P
Ã
iNE t; R; a
ð
Þ¼aP iNE t; R
ð Þ
ð17Þ
P
Ã
iNa t; R; a
ð
Þ¼aP iNa t; R
ð Þ
ð18Þ
2.4 Viscoelastic Free Energy of Deformation for Polymeric
Suspensions [8]
Song described the total free energy of viscoelasticity for polymeric suspensions as
Eqs. (19) and (20) [8]:
ΔF T ¼ kTN Ta
Â
e
τ γ _
γ
1
Γ m a
ð Þ
t À t
0
maÀ1
e
Àμ a tÀt
0
ð Þ C
a
100 I I þ C
a
020 I II þ C
a
200 I III
À
Á
þ N TE
1
Γ mE
ð Þ
t À t
0
mEÀ1
e
Àμ E tÀt
0
ð Þ C
E
100 I I þ C
E
020 I II þ C
E
200 I III
À
Á Ã ð19Þ
I I ¼ a
2
x þ a
2
y þ a
2
z , I II ¼ ln
Â
1=3
À
a
2
x þ a
2
y þ a
2
z
Ã
,
I III ¼ a
2
x þ a
2
y þ a
2
z
À 9
ð20Þ
where, C
E
100 ¼ 1/2kTξ E B
E
100 , C
E
200 ¼ 1/2kTξ E C
E
200 , C
E
020 ¼ 1/2kTξ E D
E
020 , C
a
100 ¼ 1/
2kTξ a B
a
100 , C
a
200 ¼ 1/2kTξ E C
a
200 , C
a
020 ¼ 1/2kTξ a D
a
020 . These parameters are related
to the structure of polymers and filled particles. Their meanings and expressions can
be found in the literatures [11, 12].
2.5 Relationship Between Stress and Strain [8]
The relationship between stress and strain can be determined by the relation of
τ ¼ (∂ΔF T /∂ a ) N
1. Uni-axial extension τ ¼ f/A 0
τ ¼ N TN
1
Γ m E
ð Þ
t À t
0
mEÀ1
e
Àμ E tÀt
0
ð Þ
Á 2 a À a
À2
À
Á
C
E
100 þ C
E
020 a
2
þ 2=a
À
Á À1 þ 2C
E
200 a
2
þ 2=a
À
Á
h
i
þ N Ta e
τ γ _
γ
1
Γ ma
ð Þ
t À t
0
maÀ1
e
Àμ a tÀt
0
ð Þ
Á 2 a À a
À2
À
Á
C
a
100 þ C
a
020 a
2
þ 2=a
À
Á À1 þ 2C
a
200 a
2
þ 2=a
À
Á
h
i
ð21Þ
Effect of Double Networking on Non-Linear Viscoelasticity of Elastomers
169