was generalized later to the anisotropic case by [123]. However, in filled rubber,
constitutive nonlinearities turn out to be significant even at small strains because of
their internal entangled structure. In addition, when the material is able to bear large
strains, geometrical nonlinearities also become relevant. Therefore an approach
considering both constitutive and geometrical nonlinearities must be pursued.
The early stage of development, following the works of Maxwell, Kelvin and
Voight, has constituted the starting point for many other researchers. In particular in
the mid 40s different modeling strategies have actually been used to describe
nonlinear viscoelastic solids.
4.1.2 Internal-Variables Formulations
A general approach, introduced by [124], is to formulate the constitutive equation in
terms of thermodynamic state-variables: the internal energy is expressed as function of both the current values of strain (stress) and the so-called internal state
variables [125–127]. The latter may be identified with local micro-structural quantities, e.g., filler content [49]. Rate effects are introduced through evolution equations, which usually relate time rates-of-change of internal variables to
thermodynamic forces, which are the derivatives of the internal energy with respect
to each internal variable. Simo [51] proposed a constitutive equation based on an
internal variables formulation which has provided a starting point for many successive works [52, 128–130]. In Simo’s approach, the internal energy is split
according to the multiplicative decomposition of the deformation gradient into
dilatational and volume-preserving parts 2.
Even though this choice might lead to non-physical results at finite strains [113],
Simo’s model is able to reproduce the hysteretic behaviors of carbon-filled rubber,
incorporating also the Mullins effect. In this case the internal energy is split as the
sum of three different parts: (i) a volumetric term depending upon the volume
change J ¼ detF, (ii) a term depending upon the isochoric deformation F ¼ J
À1 F
and (iii) a term relying on the internal variable q representing the nonequilibrium
part of the stress. The evolution equation of q is postulated assuming the generalized force is proportional to the derivative of the internal energy with respect to the
isochoric strain. This approach is found to be computationally very efficient, and
thus adopted in many commercial finite element codes. However, Simo’s model has
not been conclusively proven to satisfy the second law of thermodynamics for all
the admissible processes. Hence, it is essentially restricted to viscoelastic response
for strain states near the elastic equilibrium.
Govindjee and Simo [52] developed a similar model on the basis of the
micromechanical structures of the carbon black particles and rubber matrices: the
relaxation processes in the material were described through stress-like internal
variables. These variables are governed by dissipative evolution equations, and
interpreted as the nonequilibrium stresses due to the interaction between the
polymer chains. Holzapfel [128] proposed a model in which the internal energy is
Modeling of Non-Linear Viscoelastic Behavior of Filled Rubbers
241
constitutive nonlinearities turn out to be significant even at small strains because of
their internal entangled structure. In addition, when the material is able to bear large
strains, geometrical nonlinearities also become relevant. Therefore an approach
considering both constitutive and geometrical nonlinearities must be pursued.
The early stage of development, following the works of Maxwell, Kelvin and
Voight, has constituted the starting point for many other researchers. In particular in
the mid 40s different modeling strategies have actually been used to describe
nonlinear viscoelastic solids.
4.1.2 Internal-Variables Formulations
A general approach, introduced by [124], is to formulate the constitutive equation in
terms of thermodynamic state-variables: the internal energy is expressed as function of both the current values of strain (stress) and the so-called internal state
variables [125–127]. The latter may be identified with local micro-structural quantities, e.g., filler content [49]. Rate effects are introduced through evolution equations, which usually relate time rates-of-change of internal variables to
thermodynamic forces, which are the derivatives of the internal energy with respect
to each internal variable. Simo [51] proposed a constitutive equation based on an
internal variables formulation which has provided a starting point for many successive works [52, 128–130]. In Simo’s approach, the internal energy is split
according to the multiplicative decomposition of the deformation gradient into
dilatational and volume-preserving parts 2.
Even though this choice might lead to non-physical results at finite strains [113],
Simo’s model is able to reproduce the hysteretic behaviors of carbon-filled rubber,
incorporating also the Mullins effect. In this case the internal energy is split as the
sum of three different parts: (i) a volumetric term depending upon the volume
change J ¼ detF, (ii) a term depending upon the isochoric deformation F ¼ J
À1 F
and (iii) a term relying on the internal variable q representing the nonequilibrium
part of the stress. The evolution equation of q is postulated assuming the generalized force is proportional to the derivative of the internal energy with respect to the
isochoric strain. This approach is found to be computationally very efficient, and
thus adopted in many commercial finite element codes. However, Simo’s model has
not been conclusively proven to satisfy the second law of thermodynamics for all
the admissible processes. Hence, it is essentially restricted to viscoelastic response
for strain states near the elastic equilibrium.
Govindjee and Simo [52] developed a similar model on the basis of the
micromechanical structures of the carbon black particles and rubber matrices: the
relaxation processes in the material were described through stress-like internal
variables. These variables are governed by dissipative evolution equations, and
interpreted as the nonequilibrium stresses due to the interaction between the
polymer chains. Holzapfel [128] proposed a model in which the internal energy is
Modeling of Non-Linear Viscoelastic Behavior of Filled Rubbers
241