allows to easily distinguish between extra (i.e. strain-induced) and intra
(i.e. morphology-induced) nonlinear viscoelasticity.
Despite their invaluable merits, the tools and concepts of the theory of linear
viscoelastic must be used with care when addressing complex polymer systems
such as rubber compounds. Careful and reproducible measurements of the various
material functions remain thus an essential step in the science and industry of “real”
polymers. The objectives of this chapter are (1) to review a few material functions
for rubber systems, (2) to describe how they can be experimentally assessed and
(3) to demonstrate a few mathematically simple but multiparametric models that
can successfully account for the measured quantities.
2 A General Approach of Material Functions
Whatever test is performed to investigate the mechanical or the rheological properties of a (polymer) material, one has to measure simultaneously elementary
mechanical or physical quantities that can be sorted out in three categories: force,
strain (or rate of strain) and temperature. Appropriate combinations of such
elementary quantities allow defining mechanical or rheological quantities, for
instance the stress (unit: N/m or Pa) and the strain (unit: none or %) in the case of
a “mechanical” test, and the stress and the strain rate (unit: s
À1 ) in the case of a
“rheometrical” test. Combinations of mechanical or rheological quantities give
access to so-called material functions whose variables are any of the mechanical
or rheological quantities. Such a concept is easily illustrated in the case of simple
shear flow with the shear viscosity function ηð_ γ, TÞ, where _
γ is the shear rate and T
the temperature and η the ratio of the shear stress over the shear rate.
Material functions must however be considered with respect to the mode of
deformation and whether the applied strain is constant or not in time. Two simple
modes of deformation can be considered: simple shear and uniaxial extension.
When the applied strain (or strain rate) is constant, then one considers steady
material functions, e.g. ηð_ γ, TÞ or η E ð_ ε, TÞ, respectively the shear and extensional
viscosity functions. When the strain (purposely) varies with time, the only material
functions that can realistically be considered from an experimental point of view
are the so-called dynamic functions, e.g. G*(ω, γ, T) and η*(ω, γ, T) or E*(ω, γ, T)
and η
Ã
E (ω, γ, T) where the complex modulus G* (and its associated complex viscosity η*) specifically refers to shear deformation, whilst E* and η
Ã
E stand for tensile
deformation. It is worth noting here that shear and tensile dynamic deformations
can be applied to “solid” systems with currently available instruments, whilst in the
case of “molten” or “fluid” systems, only shear dynamic deformation can practically be experimented. There are indeed experimental and instrumental contingencies that severely limit the study of polymer materials in the conditions of nonlinear
viscoelasticity, relevant to processing.
The set of material functions introduced above, i.e.:
276
J.L. Leblanc
(i.e. morphology-induced) nonlinear viscoelasticity.
Despite their invaluable merits, the tools and concepts of the theory of linear
viscoelastic must be used with care when addressing complex polymer systems
such as rubber compounds. Careful and reproducible measurements of the various
material functions remain thus an essential step in the science and industry of “real”
polymers. The objectives of this chapter are (1) to review a few material functions
for rubber systems, (2) to describe how they can be experimentally assessed and
(3) to demonstrate a few mathematically simple but multiparametric models that
can successfully account for the measured quantities.
2 A General Approach of Material Functions
Whatever test is performed to investigate the mechanical or the rheological properties of a (polymer) material, one has to measure simultaneously elementary
mechanical or physical quantities that can be sorted out in three categories: force,
strain (or rate of strain) and temperature. Appropriate combinations of such
elementary quantities allow defining mechanical or rheological quantities, for
instance the stress (unit: N/m or Pa) and the strain (unit: none or %) in the case of
a “mechanical” test, and the stress and the strain rate (unit: s
À1 ) in the case of a
“rheometrical” test. Combinations of mechanical or rheological quantities give
access to so-called material functions whose variables are any of the mechanical
or rheological quantities. Such a concept is easily illustrated in the case of simple
shear flow with the shear viscosity function ηð_ γ, TÞ, where _
γ is the shear rate and T
the temperature and η the ratio of the shear stress over the shear rate.
Material functions must however be considered with respect to the mode of
deformation and whether the applied strain is constant or not in time. Two simple
modes of deformation can be considered: simple shear and uniaxial extension.
When the applied strain (or strain rate) is constant, then one considers steady
material functions, e.g. ηð_ γ, TÞ or η E ð_ ε, TÞ, respectively the shear and extensional
viscosity functions. When the strain (purposely) varies with time, the only material
functions that can realistically be considered from an experimental point of view
are the so-called dynamic functions, e.g. G*(ω, γ, T) and η*(ω, γ, T) or E*(ω, γ, T)
and η
Ã
E (ω, γ, T) where the complex modulus G* (and its associated complex viscosity η*) specifically refers to shear deformation, whilst E* and η
Ã
E stand for tensile
deformation. It is worth noting here that shear and tensile dynamic deformations
can be applied to “solid” systems with currently available instruments, whilst in the
case of “molten” or “fluid” systems, only shear dynamic deformation can practically be experimented. There are indeed experimental and instrumental contingencies that severely limit the study of polymer materials in the conditions of nonlinear
viscoelasticity, relevant to processing.
The set of material functions introduced above, i.e.:
276
J.L. Leblanc