times 30 min experiments and the appropriate data handling, a mere exploitation of
well established principles of linear viscoelasticity yields a precise characterization
of a gum elastomer in the room-to-processing temperature range, and in a wide
frequency range. It is worth underlining here that, because of the arbitrary choice of
λ i values (in fact logarithmic frequency decades within the experimental window),
the number of parameters needed to drawn a linear dynamic function is remarkably
small: 6 G i and 2 WLF parameters.
Furthermore using data given in Fig. 7 and basic relationships from linear
viscoelasticity, additional information can be obtained through easy calculation.
Indeed, by definition, the dynamic viscosity is: η
0 ω
ð Þ
T
¼
G
00 ω
ð Þ
ω
T
, so that data points
in Fig. 9 are easily obtained and plotted with respect to the product ω Â a T (T ref ).
Such data points correspond to the experimental dynamic viscosity function at the
reference temperature T ref .
An attractive mathematical model for such a dynamic viscosity function is again
the Carreau-Yasuda model, i.e.:
η
0 ω
ð Þ
T
¼ η
0
0
T
 1 þ λω
j j
a
½
nÀ1
a
ð5Þ
where η
0
0 | T is the pseudo-Newtonian dynamic viscosity (at T), λ a characteristic
time of the material, n and a are parameters of the model. However, using a
nonlinear fitting algorithm to boldly apply this model to experimental data such
as given in Fig. 8 does not generally yield satisfactory results in the author’s
experience, essentially because the pseudo-Newtonian plateau is clearly out of
reach of experimental capabilities. Consequently the fitting algorithm yields η
0
0
values with a very poor degree of confidence. Considering the following relationships from the theory of linear viscoelasticity can solve the problem:
η 0 ¼ lim
_
γ!0
η _
γ
ð Þ ¼ lim
ω!0
η
0 ω
ð Þ
ð6aÞ
and
η 0 ¼
X n
i¼1
G i λ i
ð6bÞ
Indeed η
0
0 in Eq. (5) can be calculated using Eq. (6b) and the λ i , G i set obtained
through the frequency-temperature sweep experiments and the use of the generalized Maxwell model. Moreover, a fair estimate of the flow index n can be obtained
by fitting a power law to high frequency data. The non-linear fitting problem
reduces thus in estimating only two parameters, i.e. λ and a, that are in fact within
the experimental window. The dynamic viscosity function drawn in Fig. 8 illustrates the validity of this approach. The function so obtained yields information
about the likely behavior of the material when submitted to shear flow, providing
A Multiparametric Approach of the Nonlinear Viscoelasticity of Rubber Materials
287
well established principles of linear viscoelasticity yields a precise characterization
of a gum elastomer in the room-to-processing temperature range, and in a wide
frequency range. It is worth underlining here that, because of the arbitrary choice of
λ i values (in fact logarithmic frequency decades within the experimental window),
the number of parameters needed to drawn a linear dynamic function is remarkably
small: 6 G i and 2 WLF parameters.
Furthermore using data given in Fig. 7 and basic relationships from linear
viscoelasticity, additional information can be obtained through easy calculation.
Indeed, by definition, the dynamic viscosity is: η
0 ω
ð Þ
T
¼
G
00 ω
ð Þ
ω
T
, so that data points
in Fig. 9 are easily obtained and plotted with respect to the product ω Â a T (T ref ).
Such data points correspond to the experimental dynamic viscosity function at the
reference temperature T ref .
An attractive mathematical model for such a dynamic viscosity function is again
the Carreau-Yasuda model, i.e.:
η
0 ω
ð Þ
T
¼ η
0
0
T
 1 þ λω
j j
a
½
nÀ1
a
ð5Þ
where η
0
0 | T is the pseudo-Newtonian dynamic viscosity (at T), λ a characteristic
time of the material, n and a are parameters of the model. However, using a
nonlinear fitting algorithm to boldly apply this model to experimental data such
as given in Fig. 8 does not generally yield satisfactory results in the author’s
experience, essentially because the pseudo-Newtonian plateau is clearly out of
reach of experimental capabilities. Consequently the fitting algorithm yields η
0
0
values with a very poor degree of confidence. Considering the following relationships from the theory of linear viscoelasticity can solve the problem:
η 0 ¼ lim
_
γ!0
η _
γ
ð Þ ¼ lim
ω!0
η
0 ω
ð Þ
ð6aÞ
and
η 0 ¼
X n
i¼1
G i λ i
ð6bÞ
Indeed η
0
0 in Eq. (5) can be calculated using Eq. (6b) and the λ i , G i set obtained
through the frequency-temperature sweep experiments and the use of the generalized Maxwell model. Moreover, a fair estimate of the flow index n can be obtained
by fitting a power law to high frequency data. The non-linear fitting problem
reduces thus in estimating only two parameters, i.e. λ and a, that are in fact within
the experimental window. The dynamic viscosity function drawn in Fig. 8 illustrates the validity of this approach. The function so obtained yields information
about the likely behavior of the material when submitted to shear flow, providing
A Multiparametric Approach of the Nonlinear Viscoelasticity of Rubber Materials
287