/MAT/LAW77
Block Format Keyword This open cell foam material law is a generalization of LAW70. It accounts for a nonviscous compressible ideal gas flow inside of the foam and its interaction with the foam structure.
ALE simulation of the gas flow and Lagrangian simulation of the foam deformation is performed on the same elements system. Interaction between the gas flow and the structure is through Darcy law and direct application of the gas pressure to the structure.
Format
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/MAT/LAW77/mat_ID/unit_ID  
mat_title  
${\rho}_{i}$  
E_{0}  $\nu $  E_{max}  ${\epsilon}_{\mathrm{max}}$  FP_{0}  
F_{cut}  F_{smooth}  N_{L}  N_{uL}  Iflag  Shape  Hys 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

fct_ID_{L}  ${\dot{\epsilon}}_{L}$  Fscale_{L} 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

fct_ID_{uL}  ${\dot{\epsilon}}_{uL}$  Fscale_{uL} 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

${\rho}_{\mathit{gas}}$  P_{0}  $\gamma $  R  
${\rho}_{\mathit{ext}}$  P_{ext}  Iclos  Inc_gas  
$\alpha $  $\beta $  $\tau $  K  
fct_ID_{K}  fct_ID_{R} 
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) 

${\rho}_{i}$  Initial density (Real) 
$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$ 
E_{0}  Initial Young's
modulus. (Real) 
$\left[\text{Pa}\right]$ 
$\nu $  Poisson's ratio. (Real) 

E_{max}  Maximum Young's modulus. (Real) 
$\left[\text{Pa}\right]$ 
${\epsilon}_{\mathrm{max}}$  Reference strain value for the maximum
Young's modulus usage. Default = 1 (Real) 

FP_{0}  Initial foam pressure. Default = 0 (Real) 
$\left[\text{Pa}\right]$ 
F_{cut}  Cutoff frequency for strain rate
filtering. Default = 10^{20} (Real) 
$\text{[Hz]}$ 
F_{smooth}  Smooth strain rate option flag.
(Integer) 

N_{L}  Number of loading functions. Default = 0 (Integer) 

N_{uL}  Number of unloading functions. Default = 0 (Integer) 

Iflag  Unloading response control flag.
(Integer) 

Shape  Shape factor describes the "convexity"
of the unloading curve.
Default = 1.0 (Real) 

Hys  Hysteresis unloading factor. Default = 1.0 (Real) 

fct_ID_{L}  Load function
identifier. (Integer) 

${\dot{\epsilon}}_{L}$  Strain rate for load
function. (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
Fscale_{L}  Load function scale factor. Default = 1.0 (Real) 
$\left[\text{Pa}\right]$ 
fct_ID_{uL}  Unload function
identifier. (Integer) 

${\dot{\epsilon}}_{uL}$  Strain rate for unload
function. (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
Fscale_{uL}  Unload function scale factor. Default = 1.0 (Real) 
$\left[\text{Pa}\right]$ 
${\rho}_{\mathit{gas}}$  Air density. Default = 0 (Real) 
$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$ 
P_{0}  Initial pressure. Default = 0 (Real) 
$\left[\text{Pa}\right]$ 
$\gamma $  Gamma constant for the gas. Default (Real) 

${\rho}_{\mathit{ext}}$  External gas density. Default is ${\rho}_{\mathit{gas}}$ (Real) 
$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$ 
P_{ext}  External pressure. Default = P_{0} (Real) 
$\left[\text{Pa}\right]$ 
Inc_gas  Reverse flow flag.
(Integer) 

R  Initial porosity fraction of element
volume filled with the gas. (0 < R < 1)
(Real) 

$\alpha $  Linear parameter for generalized Darcy
Law. Default = 0 (Real) 

$\beta $  Quadratic parameter for generalized
Darcy Law. Default = 0 (Real) 
$\left[\frac{\text{s}}{\text{m}}\right]$ 
$\tau $  Transient parameter for generalized
Darcy Law. Default = 0 (Real) 
$\left[\text{s}\right]$ 
K  Initial foam permeability
modulus. Default = 0 (Real) 
$\left[\frac{{\text{m}}^{2}}{\text{Pa}\cdot \text{s}}\right]$ 
Iclos  Open/close the free surface of the solid
block flag (surface not connected to any solid element).
(Integer) 

fct_ID_{K}  Permeability scale factor function
(scale versus relative foam density).
(Integer) 

fct_ID_{R}  Porosity scale factor function (scale
versus relative foam density).
(Integer) 
Example (Cell Foam)
#RADIOSS STARTER
#12345678910
/UNIT/1
unit for mat
kg mm ms
#12345678910
/MAT/LAW77/1/1
Open cell foam
# RHO_I
4.5E8
# E0 NU EMAX EPS_max FP0
.1 0 5 .99
# FCUT FSMOOTH NLOAD NUNLOAD IFLAG SHAPE HYS
.1 1 1 0 3 2 1E20
# F_ID_ID SLOAD FSCALELOAD
1 0 .001
# RHO_AIR P0 GAMMA R
1.2E9 1.0E4 1.4 1.0
# RHO_EXT P_EXT ICLOSE INC_GAS
1.2E9 1.0E4 2 0
# ALPHA BETA T K
1 5 0 1e07
# F_ID_K F_ID_R
2 3
#12345678910
/FUNCT/1
LoadCurve
# X Y
.8 .11
.7 .10
.4 .05
.2 .02
0 0
.2 .004
.4 .006
.6 .01
.7 .020
.8 .050
.99 134
/FUNCT/2
Funct_2
# X Y
0.0 0.2
0.555 0.2
0.909 1.0
1.0 1.0
/FUNCT/3
Funct_3
# X Y
0.0 0.25
0.555 0.25
0.909 0.55
1.0 0.55
#12345678910
#ENDDATA
/END
Comments
 This material law can be used only with
solid nondegenerated hexa elements. This material is available only for the following
parameters in the solid property:
 I_{solid} = 1 (Belytschko)
 I_{smstr} = 1 (small strain)
 I_{frame} = 1 (non corotational)
 For stresses above the last load function, the behavior is extrapolated by using the last two load functions. In order to avoid huge stress values, it is recommended to repeat the last load function.
 We consider nonviscous flow of a perfect gas using the ALE approach.
 When ${\epsilon}_{p}$ reaches ${\epsilon}_{p}^{max}$ , in one integration point, the deviatoric stress of the corresponding integral point is permanently set to 0; however, the solid element is not deleted.
 For coupling between the foam and the gas,
use the modified Darcy's law (DupuitForchiemer):
(7) $$\alpha V+\beta {V}^{2}+\tau \frac{\partial V}{\partial t}=K\cdot Grad\left(P\right)$$Where, $V$
 Velocity of the fluid
 $P$
 Fluid pressure
 $K$
 Permeability modulus
 In order to save the structural stress,
strain and gas flow data, the following options have to be used in Engine
file:
/STATE/BRICK/STRAIN/FULL
/STATE/BRICK/STRESS/FULL
/STATE/BRICK/AUX/FULL
 Prestress of the foam and initial state of the air inside of the foam can be defined from prestress simulation and read into the model with /INIBRI/STRS_F, /INIBRI/STRA_F and /INIBRI/AUX cards.
 The following user variables are available
to output gas flow data into animation (/ANIM/BRICK/TENS) or time history (/TH/BRIC) file:
 USR1: gas density
 USR20: gas pressure
 USR21: R value
 USR22: K value
 Gas velocity vector field can be output in animation with /ANIM/VECT/GVEL.