of the pure polymer matrix towards the effect in his explanations. He assumed the
formation of a three-dimensional network structure by the filler particles (carbon
black) within the rubber matrix which significantly influences the dynamic viscoelastic properties.
Both filler structure based and debonding theories depend on specific filler
characteristics such as surface treatment and thus difficult to distinguish experimentally. The dynamic storage and loss moduli are dependent on the dynamic
strains only and independent of a simultaneous static strain [28]. The stress strain
curves are also independent of a fully equilibrated static strain. The initial modulus
in the constant strain rate test is highly rate dependent whereas the terminal
modulus is constant with strain rate. Thus the Payne effect would vanish at lower
frequencies 10
À6 –10
À5 Hz. The imposition or removal of a static strain/large
amplitude dynamic strain causes a subsequent reduction of the storage modulus at
low strain that is fully recoverable independent of the nature of disturbance. Both
cross-linked elastomers and un-cross-linked polymer melts display similar filler
effects on dynamic mechanical properties. For polymer melts the strain amplitude
corresponding to the onset of nonlinear behavior is also reduced by the addition of
filler, similar to the Payne effect. Micro sized filler particles enhance the dynamic
moduli as they could form crosslinks within the polymer. The presence of filler also
improves damping. The surface modification of the fillers, volume fraction and the
nature of strain amplitude can also affect the moduli. For instance fillers can form
weak structures or agglomerations when their sizes are too small, and it is sometimes difficult to distinguish this from filler networks. The network formation by the
filler particles can be related to the percolation phenomenon observed in composite
systems. i.e. the crosslinks are formed among the particles when the filler concentration is higher than a particular percolation concentration. Below the critical value
of loading the contribution of filler networking to low strain mechanical behavior
becomes minor. The lowest filler concentration used in Payne’s work was 28 vol%
for the carbon black whereas for nanocomposites sometimes this value drop as low
as 0.05 %. But in short neither filler agglomeration nor network formation is a
prerequisite for nonlinear behavior. In addition, the influence of the interface on
far-field matrix behavior may be a significant factor in high molecular weight
polymers.
Three main mathematical models were derived subsequently in order to explain
the Payne effect, of which the one proposed by Kraus stands first [28]. The Kraus
model assumes the formation of agglomeration of filler aggregates due to the
existing Van der waal’s interaction between the particles. The increase in strain
amplitude causes deagglomeration kinetics thereby decreasing the storage modulus. This model was further developed by Huber and Vilgis [29] who suggested the
dependence of storage modulus on fractal dimensions and the connectivity of the
network, and by Kluppel, who introduced the cluster formation [30, 31]. However
these concepts do not take in to account the temperature and the frequency
dependence of the amplitude of Payne effect. Another criticism was made by
Funt [32] who argued the non existence of continuous filler networks within the
material with the help of electrical conductivity measurements. Moreover most of
8
D. Ponnamma and S. Thomas
formation of a three-dimensional network structure by the filler particles (carbon
black) within the rubber matrix which significantly influences the dynamic viscoelastic properties.
Both filler structure based and debonding theories depend on specific filler
characteristics such as surface treatment and thus difficult to distinguish experimentally. The dynamic storage and loss moduli are dependent on the dynamic
strains only and independent of a simultaneous static strain [28]. The stress strain
curves are also independent of a fully equilibrated static strain. The initial modulus
in the constant strain rate test is highly rate dependent whereas the terminal
modulus is constant with strain rate. Thus the Payne effect would vanish at lower
frequencies 10
À6 –10
À5 Hz. The imposition or removal of a static strain/large
amplitude dynamic strain causes a subsequent reduction of the storage modulus at
low strain that is fully recoverable independent of the nature of disturbance. Both
cross-linked elastomers and un-cross-linked polymer melts display similar filler
effects on dynamic mechanical properties. For polymer melts the strain amplitude
corresponding to the onset of nonlinear behavior is also reduced by the addition of
filler, similar to the Payne effect. Micro sized filler particles enhance the dynamic
moduli as they could form crosslinks within the polymer. The presence of filler also
improves damping. The surface modification of the fillers, volume fraction and the
nature of strain amplitude can also affect the moduli. For instance fillers can form
weak structures or agglomerations when their sizes are too small, and it is sometimes difficult to distinguish this from filler networks. The network formation by the
filler particles can be related to the percolation phenomenon observed in composite
systems. i.e. the crosslinks are formed among the particles when the filler concentration is higher than a particular percolation concentration. Below the critical value
of loading the contribution of filler networking to low strain mechanical behavior
becomes minor. The lowest filler concentration used in Payne’s work was 28 vol%
for the carbon black whereas for nanocomposites sometimes this value drop as low
as 0.05 %. But in short neither filler agglomeration nor network formation is a
prerequisite for nonlinear behavior. In addition, the influence of the interface on
far-field matrix behavior may be a significant factor in high molecular weight
polymers.
Three main mathematical models were derived subsequently in order to explain
the Payne effect, of which the one proposed by Kraus stands first [28]. The Kraus
model assumes the formation of agglomeration of filler aggregates due to the
existing Van der waal’s interaction between the particles. The increase in strain
amplitude causes deagglomeration kinetics thereby decreasing the storage modulus. This model was further developed by Huber and Vilgis [29] who suggested the
dependence of storage modulus on fractal dimensions and the connectivity of the
network, and by Kluppel, who introduced the cluster formation [30, 31]. However
these concepts do not take in to account the temperature and the frequency
dependence of the amplitude of Payne effect. Another criticism was made by
Funt [32] who argued the non existence of continuous filler networks within the
material with the help of electrical conductivity measurements. Moreover most of
8
D. Ponnamma and S. Thomas