Iform = 10

Block Format Keyword Able to handle up to four materials: Three elasto-plastic materials (solid, liquid, or gas), and one high explosive material (JWL EOS).

The material law is based on a diffusive interface technique. For sharper interfaces between submaterial zone, refer to /ALE/MUSCL.

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

$Δ P$ . At each cycle:(1)

Δ P = Δ P_{1} = Δ P_{2} = Δ P 3 = Δ P_{4}

$Δ P = Δ P_{1} = Δ P_{2} = Δ P {}_{3}= Δ P_{4}$

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

P = Δ P + P_{e x t}

$P = Δ P + P_{e x t}$

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

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

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

$σ_{d e v} = {\begin{cases} G ε i f σ_{V M} \leq α \\ (α + b ε_{p}^{n}) (1 + c \ln \frac{\dot{ε}}{{\dot{ε}}_{0}}) (1 - {(\frac{T - T_{0}}{T_{m e l t} - T_{0}})}^{m}) i f σ_{V M} > α \end{cases}$

High explosive material is modeled with linear EOS if unreacted and JWL EOS for detonation products:(6)

$Δ P = {\begin{cases} C_{0} + C_{1} μ i f T < T_{\det} \\ A (1 - \frac{ω}{R_{1} V}) e^{- R_{1} V} + B (1 - \frac{ω}{R_{2} V}) e^{- R_{2} V} + ω \frac{E}{V} i f σ_{V M} > α \end{cases}$

Where, V is relative volume: $V = V o l u m e / V_{0}$ and $E$ is the internal energy per unit initial volume: $E = E_{int} / V_{0}$ . 9 to 13

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)
$α_{0}^{m a t_ 1}$	$ρ_{0}^{m a t_ 1}$	$E_{0}^{m a t_ 1}$	$Δ P_{\min}^{m a t_1}$	$C_{0}^{m a t_ 1}$
$C_{1}^{m a t_ 1}$	$C_{2}^{m a t_ 1}$	$C_{3}^{m a t_ 1}$	$C_{4}^{m a t_ 1}$	$C_{5}^{m a t_ 1}$
$G_{1}^{m a t_ 1}$	`a`^mat_1	`b`^mat_1	`n`^mat_1
`c`^mat_1	${\dot{ε}}_{0}^{m a t_ 1}$
$m_{}^{m a t_ 1}$	$T_{0}^{m a t_ 1}$	$T_{m e l t}^{m a t_ 1}$	$T_{\lim}^{m a t_ 1}$	$ρ C_{v}^{m a t_ 1}$
$ε_{p, \max}^{m a t_1}$	$σ_{\max}^{m a t_1}$	$K_{A}^{m a t_ 1}$	$K_{B}^{m a t_ 1}$

#Material2 Parameters

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

#Material3 Parameters

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

#Material4 Parameters (Explosive)

(1)	(3)	(5)	(7)	(9)
$α_{0}^{m a t_ 4}$	$ρ_{0}^{m a t_ 4}$	$E_{0}^{m a t_ 4}$	$Δ P_{\min}^{m a t_ 4}$	$C_{0}^{m a t_ 4}$
`A`	`B`	`R`₁	`R`₂	$ω$
`D`	`P`_CJ	$C_{1}^{m a t_ 4}$		`I`_BFRAC

Definitions

Field	Contents	SI Unit Example
`mat_ID`	Material identifier. (Integer, maximum 10 digits)
`unit_ID`	Unit Identifier (Integer, 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)	$[\frac{kg}{m^{3}}]$
$E_{0}^{m a t_i}$	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`_ext 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. 9 (Real)	$[Pa]$
$C_{5}^{m a t_i}$	Hydrodynamic coefficient. (Real)
$G_{1}^{m a t_i}$	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)
${\dot{ε}}_{0}^{m a t_ i}$	Reference strain rate. If $\dot{ε} \leq {\dot{ε}}_{0}^{m a t _ i}$ , no strain rate effect (Real)	$[\frac{1}{s}]$
$m_{}^{m a t_ i}$	Temperature exponent. Default = 1.00 (Real)
$T_{0}^{m a t_i}$	Initial temperature. Default = 300 K (Real)	$[K]$
$T_{m e l t}^{m a t_i}$	Melting temperature. = 0 No temperature effect Default = 10³⁰ (Real)	$[K]$
$T_{\lim}^{m a t_i}$	Maximum temperature. Default = 10³⁰ (Real)	$[K]$
$ρ C_{v}^{m a t_i}$	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)
$σ_{\max}^{m a t_i}$	Plasticity maximum stress. Default = 10³⁰ (Real)	$[Pa]$
$K_{A}^{m a t_i}$	Thermal conductivity coefficient 1. 8 (Real)	$[\frac{W}{m \cdot K}]$
$K_{B}^{m a t_i}$	Thermal conductivity coefficient 2. 8 (Real)	$[\frac{W}{m \cdot K^{2}}]$
$α_{0}^{m a t_4}$	Initial volumetric fraction of unreacted explosive. 4 (Real)
$ρ_{0}^{m a t_4}$	Initial density of unreacted. explosive (Real)	$[\frac{kg}{m^{3}}]$
$E_{0}^{m a t_4}$	Detonation energy. (Real)	$[\frac{J}{m^{3}}]$
$Δ P_{\min}^{m a t_4}$	Minimum pressure. 5 Default = $- P_{e x t}$ (Real)	$[Pa]$
$C_{0}^{m a t_4}$	Initial pressure of unreacted explosive. (Real)	$[Pa]$
`A`	JWL EOS coefficient. (Real)	$[Pa]$
`B`	JWL EOS coefficient. (Real)	$[Pa]$
`R`₁	JWL EOS coefficient. (Real)
`R`₂	JWL EOS coefficient. (Real)
$ω$	JWL EOS coefficient. (Real)
`D`	Detonation velocity.	$[\frac{m}{s}]$
`P`_CJ	Chapman-Jouget pressure. (Real)	$[Pa]$
$C_{1}^{m a t_4}$	Hydrodynamic coefficient for unreacted explosive. 9 (Real)	$[Pa]$
`I`_BFRAC	Burn Fraction Calculation flag. 11 = 0 Volumetric Compression + Burning Time = 1 Volumetric Compression only = 2 Burning Time only (Integer)

Example

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW51/99
99.99% Water + 0.01% Air-MULTIMAT:AIR+WATER+TNT,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
        10
#---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           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                1E+5 
#              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:not defined Plastic material with Johnson-Cook Yield criteria-----------------------#
#            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           SIGMA_Y_3                BB_3                 N_3
                   0                   0                   0                   0
#               CC_3     EPSILON_DOT_0_3
                   0                   0
#               CM_3                T_30             T_3MELT            T_3LIMIT             RHOCV_3
                   0                   0                   0                   0                   0
#      EPSILON_MAX_3         SIGMA_MAX_3               K_A_3               K_B_3
                   0                   0                   0                   0
#---Material#4:TNT(JWL)----------------------------------------------------------------------------#
#            ALPHA_4             RHO_0_4               E_0_4             P_MIN_4               C_0_4
                 0.0                1590              7.0E+9               1E-30             1.0E+05
#                B_1                 B_2                 R_1                 R_2                   W
           371.20E+9            3.231E+9                4.15              0.9499                 0.3
#                  D                P_CJ                C_14                       I_BFRAC
              6930.0             2.1E+10             22.5E+5                             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 $Δ P$ .(7)
$Δ P = \max {Δ P \min, C_{0} + C_{1} μ + C_{2}^{'} μ^{2} + C_{3}^{'} μ^{3} + (C_{4} + C_{5} μ) E (μ)}$

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 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:(8)
$τ = μ [(\nabla \otimes V) +^{t} (\nabla \otimes V)] + λ (\nabla V) I$

Where,

$ν = μ / ρ$

Kinematic shear viscosity flag

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

Kinematic volumetric viscosity flag
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:(9)
$P_{\min}^{m a t_i} = Δ P_{\min}^{m a t _ i} + P_{e x t}$

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.
By default, the process is adiabatic: $δ Q = 0$ . 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:(10)
$ρ C_{V}^{mat_i}, K_{A}^{mat_i}, K_{B}^{mat_i} and T_{0}^{mat_i}$

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:(11)
$K (T) = K_{A} + K_{B} T$
$C_{1}^{m a t_4}$ can be estimated ¹ with (12)
$C_{1}^{m a t_4} = ρ_{0}^{m a t_4} \cdot {(c_{0}^{u n r e a c t e d})}^{2}$

Where, $c_{0}^{u n r e a c t e d}$ is the speed of sound in the unreacted explosive and an estimation for TNT is 2000 m/s.
Explosive material ignition is made with detonator cards, /DFS/DETPOINT or /DFS/DETPLAN.
Detonation Velocity (D) and Chapman Jouget Pressure (P_CJ) are used to compute the burn fraction calculation ( $B_{f r a c} \in [0, 1]$ ). It controls the release of detonation energy and corresponds to a factor which multiplies JWL pressure.
For a given time: $P (V, E) = B_{f r a c} P_{j w l} (V, E)$ .

A detonation time T_det is computed by the Starter from the detonation velocity. During the simulation the burn fraction is computed as:(13)
$B_{f r a c} = \min (1, m a x ( B_{f 1}, B_{f 2})) $

Where, the burn fraction calculation from burning time is:(14)
$B_{f 1} = \frac{1 - V}{1 - V_{C J}} = \frac{ρ_{0} D^{2}}{P_{C J}} (1 - V)$

and the burn fraction calculation from volumetric compression is:(15)
$B_{f 2} = {\begin{matrix} 0, \begin{matrix} \end{matrix} x < 0 \\ \frac{T - T_{d e t}}{1.5 Δ x}, \begin{matrix} \end{matrix} x \geq 0 \end{matrix}$

It can take several cycles for the burn fraction to reach its maximum value of 1.00.

Burn fraction calculation can be changed defining the I_BFRAC flag:

I_BFRAC = 1: $B_{f r a c} = min (1, B_{f 1})$

I_BFRAC = 2: $B_{f r a c} = min (1, B_{f 2})$
As of version 11.0.240, Time Histories for Detonation time and burn fraction are available through /TH/BRIC with BFRAC keyword. This allows to output a function $f$ whose first value is detonation time (with opposite sign) and positive values corresponds to the burn fraction evolution.(16)
$\begin{array}{l} T_{\det} = - f (0) \\ B_{f r a c} (t) = {\begin{cases} 0, f (t) < 0 \\ f (t), f (t) \geq 0 \end{cases} \end{array}$
Detonation times can be written in the Starter output file for each JWL element. The printout flag (Ipri) must be greater than or equal to 3 (/IOFLAG).
Material tracking is possible through animation files:
/ANIM/BRICK/VFRAC (volumetric fractions for all materials)

Hayes, B. "Fourth Symposium (International) on Detonation." Proceedings, Office of Naval Research, Department of the Navy, Washington, DC (1965): 595-601