of glassy bridges in a direction normal to the stress. They explain the strain
dependence of the elastic modulus by the local lowering of the glass transition
due to the amplification of the stress in the vicinity of the aggregates. This
plasticizing effect induces the yielding of the carbon-filled rubber can reasonably
be assumed isotropic, only isotropic constitutive relations are considered. Uniaxial
stretch histories are investigated by reducing the general three-dimensional model
to the one-dimensional case. In this context, the concept of dynamic moduli,
introduced in linear viscoelasticity and referred to as storage and loss moduli, is
applied, in a consistent manner, to nonlinear constitutive equations: it is proved that
for most of the integral models used in the literature, the sensitivity of the storage
modulus with respect to the frequency vanishes when assessed at low frequency.
The constitutive behavior of filled-elastomers is highly nonlinear for what concerns
both the quasi-static and the dynamic response. Fosdick and Yu, Hallquist’s,
Yang’s and Shime’s models are able to qualitatively describe the behavior of the
rubber-like material under investigation, even if they cannot properly catch the
dissipated energy undergoing the deformation process, especially at lower strain
rate. Fung’s model firstly introduced to describe the behavior of soft biological
tissues, has revealed completely unable to describe the dissipation properties both at
higher and lower strain rates; Hibbet’s model, instead, has produced a fitting model
which underestimates the material stiffness at very low strain in all the considered
experimental cases. For silica-filled rubber Payne effect becomes more pronounced.
Since, in many engineering applications, the material is subjected to a strain lower
than 10 %, a relevant stiffness error within this range should be consider a serious
drawback for the model applied. This allowed the stiffness around the undeformed
configuration to be evaluated in detail. The quasi-static experimental results also
allowed the influence of the Mullins effect on the quasi-static response to be
investigated: during the loading cycles, there is a significant reduction in the stress
at a given level of strain, which is a consequence of the internal material
rearrangement, i.e., the Mullins effect. This damage phenomenon is sometimes
reported to induce transverse isotropy in the material, which is usually assumed to
be isotropic.
2 Rubber Phenomenology
The behavior of carbon black-filled rubber in relation to quasi-static and dynamic
responses is examined in detail. In particular, the main features of the microstructure of the material and their influence on the macro-mechanical response
are highlighted. The effects of strain, strain-rate and temperature on the constitutive
response are discussed. Mullins and Payne effects, which are peculiar in the
behavior of filled elastomers, are reviewed and new results are shown.
Owing to its unique physical properties, rubber plays a key role in countless
industrial applications. Tyres, vibration absorbers and shoe soles are only but a few
of the myriad uses of rubber in an industry which in 2009 had an estimated market
196
G. Markovic ´ et al.
dependence of the elastic modulus by the local lowering of the glass transition
due to the amplification of the stress in the vicinity of the aggregates. This
plasticizing effect induces the yielding of the carbon-filled rubber can reasonably
be assumed isotropic, only isotropic constitutive relations are considered. Uniaxial
stretch histories are investigated by reducing the general three-dimensional model
to the one-dimensional case. In this context, the concept of dynamic moduli,
introduced in linear viscoelasticity and referred to as storage and loss moduli, is
applied, in a consistent manner, to nonlinear constitutive equations: it is proved that
for most of the integral models used in the literature, the sensitivity of the storage
modulus with respect to the frequency vanishes when assessed at low frequency.
The constitutive behavior of filled-elastomers is highly nonlinear for what concerns
both the quasi-static and the dynamic response. Fosdick and Yu, Hallquist’s,
Yang’s and Shime’s models are able to qualitatively describe the behavior of the
rubber-like material under investigation, even if they cannot properly catch the
dissipated energy undergoing the deformation process, especially at lower strain
rate. Fung’s model firstly introduced to describe the behavior of soft biological
tissues, has revealed completely unable to describe the dissipation properties both at
higher and lower strain rates; Hibbet’s model, instead, has produced a fitting model
which underestimates the material stiffness at very low strain in all the considered
experimental cases. For silica-filled rubber Payne effect becomes more pronounced.
Since, in many engineering applications, the material is subjected to a strain lower
than 10 %, a relevant stiffness error within this range should be consider a serious
drawback for the model applied. This allowed the stiffness around the undeformed
configuration to be evaluated in detail. The quasi-static experimental results also
allowed the influence of the Mullins effect on the quasi-static response to be
investigated: during the loading cycles, there is a significant reduction in the stress
at a given level of strain, which is a consequence of the internal material
rearrangement, i.e., the Mullins effect. This damage phenomenon is sometimes
reported to induce transverse isotropy in the material, which is usually assumed to
be isotropic.
2 Rubber Phenomenology
The behavior of carbon black-filled rubber in relation to quasi-static and dynamic
responses is examined in detail. In particular, the main features of the microstructure of the material and their influence on the macro-mechanical response
are highlighted. The effects of strain, strain-rate and temperature on the constitutive
response are discussed. Mullins and Payne effects, which are peculiar in the
behavior of filled elastomers, are reviewed and new results are shown.
Owing to its unique physical properties, rubber plays a key role in countless
industrial applications. Tyres, vibration absorbers and shoe soles are only but a few
of the myriad uses of rubber in an industry which in 2009 had an estimated market
196
G. Markovic ´ et al.