The Mullins effect (TB,CDM) is a phenomenon typically observed in compliant filled polymers. It is characterized by a decrease in material stiffness during loading and is readily observed during cyclic loading as the material response along the unloading path differs noticeably from the response that along the loading path. Although the details about the mechanisms responsible for the Mullins effect have not yet been settled, they might include debonding of the polymer from the filler particles, cavitation, separation of particle clusters, and rearrangement of the polymer chains and particles.
In the body of literature that exists concerning this phenomenon, a number of methods have been proposed as constitutive models for the Mullins effect. The model is a maximum load modification to the nearly- and fully-incompressible hyperelastic constitutive models already available. In this model, the virgin material is modeled using one of the available hyperelastic potentials, and the Mullins effect modifications to the constitutive response are proportional to the maximum load in the material history.
The pseudo-elastic model of the Mullins effect [375] is a modification of the standard thermodynamic formulation for hyperelastic materials and is given by:
 | (4–233) |
where
= virgin material deviatoric strain-energy potential
= evolving scalar damage variable
= damage function
The arbitrary limits 0 <
1 are imposed with = 1 defined as the state of the material without
any changes due to the Mullins effect. Then, along with equilibrium,
the damage function is defined by:
 | (4–234) |
which implicitly defines the damage parameter
. Using Equation 4–234, the
deviatoric part of the second Piola-Kirchhoff stress tensor is
then:
 | (4–235) |
The modified Ogden-Roxburgh damage function [376] has the following functional form of the damage variable:
 | (4–236) |
where 
, 
, and 
are material parameters and
Wm is the maximum virgin potential
over the time interval 
:
 | (4–237) |
The tangent stiffness tensor 
for a constitutive model defined by Equation 4–233 is expressed as follows:
 | (4–238) |
The derivative for 
in Equation 4–236 is:
 | (4–239) |
For user-defined pseudo-elastic Mullins
effects, define the damage function and the derivative 
via the user-programmable subroutine userMullins.