Iform = 1

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

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_{e x t}

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 W + δ Q = - (Δ P + P_{e x t}) d V + ​ δ Q

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

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

Deviatoric stresses are computed with a Johnson-Cook model:(5)

σ_{d e v} = {\begin{cases} G ε if in elastic domain \\ (α + b ε_{p}^{n}) (1 + c \ln \frac{\dot{ε}}{{\dot{ε}}_{0}}) (1 - {(\frac{T - T_{0}}{T_{m e l t} - T_{0}})}^{m}) others \end{cases}

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)
$α_{0}^{mat_1}$	$ρ_{0}^{mat_1}$	$E_{0}^{mat_1}$	$Δ P_{\min}^{mat_1}$	$C_{0}^{mat_1}$
${C_{1}}^{mat_1}$	${C_{2}}^{mat_1}$	${C_{3}}^{mat_1}$	${C_{4}}^{mat_1}$	${C_{5}}^{mat_1}$
$G_{1}^{mat_1}$	$a_{}^{mat_1}$	$b_{}^{mat_1}$	$n_{}^{mat_1}$
$c_{}^{mat_1}$	${\dot{ε}}_{0}^{mat_1}$
$m_{}^{mat_1}$	$T_{0}^{mat_1}$	$T_{melt}^{mat_1}$	${T_{lim}}^{mat_1}$	$ρ C_{v}^{m a t_ 1}$
$ε_{p, \max}^{m a t_1}$	$σ_{\max}^{mat_1}$	$K_{A}^{mat_1}$	$K_{B}^{mat_1}$

#Material2 Parameters

(1)	(3)	(5)	(7)	(9)
$α_{0}^{mat_2}$	$ρ_{0}^{mat_2}$	$E_{0}^{mat_2}$	$Δ P_{\min}^{mat_2}$	$C_{0}^{mat_2}$
$C_{1}^{mat_2}$	$C_{2}^{mat_2}$	$C_{3}^{mat_2}$	$C_{4}^{mat_2}$	$C_{5}^{mat_2}$
$G_{1}^{mat_2}$	$a_{}^{mat_2}$	$b_{}^{mat_2}$	$n_{}^{mat_2}$
$c_{}^{mat_2}$	${\dot{ε}}_{0}^{mat_2}$
$m_{}^{mat_2}$	$T_{0}^{mat_2}$	$T_{melt}^{mat_2}$	$T_{lim}^{mat_2}$	$ρ C_{v}^{mat_2}$
$ε_{p, \max}^{m a t_2}$	$σ_{max}^{mat_2}$	$K_{A}^{mat_2}$	$K_{B}^{mat_2}$

#Material3 Parameters

(1)	(3)	(5)	(7)	(9)
$α_{0}^{mat_3}$	$ρ_{0}^{mat_3}$	$E_{0}^{mat_3}$	$Δ P_{min}^{mat_3}$	$C_{0}^{mat_3}$
$C_{1}^{mat_3}$	$C_{2}^{mat_3}$	$C_{3}^{mat_3}$	$C_{4}^{mat_3}$	$C_{5}^{mat_3}$
$G_{1}^{mat_3}$	$a_{}^{mat_3}$	$b_{}^{mat_3}$	$n_{}^{mat_3}$
$c_{}^{mat_3}$	${\dot{ε}}_{0}^{mat_3}$
$m_{}^{mat_3}$	$T_{0}^{mat_3}$	$T_{m e l t}^{mat_3}$	$T_{lim}^{mat_3}$	$ρ C_{v}^{mat_3}$
$ε_{p, \max}^{m a t_3}$	$σ_{max}^{mat_3}$	$K_{A}^{mat_3}$	$K_{B}^{mat_3}$

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)
`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)	$[\frac{m^{2}}{s}]$
$α_{0}^{mat_i}$	Initial volumetric fraction. 4 (Real)
$ρ_{0}^{mat_i}$	Initial density. (Real)	$[\frac{kg}{m^{3}}]$
$E_{0}^{mat_i}$	Initial energy per unit volume. (Real)	$[\frac{J}{m^{3}}]$
${Δ P}_{min}^{mat_i}$	Hydrodynamic cavitation pressure. 5 If fluid material ( $G_{1}^{mat_i} = 0$ ), then default = $- P_{e x t}$ . If solid material ( $G_{1}^{mat_i} \neq 0$ ), then default = -1e30. (Real)	$[Pa]$
$C_{0}^{mat_i}$	Initial pressure. (Real)	$[Pa]$
$C_{1}^{mat_i}$	Hydrodynamic coefficient. (Real)	$[Pa]$
$C_{2}^{mat_i}$	Hydrodynamic coefficient. (Real)	$[Pa]$
$C_{3}^{mat_i}$	Hydrodynamic coefficient. (Real)	$[Pa]$
$C_{4}^{mat_i}$	Hydrodynamic coefficient. (Real)
$C_{5}^{mat_i}$	Hydrodynamic coefficient. (Real)
$G_{1}^{mat_i}$	Elasticity shear modulus. = 0 (Default) Fluid material (Real)	$[Pa]$
$a_{}^{m a t_i}$	Plasticity yield stress. (Real)	$[Pa]$
$b_{}^{m a t_i}$	Plasticity hardening parameter. (Real)	$[Pa]$
$n_{}^{m a t_i}$	Plasticity hardening exponent. Default = 1.0 (Real)
$c_{}^{m a t_i}$	Strain rate coefficient. = 0 No strain rate effect Default = 0.00 (Real)
${\dot{ε}}_{0}^{mat_i}$	Reference strain rate. If $\dot{ε} \leq {\dot{ε}}_{0}^{mat_j}$ , no strain rate effect (Real)	$[\frac{1}{s}]$
$m_{}^{m a t_i}$	Temperature exponent. Default = 1.00 (Real)
$T_{0}^{mat_i}$	Initial temperature. Default = 300 K (Real)	$[K]$
$T_{m e l t}^{mat_i}$	Melting temperature. = 0 No temperature effect Default = 10³⁰ (Real)	$[K]$
$T_{lim}^{mat_i}$	Maximum temperature. Default = 10³⁰ (Real)	$[K]$
$ρ C_{v}^{mat_i}$	Specific heat per unit of volume. 7 (Real)	$[\frac{J}{m^{3} \cdot K}]$
$ε_{p, \max}^{m a t_i}$	Failure plastic strain. Default = 10³⁰ (Real)
$σ_{max}^{mat_i}$	Plasticity maximum stress. Default = 10³⁰ (Real)	$[Pa]$
$K_{A}^{mat_i}$	Thermal conductivity coefficient 1. 8 (Real)	$[\frac{W}{m \cdot K}]$
$K_{B}^{mat_i}$	Thermal conductivity coefficient 2. 8 (Real)	$[\frac{W}{m \cdot K^{2}}]$

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----|

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,

$ν = μ / ρ$

Cinematic shear viscosity flag

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

Cinematic volumetric viscosity flag
Volumetric fractions enable the sharing of elementary volume within the three different materials.
For each material $α_{0}^{mat_i}$ must be defined between 0 and 1.

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

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

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

For solid materials, default value $Δ P_{\min}^{mat_i}$ = 1e-30 is suitable but may be modified.
Heat contribution is computed only if the thermal card is associated to the material law (/HEAT/MAT).
In this case, $δ Q = ρ C_{V} V d T$ and the parameters for thermal diffusion are read for each material:(9)
$ρ C_{V}^{mat_i}, K_{A}^{mat_i}, K_{B}^{mat_i} and T_{0}^{mat_i}$

For solids and liquids, $C_{ν} \approx C_{p}$ for perfect gas: $γ = C_{p} / C_{ν}$
The temperature evolution in the Johnson-Cook model is computed with the flag $ρ C_{V}^{mat_i}$ , even if the thermal card (/HEAT/MAT) is not defined.
Thermal conductivity, $K$ , is linearly dependent on the temperature:(10)
$K (T) = K_{A} + K_{B} T$
Material tracking is possible through animation files:
/ANIM/BRIC/VFRAC (All material volumetric fractions)