weak polymer À filler network, the chemical network, and the entanglement network. The model of variable network density proposed by Maier and Goritz has
been applied to explain the nonlinear behavior. The activation energy of desorption
was calculated and found to be within the range of Van der Waal’s interaction
energy. The model fits well with the experimental results large deformation continuum theory (Pseudo Stress Models and ABAQUS model) should be used instead
of geometrically linear theory [192].
5 Conclusions and Perspectives
If the viscoelastic kernel, i.e., the material memory, decreases too fast in time, the
sensitivity of the resulting storage modulus vanishes at low frequency, in contrast to
the experimental evidence collated. An integrability condition has been introduced
to discriminate, among the kernels, those with a sufficiently slow rate of decay.
Luckily, this is not in contrast with the Principle of Fading Memory which, instead,
imposes a sufficiently fast decay of the material’s memory. Indeed, we have shown
two examples of kernels satisfying both requests: the well-known fractional kernel,
derived from fractional rheological models, and the hypergeometric kernel, an
original proposal. The standard linear viscoelastic model endowed with such
kernels is able to match accurately the experimental data. With respect to the
fractional kernel, the hypergeometric one requires a lower computational effort
and could encompass a finite sensitivity of the storage modulus when ω ! 0.
Moreover, while the fractional kernel is obtained from the solution of a fractional
differential equation, which necessarily involves time integrals, the possibility of
representing the hypergeometric kernel as the solution of evolution equations of
selected internal variables should be carefully investigated. This possibility could
be a crucial ingredient to implement the hypergeometric model into finite element
codes: implementations of kernels in terms of evolution equations are
drammatically less time-consuming and memory-expensive.
The dynamic viscoelastic properties of nanosilica-filled natural rubber composites was investigated. The objective of the present study was to look at the nonlinear
viscoelastic behavior of natural rubber filled with commercially used nanosilica.
The Payne effect is assumed to arise from the elementary mechanism consisting of
adsorption-desorption of macromolecular chains from the filler surface. It was
found that because of the small particle size and high specific surface area,
nanosilica forms stronger and more developed filler-filler network and the breakdown of these networks results in larger Payne effect. Also, the amount and
morphology of the fillers played a major role on the Payne effect.
At low loading, there is not much variation in storage modulus, loss modulus,
and loss tangent compared to gum vulcanizates. But at higher loading, a pronounced effect has been observed. This is due to the breakage of weak polymerfiller linkages and filler-filler networks at higher strain amplitude. But surprisingly,
enhanced Payne-like behavior has been observed for gum vulcanizates at room
temperature where there are no filler-filler and no filler-polymer interactions, which
Modeling of Non-Linear Viscoelastic Behavior of Filled Rubbers
261
been applied to explain the nonlinear behavior. The activation energy of desorption
was calculated and found to be within the range of Van der Waal’s interaction
energy. The model fits well with the experimental results large deformation continuum theory (Pseudo Stress Models and ABAQUS model) should be used instead
of geometrically linear theory [192].
5 Conclusions and Perspectives
If the viscoelastic kernel, i.e., the material memory, decreases too fast in time, the
sensitivity of the resulting storage modulus vanishes at low frequency, in contrast to
the experimental evidence collated. An integrability condition has been introduced
to discriminate, among the kernels, those with a sufficiently slow rate of decay.
Luckily, this is not in contrast with the Principle of Fading Memory which, instead,
imposes a sufficiently fast decay of the material’s memory. Indeed, we have shown
two examples of kernels satisfying both requests: the well-known fractional kernel,
derived from fractional rheological models, and the hypergeometric kernel, an
original proposal. The standard linear viscoelastic model endowed with such
kernels is able to match accurately the experimental data. With respect to the
fractional kernel, the hypergeometric one requires a lower computational effort
and could encompass a finite sensitivity of the storage modulus when ω ! 0.
Moreover, while the fractional kernel is obtained from the solution of a fractional
differential equation, which necessarily involves time integrals, the possibility of
representing the hypergeometric kernel as the solution of evolution equations of
selected internal variables should be carefully investigated. This possibility could
be a crucial ingredient to implement the hypergeometric model into finite element
codes: implementations of kernels in terms of evolution equations are
drammatically less time-consuming and memory-expensive.
The dynamic viscoelastic properties of nanosilica-filled natural rubber composites was investigated. The objective of the present study was to look at the nonlinear
viscoelastic behavior of natural rubber filled with commercially used nanosilica.
The Payne effect is assumed to arise from the elementary mechanism consisting of
adsorption-desorption of macromolecular chains from the filler surface. It was
found that because of the small particle size and high specific surface area,
nanosilica forms stronger and more developed filler-filler network and the breakdown of these networks results in larger Payne effect. Also, the amount and
morphology of the fillers played a major role on the Payne effect.
At low loading, there is not much variation in storage modulus, loss modulus,
and loss tangent compared to gum vulcanizates. But at higher loading, a pronounced effect has been observed. This is due to the breakage of weak polymerfiller linkages and filler-filler networks at higher strain amplitude. But surprisingly,
enhanced Payne-like behavior has been observed for gum vulcanizates at room
temperature where there are no filler-filler and no filler-polymer interactions, which
Modeling of Non-Linear Viscoelastic Behavior of Filled Rubbers
261