/MAT/LAW77

Block Format Keyword This open cell foam material law is a generalization of LAW70. It accounts for a non-viscous 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)	(3)	(4)	(5)	(6)	(7)	(9)
/MAT/LAW77/`mat_ID`/`unit_ID`
`mat_title`
$ρ_{i}$
`E`₀	$ν$		`E`_max		$ε_{\max}$	`FP`₀
`F`_cut	`F`_smooth	`N`_L	`N`_uL	`Iflag`	`Shape`	`Hys`

If N_L ≠ 0, each loading function per line

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
`fct_ID`_L	${\dot{ε}}_{L}$		`Fscale`_L

If N_uL ≠ 0, each unloading function per line

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
`fct_ID`_uL	${\dot{ε}}_{u L}$		`Fscale`_uL

The cards for the gas and external gas

(1)	(2)	(3)	(5)	(6)	(7)	(9)
$ρ_{gas}$		`P`₀	$γ$			`R`
$ρ_{ext}$		`P`_ext	`Iclos`	`Inc_gas`
$α$		$β$	$τ$		`K`
`fct_ID`_K	`fct_ID`_R

Definition

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)
$ρ_{i}$	Initial density (Real)	$[\frac{kg}{m^{3}}]$
`E`₀	Initial Young's modulus. (Real)	$[Pa]$
$ν$	Poisson's ratio. (Real)
`E`_max	Maximum Young's modulus. (Real)	$[Pa]$
$ε_{\max}$	Reference strain value for the maximum Young's modulus usage. Default = 1 (Real)
`FP`₀	Initial foam pressure. Default = 0 (Real)	$[Pa]$
`F`_cut	Cutoff frequency for strain rate filtering. Default = 10²⁰ (Real)	$[Hz]$
`F`_smooth	Smooth strain rate option flag. = 0 (Default) No strain rate smoothing. = 1 Strain rate smoothing active. (Integer)
`N`_L	Number of loading functions. Default = 0 (Integer)
`N`_uL	Number of unloading functions. Default = 0 (Integer)
`Iflag`	Unloading response control flag. = 0 The material behavior follows the defined curves for loading and unloading. = 1 (Default) The loading curves are used for both loading and unloading behavior. For unloading the deviatoric stress is damaged by using the quasi-static unloading curve:(1) $σ = (1 - D) (σ + P) - P$ Where, `D` is computed by respecting the quasi-static unloading curve,(2) $D = (\frac{σ_{unloading}}{σ_{quasi - static}})$ are the current stresses computed respectively from the unloading and quasi-static curves. = 2 The loading curves are used for both loading and unloading behavior. For unloading the tensor stress is reduced by using the quasi-static unloading curve:(3) $σ = (1 - D) σ$ where, `D` is computed by respecting the quasi-static unloading curve,(4) $D = (\frac{σ_{unloading}}{σ_{quasi - static}})$ are the current stresses computed respectively from the unloading and quasi-static curves. = 3 The loading curves are used for both loading and unloading behavior. The deviatoric unloading stress is reduced by:(5) $σ = (1 - D) (σ + P) - P$ (6) $D = (1 - Hys) (1 - {(\frac{W_{cur}}{W_{\max}})}^{Shape})$ Where, `W`_cur and `W`_max are current and maximum energy. If `Iflag` = 3, the unloading curves are not used. (Integer)
`Shape`	Shape factor describes the "convexity" of the unloading curve. = 1 Corresponds to linear unloading curve. < 1 Convex unloading curve. Default = 1.0 (Real)
`Hys`	Hysteresis unloading factor. Default = 1.0 (Real)
`fct_ID`_L	Load function identifier. (Integer)
${\dot{ε}}_{L}$	Strain rate for load function. (Real)	$[\frac{1}{s}]$
`Fscale`_L	Load function scale factor. Default = 1.0 (Real)	$[Pa]$
`fct_ID`_uL	Unload function identifier. (Integer)
${\dot{ε}}_{u L}$	Strain rate for unload function. (Real)	$[\frac{1}{s}]$
`Fscale`_uL	Unload function scale factor. Default = 1.0 (Real)	$[Pa]$
$ρ_{gas}$	Air density. Default = 0 (Real)	$[\frac{kg}{m^{3}}]$
`P`₀	Initial pressure. Default = 0 (Real)	$[Pa]$
$γ$	Gamma constant for the gas. Default (Real)
$ρ_{ext}$	External gas density. Default is $ρ_{gas}$ (Real)	$[\frac{kg}{m^{3}}]$
`P`_ext	External pressure. Default = `P`₀ (Real)	$[Pa]$
`Inc_gas`	Reverse flow flag. = 0 (Default) No reverse flow of outside air into the foam. = 1 Reverse flow is admitted. (Integer)
`R`	Initial porosity fraction of element volume filled with the gas. (0 < `R` < 1) = 0.0 No gas in element, no gas flow. = 1.0 Complete element volume is used for gas flow. (Real)
$α$	Linear parameter for generalized Darcy Law. Default = 0 (Real)
$β$	Quadratic parameter for generalized Darcy Law. Default = 0 (Real)	$[\frac{s}{m}]$
$τ$	Transient parameter for generalized Darcy Law. Default = 0 (Real)	$[s]$
`K`	Initial foam permeability modulus. Default = 0 (Real)	$[\frac{m^{2}}{Pa \cdot s}]$
`Iclos`	Open/close the free surface of the solid block flag (surface not connected to any solid element). = 0 (Default) Free outflow on free surface (open). = 1 No outflow on free surface (close). = 2 No gas outflow through external surfaces which are in contact. Parameter `Ibag`= 1 should be activated in corresponding contact. (Integer)
`fct_ID`_K	Permeability scale factor function (scale versus relative foam density). = 0 (Default) The permeability modulus is constant. (Integer)
`fct_ID`_R	Porosity scale factor function (scale versus relative foam density). = 0 (Default) The porosity is constant and `fct_ID`_R= `fct_ID`_K. (Integer)

Example (Cell Foam)

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
                  kg                  mm                  ms
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW77/1/1
Open cell foam
#              RHO_I
              4.5E-8
#                 E0                  NU                EMAX             EPS_max                 FP0
                  .1                   0                   5                 .99
#               FCUT  FSMOOTH     NLOAD   NUNLOAD     IFLAG               SHAPE                 HYS
                  .1         1         1         0         3                   2               1E-20
#  F_ID_ID               SLOAD          FSCALELOAD
         1                   0                .001
#            RHO_AIR                  P0               GAMMA                                       R
              1.2E-9              1.0E-4                 1.4                                     1.0
#            RHO_EXT               P_EXT    ICLOSE   INC_GAS
              1.2E-9              1.0E-4         2         0
#              ALPHA                BETA                   T                   K
                   1                   5                   0                1e07
#   F_ID_K    F_ID_R
         2         3
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|		 
/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    
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END

Comments

This material law can be used only with solid non-degenerated 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 co-rotational)
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 non-viscous flow of a perfect gas using the ALE approach.
When $ε_{p}$ reaches $ε_{p}^{m a x}$ , 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 (Dupuit-Forchiemer):(7)
$α V + β V^{2} + τ \frac{\partial V}{\partial t} = - K \cdot G r a d (P)$

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
Pre-stress of the foam and initial state of the air inside of the foam can be defined from pre-stress 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.