 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)

Iform Formulation flag.

(Integer)

Pext External pressure. 2

Default = 0 (Real)

$\left[\text{Pa}\right]$
$\nu$ Kinematic viscosity shear $\nu =\mu /\rho$ . 3

Default = 0 (Real)

$\left[\frac{{\text{m}}^{\text{2}}}{\text{s}}\right]$
${\nu }_{vol}$ Kinematic viscosity (volumetric), ${\nu }_{vol}=\frac{3\lambda +2\mu }{\rho }$ which corresponds to Stokes Hypothesis. 3

Default = 0 (Real)

$\left[\frac{{\text{m}}^{\text{2}}}{\text{s}}\right]$
${\alpha }_{0}^{mat\text{ }_\text{​}\text{ }i}$ Initial volumetric fraction. 4

(Real)

${\rho }_{0}^{mat\text{ }_\text{​}\text{ }i}$ Initial density.

(Real)

$\left[\frac{\text{kg}}{{\text{m}}^{2}}\right]$
${E}_{0}^{mat\text{ }_\text{​}\text{ }i}$ Initial energy per unit volume.

(Real)

$\left[\frac{\text{J}}{{\text{m}}^{\text{3}}}\right]$
$\text{Δ}{P}_{\mathrm{min}}^{mat\text{ }_i}$ Hydrodynamic cavitation pressure. 5

If fluid material ( ${G}_{1}^{mat\text{ }_\text{​}\text{ }i}=0$ ), then default = $-{P}_{ext}$ .

If solid material ( ${G}_{1}^{mat\text{ }_\text{​}\text{ }i}\ne 0$ ), then default = -1e30.

(Real)

$\left[\text{Pa}\right]$
${C}_{0}^{mat\text{ }_i}$ Initial pressure.

(Real)

$\left[\text{Pa}\right]$
${C}_{1}^{mat\text{ }_i}$ Hydrodynamic coefficient.

(Real)

$\left[\text{Pa}\right]$
${C}_{2}^{mat\text{ }_i}$ Hydrodynamic coefficient.

(Real)

$\left[\text{Pa}\right]$
${C}_{3}^{mat\text{ }_i}$ Hydrodynamic coefficient.

(Real)

$\left[\text{Pa}\right]$
${C}_{4}^{mat\text{ }_i}$ Hydrodynamic coefficient.

(Real)

${C}_{5}^{mat\text{ }_i}$ Hydrodynamic coefficient.

(Real)

${G}_{1}^{mat\text{ }_i}$ Elasticity shear modulus.
= 0 (Default)
Fluid material

(Real)

$\left[\text{Pa}\right]$

## 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----|

1. 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
2. Radioss computes and outputs a relative pressure $\text{Δ}P$ .(6)
$\text{Δ}P=\mathrm{max}\text{ }\left\{\text{Δ}P\text{​}\mathrm{min},\text{ }\text{\hspace{0.17em}}{C}_{0}+{C}_{1}\mu +{C}_{2}^{\text{'}}{\mu }^{2}+{C}_{3}^{\text{'}}{\mu }^{3}+\left({C}_{4}+{C}_{5}\mu \right)E\left(\mu \right)\right\}$

However, total pressure is essential for energy integration ( $d{E}_{\mathrm{int}}=-PdV$ ). It can be computed with the external pressure flag Pext.

$P=\text{Δ}P+{P}_{ext}$ leads to $d{E}_{\mathrm{int}}=-\left({P}_{ext}+\text{Δ}P\right)dV$ .

This means that if Pext = 0, the computed pressure $\text{Δ}P$ is also the total pressure: $\text{Δ}P=P$

3. Kinematic viscosities are global and is not specific to each material. It allows computing viscous stress tensor:(7)
$\tau =\mu \left[\left(\nabla \otimes V\right)+{\text{\hspace{0.17em}}}^{t}\text{​}\left(\nabla \otimes V\right)\right]+\lambda \left(\nabla V\right)I$
Where,
$\nu =\mu /\rho$
Kinematic viscosity in shear
${\nu }_{vol}=\frac{3\left(\lambda +\frac{2\mu }{3}\right)}{\rho }$
Kinematic volumetric viscosity
4. Volumetric fractions enable the sharing of elementary volume within the three different materials.

For each material ${\alpha }_{0}^{mat\text{ }_\text{​}\text{ }i}$ must be defined between 0 and 1.

Sum of initial volumetric fractions ${\sum }_{i=1}^{3}{\alpha }_{0}^{mat\text{​}_\text{​}i}$ must be equal to 1.

For automatic initial fraction of the volume, refer to the /INIVOL card.

5. $\text{Δ}{P}_{\mathrm{min}}^{mat\text{ }_i}$ flag is the minimum value for the computed pressure $\text{Δ}P$ . It means that total pressure is also bounded to:(8)
${P}_{\mathrm{min}}^{mat\text{ }_i}=\text{Δ}{P}_{\mathrm{min}}^{mat\text{​}_\text{​}i}+{P}_{ext}$

For fluid materials and detonation products, ${P}_{\mathrm{min}}^{mat\text{ }_i}$ must remain positive to avoid any tensile strength so $\text{Δ}{P}_{\mathrm{min}}^{mat\text{ }_i}$ must be set to $-{P}_{ext}$ .

For solid materials, default value $\text{Δ}{P}_{\mathrm{min}}^{mat\text{ }_i}$ = 1e-30 is suitable but may be modified.

6. Material tracking is possible through animation files:

/ANIM/BRIC/VFRAC (All material volumetric fractions)