Iform = 11

Block Format Keyword Able to handle up to four materials: Three elasto-plastic materials with polynomial EOS, following the available yield criteria: Johnson-Cook or Drücker-Prager, and one high explosive material with 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

. At each cycle:(1)

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}

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, $E = E_{int} / V_{0}, C_{2}^{'} = C_{2} δ_{μ \geq 0} and C_{3}^{'} = C_{3} δ_{μ \geq 0}$ mean that 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 can be computed with either Johnson-Cook model or Drücker-Prager.

Johnson-Cook:(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}

Drücker-Prager:(6)

J_{2} = (A_{0} + A_{1} P + A_{2} P^{2})

High explosive material is modeled with linear EOS if unreacted (for equilibrium purpose) and JWL EOS for detonation products:(7)

Δ P = {\begin{array}{c} C_{0} + C_{1} μ & if 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} & if T \geq T_{\det} \end{array}

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 12

Format

(1)	(3)	(4)	(5)	(6)	(7)	(9)
/MAT/LAW51/`mat_ID`/`unit_ID`
`mat_title`
Blank Format
`I`_form		`I`_{pla_1}	`I`_{pla_2}	`I`_{pla_3}
#Global parameters
`P`_ext	`v`		$ν_{v o l}$
#SubMaterial_1 parameters
(Input depends on `I`_{pla_1} flag, see below)
#SubMaterial_2 parameters
(Input depends on `I`_{pla_2} flag, see below)
#SubMaterial_3 parameters
(Input depends on `I`_{pla_3} flag, see below)
#SubMaterial_4 parameters (Necessarily Jones-Wilkins-Lee material law)
$α_{0}^{mat_ 4}$	$ρ_{0}^{mat_ 4}$		$E_{0}^{mat_ 4}$		$Δ P_{\min}^{mat_4}$	$C_{0}^{mat_ 4}$
`A`	`B`		`R`₁		`R`₂	$ω$
`D`	`P`_CJ		$C_{1}^{mat_ 4}$			`I`_BFRAC

Specific input for sub-material j (j= 1, 2, or 3) parameters.

If I_{pla_j} = 0 (no Yield criteria) sub-material input:

$α_{0}^{mat_ j}$	$ρ_{0}^{mat_ j}$	$E_{0}^{mat_ j}$	$Δ P_{\min}^{mat_j}$	$C_{0}^{mat_ j}$
$C_{1}^{mat_ j}$	$C_{2}^{mat_ j}$	$C_{3}^{mat_ j}$	$C_{4}^{mat_ j}$	$C_{5}^{mat_ j}$
$G_{1}^{mat_ j}$
	$T_{0}^{mat_ j}$	$T_{m e l t}^{mat_ j}$	$T_{limit}^{mat_ j}$	$ρ C_{v}^{mat_ j}$
$ε_{p, \max}^{mat_1}$	$σ_{\max}^{mat_1}$	$K_{A}^{mat_ 1}$	$K_{B}^{mat_ 1}$

If I_{pla_j}= 1 (Johnson-Cook Yield criteria) sub-material input:

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

If I_{pla_j} = 2 (Drücker-Prager yield criteria) sub-material input:

$α_{0}^{mat_ j}$	$ρ_{0}^{mat_ j}$	$E_{0}^{mat_ j}$	$Δ P_{\min}^{mat_j}$	$C_{0}^{mat_ j}$
$C_{1}^{mat_ j}$	$C_{2}^{mat_ j}$	$C_{3}^{mat_ j}$	$C_{4}^{mat_ j}$	$C_{5}^{mat_ j}$
$A_{0}^{mat_ j}$	$A_{1}^{mat_ j}$	$A_{2}^{mat_ j}$	$A_{\max}^{mat_ j}$
`E`^mat_j	`v`^mat_j
	$T_{0}^{mat_ j}$	$T_{m e l t}^{mat_ j}$	$T_{limit}^{mat_ j}$	$ρ C_{v}^{mat_ j}$
$ε_{p, \max}^{mat_1}$	$σ_{\max}^{mat_1}$	$K_{A}^{mat_ 1}$	$K_{B}^{mat_ 1}$

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)
`I`_{pla_j}	Yield criteria flag (sub-material index j could be 1, 2, or 3). = 0 (Default) No yield criteria = 1 Johnson-Cook = 2 Drücker-Prager (Integer)
#Global parameters
`P`_ext	External pressure. 2 Default = 0 (Real)	$[Pa]$
`v`	Global Kinematic viscosity (shear) $ν = μ / ρ$ . 3 Default = 0 (Real)	$[\frac{m^{2}}{s}]$
$ν_{vol}$	Global Kinematic viscosity (volumetric) $ν_{vol} = \frac{3 λ + 2 μ}{ρ}$ . 3 Default = 0 (Real) (Stokes Hypothesis)	$[\frac{m^{2}}{s}]$
#submaterial parameters for Polynomial EOS
$α_{0}^{mat_j}$	Initial volumetric fraction. 4 (Real)
$ρ_{0}^{mat_j}$	Initial density. (Real)	$[\frac{kg}{m^{3}}]$
$E_{0}^{mat_j}$	Initial energy per unit volume. (Real)	$[\frac{J}{m^{3}}]$
$Δ P_{\min}^{mat_j}$	Hydrodynamic cavitation pressure. 5 Default = -10^-30 (Real)	$[Pa]$
$C_{0}^{mat_j}$	Initial pressure. (Real)	$[Pa]$
$C_{1}^{mat_j}$	Hydrodynamic coefficient. (Real)	$[Pa]$
$C_{2}^{mat_j}$	Hydrodynamic coefficient. (Real)	$[Pa]$
$C_{3}^{mat_j}$	Hydrodynamic coefficient. (Real)	$[Pa]$
$C_{4}^{mat_j}$	Hydrodynamic coefficient. (Real)
$C_{5}^{mat_j}$	Hydrodynamic coefficient. (Real)
$G_{1}^{mat_j}$	Elasticity shear modulus. (Real)	$[Pa]$
#submaterial parameters specific to Johnson-Cook yield criteria
`a`^mat_j	Plasticity yield stress. (Real)	$[Pa]$
`b`^mat_j	Plasticity hardening parameter. (Real)	$[Pa]$
`n`^mat_j	Plasticity hardening exponent. Default = 1.0 (Real)
`c`^mat_j	Strain rate coefficient. = 0 No strain rate effect Default = 0.00 (Real)
${\dot{ε}}_{0}^{mat_ j}$	Reference strain rate. If $\dot{ε} \leq {\dot{ε}}_{0}^{mat _ j}$ means no strain rate effect (Real)	$[\frac{1}{s}]$
`m`^mat_j	Temperature exponent. = 0 No temperature effect (Real)
#submaterial parameters specific to thermal behavior
$T_{0}^{mat_j}$	Initial temperature. Default = 300 K (Real)	$[K]$
$T_{m e l t}^{mat_j}$	Melting temperature. Default = 10³⁰ (Real)	$[K]$
$T_{limit}^{mat_j}$	Maximum temperature. Default = 10³⁰ (Real)	$[K]$
$ρ C_{v}^{mat_j}$	Specific heat per unit of volume. 7 (Real)	$[\frac{J}{K \cdot m^{3}}]$
$ε_{p, \max}^{mat_j}$	Failure plastic strain. Default = 10³⁰ (Real)
$σ_{\max}^{mat_j}$	Plasticity maximum stress. Default = 10³⁰ (Real)	$[Pa]$
$K_{A}^{mat_j}$	Thermal conductivity coefficient 1. 8 (Real)	$[\frac{W}{m \cdot K}]$
$K_{B}^{mat_i}$	Thermal conductivity coefficient 2. 8 (Real)	$[\frac{W}{m \cdot K^{2}}]$
#submaterial parameters specific to Drücker-Prager criteria
`E`^mat_j	Young's modulus. (Real)	$[Pa]$
`v`^mat_j	Poisson's ratio. (Real)
$A_{0}^{mat_ j}$	Yield coefficient. (Real)	$[P a^{2}]$
$A_{1}^{mat_ j}$	Yield coefficient. (Real)	$[Pa]$
$A_{2}^{mat_ j}$	Yield coefficient. (Real)
$A_{\max}^{mat_ j}$	Yield coefficient. (Real)	$[P a^{2}]$
#submaterial parameters specific to Jones-Wilkins-Lee EOS
$α_{0}^{mat_4}$	Initial volumetric fraction of unreacted explosive. 4 (Real)
$ρ_{0}^{mat_4}$	Initial density of unreacted explosive. (Real)	$[\frac{kg}{m^{3}}]$
$E_{0}^{mat_4}$	Detonation energy. (Real)	$[\frac{J}{m^{3}}]$
$Δ P_{\min}^{mat_4}$	Minimum pressure. 5 Default = -10^-30 (Real)	$[Pa]$
$C_{0}^{mat_4}$	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)	$[Pa]$
$C_{1}^{mat_4}$	Bulk modulus 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/LAW51/1
Underground explosion in Sand with Air,units{kg,m,s,Pa}
#---------------------------------------------------------------------------------------------------
#                    Material Law No 51. ALE MULTI-MATERIAL SOLID LIQUID GAS
#---------------------------------------------------------------------------------------------------
            
#    IFORM                        IPLA_1    IPLA_2    IPLA_3
        11                             0         2         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
                 0.0                 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 
#                                   T_10             T_1MELT            T_1LIMIT             RHOCV_1
                                       0                   0                   0                   0
#      EPSILON_MAX_1         SIGMA_MAX_1               K_A_1               K_B_1
                   0                   0                   0                   0
#---Material#2:SAND Plastic material with Drucker-Prager Yield Criteria----------------------------#
#            ALPHA_2             RHO_0_2               E_0_2             P_MIN_2               C_0_2
                 1.0                1370                   0                   0             1.0E+05
#              C_1_2               C_2_2               C_3_2               C_4_2               C_5_2 
             1.0E+09             2.5E+09             3.0E+10                   0                   0
#               A0_2                A1_2                A2_2             A_MAX_2
                   0                   0                0.25                   0
#                E_2                NU_2             B_MAT_2            MU_MAX_2
             3.4E+09                 0.3                   0                   0
#                                   T_20             T_2MELT            T_2LIMIT             RHOCV_2
                                       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                1638             7.0E+09                   0             1.0E+05
#                B_1                 B_2                 R_1                 R_2                   W
              371199                3231                4.15              0.9499                 0.3
#                  D                P_CJ                C_14                       I_BFRAC
              6930.0             2.1E+10             4.0E+09                             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 $Δ P$ .(8)
$Δ P = \max {Δ P_{\min}, C_{0} + C_{1} μ + C_{2}^{'} μ^{2} + C_{3}^{'} μ^{3} + (C_{4} + C_{5} μ) E (μ)}$

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.

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

This means if $P_{e x t} = 0$ , the computed pressure $Δ P$ is also the total pressure $Δ P = P$ .
Kinematic viscosities are global and not specific to each material for computing viscous stress tensor:(9)
$τ = μ [(\nabla \otimes V) +^{t} (\nabla \otimes V)] + λ (\nabla V) I$

Where,

$ν = μ / ρ$

Cinematic shear viscosity flag

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

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

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

For automatic initial fraction of the volume, refer to the /INIVOL card.
$Δ P_{\min}^{mat_j}$ flag is the minimum value for the computed pressure $Δ P$ . It means that total pressure is also bounded to:
$P_{\min}^{mat_j} = Δ P_{\min}^{mat _ j} + P_{ext}$

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

For solid materials, default value $Δ P_{\min}^{mat_j}$ = -1e30 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:(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, refer to /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_{frac} = \min (1, \max ( B_{f 1}, B_{f 2})) $

Where, (14)
$B_{f 2} = \frac{1 - V}{1 - V_{C J}} = \frac{ρ_{0} D^{2}}{P_{C J}} (1 - V)$
the burn fraction calculation from burning time.(15)
$B_{f 1} = \begin{matrix} \frac{(T - T_{\det})}{1.5 Δ x} & T \geq T_{\det} \\ 0 & T < T_{\det} \end{matrix}$

is the burn fraction calculation from volumetric compression.

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: $B_{frac} = min (1, B_{f 1})$

I_BFRAC = 2: $B_{frac} = min (1, B_{f 2})$

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)
$\begin{array}{l} T_{\det} = - f (0) \\ B_{frac} (t) = {\begin{array}{r} 0, f (t) < 0 \\ f (t), f (t) \geq 0 \end{array} \end{array}$
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).

¹ Hayes, B. "Fourth Symposium (International) on Detonation." Proceedings, Office of Naval Research, Department of the Navy, Washington, DC (1965): 595-601