This stress softening at small deformations, called Payne-effect [10, 47], plays
an important role in the understanding of reinforcement mechanism of filled
rubber samples [48].
Perhaps the most general nonlinear viscoelastic theory using a single convolution integral instead of multiple integrals was introduced by Schapery [49] based on
irreversible thermodynamics. These models have the form of the convolution
integrals of linear viscoelasticity with the nonlinearities appearing only in the
measures of stress and strain and in the reduced time. So far most of these models
were established for high-modulus (e.g. glassy) polymers, using the usual stress and
strain measures of infinitesimal deformation. However, for rubbery polymers,
strong geometrical nonlinearities such as finite strains and finite rotations are
present, making different stress and strain measures possible for constitutive
modeling. One of the main restrictions in selecting these stress and strain tensors
is to make use of conjugate pairs, such as the second Piola–Kirchhoff stress and
Green strain or the Piola stress and deformation gradients (see Ogden [50] for the
other conjugate pairs). Another restriction is that the constitutive laws should
satisfy the principle of objectivity (material frame-indifference) to ensure that the
stress-strain response is not altered by any superposed rigid-body motions.
The range of time-dependent, finite strain constitutive models for rubbery
materials is quite limited. Perhaps the most simple and effective models are the
so called pseudo stress and pseudo strain models introduced by Schapery [49].
Basically, the time dependence of the nonlinear behavior is considered to be in a
separable form, where the viscoelasticity is accounted for by a relaxation function
that is independent of stress or strain, while the effects of large deformations are
incorporated in a reference potential. Simo [51] developed a nonlinear viscoelastic
model based on a free energy with uncoupled volumetric and deviatoric parts. The
time-dependent effects are contained in the deviatoric stress component, while
volumetric stress response is assumed to be elastic.
Govindjee and Simo [52] combined micromechanical and phenomenological
approaches to develop a continuum damage model for carbon black filled elastomers, where the softening effect is considered as the detachment of carbon particles
from the elastomer matrix.
The nonlinear constitutive models used in this study are the so-called pseudo
stress and pseudo strain models of Schapery [49]. They were quite easily
implemented and the former yielded a reasonable representation of the rubber
used in this work. The time-dependent, finite strain constitutive model from the
finite element code, ABAQUS, was also used in this study.
Modeling of Non-Linear Viscoelastic Behavior of Filled Rubbers
201
an important role in the understanding of reinforcement mechanism of filled
rubber samples [48].
Perhaps the most general nonlinear viscoelastic theory using a single convolution integral instead of multiple integrals was introduced by Schapery [49] based on
irreversible thermodynamics. These models have the form of the convolution
integrals of linear viscoelasticity with the nonlinearities appearing only in the
measures of stress and strain and in the reduced time. So far most of these models
were established for high-modulus (e.g. glassy) polymers, using the usual stress and
strain measures of infinitesimal deformation. However, for rubbery polymers,
strong geometrical nonlinearities such as finite strains and finite rotations are
present, making different stress and strain measures possible for constitutive
modeling. One of the main restrictions in selecting these stress and strain tensors
is to make use of conjugate pairs, such as the second Piola–Kirchhoff stress and
Green strain or the Piola stress and deformation gradients (see Ogden [50] for the
other conjugate pairs). Another restriction is that the constitutive laws should
satisfy the principle of objectivity (material frame-indifference) to ensure that the
stress-strain response is not altered by any superposed rigid-body motions.
The range of time-dependent, finite strain constitutive models for rubbery
materials is quite limited. Perhaps the most simple and effective models are the
so called pseudo stress and pseudo strain models introduced by Schapery [49].
Basically, the time dependence of the nonlinear behavior is considered to be in a
separable form, where the viscoelasticity is accounted for by a relaxation function
that is independent of stress or strain, while the effects of large deformations are
incorporated in a reference potential. Simo [51] developed a nonlinear viscoelastic
model based on a free energy with uncoupled volumetric and deviatoric parts. The
time-dependent effects are contained in the deviatoric stress component, while
volumetric stress response is assumed to be elastic.
Govindjee and Simo [52] combined micromechanical and phenomenological
approaches to develop a continuum damage model for carbon black filled elastomers, where the softening effect is considered as the detachment of carbon particles
from the elastomer matrix.
The nonlinear constitutive models used in this study are the so-called pseudo
stress and pseudo strain models of Schapery [49]. They were quite easily
implemented and the former yielded a reasonable representation of the rubber
used in this work. The time-dependent, finite strain constitutive model from the
finite element code, ABAQUS, was also used in this study.
Modeling of Non-Linear Viscoelastic Behavior of Filled Rubbers
201