shift between the applied displacement and the load [70]. This shift is proportional
to the viscous losses.
In order to explain thoroughly elastomers behavior under oscillatory deformation, let be the longitudinal displacement in an uniaxial deformation from which the
nonlinear Lagrangian strain follows as:
u t
ð Þ ¼ u 0 þ Δu sin ϖt
ð Þ
ð1Þ
∈ t
ð Þ ¼ ∈ 0 þ Δ ∈ 1 sin ωt
ð Þ
ð2Þ
obtained by dividing u by the length l 0 of the undeformed specimen. The imposed
strain function (2) implies, in the nonlinear case, the time-dependent nominal stress
response σ (t), i.e., force applied to the specimen divided by the initial area, whose
steady state response is assumed to have the Fourier series
σ t
ð Þ ¼
b 0
2
þ
X 1
k¼1
a k sin kωt
ð Þþb k cos kωt
ð Þ
½
ð 3Þ
Here
S ∈ 0 , ω, Δ ∈ 1
ð
Þ:¼
1
Δ ∈
a 1 ∈ 0 , ω, Δ ∈ 1
ð
Þ
ð 4Þ
L ∈ 0 ; ω; ∈ 1
ð
Þ:¼
1
Δ ∈ 1
b 1 ∈ 0 ; ω; ∈ 1
ð
Þ
ð 5Þ
are the storage and loss moduli, also generically referred to as complex moduli.
In general, neither S nor L depend on Δ ∈ 1 if |Δ ∈ 1 | is small (small strain). On
the contrary, the aforementioned moduli for carbon black-reinforced rubber show a
rather strong dependence on Δ ∈ 1 in the case |Δ ∈ 1 | is large. This nonlinear
amplitude dependence is called the Payne effect (see Sect. 1.3).
The storage and loss moduli frequency dependence bears no special name, but it
is of fundamental importance to understand the dynamic behavior of elastomers.
Figure 10 outlines the dynamic moduli as function of the frequency ω for
different values of static prestrain ∈ 0 [71]. At lower frequencies (ω ! 0) the
storage modulus tends to a finite nonzero value with a nonzero derivative. This
behavior cannot be described by (linear or nonlinear) standard viscoelastic constitutive equations. The data collated by [71] suggest a non-monotonic dependence of
the storage modulus upon the static prestrain ∈ 0 : from ∈ 0 ¼ 0.65 to ∈ 0 ¼ 0.75,
the storage modulus S considerably decreases, but it increases again at ∈ 0 ¼ 0.95.
A similar, but less accentuated, trend is shown by the loss modulus. Experiments
collated in [72, 73] and more recently in [74] are in agreement with Lee and Kim’s
results.
As in the static case, the dynamic behavior of elastomers also exhibits very
strong temperature dependence. This effect is much more pronounced than in the
comparable types of tests conducted upon metals, where the mechanical properties
206
G. Markovic ´ et al.
to the viscous losses.
In order to explain thoroughly elastomers behavior under oscillatory deformation, let be the longitudinal displacement in an uniaxial deformation from which the
nonlinear Lagrangian strain follows as:
u t
ð Þ ¼ u 0 þ Δu sin ϖt
ð Þ
ð1Þ
∈ t
ð Þ ¼ ∈ 0 þ Δ ∈ 1 sin ωt
ð Þ
ð2Þ
obtained by dividing u by the length l 0 of the undeformed specimen. The imposed
strain function (2) implies, in the nonlinear case, the time-dependent nominal stress
response σ (t), i.e., force applied to the specimen divided by the initial area, whose
steady state response is assumed to have the Fourier series
σ t
ð Þ ¼
b 0
2
þ
X 1
k¼1
a k sin kωt
ð Þþb k cos kωt
ð Þ
½
ð 3Þ
Here
S ∈ 0 , ω, Δ ∈ 1
ð
Þ:¼
1
Δ ∈
a 1 ∈ 0 , ω, Δ ∈ 1
ð
Þ
ð 4Þ
L ∈ 0 ; ω; ∈ 1
ð
Þ:¼
1
Δ ∈ 1
b 1 ∈ 0 ; ω; ∈ 1
ð
Þ
ð 5Þ
are the storage and loss moduli, also generically referred to as complex moduli.
In general, neither S nor L depend on Δ ∈ 1 if |Δ ∈ 1 | is small (small strain). On
the contrary, the aforementioned moduli for carbon black-reinforced rubber show a
rather strong dependence on Δ ∈ 1 in the case |Δ ∈ 1 | is large. This nonlinear
amplitude dependence is called the Payne effect (see Sect. 1.3).
The storage and loss moduli frequency dependence bears no special name, but it
is of fundamental importance to understand the dynamic behavior of elastomers.
Figure 10 outlines the dynamic moduli as function of the frequency ω for
different values of static prestrain ∈ 0 [71]. At lower frequencies (ω ! 0) the
storage modulus tends to a finite nonzero value with a nonzero derivative. This
behavior cannot be described by (linear or nonlinear) standard viscoelastic constitutive equations. The data collated by [71] suggest a non-monotonic dependence of
the storage modulus upon the static prestrain ∈ 0 : from ∈ 0 ¼ 0.65 to ∈ 0 ¼ 0.75,
the storage modulus S considerably decreases, but it increases again at ∈ 0 ¼ 0.95.
A similar, but less accentuated, trend is shown by the loss modulus. Experiments
collated in [72, 73] and more recently in [74] are in agreement with Lee and Kim’s
results.
As in the static case, the dynamic behavior of elastomers also exhibits very
strong temperature dependence. This effect is much more pronounced than in the
comparable types of tests conducted upon metals, where the mechanical properties
206
G. Markovic ´ et al.