k t
ð Þ ¼ 1 þ
1
Γ 1 À α
ð
Þ
ð t
0
t À s
ð
Þ
Àα
η s
ð Þds
ð97Þ
Where 0 < α < 1 , Γ(x) ¼
Ð
0
1
z
x À 1 e
À z dz is the Eulerian Gamma function and
η(s) is a suitable function such that lim t ! 1 k(t) < 1. The time derivative of k
becomes the left-sided Riemann-Liouville fractional derivative, i.e.,
0 D
0
t η t
ð Þ ¼
1
Γ 1 À α
ð
Þ
d
dt
ð t
0
t À s
ð
Þ
Àα
η s
ð Þds
ð98Þ
The behavior of the dynamic moduli arising from a fractional order viscoelastic
kernel, like (96), has been studied in Rogers [170].
Another way to introduce fractional derivatives is through rheological models of
fractional order. In particular, the fractional Maxwell element corresponds to a
spring in series with a fractional damper. The one-dimensional linear stress, σ,
versus strain, ∈ , relation of a spring in parallel with the fractional Maxwell
element can expressed in terms of fractional derivatives [171], e.g.,
σ t
ð Þ ¼
ð t
0
μ eq þ μ 0v E β À
t À s
ð
Þ
β
ς β
!
"
#
∈ s
ð Þds
ð99Þ
Where the kelner
E β t
ð Þ ¼
X 1
i¼0
t
i
Γ 1 þ βi
ð
Þ
ð100Þ
is the Mittag-Leffler function ([142, 171], and references therein). Since E1
(t) equals the exponential function, the Mittag Leffler function is also known as
the fractional exponential function.
Fractional order models present a relevant drawback due to the difficulty in
handling numerically constitutive equations of fractional order, in particular of
differential type [158]. Moreover, the identification of the constitutive parameters
relies on a strongly ill-conditioned minimization problem. For this reason, they are
rarely implemented in commercial codes and therefore their use is very limited.
4.1.6 Quasi-Linear Viscoelasticity
Because of the inherent nonlinear behavior exhibited by most of carbon black-filled
rubber, the linear formulation is not applicable in general.
In the context of nonlinear viscoelasticity, one of the simplest model is the
Quasi-Linear viscoelastic model proposed by Fung [164]. In one dimension, he
Modeling of Non-Linear Viscoelastic Behavior of Filled Rubbers
247
ð Þ ¼ 1 þ
1
Γ 1 À α
ð
Þ
ð t
0
t À s
ð
Þ
Àα
η s
ð Þds
ð97Þ
Where 0 < α < 1 , Γ(x) ¼
Ð
0
1
z
x À 1 e
À z dz is the Eulerian Gamma function and
η(s) is a suitable function such that lim t ! 1 k(t) < 1. The time derivative of k
becomes the left-sided Riemann-Liouville fractional derivative, i.e.,
0 D
0
t η t
ð Þ ¼
1
Γ 1 À α
ð
Þ
d
dt
ð t
0
t À s
ð
Þ
Àα
η s
ð Þds
ð98Þ
The behavior of the dynamic moduli arising from a fractional order viscoelastic
kernel, like (96), has been studied in Rogers [170].
Another way to introduce fractional derivatives is through rheological models of
fractional order. In particular, the fractional Maxwell element corresponds to a
spring in series with a fractional damper. The one-dimensional linear stress, σ,
versus strain, ∈ , relation of a spring in parallel with the fractional Maxwell
element can expressed in terms of fractional derivatives [171], e.g.,
σ t
ð Þ ¼
ð t
0
μ eq þ μ 0v E β À
t À s
ð
Þ
β
ς β
!
"
#
∈ s
ð Þds
ð99Þ
Where the kelner
E β t
ð Þ ¼
X 1
i¼0
t
i
Γ 1 þ βi
ð
Þ
ð100Þ
is the Mittag-Leffler function ([142, 171], and references therein). Since E1
(t) equals the exponential function, the Mittag Leffler function is also known as
the fractional exponential function.
Fractional order models present a relevant drawback due to the difficulty in
handling numerically constitutive equations of fractional order, in particular of
differential type [158]. Moreover, the identification of the constitutive parameters
relies on a strongly ill-conditioned minimization problem. For this reason, they are
rarely implemented in commercial codes and therefore their use is very limited.
4.1.6 Quasi-Linear Viscoelasticity
Because of the inherent nonlinear behavior exhibited by most of carbon black-filled
rubber, the linear formulation is not applicable in general.
In the context of nonlinear viscoelasticity, one of the simplest model is the
Quasi-Linear viscoelastic model proposed by Fung [164]. In one dimension, he
Modeling of Non-Linear Viscoelastic Behavior of Filled Rubbers
247