# /MAT/LAW51 (Iform = 1) (Obsolete)

Block Format Keyword This material is able to handle up to three elasto-plastic materials (solid, liquid, or gas). The material law is based on a diffusive interface technique.

To get sharper interfaces between submaterial zones, refer to /ALE/MUSCL.
Note: It is not recommended to use this law with Radioss single precision engine.

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:

$\Delta P=\Delta P{\text{ }}_{1}=\Delta P{\text{ }}_{2}=\Delta {P}_{3}$

Total pressure can be calculated with external pressure:

$P=\Delta P+{P}_{ext}$

Where,
P
Positive for a compression and negative for traction.

Hydrostatic stresses are computed from Polynomial EOS:

$-{\sigma }_{m}=\Delta P={C}_{0}+{C}_{1}\mu +{C}_{2}^{\text{'}}{\mu }^{2}+{C}_{3}^{\text{'}}{\mu }^{3}+\left({C}_{4}+{C}_{5}\mu \right)E\left(\mu \right)$
$d{E}_{\mathrm{int}}=\delta W+\delta Q=-\left(\Delta P+{P}_{ext}\right)dV+\text{​}\delta Q$

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

By default process is adiabatic $\delta Q=0$ . To enable thermal computation, refer to 6.

Deviatoric stresses are computed with a Johnson-Cook model:

## 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 $\text{ν}$ ${\nu }_{vol}$
#Material1 Parameters
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
${\alpha }_{0}^{\mathit{mat}_1}$ ${\rho }_{0}^{\mathit{mat}_1}$ ${E}_{0}^{\text{mat}_1}$ $\Delta {P}_{\mathrm{min}}^{\mathit{mat}_1}$ ${\text{C}}_{0}^{\text{mat}_1}$
${{\text{C}}_{1}}^{\text{mat}_1}$ ${{\text{C}}_{2}}^{\text{mat}_1}$ ${{\text{C}}_{3}}^{\text{mat}_1}$ ${{\text{C}}_{4}}^{\text{mat}_1}$ ${{\text{C}}_{5}}^{\text{mat}_1}$
${\text{G}}_{1}^{\text{mat}_1}$ ${\text{a}}_{}^{\text{mat}_1}$ ${\text{b}}_{}^{\text{mat}_1}$ ${\text{n}}_{}^{\text{mat}_1}$
${\text{c}}_{}^{\text{mat}_1}$ ${{\stackrel{˙}{\epsilon }}_{0}}^{\text{mat}_1}$
${\text{m}}_{}^{\text{mat}_1}$ ${\text{T}}_{0}^{\text{mat}_1}$ ${\text{T}}_{\text{melt}}^{\text{mat}_1}$ ${{\text{T}}_{\text{lim}}}^{\text{mat_1}}$ $\rho {C}_{v}^{mat\text{ }_\text{​}\text{ }1}$
${\epsilon }_{p,\mathrm{max}}^{mat\text{ }_1\text{ }}$ ${\sigma }_{\mathrm{max}}^{\mathit{mat}_1}$ ${\text{K}}_{\text{A}}^{\text{mat}_1}$ ${\text{K}}_{\text{B}}^{\text{mat}_1}$
#Material2 Parameters
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
${\alpha }_{0}^{\mathit{mat}_2}$ ${\rho }_{0}^{\mathit{mat}_2}$ ${E}_{0}^{\text{mat}_2}$ $\Delta {P}_{\mathrm{min}}^{\mathit{mat}_2}$ ${\text{C}}_{0}^{\text{mat}_2}$
${\text{C}}_{1}^{\text{mat}_2}$ ${\text{C}}_{2}^{\text{mat}_2}$ ${\text{C}}_{3}^{\text{mat}_2}$ ${\text{C}}_{4}^{\text{mat}_2}$ ${\text{C}}_{5}^{\text{mat}_2}$
${\text{G}}_{1}^{\text{mat}_2}$ ${\text{a}}_{}^{\text{mat}_2}$ ${\text{b}}_{}^{\text{mat}_2}$ ${\text{n}}_{}^{\text{mat}_2}$
${\text{c}}_{}^{\text{mat}_2}$ ${\stackrel{˙}{\epsilon }}_{0}^{\text{mat}_2}$
${\text{m}}_{}^{\text{mat}_2}$ ${\text{T}}_{0}^{\text{mat}_2}$ ${\text{T}}_{\text{melt}}^{\text{mat}_2}$ ${\text{T}}_{\text{lim}}^{\text{mat}_2}$ $\rho {C}_{v}^{\mathit{mat}_2}$
${\epsilon }_{p,\mathrm{max}}^{mat\text{ }_2\text{ }}$ ${\text{σ}}_{\text{max}}^{\mathit{mat}_2}$ ${\text{K}}_{\text{A}}^{\text{mat}_2}$ ${\text{K}}_{\text{B}}^{\text{mat}_2}$
#Material3 Parameters
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
${\text{α}}_{0}^{\mathit{mat}_3}$ ${\text{ρ}}_{0}^{\mathit{mat}_3}$ ${\text{E}}_{0}^{\text{mat}_3}$ $\text{Δ}{P}_{\text{min}}^{\mathit{mat}_3}$ ${C}_{0}^{\text{mat}_3}$
${\text{C}}_{1}^{\text{mat}_3}$ ${\text{C}}_{2}^{\text{mat}_3}$ ${\text{C}}_{3}^{\text{mat}_3}$ ${\text{C}}_{4}^{\text{mat}_3}$ ${\text{C}}_{5}^{\text{mat}_3}$
${\text{G}}_{1}^{\text{mat}_3}$ ${\text{a}}_{}^{\text{mat}_3}$ ${\text{b}}_{}^{\text{mat}_3}$ ${\text{n}}_{}^{\text{mat}_3}$
${\text{c}}_{}^{\text{mat}_3}$ ${\stackrel{˙}{\epsilon }}_{0}^{\text{mat}_3}$
${\text{m}}_{}^{\text{mat}_3}$ ${\text{T}}_{0}^{\text{mat}_3}$ ${\text{T}}_{\text{m}\text{e}\text{l}\text{t}}^{\text{mat}_3}$ ${\text{T}}_{\text{lim}}^{\text{mat}_3}$ $\rho {C}_{\text{v}}^{\mathit{mat}_3}$
${\epsilon }_{p,\mathrm{max}}^{mat\text{ }_3\text{ }}$ ${\text{σ}}_{\text{max}}^{\mathit{mat}_3}$ ${\text{K}}_{\text{A}}^{\text{mat}_3}$ ${\text{K}}_{\text{B}}^{\text{mat}_3}$

## Definition

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)

$\left[\text{Pa}\right]$
$\text{ν}$ Kinematic viscosity shear $\text{ν}=\text{μ}/\text{ρ}$ . 3

Default = 0 (Real)

$\left[\frac{{\text{m}}^{\text{2}}}{\text{s}}\right]$
${\text{ν}}_{\text{v}\text{o}\text{l}}$ Kinematic viscosity (volumetric), ${\text{ν}}_{\text{v}\text{o}\text{l}}=\frac{3\text{λ}+2\text{μ}}{\text{ρ}}$ which corresponds to Stokes Hypothesis. 3

Default = 0 (Real)

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

(Real)

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

(Real)

$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$
${\text{E}}_{0}^{\text{mat}_\text{i}}$ Initial energy per unit volume.

(Real)

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

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

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

(Real)

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

(Real)

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

(Real)

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

(Real)

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

(Real)

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

(Real)

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

(Real)

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

(Real)

$\left[\text{Pa}\right]$
${a}_{}^{mat\text{ }_i}$ Plasticity yield stress.

(Real)

$\left[\text{Pa}\right]$
${b}_{}^{mat\text{ }_i}$ Plasticity hardening parameter.

(Real)

$\left[\text{Pa}\right]$
${n}_{}^{mat\text{ }_i}$ Plasticity hardening exponent.

Default = 1.0 (Real)

${c}_{}^{mat\text{ }_i}$ Strain rate coefficient.
= 0
No strain rate effect

Default = 0.00 (Real)

${\stackrel{˙}{\epsilon }}_{0}^{\text{mat}_\text{i}}$ Reference strain rate.

If $\stackrel{˙}{\epsilon }\le {\stackrel{˙}{\epsilon }}_{0}^{\text{mat}_\text{j}}$ , no strain rate effect

(Real)

$\left[\frac{\text{1}}{\text{s}}\right]$
${m}_{}^{mat\text{ }_i}$ Temperature exponent.

Default = 1.00 (Real)

${\text{T}}_{0}^{\text{mat}_\text{i}}$ Initial temperature.

Default = 300 K (Real)

$\left[\text{K}\right]$
${\text{T}}_{\text{m}\text{e}\text{l}\text{t}}^{\text{mat}_\text{i}}$ Melting temperature.
= 0
No temperature effect

Default = 1030 (Real)

$\left[\text{K}\right]$
${\text{T}}_{\text{lim}}^{\text{mat}_\text{i}}$ Maximum temperature.

Default = 1030 (Real)

$\left[\text{K}\right]$
$\rho {C}_{\text{v}}^{\mathit{mat}_\text{i}}$ Specific heat per unit of volume. 7

(Real)

$\left[\frac{\text{J}}{{\text{m}}^{3}\cdot \text{K}}\right]$
${\epsilon }_{p,\mathrm{max}}^{mat\text{ }_i\text{ }}$ Failure plastic strain.

Default = 1030 (Real)

${\text{σ}}_{\text{max}}^{\text{mat}_\text{i}}$ Plasticity maximum stress.

Default = 1030 (Real)

$\left[\text{Pa}\right]$
${\text{K}}_{\text{A}}^{\text{mat}_\text{i}}$ Thermal conductivity coefficient 1. 8

(Real)

$\left[\frac{\text{W}}{\text{m}\cdot \text{K}}\right]$
${\text{K}}_{\text{B}}^{\text{mat}_\text{i}}$ Thermal conductivity coefficient 2. 8

(Real)

$\left[\frac{\text{W}}{\text{m}\cdot {\text{K}}^{2}}\right]$

## Example

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW51/1
99.99% Water + 0.01% Air-MULTIMAT:AIR+WATER+COPPER,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
1
#---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           SIGMA_Y_1                BB_1                 N_1
0                   0                   0                   0
#               CC_1     EPSILON_DOT_0_1
0                   0
#               CM_1                T_10             T_1MELT            T_1LIMIT             RHOCV_1
0                   0                   0                   0                   0
#      EPSILON_MAX_1         SIGMA_MAX_1               K_A_1               K_B_1
0                   0                   0                   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           SIGMA_Y_2                BB_2                 N_2
0                   0                   0                   0
#               CC_2     EPSILON_DOT_0_2
0                   0
#               CM_2                T_20             T_2MELT            T_2LIMIT             RHOCV_2
0                   0                   0                   0                   0
#      EPSILON_MAX_2         SIGMA_MAX_2               K_A_2               K_B_2
0                   0                   0                   0
#---Material#3:OFHC COPPER(elastic plastic solid:Mie_Gruneisen+JCook)------------------------------#
#            ALPHA_3             RHO_0_3               E_0_3             P_MIN_3               C_0_3
0.0              8930.0                   0                   0                   0
#              C_1_3               C_2_3               C_3_3               C_4_3               C_5_3
1.389E+11           1.379E+11          -0.351E+11                0.97                0.97
#                G_3           SIGMA_Y_3                BB_3                 N_3
47.7E+9              120E+6              292E+6                0.31
#               CC_3     EPSILON_DOT_0_3
0.025                   1
#               CM_3                T_30             T_3MELT            T_3LIMIT             RHOCV_3
1.09                 300                1790                   0          3.42019E+6
#      EPSILON_MAX_3         SIGMA_MAX_3               K_A_3               K_B_3
0              1.2E+9                   0                   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 $\text{Δ}P$ .
$\text{Δ}\text{P}=\mathrm{max}\left\{\Delta P\mathrm{min}, {C}_{0}+{C}_{1}\mu +{C}_{2}^{\text{'}}{\mu }^{2}+{C}_{3}^{\text{'}}{\mu }^{3}+\left({C}_{4}+{C}_{5}\mu \right)E\left(\mu \right)\right\}$

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

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

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:
$\tau =\mu \left[\left(\nabla \otimes V\right)+{\text{\hspace{0.17em}}}^{t}\text{​}\left(\nabla \otimes V\right)\right]+\lambda \left(\nabla V\right)I$
Where,
$\nu =\mu /\rho$
Cinematic shear viscosity flag
${\nu }_{vol}=\frac{3\left(\lambda +\frac{2\mu }{3}\right)}{\rho }$
Cinematic volumetric viscosity flag
4. Volumetric fractions enable the sharing of elementary volume within the three different materials.

For each material ${\alpha }_{0}^{\mathit{mat}_i}$ must be defined between 0 and 1.

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

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

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

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

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

6. Heat contribution is computed only if the thermal card is associated to the material law (/HEAT/MAT).

In this case, $\delta Q=\rho {C}_{V}VdT$ and the parameters for thermal diffusion are read for each material:

$\rho {C}_{V}^{\mathit{mat}_i},{K}_{A}^{\mathit{mat}_i},{K}_{B}^{\mathit{mat}_i} \mathrm{and} {T}_{0}^{\mathit{mat}_i}$

For solids and liquids, ${C}_{\nu }\approx {C}_{p}$ for perfect gas: $\gamma ={C}_{p}/{C}_{\nu }$

7. The temperature evolution in the Johnson-Cook model is computed with the flag $\rho {C}_{V}^{\mathit{mat}_i}$ , even if the thermal card (/HEAT/MAT) is not defined.
8. Thermal conductivity, $\mathrm{K}$ , is linearly dependent on the temperature:
$\mathrm{K}\left(T\right)={K}_{A}+{K}_{B}T$
9. Material tracking is possible through animation files:

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

10. As of version 2023, this option is obsolete and should be replaced by Iform=12.