in the room-to-curing temperature range (i.e. 60 to 180
C) on several SBR 1500
based materials using a RPA
® 2000 (Alpha Technologies). As can be seen the gum
rubber and the no black compound (upper graphs) exhibit a linear viscoelastic
behavior (i.e., flat G* plateau) up to about 20–30 % strain, then a typical strain
thinning behavior that marks the nonlinear response. The filled compounds (lower
graphs) do not show any linear behavior within the experimental window and the
higher the carbon black loading the severer the nonlinearity.
For a number of rubber systems (gums and compounds) it has been shown [14,
23, 24] that, at constant frequency and temperature, the (complex) modulus variation with strain amplitude is well modeled with the following relationship:
G
Ã
γ
ð Þ ¼ G
Ã
f þ
G
Ã
0 À G
Ã
f
1 þ
γ
γ md
B
2
6
4
3
7
5 or G Ã γ
ð Þ ¼
G
Ã
0 þ G
Ã
f Â
γ
γ md
B
1 þ
γ
γ md
B
ð9Þ
where γ is the set strain amplitude (%), G
Ã
0 the modulus in the linear region, G
Ã
f the
final modulus, γ md the strain for reaching the mid-modulus value, i.e. (G
Ã
0 + G
Ã
f )/2,
and B a parameter related to the strain sensitivity of the material.
Equation (9) has an empirical origin but a theoretical foundation can be proposed
as follows. Indeed, quite a common assumption in many approaches of nonlinear
viscoelasticity consists in considering time-strain separability (or factorability).
Such an assumption readily means that the nonlinear relaxation modulus function
G(t, γ) can be separated into a time-dependent and a strain-dependent contributions,
so that:
G t; γ
ð Þ ¼ G t
ð Þ Â h γ
ð Þ
ð10Þ
where h(γ) is the so-called “damping function”. Various mathematical forms (see
[25] for a recent review on damping functions) have been proposed for h(γ), e.g., in
shear flow h(γ) ¼ A Á γ
À2 (Doi-Edward prediction [26]; see also [27]), h(γ) ¼ B Á exp
(ÀC Á γ), h γ
ð Þ ¼
1
1þDÁγ 2 , etc., to mention a few of the simples ones, where A, B, C and
D are parameters. All these mathematical forms correspond to decaying curves and
the last one is particularly interesting because if D ¼ A, the damping function at
sufficiently high strain corresponds exactly to the Doi-Edwards prediction. Let us
now rewrite this equation in the following form:
h γ
ð Þ ¼
1
1 þ
γ
ffiffiffi γ c
p
2
ð11Þ
so that the adjustable parameter appears as a critical strain that obviously corresponds to the intersection between a horizontal line logh(γ) ¼ 1 and the high strain
asymptote, i.e., lim
γ!1
γ c Á γ
À2
ð
Þ. A simpler but similar form of this equation can be
written as:
294
J.L. Leblanc
C) on several SBR 1500
based materials using a RPA
® 2000 (Alpha Technologies). As can be seen the gum
rubber and the no black compound (upper graphs) exhibit a linear viscoelastic
behavior (i.e., flat G* plateau) up to about 20–30 % strain, then a typical strain
thinning behavior that marks the nonlinear response. The filled compounds (lower
graphs) do not show any linear behavior within the experimental window and the
higher the carbon black loading the severer the nonlinearity.
For a number of rubber systems (gums and compounds) it has been shown [14,
23, 24] that, at constant frequency and temperature, the (complex) modulus variation with strain amplitude is well modeled with the following relationship:
G
Ã
γ
ð Þ ¼ G
Ã
f þ
G
Ã
0 À G
Ã
f
1 þ
γ
γ md
B
2
6
4
3
7
5 or G Ã γ
ð Þ ¼
G
Ã
0 þ G
Ã
f Â
γ
γ md
B
1 þ
γ
γ md
B
ð9Þ
where γ is the set strain amplitude (%), G
Ã
0 the modulus in the linear region, G
Ã
f the
final modulus, γ md the strain for reaching the mid-modulus value, i.e. (G
Ã
0 + G
Ã
f )/2,
and B a parameter related to the strain sensitivity of the material.
Equation (9) has an empirical origin but a theoretical foundation can be proposed
as follows. Indeed, quite a common assumption in many approaches of nonlinear
viscoelasticity consists in considering time-strain separability (or factorability).
Such an assumption readily means that the nonlinear relaxation modulus function
G(t, γ) can be separated into a time-dependent and a strain-dependent contributions,
so that:
G t; γ
ð Þ ¼ G t
ð Þ Â h γ
ð Þ
ð10Þ
where h(γ) is the so-called “damping function”. Various mathematical forms (see
[25] for a recent review on damping functions) have been proposed for h(γ), e.g., in
shear flow h(γ) ¼ A Á γ
À2 (Doi-Edward prediction [26]; see also [27]), h(γ) ¼ B Á exp
(ÀC Á γ), h γ
ð Þ ¼
1
1þDÁγ 2 , etc., to mention a few of the simples ones, where A, B, C and
D are parameters. All these mathematical forms correspond to decaying curves and
the last one is particularly interesting because if D ¼ A, the damping function at
sufficiently high strain corresponds exactly to the Doi-Edwards prediction. Let us
now rewrite this equation in the following form:
h γ
ð Þ ¼
1
1 þ
γ
ffiffiffi γ c
p
2
ð11Þ
so that the adjustable parameter appears as a critical strain that obviously corresponds to the intersection between a horizontal line logh(γ) ¼ 1 and the high strain
asymptote, i.e., lim
γ!1
γ c Á γ
À2
ð
Þ. A simpler but similar form of this equation can be
written as:
294
J.L. Leblanc