degree of non-linearity seems to be constant for diluted solutions whatever the
concentration in silica (Fig. 27c) and concentrations in polymer below φ c . Furthermore, a singular behaviour is observed at strains above 10 % for diluted solution of
EVA in xylene. Usually, a sigmodal decline is observed leading to a lower plateau
at high deformations. Figure 27c shows that for diluted solution the storage
modulus slightly increases for deformation higher than 100 % and goes through a
maximum around 300 % before decreasing again. However, this behaviour is less
pronounced for concentrated solution (φ ! φ c ) as the amplitude dependence of the
storage modulus only exhibits a shoulder around 300 % (Fig. 27b). It can be pointed
out that same trends were observed for the variation of the loss modulus. This
singular behaviour could be explained by a re-arrangement of the filler particles in
the flow direction after the break down of the equilibrium filler network. However,
another explanation could be the stress-induced debonding of polymer chains from
the filler surface at high deformation.
The dynamic behavior of filled rubbers is of great importance in the performance
of rubber engineering components and is essential in tire applications.
The modulus drop (ΔG ¼ G
0
0 À G
0
1 ) with strain amplitude, showing a typical
nonlinear behavior, represents the Payne effect. At larger deformations, the difference between unfilled and filled rubber (G¤) contains the contribution arising from
the inclusion of rigid particles (accounted for by the Guth and Gold expression) and
also the contribution of the polymer–filler cross links to the network structure31
(Fig. 28a). The Payne effect is influenced by filler parameters: concentration,
surface area, distribution, surface characteristics, and temperature. As seen in
Fig. 28a, the chemical modification of the silica particles by means of the coupling
agent reduces the amplitude of the Payne effect (modulus drop with strain amplitude), the loss modulus, and tgδ, which is an important parameter in the rolling
resistance of tires.
The main aspects of the nonlinear theory of elasticity are presented. As nonlinear
elasticity, and, in particular, hyperelasticity, is such a useful tool in the description
of the behavior of carbon black-filled rubber undergoing quasi-static loadings, the
main methodologies for describing the behavior of materials subjected to large
strains are introduced. Some of the results herein presented will be applied to
nonlinear viscoelastic constitutive models and discussed in subsequent review.
3 Nonlinear Elasticity
3.1 Kinematics
In the following section, the basic concepts used to describe the (finite) deformation
of a simple material are briefly presented. A comprehensive introduction of finite
elasticity can be found, for instance, in [97, 102, 103].
Modeling of Non-Linear Viscoelastic Behavior of Filled Rubbers
225
concentration in silica (Fig. 27c) and concentrations in polymer below φ c . Furthermore, a singular behaviour is observed at strains above 10 % for diluted solution of
EVA in xylene. Usually, a sigmodal decline is observed leading to a lower plateau
at high deformations. Figure 27c shows that for diluted solution the storage
modulus slightly increases for deformation higher than 100 % and goes through a
maximum around 300 % before decreasing again. However, this behaviour is less
pronounced for concentrated solution (φ ! φ c ) as the amplitude dependence of the
storage modulus only exhibits a shoulder around 300 % (Fig. 27b). It can be pointed
out that same trends were observed for the variation of the loss modulus. This
singular behaviour could be explained by a re-arrangement of the filler particles in
the flow direction after the break down of the equilibrium filler network. However,
another explanation could be the stress-induced debonding of polymer chains from
the filler surface at high deformation.
The dynamic behavior of filled rubbers is of great importance in the performance
of rubber engineering components and is essential in tire applications.
The modulus drop (ΔG ¼ G
0
0 À G
0
1 ) with strain amplitude, showing a typical
nonlinear behavior, represents the Payne effect. At larger deformations, the difference between unfilled and filled rubber (G¤) contains the contribution arising from
the inclusion of rigid particles (accounted for by the Guth and Gold expression) and
also the contribution of the polymer–filler cross links to the network structure31
(Fig. 28a). The Payne effect is influenced by filler parameters: concentration,
surface area, distribution, surface characteristics, and temperature. As seen in
Fig. 28a, the chemical modification of the silica particles by means of the coupling
agent reduces the amplitude of the Payne effect (modulus drop with strain amplitude), the loss modulus, and tgδ, which is an important parameter in the rolling
resistance of tires.
The main aspects of the nonlinear theory of elasticity are presented. As nonlinear
elasticity, and, in particular, hyperelasticity, is such a useful tool in the description
of the behavior of carbon black-filled rubber undergoing quasi-static loadings, the
main methodologies for describing the behavior of materials subjected to large
strains are introduced. Some of the results herein presented will be applied to
nonlinear viscoelastic constitutive models and discussed in subsequent review.
3 Nonlinear Elasticity
3.1 Kinematics
In the following section, the basic concepts used to describe the (finite) deformation
of a simple material are briefly presented. A comprehensive introduction of finite
elasticity can be found, for instance, in [97, 102, 103].
Modeling of Non-Linear Viscoelastic Behavior of Filled Rubbers
225