/MAT/LAW70 (FOAM_TAB)

Block Format Keyword This law describes the visco-elastic foam tabulated material. This material law can be used only with solid elements.

Format

(1)	(3)	(4)	(5)	(6)	(7)	(9)
/MAT/LAW70/`mat_ID`/`unit_ID` or /MAT/FOAM_TAB/`mat_ID`/`unit_ID`
`mat_title`
$ρ_{i}$
`E`₀	`v`		`E`_max		$ε_{\max}$	`Itens`
`F`_cut	`F`_smooth	`N`_L	`N`_uL	`Iflag`	`Shape`	`Hys`

N_{L} > 0

, define

N_{L}

loading function per line

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

N_{u L} > 0

, define

N_{u L}

unloading function per line

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

If Itens = 1

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
`fct_ID`_T	`Fscale`_T

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)
$ρ_{i}$	Initial density (Real)	$[\frac{kg}{m^{3}}]$
`E`₀	Initial Young's modulus. 3 (Real)	$[Pa]$
`v`	Poisson's ratio. (Real)
`E`_max	Maximum Young's modulus. 3 = 0 `E`_max is equal to `E`₀ (default). (Real)	$[Pa]$
$ε_{\max}$	Reference strain value for the maximum Young's modulus usage. Default = 1 (Real)
`Itens`	Flag to activate different behavior between tensile and compression. = 0 (Default) Same behavior between the compression tensile. = 1 Different behavior between compression tensile. The tensile behavior is the compression curve multiply by Scale factor, which is defined in `fct_ID`_T. (Integer)
`F`_cut	Cutoff frequency for strain rate filtering. Default = 10³⁰ (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. 2 (Integer)
`N`_uL	Number of unloading functions. 2 (Integer)
`Iflag`	Flag to control the unloading response. 2 = 0 (Default) The material behavior follows `N`_L loading curves and `N`_uL unloading curves. = 1 The material behavior follows `N`_L loading curves (only first curve is used for quasi-static) and `N`_uL loading curves. For unloading the deviatoric stress is reduced by using the quasi-static unloading curve:(1) $σ = (1 - D) (σ + P) - P$ with $D = (\frac{σ_{unloading}}{σ_{quasi-static}})$ = 2 The material behavior follows `N`_L loading curves (only first curve used for quasi-static) and `N`_uL unloading curves. For unloading the tensor stress is reduced by using the quasi-static unloading curve:(2) $σ = (1 - D) σ$ with, $D = (\frac{σ_{unloading}}{σ_{quasi-static}})$ 3 The loading curves are used for both loading and unloading behavior. The deviatoric unloading stress is reduced by:(3) $σ = (1 - D) (σ + P) - P$ with $D = (1 - Hys) (1 - {(\frac{W_{cur}}{W_{\max}})}^{Shape})$ = 4 The loading curves are used for both loading and unloading behavior and the tensor unloading tensor stress is reduced by:(4) $σ = (1 - D) σ$ with $D = (1 - Hys) (1 - {(\frac{W_{cur}}{W_{\max}})}^{Shape})$ (Integer)
`Shape`	Shape factor. Default = 1.0 (Real)
`Hys`	Hysteresis unloading factor. Default = 1.0 (Real)
`fct_ID`_L	Load function (in compression) identifier. The first function must define the ${\dot{ε}}_{L} = 0$ strain rate. (Integer)
${\dot{ε}}_{L}$	Strain rate for load function. (Real)	$[\frac{1}{s}]$
`Fscale`_L	Load function scale factor. (Real)	$[Pa]$
`fct_ID`_uL	Unload function (in compression) identifier. The first function must define the ${\dot{ε}}_{u L} = 0$ strain rate. (Integer)
${\dot{ε}}_{u L}$	Strain rate for unload function. (Real)	$[\frac{1}{s}]$
`Fscale`_uL	Unload function scale factor. (Real)	$[Pa]$
`fct_ID`_T	Scale factor function between tensile and compression according strain. (Integer)
`Fscale`_T	Ordinate scale. (Real)

Example (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----|
#-  2. MATERIALS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW70/1/1
Foam 
#              RHO_I
                5E-8
#                 EO                  NU               E_max             EPS_max    Itens
                 .01                   0                  10                  .8        0
#              F_cut  F_smooth       N_L      N_ul     Iflag               Shape                 Hys
                  .1         1         4         0         4                   2               1E-20
#  fctID_L             Eps_._L            Fscale_L
         1                   0                .001
         2                 .01               .0015
         3                  .1                .002
         3                   1                .003
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  3. FUNCTIONS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/1
Foam
#                  X                   Y
                   0                   0 
                 .03                .002 
                 .04                .003 
                 .14                .005
                 .46                .008 
                 .63                 .01 
                 .82                 .07 
                 .83                 .08 
                 .93                 1.4 
                 .94                 2.0 
                 .95                 3.0 
                 .96                   6 
                 .97                  10 
                 .98                  35 
                 .99                 300 
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/2
Foam
#                  X                   Y
                   0                   0
                 .03                .002
                 .04                .003
                 .14                .005
                 .46                .008 
                 .63                 .01 
                 .82                 .07 
                 .83                 .08 
                 .93                 1.4 
                 .94                 2.0 
                 .95                 3.0 
                 .96                   6 
                 .97                  10 
                 .98                  35 
                 .99                 300 
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/3
Foam
#                  X                   Y
                   0                   0 
                 .03                .002 
                 .04                .003 
                 .14                .005
                 .46                .008 
                 .63                 .01 
                 .82                 .07 
                 .83                 .08 
                 .93                 1.4 
                 .94                 2.0 
                 .95                 3.0 
                 .96                   6 
                 .97                  10 
                 .98                  35 
                 .99                 300 
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

This material is available for the following parameters in the solid property:

For Hexas:

Element	`I`_solid	`I`_smstr	`I`_frame
Hexa	1	1	1
	1	1	2
	1	11	1
	1	11	2
	17	11	1
	17	11	2
	14	11	n/a

Choice of formulation depends on particular load case. It is advised to use I_solid= 1, I_smstr=11 and I_frame= 1 or 2. When hourglass appears, then fully-integrated solid elements with I_solid=14, I_smstr=11 or I_solid= 17, I_smstr= 11, I_frame= 1 or 2 can be used.

For Tetras:

Element	`I`_solid	`I`_smstr	`I`_frame
Tetra	1	1	1
Tetra	1	11	1

Flag to control the unloading response Iflag.
- If Iflag = 0, then N_L and N_uL must be greater than 0 (N_L ≥ 1 and N_uL ≥ 1).
- If Iflag = 1 or 2:
  - N_L and N_uL must be greater than 0 (N_L ≥ 1 and N_uL ≥ 1)
  - The first loading curve used for quasi-static
  - D is computed as below:(5)
    $D = (\frac{σ_{unloading}}{σ_{quasi-static}})$
    
    Where, $σ_{unloading}$ and $σ_{quasi-static}$ are the current stresses computed, respectively.
  - P is the pressure $P = - \frac{1}{3} (σ_{xx} + σ_{yy} + σ_{zz})$
- If Iflag = 3 or 4:
  - N_uL could be 0, because unloading curves are not used.
  - D is computed as:(6)
    $D = (1 - Hys) (1 - {(\frac{W_{cur}}{W_{\max}})}^{Shape})$
    
    Where, W_curv and W_max are current and maximum energy.
When $ε_{\max}$ is reached, E_max is used whatever the curve definition is.
E₀ and E_max used to calculate the current time step. According to current value of strain, Radioss interpolates Young's modulus between E₀ and E_max linearly, where E₀ is also used to calculate contact stiffness. Radioss automatically modifies E₀ if it is less than the initial value according to the input stress/strain curves tangents.
- If E₀ is not specified, use maximal initial slope of all stress strain loading curves as E₀.
- If E_max is not specified (or set default), use E_max as E₀. Specified value of E_max should be greater than E₀, otherwise also take = E₀ as E_max.
- If $ε_{\max}$ is not specified (or set default), take the strain where, E_max is reached for the first time on one of the loading curves.
- If both $ε_{\max}$ and E_max are specified, take $ε_{\max}$ where, E_max is reached for the first time on one of the loading curves.
For stresses above the last load function, the behavior is extrapolated by using the two last load functions. Then, in order to avoid huge stress values, it is recommended to repeat the last load function.
All curves need to be defined as positive abscissa and ordinate.
Function fct_ID_T is used to scale specified stress strain curve in compression. Product of this function and specified stress strain function in compression gives the stress strain function in tension. Note that stress strain function in compression can be specified only until strain is equal to 1, which corresponds to full contraction of the foam. Therefore, the stress strain function in tension can be defined only until the tensile strain of 1.
In order to recover the stress and strain the initial state file, the following options have to be saved in the ASCII Output File (STY-File):
- /OUTP/STRESS/FULL
- /OUTP/STRAIN/FULL
- /OUTP/USERS/FULL
Specific material output variables:
- USR1: Equivalent strain* ( $ε_{eq}^{*} = ε_{eq} - \frac{σ_{y}}{E}$ )
- USR2: Max of internal energy
- USR3: Current Young's modulus
- USR4: Equivalent strain $ε_{eq}$
- USR5: Status (1=loading; -1=unloading)
- USR6: Stress
- USR7: Strain rate
- USR8: Internal energy
/VISC/PRONY can be used with this material law to include viscous effects.