The real behavior of filled rubber elastomers under cyclic loading conditions is quite complex as shown in Figure 23.7.1–1.

Idealized material behavior

From Figure 23.7.1–1 it is clear that the observed permanent set is different for each cycle, but the material has a tendency to stabilize after a number of cycles of loading between zero stress and a given level of strain. For a given load level along the primary loading path shown with the dashed line in Figure 23.7.1–1, the idealized representation of permanent set will be a single strain value after unloading has taken place. Since rate and time effects are ignored in this model, idealized loading and unloading take place along the same path, whether Mullins effect is included or not.

The permanent set behavior is captured by isotropic hardening Mises plasticity with an associated flow rule. In the context of finite elastic strains associated with the underlying rubberlike material, plasticity is modeled using a multiplicative split of the deformation gradient into elastic and plastic components:

where is the elastic part of the deformation gradient (representing the hyperelastic behavior) and is the plastic part of the deformation gradient (representing the stress-free intermediate configuration).

An example of modeling permanent set along with Mullins effect for a rubberlike material can be found in “Analysis of a solid disc with Mullins effect and permanent set,” Section 3.1.7 of the Abaqus Example Problems Guide.

The primary hyperelastic behavior can be defined by using any of the hyperelastic material models (see “Hyperelastic behavior of rubberlike materials,” Section 22.5.1). If test data input is used to define the hyperelastic response of the material, the data must be specified with respect to the stress-free intermediate configuration after unloading has taken place.

Permanent set can be defined through an isotropic hardening function in terms of the yield stress and the equivalent plastic strain. In this case the yield stress is the (effective) Kirchoff stress on the primary loading path from which unloading takes place, and the equivalent plastic strain is the corresponding logarithmic permanent set observed in the material. If is the true (Cauchy) stress, Kirchoff stress is defined as , where is the determinant of .

Depending on what is being modeled, permanent set may be defined as the true permanent set seen in the material after recovery of viscoelastic strains or it may include viscoelastic strains. In either case, an initial yield stress is required, below which there will be no permanent set and the behavior of the material will be fully elastic. In the case of filled rubbers this initial yield stress may correspond to a small nonzero stress; whereas for the family of thermoplastic materials, there may be a more marked value of initial yield stress.

*PLASTIC, HARDENING=ISOTROPIC

The model is intended to capture permanent set under multiaxial stress states and mild reverse loading conditions, as illustrated by Govindarajan, Hurtado, and Mars (2007). This model is not intended to capture deformation under complete reverse loading. Any rate effects apply only to the plastic part of the material definition.

The real behavior of filled rubber elastomers under cyclic loading conditions is quite complex as shown in Figure 23.7.1–1.

Your query was poorly formed. Please make corrections.

Idealized material behavior

From Figure 23.7.1–1 it is clear that the observed permanent set is different for each cycle, but the material has a tendency to stabilize after a number of cycles of loading between zero stress and a given level of strain. For a given load level along the primary loading path shown with the dashed line in Figure 23.7.1–1, the idealized representation of permanent set will be a single strain value after unloading has taken place. Since rate and time effects are ignored in this model, idealized loading and unloading take place along the same path, whether Mullins effect is included or not.

The permanent set behavior is captured by isotropic hardening Mises plasticity with an associated flow rule. In the context of finite elastic strains associated with the underlying rubberlike material, plasticity is modeled using a multiplicative split of the deformation gradient into elastic and plastic components:

where is the elastic part of the deformation gradient (representing the hyperelastic behavior) and is the plastic part of the deformation gradient (representing the stress-free intermediate configuration).

An example of modeling permanent set along with Mullins effect for a rubberlike material can be found in “Analysis of a solid disc with Mullins effect and permanent set,” Section 3.1.7 of the Abaqus Example Problems Guide.

Your query was poorly formed. Please make corrections.

The primary hyperelastic behavior can be defined by using any of the hyperelastic material models (see “Hyperelastic behavior of rubberlike materials,” Section 22.5.1). If test data input is used to define the hyperelastic response of the material, the data must be specified with respect to the stress-free intermediate configuration after unloading has taken place.

Permanent set can be defined through an isotropic hardening function in terms of the yield stress and the equivalent plastic strain. In this case the yield stress is the (effective) Kirchoff stress on the primary loading path from which unloading takes place, and the equivalent plastic strain is the corresponding logarithmic permanent set observed in the material. If is the true (Cauchy) stress, Kirchoff stress is defined as , where is the determinant of .

Depending on what is being modeled, permanent set may be defined as the true permanent set seen in the material after recovery of viscoelastic strains or it may include viscoelastic strains. In either case, an initial yield stress is required, below which there will be no permanent set and the behavior of the material will be fully elastic. In the case of filled rubbers this initial yield stress may correspond to a small nonzero stress; whereas for the family of thermoplastic materials, there may be a more marked value of initial yield stress.

*PLASTIC, HARDENING=ISOTROPIC

The model is intended to capture permanent set under multiaxial stress states and mild reverse loading conditions, as illustrated by Govindarajan, Hurtado, and Mars (2007). This model is not intended to capture deformation under complete reverse loading. Any rate effects apply only to the plastic part of the material definition.