By projecting σ s (t) over sin(ωt) and cos(ωt), Eqs. (3.94)-(3.95) of the storage and
loss moduli are recovered.In order to evaluate the derivatives ∂S/∂ω and ∂L/∂ω,
∂H
S
i /∂ω and ∂H
C
i /∂ω can be derived from Eqs. (3.91)-(3.93), i.e.,
∂H
S
i
∂ω
¼
∂h
S
i
∂ω
ð þ1
0
_
k s
ð Þ cos iωs
ð Þds À h
S
i
ð þ1
0
is _
k s
ð Þ sin iωs
ð Þds
À
∂h
C
i
∂ω
ð þ1
0
_
k s
ð Þ sin iωs
ð Þds À h
C
i
ð þ1
0
is _
k s
ð Þ cos iωs
ð Þds
ð164Þ
∂H
S
i
∂ω
¼
∂h
S
i
∂ω
ð þ1
0
_
k s
ð Þ sin iωs
ð Þds À h
S
i
ð þ1
0
is _
k s
ð Þ cos iωs
ð Þds
À
∂h
C
i
∂ω
ð þ1
0
_
k s
ð Þ cos iωs
ð Þds À h
C
i
ð þ1
0
is _
k s
ð Þ sin iωs
ð Þds
ð165Þ
provided that the integrability condition (3.96) holds. As a consequence, when
assessed at low frequencies, the result is
∂H
S
i
∂ω
ω¼0
j
¼ k 1 À k 0
ð
Þ
∂h
S
i
∂ω
ω¼0
j
À h
C
i ω¼0
j i
ð þ1
0
s _
k s
ð Þds
ð166Þ
∂H
C
i
∂ω
ω¼0
j
¼ k 1 À k 0
ð
Þ
∂h
C
i
∂ω
ω¼0
j
À h
S
i ω¼0
j i
ð þ1
0
s _
k s
ð Þds
ð167Þ
and, therefore, these sensitivities depend upon the sensitivities of the constitutive
functions h
S
i and h
C
i , respectively. To evaluate these quantities, let us assume that
the functions f, l and h are analytic with respect to E 1 . By expanding h in Taylor
series, as in Eq. (156), and by projecting over sin(iωt) and cos(iωt), the Fourier
coefficients h
C
i and h
S
i are obtained, e.g.,
h
S
i ¼
X 1
p¼0
X p
q¼0
ω
pÀq E
p
1
q! p À q
ð
Þ!
h
q, pÀq
ð
Þ E 0
ð Þ
ð 1
À1
sin πs
ð Þ
q cos πs
ð Þ
pÀq sin iπs
ð Þds ð168Þ
h
C
i ¼
X 1
p¼0
X p
q¼0
ω
pÀq E
p
1
q! p À q
ð
Þ!
h
q, pÀq
ð
Þ E 0
ð Þ
ð 1
À1
sin πs
ð Þ
q cos πs
ð Þ
pÀq cos iπs
ð Þds ð169Þ
When assessing h
C
i and h
S
i at ω ! 0, the only term not vanishing in the infinite
summations (168) and (169) are those corresponding to n ¼ m, that is
h
S
i ω¼0
j
¼
X 1
p¼0
E
p
1
p!
h
p;0
ð Þ E 0
ð Þ
ð 1
À1
sin πs
ð Þ
p sin iπs
ð Þds
¼ 0, iEven
6 ¼ 0, iOdd
&
ð170Þ
Modeling of Non-Linear Viscoelastic Behavior of Filled Rubbers
263