1
Basic characteristics of uniaxial extension rheology:
Comparing monodisperse and bidisperse polymer melts
Yangyang Wang, Shiwang Cheng, and Shi-Qing Wanga)
Department of Polymer Science and Maurice Morton Institute of Polymer Science, The
University of Akron, Akron, Ohio 44325-3909
Synopsis
We have carried out continuous and step uniaxial extension experiments on monodisperse
and bidisperse styrene-butadiene random copolymers (SBR) to demonstrate that their
nonlinear rheological behavior can be understood in terms of yielding through
disintegration of the chain entanglement network and rubber-like rupture via
non-Gaussian chain stretching leading to scission. In continuous extension, the sample
with bidisperse molecular weight distribution showed greater resistance, due to
double-networking, against the yielding-initiated failure. An introduction of 20 % high
molecular weight (106 g/mol) SBR to a SBR matrix (2.4×10
5 g/mol) could postpone the
onset of non-uniform extension by as much as two Hencky strain units. In step
extension, the bidisperse blends were also found to be more resistant to elastic breakup
than the monodisperse matrix SBR. Rupture in both monodisperse and bidisperse SBR
samples occurs when the finite chain extensibility is reached at sufficiently high rates. It
is important to point out here that these results along with the concept of yielding allow
us to clarify the concept of strain hardening in extensional rheology of entangled
polymers for the first time.
a) Electronic mail: [email protected]
2
I. INTRODUCTION
Extensional rheological behavior of entangled polymer melts has been studied for
several decades. A clear understanding of failure behavior and material cohesive
strength in extensional deformation is important for material designs. In the 1970s and
80s, Vinogradov and coworkers carried out extensive studies on the failure behavior of
monodisperse entangled polymer melts in uniaxial extension [Vinogradov (1975);
Vinogradov et al. (1975a, 1975b); Vinogradov (1977); Vinogradov and Malkin (1980);
Malkin and Vinogradov (1985)]. During the same period, the failure behavior of
various commercial polydisperse polymers were also studied by several teams [Takaki
and Bogue (1975); Ide and White (1977, 1978); Pearson and Connelly (1982)]. It has
been realized that polydispersity in the molecular weight distribution significantly affects
the failure behavior in uniaxial extension [Takaki and Bogue (1975)]. Since most of the
commercial synthetic polymers are polydisperse, a general understanding of the influence
of polydispersity on failure behavior is of great industrial value.
A first step towards a better understanding of the molecular weight distribution
effect is to study bimodal blends where the behavior and dynamics of each individual
component can be readily established. Most of the previous investigations of such
systems [Minegishi et al. (2001); Ye et al. (2003); Nielsen et al. (2006)] focused on the
"strain-hardening" characteristic during uniform extension, leaving the failure phenomena
largely unexplored.
Our recent studies on a series of monodisperse linear SBR melts [Wang et al.
(2007b); Wang and Wang (2008)] have suggested that certain failure behavior of highly
entangled polymers in rapid uniaxial extension is analogous to the shear inhomogeneity
revealed by particle-tracking velocimetry [Tapadia and Wang (2006); Wang (2007)], and
can be understood in terms of the disintegration of the chain entanglement network.
Specifically, in both startup shear and extension, entangled polymers exhibit the same
scaling characteristics associated with the yield point in the elastic deformation regime
[Wang et al. (2007b)], which is the point where the shear stress and engineering stress
peak. Beyond the yield point, structural inhomogeneity develops in the form of
3
non-uniform spatial distribution of chain entanglement. The purpose of this study is to
demonstrate that the notion of yielding can be readily applied to explain some basic
rheological characteristics of entangled bimodal blends in uniaxial extension, including
their failure behaviors. Moreover, this concept of yielding allows us to clarify when
strain hardening really happens in uniaxial extension of entangled polymers.
II. MATERIALS AND METHODS
The bimodal blends in the current investigation were made from three of
near-monodisperse linear styrene-butadiene random copolymers (SBR) provided by Dr.
Xiaorong Wang at the Bridgestone Americas Center for Research and Technology. The
molecular characteristics of three SBR melts can be derived from small amplitude
oscillatory shear measurements using a Physica MCR 301 rotational rheometer equipped
with 25mm parallel plates. Two samples involve 20 wt. % of SBR1M in two different
"matrices" of SBR240K and SBR70K respectively, and are labeled as 240K/1M (80:20)
and 70K/1M (80:20) respectively. A third mixture has the composition of SBR240K
and SBR1M given by 240K/1M (90:10). The small amplitude oscillatory measurements
of the bimodal blends are shown in Fig. 1, from which some basic information is
obtained as shown in Table I, where the equilibrium melt shear modulus Geq is
determined as the value of G' at the frequency, at which G" shows a minimum. The
Rouse relaxation time τR of the sample was estimated as τ/3Z, according to the tube
model [Doi and Edwards (1986)]. Due to their slight microstructure and polydispersity
differences, the terminal relaxation time τ of these samples does not scale with the
molecular weight M as: τ ~ M3.4
.
TABLE I. The Molecular Characteristics of SBR Melts
Sample Mn (kg/mol) Mw/Mn Geq (MPa) Z τ (s) τR (s)
SBR70K 70 1.05 0.74 25 0.67 0.0089
SBR240K 241 1.10 0.82 98 34 0.12
SBR1M 1068 1.23 0.85 510 11000 7.2
4
103
104
105
106
0.01 0.1 1 10 100
240K/1M (80:20) G'
240K/1M (80:20) G"
70K/1M (80:20) G'
70K/1M (80:20) G"
Sto
rage/
loss
modulu
s (
Pa)
Angular frequency (rad/s)
0.002
Figure 1 Small amplitude oscillatory shear measurements of the bimodal blends at room
temperature. The storage modulus and the loss modulus are represented by the open and filled
symbols respectively.
The uniaxial extension experiments were carried out using a first generation SER
fixture [Sentmanat (2004); Sentmanat et al. (2005)] mounted on the Physica MCR 301
rotational rheometer. The failure behaviors of pure melts and blends were investigated
in two different types of testing. One was the failure during startup continuous uniaxial
extension at a constant Hencky strain rate . The other was the failure during step
extension where the sample was suddenly subjected to a certain amount of strain. The
specimen failure is video-recorded to allow post-experiment analyses.
III. RESULTS
A. Continuous Extension
Two types of failure mode were observed during startup continuous extension of the
pure SBR melts. During low-rate extension of SBR240K and SBR1M melts the sample
breakage is found to initiate from non-uniform extension. At sufficiently high rates
rubber-like rupture was found for SBR1M when the sample broke sharply and the
original cross-sectional dimensions returned after rupture, indicating that no part of the
specimen suffered much irrecoverable deformation (i.e., yielding or flow)
Fig. 2(a) and Fig. 3 present the stress-strain curves of SBR240K and SBR1M melts
at various rates. The experiments ending in rupture are represented by open symbols.
In filled symbols we see that the engineering stress engr always exhibits a maximum such
that (engr)max is linearly proportional to max. This characteristic has been reported
5
before and suggested to be a signature of yielding [Wang et al. (2007b), Wang and Wang
(2008)], beyond which chains mutually slide past one another [Wang and Wang (2009)].
The engineering stress is also presented as a function of time in Fig. 2(b) for
SBR240K. It is easy to see that the previously reported scaling behavior in the elastic
deformation regime σengr ~ t1/2
is valid for SBR240K as well [Wang et al. (2007b)].
The end of each stress-strain curve corresponds to the onset of non-uniform extension by
visual inspection, i.e., by examining the recorded images of the stretched specimen. On
the other hand, at high rates rupture of SBR1M truncates the monotonic rise in the
engineering stress engr. In other words, engr never had a chance to decline before
rupture. Note that the data at 1 s-1
in Fig. 3 approaches the borderline between yielding
and rupture.
0
0.5
1
1.5
2
2.5
3
3.5
4
0 1 2 3 4
en
gr
(MP
a)
Hencky strain,
15 s-110 s
-16.0 s-1
3.0 s-1
1.0 s-1
0.3 s-1
0.1 s-1
SBR240K
max
(a)
0.01
0.1
1
10
0.01 0.1 1 10Time (s)
(b)
15 s-1
10 s-1
6.0 s-1
3.0 s-1
1.0 s-1
SBR240K
0.3 s-1 0.1 s
-1
-0.5
en
gr
(MP
a)
Figure 2 Engineering stress as a function of (a) Hencky strain and (b) time for SBR240K at
various strain rates. The straight dashed line in (a) provides an indication of how the
engineering stress increases more weakly than linearly with the Hencky strain.
0
2.5
5
7.5
10
0 1 2 3 4
6.0 s-1
3.0 s-1
2.0 s-1
1.0 s-1
0.6 s-1
0.3 s-1
Hencky strain
en
gr
(MP
a)
SBR1M
neo-Hookean
Figure 3 Engineering stress as a function of Hencky strain for SBR1M at various strain rates.
The stretching, ending in yielding-initiated failure and rupture, is represented by filled and open
symbols respectively. The dotted curve is the neo-Hookean formula of Eq. (1) with Geq = 0.85
MPa from Table 1, showing exponential growth with the Hencky strain .
6
The engineering stress-strain curves for startup continuous stretching of bimodal
SBR blends are shown in Fig. 4, Fig. 5 and Fig. 6 respectively. The shape of the
engineering stress-strain curve for bimodal SBR blends is qualitatively different from that
of the pure SBR melts. At various rates, all three blends, i.e., 240K/1M (90:10),
240K/1M (80:20) and 70K/1M (80:20), exhibited an engineering stress-strain curve
containing two maxima and only showed non-uniform extension beyond the second
maximum. At sufficiently high rates, the uniaxial extension of 240K/1M (80:20) and
70K/1M (80:20) was terminated abruptly without displaying a second maximum in the
engineering stress when the specimens ruptured. The rate dependence of strains at
yielding-initiated failure and rupture for SBR240K, 240K/1M (90:10), and 240K/1M
(80:20) is shown in Fig. 7. The onset of both failures in the bimodal blends is
significantly postponed at the highest four rates relative to those of the pure components.
0
0.5
1
1.5
2
0 1 2 3 4 5
15 s-1
10 s-1
6.0 s-1
3.0 s-1
1.0 s-1
0.3 s-1
0.1 s-1
240K/1M (90:10)
Hencky strain
en
gr
(MP
a)
Figure 4 Engineering stress as a function of Hencky strain at various strain rates for the
240K/1M (90:10) blend.
0
0.5
1
1.5
2
2.5
3
0 1 2 3 4 5
240K/1M (80:20) 15 s-1
10 s-1
6.0 s-1
3.0 s-1
2.0 s-1
1.0 s-1
0.6 s-1
0.3 s-1
Hencky strain
en
gr
(MP
a)
Figure 5 Engineering stress as a function of Hencky strain at various strain rates for the
240K/1M (80:20) blend. The stretching, yielding-initiated failure and rupture, is represented by
filled and open symbols respectively.
7
0
0.5
1
1.5
2
0 1 2 3 4 5
70K/1M (80:20)
15 s-1
10 s-1
6.0 s-1
3.0 s-1
2.0 s-1
1.0 s-1
0.6 s-1
Hencky strain
en
gr
(MP
a)
Figure 6 Engineering stress as a function of Hencky strain at various strain rates for the
70K/1M (80:20) blend. The stretching, yielding-initiated failure and rupture, is represented by
filled and open symbols respectively.
1
10
0.1 1 10 100
SBR240K
240K/1M (90:10)
240K/1M (80:20)
Elastic rupture (80:20)
Elastic rupture (0:100)
Fail
ure
Hencky
str
ain
Strain rate (1/s)
Figure 7 Failure Hencky strain as a function of applied strain rate for SBR(240K) and the
two bimodal blends. The solid symbols represent ductile failure through yielding. The
dashed lines show the borderline between viscoelastic and elastic regimes for the two blends.
The open symbols denote strains at rupture for the pure SBR1M (triangles) and the 240K/1M
(80:20) blend (diamond).
B. Step Extension
The engineering stress as a function of time during and after step extension of the
pure SBR1M melt is first presented in Fig. 8 at three different amplitudes all high enough
to produce elastic yielding, i.e., failure during relaxation. The induction times, before
which the specimens appear intact, are all longer than the Rouse relaxation time R of 7.2
s. After this period, one portion of the specimen started to shrink in its dimensions,
leading to the sample breakage. Fig. 9(a) shows the elastic breakup behavior of not only
the pure SBR240K in solid symbols but also the bimodal SBR blends of 240K/1M
8
(90:10). Here the critical Hencky strain for the breakup is just beyond 0.6 for the pure
SBR240K, which is slightly lower than the vinyl-rich SBR melts previously reported
[Wang et al. (2007b)]. The critical strain for SBR 240K/1M (90:10) is apparently the
same as the pure SBR240K melt, although the induction time to break is markedly longer.
0.01 0.1 1 10 100
1.20.90.75
en
gr
(MP
a)
Applied rate: 15 s-1
strain
SBR1M10
0.1
1
0.01
Time (s)
Figure 8 Engineering stress as a function of time in step relaxation experiments for the
SBR240K melt and the 240K/1M (90:10) blend.
0.01
0.1
1
10
0.01 0.1 1 10 100
0.9
0.75
0.9
0.75
240K
240K/1M
(90:10)
Applied rate: 15 s-1
Time (s)
strain
en
gr
(MP
a)
(a)
0.1
1
0.01 0.1 1 10 100 1000
1.2
0.9
0.6
240K/1M (80:20)
Strain
Time (s)
Applied rate: 15 s-1
No breakup
Elastic breakup
2
en
gr
(MP
a)
(b)
Figure 9 Engineering stress as a function of time in step relaxation experiments for (a) the
SBR240K melt and the 240K/1M (90:10) blend, and (b) 240K/1M (80:20) blend.
In contrast, the critical Hencky strain c for SBR 240K/1M (80:20) shifted to c = 0.9,
significantly higher than c ~ 0.6 for the pure SBR240K. Thus, the bimodal blends are
9
also more resistant to breakup after a step extension than the corresponding pure
monodisperse matrix.
IV. DISCUSSION
A. Continuous Extension
A.1 Yielding and rupture of Monodisperse Melts
Yielding and non-uniform extension
During startup deformation at rates involving Weissenberg number Wi = , after
the initial elastic deformation, the entanglement network is forced to disintegrate when an
imbalance emerges between the elastic retraction force and intermolecular locking force
[Wang et al. (2007a), Wang and Wang (2009)]. In other words, the tensile force
represented by engr is expected to display a maximum. The present study confirms that
pure monodisperse melts exhibit yielding in constant-rate uniaxial extension. The
yielding is signified by the emergence of a maximum in the engineering stress as shown
in Fig. 2(a). Consistent with the reported scaling behavior of yielding in simple shear
[Boukany et al. (2009a)], the solid symbols of Fig. 2(a) and Fig. 3 also show a shift of the
yielding strain (corresponding to the engineering stress maximum) εy to higher values
upon increasing the applied rate. The broad maxima at lower rates in Fig. 3 are due to
the large polydispersity in the molecular weight distribution of SBR1M.
There is strong evidence that the observed sample necking is not due to an elastic
Considère-type instability, advocated recently by Hassager and coworkers [Lyhne et al.
(2009); Hassager et al. (2010)]: When set to a stress-free state after the engineering
stress maximum, the specimen cannot return to its original dimensions [Wang and Wang
(2008)]. The plastic deformation and the corresponding decline in the engineering
stress are a result of the entanglement network disintegration in presence of the
continuing extension. Apparently, the eventual outcome is, not surprisingly, that one
portion of the stretched specimen reaches a point of network disintegration and
irreversible deformation before the rest of the specimen does, resulting in necking or lack
of uniform extension. It is interesting to note from Fig. 2(a) that the specimen can
extend uniformly well after the yield point for the group of data involving the higher rates
10
of 6.0 s-1
and above. This dividing line coincides with the boundary between the
viscoelastic and elastic deformation regimes, given by the condition of the Rouse
Weissenberg number WiRouse = τR ~ 1 [Wang and Wang (2008)]. In the viscoelastic
regime, the entanglement network could undergo irreversible deformation even before
reaching the engineering stress maximum, so that the specimen is more ready to fall apart
beyond the stress maximum. Conversely, in the elastic deformation regime, the network
would be largely intact and fully recoverable up to the yield point (engineering stress
maximum) [Wang and Wang (2008)]. As a consequence, the onset of structural collapse
is delayed markedly beyond the yield point.
We close this subsection by emphasizing that nearly monodisperse linear melts such
as SBR240K only show significant strain softening in the rate range bounded by WiRouse <
10. Fig. 2(a) shows, consistent with previous analyses [Wang et al. (2007b), Wang and
Wang (2008)], that engr initially grows with the Hencky strain = t linearly. This may
be expected because even the neo-Hookean model prescribes such a linear response at
small extensions:
engr = G(1/2),
which has a limiting form of engr ~ 3Gfor << 1 where the stretching ratio =exp()
and G is the elastic shear modulus. engr ~ is simply the straight dashed line in Fig.
2(a). However, the actual engr soon bends downward in Fig. 2(a), which is a linear
scale plot of engr versus Hencky strain . Thus, the engineering stress build-up is even
weaker than the linear relation of Eq. (1) at higher strains. We have taken this behavior
to imply strain softening. Although this strain softening may just reflect weakening of
the entanglement network, we note that chain relaxation during stretching can also
partially contribute to this apparent strain softening.
Rubber-like Rupture
For monodisperse entangled linear melts, the transition from yielding to rubber-like
rupture typically takes place at a critical extensional rate corresponding to Rouse
Weissenberg number WiRouse ~ 10 [Wang and Wang (2010a)]. This transition is what
Malkin and Petrie (1997) referred to as the rubber-to-glass transition connecting regime
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III and IV. However, our recent study [Wang and Wang (2010a)] has revealed that for
highly entangled polymers, such a yielding-to-rupture transition is located in the middle
of the rubbery plateau region, and does not involve any transformation to the glassy state.
In other words, it would be misleading to associate the rupture with the term “glass-like
zone”, as classified in the Malkin-Petrie master curve.
The ductile failure arises from yielding of the entanglement network, during which
linear chains mutually slide past one another [Wang et al. (2007a); Wang and Wang
(2009)], whereas the origin of the rupture could have something to do with chain scission.
At the very high rates, the mode of yielding, i.e., chain mutual sliding, is no longer
available when sufficient tension in the chains can build up to cause chain scission. In
other words, chain scission would occur before chains have reached the point of force
imbalance [Wang and Wang (2009) to allow mutual sliding. This appears to occur in
SBR1M at an extensional rate = 2.0 s-1
and above as shown in Fig. 3 where the tensile
force (i.e., the engineering stress) rises monotonically. This rapid rise in the engineering
stress at higher rates is plausibly due to non-Gaussian stretching [Wang and Wang
(2010a)]. In the explored rate range, SBR240K cannot be stretched fast enough at room
temperature to produce the non-Gaussian stretching that leads to the upward rise in
tensile force and ends in rupture via chain scission. In passing, we note that some
alternative theoretical explanation for specimen failure during uniaxial extension has
been made in the literature [Joshi and Denn (2003, 2004)]. We do not elaborate on this
theoretical study because it appears to have oversimplified the process leading to the
yielding-initiated failure.
A.2 Failure Behaviors of the Bimodal Blends
Effect of the high molecular weight component on yielding
We have seen that the incorporation of a small amount of SBR1M to SBR240K
significantly delays the sample failure. For example, at = 6.0 s-1
, the SBR240K melt
would yield at ≈ 1.4 and eventually undergo ductile failure at ≈ 2.6. The presence of
10% or 20% 1M SBR postpones the onset of the ductile failure, as shown in Fig. 4 and
Fig. 5, to much higher strains. A closer examination shows that there exists a critical
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rate, beyond which the strength of the blends is greatly improved. As evident from Fig.
7, the failure behavior of the 240K/1M (90:10) blend is not so different until reaching c
~ 6.0 s-1
. Similarly, c ~ 2.0 s-1
can be found for the 240K/1M (80:20) blend.
The relaxation times of the two individual components in the 240K/1M blends are
widely separated. As indicated in a preceding subsection, the Rouse relaxation times of
240K and 1M are different by a factor of 60, whereas the difference in their terminal
relaxation times is even larger, approaching 330. The wide separation of relaxation
times causes the high and the low molecular weight components to stay in different
deformation regimes under a given applied strain rate. In other words, the incorporated
SBR1M chains actually formed a second entanglement network, which is highly
plausible given the huge separation of its relaxation time scale from that of the "matrix"
SBR240K. Specifically, the number of entanglements per chain in such a second
network can be estimated according to the empirical scaling law for the entanglement
molecular weight: Me() ~ Me,0-1.3
[Yang et al. (1999)]. At the volume fractions of 0.1
and 0.2, the second network involves respectively 32 and 73 entanglement points per
SBR1M chain.
Our previous study on the scaling characteristics of yielding [Wang and Wang
(2009)] has revealed, as also demonstrated by Fig. 2(a), that the onset of yielding moves
to higher strains with increasing Rouse Weissenberg Number WiRouse. Thus, at a given
rate, the second entanglement network formed by the SBR1M would yield at a much
higher strain. The blend could retain its integrity well beyond the first engineering
stress maximum where the (first) matrix network has collapsed. This evidently
indicates that the second entanglement network made of SBR1M did not give in until
much higher strains. Since the second network involves much greater entanglement
spacing, the non-Gaussian stretching sets in at significantly higher strains around a
Hencky strain of 3.0 as read from Fig. 5 [Wang and Wang (2010b)] than a Hencky strain
of 2.0 read from Fig. 3 for the neat SBR1M. More discussion on non-Gaussian
stretching is deferred to IV.D.4.
13
Rupture
With only 10% SBR1M, the first blend did not undergo rupture in the explored rate
range as shown in Fig. 4. It appears that at these rates the second network associated
with 10% SBR1M never got extended sufficiently to suffer chain scission. As a
consequence, it yields in the form of mutual chain sliding.
It is truly remarkable that the two blends of 240K/1M (80:20) and 70K/1M (80:20)
containing only 20 % of SBR1M would undergo rupture in the range of Hencky rates
where the pure "matrices" only show yielding-initiated non-uniform extension at the
same rates. Apparently at these rates there is not sufficient intermolecular gripping
force to produce full chain extension and enough chain tension to cause chain scission in
either pure SBR240K or SBR70K. Thus, the 80% matrix chains only yield in the form
of mutual chain sliding. On the other hand, the second network due to SMR1M can be
fully stretched to reach the point of chain scission. Apparently, the 80% matrix chains
at the point of rupture have reached such a fully disengaged state that the creation of the
new surface during rupture costs no more energy than that estimated from the surface
tension. In other words, the incorporation of 20% SBR1M as a second network
provided so much structural stability that the matrix, i.e., the first network made of
SBR240K or SBR70K, was able to disintegrate fully.
The rupture occurred at much higher strains in the blends than in the neat SBR1M
because at the volume fraction of 0.2, a strand (made of SBR1M) between two
neighboring entanglement points of the second network is of a much longer chain length.
Our recent study has shown that a higher stretching ratio is required to produce full chain
extension, which appears to be a necessary condition for chain scission [Wang and Wang
(2010b)]. The difference between the open diamond and triangles in Fig. 7 indicates
rupture at rather different stretching ratios due to the difference in entanglement density.
Finally, it is more than interesting to note that the blend of 240K/1M (80:20) barely
shows rupture at = 15 s-1
, whereas the blend with SBR70K as the matrix would
undergo rupture at a rate as low as = 6 s-1
. We know that at the same rate SBR70K
yields at a lower strain than SBR240K because the yield strain decreases with lowering
WiRouse, which is lower for SBR70K. Consequently, SBR70K can reach a fully
14
disengaged state at a lower strain. As suggested above, the disappearance of any
entanglement networking in the matrix appears to be a prerequisite for rupture.
Apparently at = 6 s-1
, when SBR1M suffers chain scission, the matrix is also already in
a state of full disengagement, allowing rupture to take place in 70K/1M (80:20).
Conversely, in 240K/1M (80:20) that is being extended at = 6 s-1
, even when a
significant fraction of SBR1M chains have become fully extended, evidenced by the
upturn in the engr vs. curve of Fig. 5, and reached the condition for chain scission, and
even if these chains do suffer scission, the matrix apparently has not fully collapsed at
these strains. The engineering stress engr could further build up until most SBR1M
chains have undergo chain scission. Beyond this point, the matrix chains further
mutually slide past one another until the system lose uniform structural support from
chain entanglement. This appears to be what is happening in the blend involving the
stronger first network made of SBR240K.
B. Step Extension
Recent studies [Wang et al. (2007b); Wang and Wang (2008)] on the relaxation
behavior of monodisperse entangled linear polymers after step uniaxial extension have
revealed that the entanglement network suffers cohesive breakdown at large strains.
Here the cohesion refers to that due to chain entanglement [Wang et al. (2007a)]. Such
elastic breakup is analogous to those disclosed by the particle-tracking velocimetric
observations of entangled polymer solutions [Wang et al. (2006); Ravindranath and Wang
(2007)] and melts [Boukany et al. (2009b)] in startup simple shear. The dynamics of
such elastic yielding appear to depend on the molecular weight of the polymer melt and
the amplitude of the step strain. For both uniaxial extension [Wang et al. (2007b); Wang
and Wang (2008)] and simple shear [Boukany et al. (2009b)], the elastic yielding takes
place faster with increasing step strain amplitude. For the same amplitude of step strain,
a sample with higher molecular weight takes a longer time to lose the cohesive integrity
[Boukany et al. (2009b)]. In fact, the elastic yielding appears to be related to the Rouse
chain dynamics in the sense that the induction time for the breakdown scales with the
longest Rouse relaxation time [Wang et al. (2007a), Boukany et al. (2009b)]. Because
15
the Rouse relaxation time of SBR1M is 60 times of SBR240K (note that this ratio is far
greater than Z2 ~ 25, due to the polydispersity of SBR1M), the incorporation of
incorporation of SBR1M may delay the specimen breakup after step extension as shown
in Fig. 9(a) for the 240K/1M (90:10) blend. Upon a step extension of Hencky strain of
0.75 and 0.9, the pure SBR240K undergoes elastic yielding that leads to the sample
failure at 3.5 and 10 s respectively as shown by the filled symbols in Fig. 9(a). The
blend (open symbols) fails at appreciably later times of 6.6 and 13 s respectively. It
appears that the specimen would not undergo failure until the second network associated
with the high molecular weight SBR1M also collapse on some longer time scale.
Something more dramatic occurs in the second blend containing 20% SBR1M.
Here the presence of the SBR1M can prevent the sample from undergoing the cohesive
failure. For a step extension of Hencky strain = 0.9, the blend does not suffer failure.
Perhaps before the second network made of SBR1M undergoes any disentanglement the
first network has already recovered its entanglement structure after experiencing elastic
yielding. At the higher strain of 1.2, the disintegration of the second network perhaps
occurred so quickly that the first network had no chance to return to its entangled state to
provide any adequate structural support. Consequently, the sample failure was observed
at this higher strain. Fig. 8 indeed shows that the breakup of the pure SBR1M
accelerates from an induction of 28 s for the step extension of =0.9 to 11s for = 1.2.
C. Elastic Instability or Yielding
Over the past a few decades, there have been extensive studies of entangled polymer
melts and solutions in extensional deformation, because of their theoretical and practical
importance. Among all these studies, significant efforts have been made toward
understanding the nature of specimen failure. Vincent (1960) was the first to use the
Considère criterion [Considère (1885)] to analyze the necking instability in elongation
and cold flow of solid plastics (PE and unplasticized PVC). Such an analysis was
subsequently extended to polymer melts in the rubbery state [Cogswell and Moore (1974);
Pearson and Connelly (1982)] and polymer solutions [Hassager et al. (1998); Yao et al.
(1998)]. Doi and Edwards (1979) also suggested an elastic necking instability in their
16
original tube theory based on the Considère reasoning. McKinley and Hassager (1999)
applied the Doi-Edwards theory and pom-pom model [McLeish and Larson (1998)] to
predict the critical Hencky strain for failure in linear and branched polymer melts during
fast stretching.
0
0.5
1
1.5
2
0 1 2 3 4 5
15 s-1
10 s-1
6.0 s-1
3.0 s-1
15 s-1
10 s-1
6.0 s-1
3.0 s-1
Hencky strain
240K
240K/1M
(90:10)
en
gr
(MP
a)
Figure 10 Comparison of the engineering stress-strain curves for the SBR240K melt and the
240K/1M (90:10) blend.
In our view, the emergence of a maximum in the engineering stress implies
weakening of the underlying structure. Beyond the maximum, the material is no longer
the original elastic body. In rigid solids such as metals studied originally by Considère
(1885), when the measured force passes a maximum, non-uniform extension take place
immediately due to the very limiting amount of extensibility. The debate in the
literature about how to apply the Considère criterion has been around the issue of whether
necking could occur immediately after the force maximum [Barroso et al. (2010); Petrie
(2009); Joshi and Denn (2004b)]. For rubbery materials including entangled melts, the
force maximum occurs at a significant stretching ratio, and non-uniform extension
usually does not occur right after the force maximum. Fig. 2(a) shows that the
maximum occurs for = 15 s-1
at a Hencky strain of = 1.8, and the failure strain is at
3.4. This behavior renders the application of the Considère criterion irrelevant. The
force peak is the yield point when the sample is weakening in its resistance against
further external deformation. The entanglement network takes some further extension
before disintegration [Wang and Wang (2009)]. In other words, uniform extension
could persist for a while until the network structure of the sample becomes
inhomogeneous. The eventual "necking" and failure are anything but a mechanical
17
(elastic) instability in our judgment. So far no theory can describe quantitatively how
such a localization of yielding takes place. Any continuum mechanical depiction based
on a non-monotonic relation between the tensile force and degree of extension would
have to first explain the microscopic physics responsible for the tensile force decline with
increasing stretching.
More specifically, the phenomenon of the specimen failure after yielding is a result
of localized cohesive failure, having to do with how the initial elastic deformation turns
into plastic deformation in an inhomogeneous manner. Introduction of 10 or 20 %
SBR1M long chains into the SBR240K produces similar engineering stress versus
Hencky strain characteristics as the pure SBR240K up to some significant strains well
beyond the maxima, as shown in Fig. 10. For example, at = 6.0 s-1
, both the pure
SBR240K and the blend show nearly identical stress-strain curve till = 2.7. A
continuum mechanical analysis based on such curves would have predicted the onset of
necking instability, i.e., non-uniform extension, at the same strain for both the pure
SBR240K and its blend with SBR1M. This is far from truth: the presence of only 10 %
SBR1M long chains significantly extended the range of uniform stretching till = 4.0,
well beyond the strain where the necking and failure took place in the pure SBR240K.
We assert that this suppression of non-uniform extension by the second entanglement
network is something well beyond any existing theoretical description based on
continuum mechanical calculations.
More recently, Hassager and coworkers also tried to explain the delayed failure after
rapid uniaxial extension (i.e., the elastic yielding according to Wang et al. (2007b)) on the
basis of the Doi-Edwards model and a Considère-type analysis [Lyhne et al. (2009);
Hassager et al. (2010)]. Since the observed elastic-yielding-initiated failures after step
extension only involved a level of extension well below the tensile force maxima that
occur around = 2.0, it is rather difficult to understand that this failure discussed in Fig.
9(a)-(b) has something to do with the emergence of the non-monotonic engineering stress
vs. strain curves shown in Fig. 2(a) and 5.
18
D. Where Is Strain Hardening
Since strain hardening is a frequently invoked expression of extensional rheological
behavior of polymer melts, we will discuss thoroughly this idea in the present context.
In particular, we will review and comment on four ways in which the concept of "strain
hardening" may get used to describe uniaxial extension of polymers.
10-1
100
101
102
103
104
0 1 2 3 4 5
106.03.02.0
1.00.60.3
Str
ess
(M
Pa)
Hencky strain
Strain rate (s-1)
240K/1M
(80:20)
neo-Hookean
Figure 11 Cauchy stress as a function of Hencky strain for the 240K/1M (80:20) blend. The
dotted line represents the stress-strain curve from the neo-Hookean model of Eq. (1) with Geq =
0.82 MPa from Table 1.
D.1 Strain hardening with stress-strain curve above neo-Hookean line
One of the earlier references to strain hardening came from high extension of natural
rubbers when the stress-strain curve appears above the prediction of the classical rubber
elastic theory, or the neo-Hookean model for a network of Gaussian chains [Treloar
(1944)]. This strain hardening at high stretching ratios is partially due to the finite chain
extensibility, i.e., the system has reached the point of non-Gaussian stretching.
Moreover, the strain-induced crystallization in natural rubber can also enhance strain
hardening [Smith et al. (1964)]. However, entangled melts that do not undergo
strain-induced crystallization, such as the present SBR melts and blends, only appear to
show stress-strain curves beneath the neo-Hookean model prediction as shown in Fig. 3
for the SBR1M and Fig. 11 for the 240K/1M (80:20) blend. Entangled polymer melts,
in absence of chemical cross-linking, apparently always suffer so much loss of
entanglements during extension that the overall stress-strain curve could not rise above
the neo-Hookean line even when non-Gaussian stretching takes place. In other words, a
19
significant fraction of entangled points in the network reach the point of force imbalance
and become ineffective to bear the load during continuous extension of entangled melts,
which does not occur in the neo-Hookean model. More discussion on this point is given
in the following IV.D.4
D.2 Monotonic increase of stress in startup extension: false strain hardening
At high Weissenberg numbers, startup deformation is known to produce stress
overshoot in simple shear, which has recently been suggested to indicate the onset of
yielding [Wang et al. (2007a), Wang and Wang (2009)]. Beyond the yield point, the
shear stress or transient viscosity decreases because the elastic deformation ceases and
disintegration of the entanglement network leads to reduced elastic resistance. In
startup uniaxial extension, the so called true stress E, i.e., the product of the tensile force
and stretching ratio = exp( t), would usually only monotonically grow with continuing
extension until the point of non-uniform extension. In other words, in uniaxial
extension, the extensional (Cauchy) stress E does not exhibit overshoot, and thus the
transient viscosity E =/ at a given Hencky strain rate only monotonically rises.
This contrast between simple shear and uniaxial extension has caused Münstedt and
Kurzbeck (1998) to declare that "The viscosity increase as a function of time or strain,
respectively, is called strain hardening. It is a special feature of elongational
deformation of polymer melts."
In our view, this difference between simple shear and uniaxial extension should be
looked at in a different light. Upon startup deformation the transient shear viscosity also
initially increases with time, which is just an indication that the elastic deformation is
dominant at the beginning of startup shear. The elastic deformation cannot ensue
indefinitely, and yielding must take place, leading to the observed decrease of the shear
stress as well as viscosity over time. In contrast, during startup continuous extension at
Hencky rate the dominant reason for the continuous rise in E is the simple geometric
factor of the exponentially shrinking cross-sectional area A(t):
= F/A(t) = engr = engr(t) exp( t), (2)
where F is the total tensile force, and
20
A(t)=A0 exp( t) (3)
is the time-dependent cross-sectional area with A0 being the original area. Even if
yielding occurs, i.e., engr (t) starts to decline, and E would still only increase with
time until the point of non-uniform extension and specimen failure, provided that the
decline of engr due to yielding is not as fast as the exponential areal shrinkage, which is
often the case. Thus, the continuous increase of E and E does not mean that the
uniaxial extension has not experienced the same yielding as seen in simple shear. In
other words, such increases do not mean that the sample’s entanglement structure is not
deteriorating and becoming less resistant to the continuous extension. Actually, strain
softening not hardening has taken place as long as the tensile force is declining with
continuing extension. As far as we can tell, the point of the entanglement network
weakening occurs at the engineering stress maximum and cannot be discerned readily
from such quantities as the Cauchy stress and extensional viscosity.
D.3 Upward deviation of transient viscosity from zero-rate limit
There is yet another way in which “strain hardening” is referred to in uniaxial
extensions of polymer melts. It refers to a specific phenomenon observed during startup
uniaxial extension when the transient elongational viscosity shows upward deviation
from the limiting zero-rate-viscosity vs. time curve, in contrast to the phenomenon in
simple shear where the transient shear viscosity function is always below the zero-shear
viscosity function. This is perhaps the most widely recognized signature for “strain
hardening”. The phrase "strain hardening" might have first been used by Meissner
(1975) in describing the uniaxial extensional behavior of low-density polyethylenes
(LDPE) [Meissner (1971, 1975); Laun and Münstedt (1978)]. Linear melts with
bidispersity in molecular weight distribution [Münstedt (1980); Koyama (1991);
Münstedt and Kurzbeck (1998); Minegishi et al. (2001); Wagner et al. (2005); Nielsen et
al. (2006)] and even monodisperse melts –stretched at high enough rates– [Wang and
Wang (2008), and figures below] could also produce this upward deviation. Since
LDPE shows the most pronounced upward trend, long chain branching has been thought
21
to play some peculiar role in producing this "strain hardening" [McLeish (2008); van
Ruynbeke et al (2010)].
In our view, this noted difference between simple shear and uniaxial extension is
perhaps superficial rather than fundamental for entangled polymer melts. Entangled
polymer melts have been found to exhibit only strain softening in simple shear as a
consequence of yielding during startup deformation. The maximum shear stress max
has been found to grow with the applied rate more weakly than linearly [Boukany et
al. (2009a)] so that the peak transient shear viscosity +
max= max/ can only decrease
with . Moreover, the steady shear viscosity is always lower than the zero-shear
steady-state viscosity: In the zero-rate limit, the entanglement network is intact, and it is
the Brownian diffusion that brings the chains past one another. There is maximum
viscous resistance to terminal flow because the equilibrium state of chain entanglement is
preserved.
In many cases, uniaxial extension is different in appearance only because the
cross-sectional area A(t) of the sample keeps shrinking exponentially with time at a given
Hencky strain rate as shown in Eq. (3). If one insists on representing the transient
elastic response in terms of the Cauchy extensional stress of Eq. (2) and the transient
extensional viscosity
E (t) =/ = engr(t)exp( t)/ , (4)
then the continuous increase of and E with time largely originates from the
exponential factor associated with the shrinking cross-sectional area. Let consider the
extension in the beginning in the sense that t < In the zero-rate limit, i.e., when
Wi, we can estimate Eq. (4) by employing Eq. (1). When engr ~ Eand exp( t) =
1, Eq. (4) turns into
E (t)|0 = Et, for t <<1, (5)
where |0 denotes the zero-rate limit and E = 3G. At very high rates, i.e., Wi >>1, E (t)
is also given by Eq. (5) at short times. But at longer times, the exponential factor exp(
22
t) in Eq. (4) is a rapidly rising function of time. This causes Eq. (4) to grow above Eq.
(5) by the same exponential factor had engr(t) continued to rise linearly with the Hencky
strain = t as shown by the dashed line in Fig. 2(a). In reality, the sample yields
eventually. Thus, before the yield point when engr has not declined, the exponential
factor in Eq. (4) kicks in to produce a higher E than the zero-rate curve. In other
words, there is a small window of extension where the transient viscosity in Eq. (4) ticks
upward as shown in Fig. 12. This upward deviation is largely due to the geometrical
shrinkage of the transverse dimensions and is not true strain hardening.
The less monodisperse sample of SBR1M can extend more before non-uniform
extension terminates the experiments. In Fig. 3, although engr only grows by 40 % at
= 1.0 s-1
from = 3.0 to the point of failure, Fig. 13(a) shows a much stronger rise in
E (t) because of the exponential decreasing function A(t) of Eq. (3) in the dominator of
Eq. (2) for the Cauchy stress E. As a consequence, the data rise significantly above the
zero-rate envelope. Actually, whenever significant extension occurs without sample
failure, there would be a great contribution to E from the cross-sectional areal
shrinkage that has little to do with strain hardening. Therefore, from now on we shall
call this upward deviation of the transient elongational viscosity from its zero-rate curve
"pseudo strain hardening".
0.01
0.1
1
10
100
0.01 0.1 1 10 100 1000
15
10
6.0
3.0
1.0
0.3
0.1
Time (s)
3|*(1/)|
Strain rate (s-1)
SBR240K
+ (
MP
a.s)
E
Fig. 12 Transient extensional viscosity as a function of time for the SBR240K melt, where the
"linear response" data given by ηE+ = 3 |η*(1/ω)| from the small amplitude oscillatory shear
measurements are also presented as the reference [Gleissle (1980)].
23
Long chain branching (LCB) in entangled melts delays the onset of non-uniform
extension. For example, LCB in LDPE allows such materials to display a very shallow
maximum in the engineering stress. In other words, the engineering stress only
decreases gradually beyond its maximum. This gives the geometrical exponential factor
a large range of time or strain to boost the transient viscosity in Eq. (4) above the limiting
zero-rate curve, and caused the phrase "strain hardening" to be invoked to differentiate
the extensional rheological behavior of LDPE from that of other linear polymer melts
such as high-density polyethylene [McLeish (2008)]. It is clear that LCB plays a critical
role to prolong uniform extension relative to entangled melts made of linear chains.
However, engr does typically decrease with increasing extension in LDPE. In other
words, there is only evidence of yielding and strain softening and little sign of strain
hardening. Actually, there is explicit evidence from birefringence measurements that
LDPE does not suffer non-Gaussian chain stretching during extension in the typically
explored range of strain rates and temperatures [Koyama and Ishizuka (1989); Okamoto
et al. (1998)].
Finally, we need to explain why most literature data on linear melts show little
upward deviation from the limiting zero-rate curve, a fact that makes LDPE look
somehow special. Most experiments on uniaxial extension of linear polymer melts have
been conducted in the moderately high (rather than extremely high) rate regime, and few
have been based on monodisperse samples. For example, up to = 3.0 s-1
,
corresponding to Wi = 100, the data in Fig. 12 hardly tilted above the limiting curve, and
some unimpressive upward deviation shows up only at the higher rates. As indicated
above, E (t) would deviate exponentially fast above the zero-rate viscosity function if
the sample would maintain linear growth of engr with the Hencky strain as shown by
the dashed line in Fig. 2(a) In reality, the sample yields. The deviation of the actual
data in Fig. 2(a) from this linear growth engr ~ E, i.e., the dashed straight line increases
sharply with time. Beyond the engineering stress maximum, the deviation actually
increases approximately exponentially, cancelling the exponential factor of exp( t) in Eq.
(4) associated with the area shrinkage. Thus, at these intermediate rates, one can hardly
24
see any upward deviation in linear melts that readily suffers yielding. It is the yielding
and the resulting non-uniform extension that makes it impossible to collect data points at
longer times. As a consequence, it has been impractical to obtain the extensional
viscosity in the fully developed steady flow state during startup continuous extension as
noted by Petrie (2006). But Petrie did not know why it is almost impossible to reach the
flow state [Petrie (2008)], which was due to uniform yielding as explained by Wang and
Wang (2008).
D.4 A case of “entanglement strain hardening”: non-Gaussian stretching
The transition from elastic extension to flow, known as yielding in engineering
terms, occurs during startup uniaxial extension in a wide range of rates when further
overall elastic deformation of the entanglement network is no longer possible and the
chains mutually slide past one another. At higher rates, chain extension could continue
until a fraction of the chains in the sample approaches the finite chain extensibility limit.
Chains at such a high stretching ratio appear stiffer and non-Gaussian. This is
non-obvious from a conventional plot like Fig. 13(a). The mechanical evidence of this
non-Gaussian stretching comes from further analysis of data such as those in Fig. 3. It
is obvious that some level of yielding, i.e., mutual chain sliding, can and does occur
before a fraction of the yield-surviving entanglement strands reaches the finite chain
extensibility limit. Fig. 13(b) shows that when the shear modulus G in Eq. (1) is
reduced from its equilibrium value of 0.85 MPa to 0.424 MPa, the neo-Hookean would
emerge onto the data at the applied Hencky strain rate of 6.0 s-1
at a stretching ratio of nG
= 8.5, where the subscript nG stands for non-Gaussian. Beyond this turning point, the
data deviate upward from the neo-Hookean curve, which can be taken as a sign of
non-Gaussian stretching. This upward deviation is true strain hardening at the chain
level, which we shall call "entanglement strain hardening" to differentiate from the strain
hardening in vulcanized rubbers that produces a stress-strain curve above the
neo-Hookean limit as discussed in IV.D.1. We see this behavior as shown in the open
symbols in Fig. 3 for the SBR1M melt and in Fig. 5 and 6 for the binary mixtures. The
full chain extension leads to chain scission and rupture during startup continuous
25
extension. One key characteristic for this strain hardening is the emergence of the
upturn seen in Fig. 3 in open symbols in the pure SBR1M, and more instructively in Fig.
13(a).
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
Time (s)
+ (
MP
a.s)
E
3|*(1/)|
SBR1M
(a) = 1 s-1
0
1
2
3
4
5
6
7
4 8 12 16 20
en
gr
(MP
a)
1
SBR1M
6.0 s-1
neo-Hookean
(b)
nG
Figure 13 (a) Transient extensional viscosity as a function of time for the SBR240K melt, where
the "linear response" data given by ηE+ = 3 |η*(1/ω)| from the small amplitude oscillatory shear
measurements are also presented as a reference [Gleissle (1980)].
(b) Engineering stress engr versus the stretching ratio at = 6.0 s-1
, relative to a neo-Hookean
curve of Eq. (1) based on G = 0.424 MPa = Geq/2, i.e, half of the equilibrium shear modulus,
implying that half of the strands in the equilibrium entanglement network are lost at the
stretching ratio nG ~ 8.5.
10-1
100
101
102
0.01 0.1 1 10 100 103
106.03.02.01.00.60.3
240K/1M
(80:20)
3|*(1/)|
Strain rate (s-1)
Time (s)
+ (
MP
a.s)
E
(a)
0
0.5
1
1.5
2
2.5
10 20 30 40 50
10
6.03.02.0
70K/1M (80:20)
Strain rate (s-1
)
en
gr
(MP
a)
neo-Hookean
(b)
nG
1
Figure 14 (a) Transient extensional viscosity as a function of time for the 240K/1M (80:20)
blend, where the "linear response" data given by ηE+ = 3 |η*(1/ω)| from the small amplitude
oscillatory shear measurements are also presented as a reference [Gleissle (1980)].
(b) Engineering stress engr versus the stretching ratio at the various rates for the 70K/1M
(80:20) blend, relative to a neo-Hookean curve of Eq. (1) based on G = Geq2.2
= 0.025 MPa
where Geq = 0.85 MPa and = 0.2, where nG ~ 23, far higher than that of 8.5 for the pure
SBR1M in Fig. 13(b).
This upturn also shows up strongly in Fig. 5 and 6 that depict the two blends with
the 20% long chains of SBR1M. It occurs whenever the limit of finite chain
extensibility is approached to cause non-Gaussian stiffening of the entanglement network.
26
When expressed in terms of the transient viscosity E (t) as shown in Fig. 14(a),
significant upward deviation from the zero-rate curve shows up, reminiscent of the data
of LDPE. This strong upward deviation arises from both the exponential factor in Eq.
(4) and the “entanglement strain hardening”, i.e., non-Gaussian stretching. For LDPE,
there is little evidence of non-Gaussian stretching [Koyama and Ishizuka (1989);
Okamoto et al. (1998)], yet the similar behavior is observed for the following reason:
LDPE typically can undergo significant extension without encountering non-uniform
extension despite the occurrence of yielding, signified by the emergence of a
non-monotonic relation between the engineering stress and strain. The prolonged
uniform extension allows the geometric exponential factor in Eq. (4) to produce the
upward deviation that is well documented in the literature [Ferry (1980); Laun and
Schuch (1989)].
Actually, it is again more instructive to present the evidence of non-Gaussian
stretching in the blends by referring to a hypothetical neo-Hookean behavior of the
second network formed by SBR1M chains. The second network at a weight fraction of
20 % has a shear modulus given by G = G(=1)2.2
. Fig. 14 (b) shows that the data at
high rates show significant upward deviation from the neo-Hookean curve. The
deviation occurs at nG = 23, which significantly higher than the degree of extension
given by nG = 8.5 for the pure SBR1M in Fig. 13(b). The separation between the blend
and SBR1M is once again exactly a factor of 1 Hencky strain, as noted above due to the
difference in entanglement spacing. The data at = 3.0 s-1
and below are below the
neo-Hookean curve, indicating that there is significant loss of entanglement due to
yielding of the second network. The strands at these lower rates can hardly reach the
fully extended chain limit to cause chain scission. The sample eventually fails by
mutual chain sliding to reach a state of disengagement.
In summary, the transient viscosity or Cauchy stress involves an exponentially
decreasing area in the dominator of its definition so that it tends to grow in time and
becomes greater than its value in the zero-rate limit, in contrast to the counterpart in
simple shear where the sheared area stays constant. This geometric difference has
27
caused considerable confusion in the literature. We have examined four situations
where the phase “strain hardening” may have emerged to characterize the extensional
rheological behavior of entangled polymers with either linear or branched chain
architecture. In all cases, the definition of the extensional transient viscosity permits the
exponentially shrinking cross-sectional area to mask the origins of the physical
phenomena. Systems that resist yielding and structural failure, such as long-chain
branched LDPE and samples with bimodal molecular weight distribution, simply show
greater upward deviation from its zero-rate linear response because at the same moment
during extension the high rate test is in a more stretched state with a thinner cross-section
than a limiting zero-rate test. This upward deviation may not imply strain hardening at
all. The real strain hardening involving non-Gaussian stretching amounts to having an
engineering stress that grows monotonically with increasing extension and therefore
resist any non-uniform stretching or necking.
V. CONCLUSION
Ductile failure after yielding and rupture after non-Gaussian stretching have both
been shown to occur in uniaxial extension for monodisperse and bidisperse entangled
styrene-butadiene rubber (SBR) melts. Within the accessible range of Hencky strain
rates, the internal chain dynamics of the SBR240K are too fast to allow full chain
extension and rupture, whereas the SBR1M undergoes cohesive failure in the form of
yielding and non-uniform extension at low rates, and rupture at high rates. The
incorporation of a small fraction of SBR1M into a matrix of SBR240K greatly alters the
characteristic responses to both startup extension and step extension.
In particular, we have reached the following conclusions. (A) There is evidence of
double entanglement networking. Following the disintegration of the faster network
formed by the matrix chains, the second network made of SBR1M can retain the
structural integrity of the specimen until it also subsequently yields. The onset of
structural failure is considerably extended beyond that of the pure SBR240K matrix.
The presence of the SBR1M also altered the kinetics of elastic yielding and even delayed
the onset of elastic yielding in the blend of 240/1M (80:20). (B) More surprisingly, in
28
the same range of extensional rates, under which the pure SBR240K and SBR70K only
fail through ductile yielding, the blend of 240/1M (80:20) and 70K/1M (80:20) suffer
rupture. (C) The application of the Considère criterion is irrelevant because the origin
of non-uniform extension appears to be yielding of the entanglement network, unrelated
to any type of elastic instability. (D) We confirm our previous assertion [Wang and
Wang (2008)] that entangled linear polymers cannot attain steady flow during startup
uniaxial extension. In other words, such linear chain systems as the present pure SBR
melts and their blends fail after yielding over a wide range of rates in the form of
non-uniform extension without ever reaching a fully developed flow state. (E) At
various extensional rates beyond the terminal regime, the monotonic rise of the Cauchy
stress before the onset of non-uniform extension stems from its definition that involves
the exponentially shrinking area in the denominator. This pseudo strain hardening has
little to do with the entanglement strain hardening due to the finite chain extensibility that
produces non-Gaussian stretching of the entanglement network.
Acknowledgements The authors would like to express their sincere gratitude to Dr.
Xiaorong Wang from Bridgestone-Americas Center for Research and Technology for
providing the SBR samples in this study. This work is supported, in part, by grants
(DMR-0821697 and CMMI-0926522) from the National Science Foundation.
29
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