class of constitutive restrictions because there exist materials, e.g., porous media
and functionally graded, leading to an unacceptable approximation if higher order
spatial gradients are ignored.
Other simplifications of the constitutive laws are obtained by assuming that the
material is non-aging, which means that the microscopic changes at the time-scale
of experimental test can be ignored, which indeed, complies with the experimental
observations [119]. An additional assumption, which is also corroborated by the
experimental data, is the isotropy of the material, i.e., the material at a given point in
one reference configuration is indistinguishable in its response from the same
material after it has been statically rotated into another reference configuration.
Then, the constitutive equation is simplified accordingly. As expected, the constitutive laws for isotropic materials are, by far, the simplest ones. Finally, the internal
material constraints of incompressibility is yet another way to restrict and simplify
the constitutive laws.
All the results presented are consistent with these principles.
In the following, we are motivated to develop a general nonlinear theory of
viscoelasticity because, in the practical application of tire industry, rubber materials
are used under conditions which do not comply with the infinitesimal deformation
assumptions of the linear theory. For these materials, the range of deformation
beyond which superposition and thereby linearity holds is extremely limited.
Anyway, one of the first requirements for a nonlinear constitutive law is that, for
a very small deformation, the model reduces to the corresponding linear
model [120].
In the following section a review of the constitutive equations used to model
nonlinear viscoelastic solids undergoing isothermal deformation is provided.
4.1.1 Pioneering Works
Polymeric materials, such as rubber, exhibit a mechanical response which cannot be
properly described neither by means of elastic nor viscous effects only. In particular, elastic effects account for materials which are able to store mechanical energy
with no dissipation. On the other hand, a viscous fluid in a hydrostatic stress state
dissipates energy, but is unable to store it. As the experimental results reported in
Part 1 have shown, filled rubber present both the characteristics of a viscous fluid
and of an elastic solid. Viscoelastic constitutive relations have been introduced with
the intent of describing the behavior of such materials able to both store and
dissipate mechanical energy.
The origin of the theory of viscoelasticity may be traced to various isolated
researchers in the last decades of the nineteenth Century. This early stage of
development is essentially due to the work of Maxwell, Kelvin and Voigt who
independently studied the one dimensional response of such materials. The linear
constitutive relationships introduced therein are the base of rheological models
which are still used in many applications [121]. Their works led to Boltzmann’s
[122] first formulation of three dimensional theory for the isotropic medium, which
240
G. Markovic ´ et al.
and functionally graded, leading to an unacceptable approximation if higher order
spatial gradients are ignored.
Other simplifications of the constitutive laws are obtained by assuming that the
material is non-aging, which means that the microscopic changes at the time-scale
of experimental test can be ignored, which indeed, complies with the experimental
observations [119]. An additional assumption, which is also corroborated by the
experimental data, is the isotropy of the material, i.e., the material at a given point in
one reference configuration is indistinguishable in its response from the same
material after it has been statically rotated into another reference configuration.
Then, the constitutive equation is simplified accordingly. As expected, the constitutive laws for isotropic materials are, by far, the simplest ones. Finally, the internal
material constraints of incompressibility is yet another way to restrict and simplify
the constitutive laws.
All the results presented are consistent with these principles.
In the following, we are motivated to develop a general nonlinear theory of
viscoelasticity because, in the practical application of tire industry, rubber materials
are used under conditions which do not comply with the infinitesimal deformation
assumptions of the linear theory. For these materials, the range of deformation
beyond which superposition and thereby linearity holds is extremely limited.
Anyway, one of the first requirements for a nonlinear constitutive law is that, for
a very small deformation, the model reduces to the corresponding linear
model [120].
In the following section a review of the constitutive equations used to model
nonlinear viscoelastic solids undergoing isothermal deformation is provided.
4.1.1 Pioneering Works
Polymeric materials, such as rubber, exhibit a mechanical response which cannot be
properly described neither by means of elastic nor viscous effects only. In particular, elastic effects account for materials which are able to store mechanical energy
with no dissipation. On the other hand, a viscous fluid in a hydrostatic stress state
dissipates energy, but is unable to store it. As the experimental results reported in
Part 1 have shown, filled rubber present both the characteristics of a viscous fluid
and of an elastic solid. Viscoelastic constitutive relations have been introduced with
the intent of describing the behavior of such materials able to both store and
dissipate mechanical energy.
The origin of the theory of viscoelasticity may be traced to various isolated
researchers in the last decades of the nineteenth Century. This early stage of
development is essentially due to the work of Maxwell, Kelvin and Voigt who
independently studied the one dimensional response of such materials. The linear
constitutive relationships introduced therein are the base of rheological models
which are still used in many applications [121]. Their works led to Boltzmann’s
[122] first formulation of three dimensional theory for the isotropic medium, which
240
G. Markovic ´ et al.