/FAIL/MULLINS_OR

Block Format Keyword Describes the stress softening mullins effect that is observed during a cyclic loading unloading based on the criterion proposed by Ogden and Roxburgh.

Format

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
/FAIL/MULLINS_OR/`mat_ID`/`unit_ID`
`R`		$β$ $β$		`m`

Optional line

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
`fail_ID`

Definitions

Field	Contents	SI Unit Example
`mat_ID`	Material identifier. (Integer, maximum 10 digits)
`unit_ID`	Optional unit identifier. (Integer, maximum 10 digits)
`R`	Damage parameter relative to undeformed material. 1 3 Default = 1.0 (Real)
$β$ $β$	Damage parameter. 1 3 (Real)
`m`	Damage parameter relative to deformation. 1 3 (Real)	$[J]$ $[J]$

Example

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for material and failure
                  Mg                  mm                   s
/MAT/LAW100/1/1 
Neo Hookean material
#
 1.000000000000E-09
#N_NETWORK   FLAG_HE   FLAG_CR
         0         3          
#                C10                  D1 
              0.5000
/FAIL/MULLINS_OR/1/1
#                  R                BETA                   m
                 2.0                0.02                 0.2
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

This failure model can only be used with materials /MAT/LAW92, /MAT/LAW95, and /MAT/LAW100.
The stress during the first loading process is equal to the undamaged stress. Upon unloading and reloading the stress is multiplied by a positive softening factor function:(1)
$σ = η d e v (σ) - p Ι$ $σ = η d e v (σ) - p Ι$

Where,

$d e v (σ)$ $d e v (σ)$

The deviatoric part of the stress.

$p$ $p$

Hydrostatic pressure.

$η$ $η$

Damage factor that is a function of the strain energy of the hyperelastic model which respects:

(2)
$η = {\begin{matrix} = 1, if W = W_{m a x} \\ < 1, if W < W_{m a x} \end{matrix}$ $η = {\begin{matrix} = 1, if W = W_{m a x} \\ < 1, if W < W_{m a x} \end{matrix}$

Where, $W_{m a x}$ $W_{m a x}$ is the maximum strain energy that the material has been subjected to during its loading history:(3)
$η = 1 - \frac{1}{R} e r f (\frac{W_{m a x} - W}{m + β W_{m a x}})$ $η = 1 - \frac{1}{R} e r f (\frac{W_{m a x} - W}{m + β W_{m a x}})$

Where, $e r f$ $e r f$ is the Gauss error function.
The larger the parameter $R$ $R$ , the less $η$ $η$ can depart from unity and hence less damage can occur. For small values of m, there is more damage at smaller strains. For higher values of m, there is less damage in the small strain region during the initial loading, but during reloading there is will be more damage at smaller strains. Smaller values of $β$ $β$ result in increased damage.
There is no failure or element deletion with this failure model.

Ogden, R. W., and D. G. Roxburgh, “A Pseudo-Elastic Model for the Mullins Effect in Filled Rubber,” Proceedings of the Royal Society of London, Series A, vol. 455, pp. 2861–2877, 1999