/MAT/LAW88

Block Format Keyword This law represents the behavior of a hyperelastic material with strain rate effects. This law is generally used to model incompressible rubbers, polymers, foams, and elastomers. It is defined by a family of stress vs strain curves at different strain rates.

Unloading can be represented using an unloading function or by providing hysteresis and shape factor inputs to a damage model based on energy. This law is only compatible with solid elements.

Format

(1)	(3)	(5)	(7)	(8)	(9)
/MAT/LAW88/`mat_ID`/`unit_ID`
`mat_title`
$ρ_{i}$
$ν$	`K`	`F`_cut	`F`_smooth	`N`_L
`fct_ID`_unL	`Fscale`_unL	`Hys`	`Shape`		`Tension`

If N_L > 0, define N_L functions per line

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

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)	$[\frac{kg}{m^{3}}]$
$ν$	Poisson ratio For incompressible materials 0.495 is maximum value. (Real)
`K`	Bulk modulus (Real)	$[Pa]$
`F`_cut	Cutoff frequency for strain rate filtering Default = 10³⁰ (Real)
`F`_smooth	Smooth strain rate option flag = 0 (Default) No strain rate smoothing = 1 Strain rate smoothing active (Integer)
`N`_L	Number of loading stress strain curve (Integer)
`fct`_unL	Unloading engineering stress vs engineering strain function identifier. 3 (Integer)
`Fscale`_unL	Unloading function scale factor Default = 1.0 (Real)	$[Pa]$
`Hys`	Hysteresis unloading factor. Ignore if, unloading function is used. 3 0.0 ≤ `Hys` ≤ 1.0 Default = 0.0 (Real)
`Shape`	Shape factor. Ignored if, unloading function is used. 3 Default = 1.0 (Real)
`Tension`	Unloading rate effects option flag During loading in tension or compression, the loading strain rate dependent curves are always used. For unloading, the following options are available: =0 (Default) Unloading follows the quasi-static unloading curve. =1 Unloading follows the loading strain rate dependent curves and the unloading curve is not used. =-1 Unloading in tension follows the quasi static unloading curve; Unloading in compression follow the loading strain rate dependent curves.
`fct_ID`_Li	Loading function identifier defining engineering stress vs engineering strain for i^th strain rate function. (Integer)
${\dot{ε}}_{L i}$	Strain rate for i^th loading engineering stress vs engineering strain function. (Real)	$[\frac{1}{s}]$
`Fscale`_Li	Scale factor for i^th loading function. Default = 1.0 (Real)	$[Pa]$

Example (Rubber)

#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/LAW88/1/1
rubber
#              RHO_I        
                1E-6 		
#                 NU                   K               F_cut  F_smooth       N_L      
                .495               19.93                   0                   1
#fctID_Unl                 Fscale_unload                 HYs               Shape   Tension     
         1                            1.                  0.                  0.         0
#fctID_l                     Fscale_load          Eps_._load
         1                            1.                  0.
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/1
function 1
#                  X                   Y
           -8.51E-01           -3.55E+01	
           -7.76E-01           -1.10E+01	
           -7.02E-01           -4.83E+00	
           -6.01E-01           -2.06E+00	
           -5.00E-01           -1.05E+00	
           -4.05E-01           -5.98E-01	
           -3.04E-01           -3.33E-01	
            0.00E+00            0.00E+00	
            4.05E-01            1.53E-01	
            8.50E-01            2.37E-01
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

This model utilizes an Ogden material formulation. Material parameters are directly derived from the input stress strain curves from uniaxial tests for different strain rates. The material is assumed to be nearly incompressible with Poisson’s ratio = 0.495.
Strain rate effects can be modelled by including loading engineering stress strain test data at different strain rates fct_ID_Li. This can be easier than calculating viscous parameters for traditional hyperelastic material models.
Unloading can be represented using an unloading function, Fscale_unL, or by providing hysteresis, Hys, and shape factor, Shape, inputs to a damage model based on energy.
When using the damage model, the loading curves are used for both loading and unloading and the unloading stress tensor is reduced by:(1)
$σ = (1 - D) σ$

with(2)
$D = (1 - H y s) (1 - {(\frac{W_{c u r}}{W_{\max}})}^{S h a p e})$

Where,

$W_{c u r}$

Current energy

$W_{\max}$

Maximum energy corresponding to the quasi-static behavior

If the unloading function, Fscale_unL, is entered, unloading is defined based on the unloading flag, Tension.