Strain Hardening and Its Relation to Bauschinger Effects in
Oriented Polymers
D. J. A. SENDEN, J. A. W. VAN DOMMELEN, L. E. GOVAERT
Department of Mechanical Engineering, Materials Technology Institute, Eindhoven University of Technology,
P.O. Box 513, 5600 MB, Eindhoven, The Netherlands
Received 9 January 2010; revised 1 April 2010; accepted 25 April 2010
DOI: 10.1002/polb.22056
Published online in Wiley InterScience (www.interscience.wiley.com).
ABSTRACT: The nature of strain hardening in glassy polymers is
investigated by studying the mechanical response of oriented
polycarbonate in uniaxial extension and compression. The yield
stress in extension is observed to increase strongly with predeformation, whereas it slightly decreases in compression (the socalled Bauschinger effect). Moreover, oriented specimens tend
to display increased strain hardening in extension, whereas this
nearly vanishes in compression. It is shown that these observations can be captured by the introduction of a viscous contribution
to strain hardening in terms of a deformation dependence of the
flow stress. This can originate either from a deformation-induced
change in activation volume, as observed for isotactic polypropylene, or from a deformation-induced change of the rate constant,
as observed for polycarbonate, which causes the room temperature yield kinetics of this material to shift from the into the (+)
regime. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym
Phys 48: 1483–1494, 2010
INTRODUCTION The postyield stress–strain response of glassy
rubber-elastic spring,9 a so-called “Langevin spring”. The
application of this stress decomposition formed a solid basis
for the development of a number of 3-D constitutive models, starting with the Boyce-Parks-Argon (BPA) model.10 In
its original form, the strain hardening contribution in the
BPA model was captured by the “three-chain” model of Wang
and Guth11, 12 ; later, this was replaced with the more realistic “eight-chain” model.13 Other 3-D models using different
hyperelastic strain hardening approaches followed, for example, the Oxford Glass-Rubber (OGR) model,14, 15 incorporating
the crosslink sliplink model of Edwards and Vilgis,16 and the
Eindhoven Glassy Polymer (EGP) model,17, 18 which uses a neoHookean model, equivalent to the application of the Gaussian
network theory of rubber elasticity. In this theory, the polymer
strands between entanglements never reach a fully stretched
conformation, and the elastic stress in uniaxial loading is
represented by the following:
polymers generally displays two characteristic phenomena:
strain softening, the initial decrease of true stress with strain,
and strain hardening, the subsequent upswing of the true
stress-strain curve. Localization of strain is typically induced
by intrinsic strain softening, whereas the evolution of this
localized plastic zone strongly depends on the stabilizing effect
of strain hardening. In case of insufficient strain hardening, the
material will tend to form crazes; extremely localized zones of
plastic deformation that act as precursors for cracks, and thus
induce macroscopically brittle failure.1, 2 As the latter applies
to most polymer glasses, it is evident that a fundamental
understanding of the origin of strain hardening is essential
in the molecular design of novel, ductile polymer systems.3
An important step in this direction was made by Haward and
Thackray,4 who were the first to envision strain hardening
as an entropy-elastic contribution of the entangled molecular network. Their inspiration was found in the observation
that plastic deformation of a polymer glass is (almost) fully
recovered by heating above the glass transition temperature
Tg ,5–8 which gives evidence that the entangled molecular network remains largely intact during plastic deformation. This
concept was translated into a 1-D constitutive relation in
which the post-yield stress is additively decomposed into a
viscous contribution, representing the stress-activated yield
process, and a strain hardening contribution, representing the
chain-orientation hardening modeled with a finitely extensible
KEYWORDS: Bauschinger effects; orientation; strain hardening;
thermoplastics; yielding
= Gr (2 − −1 ),
(1)
where Gr is the strain hardening modulus and is the draw
ratio. This expression proved successful from a phenomenological, descriptive point of view, because most amorphous
and semicrystalline polymers display this specific hardening
response over a large deformation range.19–26 It should be
noted that the responses of the eight-chain and the EdwardsVilgis model are indistinguishable from the neo-Hookean
model at some distance from the extensibility limit,26, 27 and
Correspondence to: L. E. Govaert (E-mail: l.e.govaert@tue.nl)
Journal of Polymer Science: Part B: Polymer Physics, Vol. 48, 1483–1494 (2010) © 2010 Wiley Periodicals, Inc.
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JOURNAL OF POLYMER SCIENCE: PART B: POLYMER PHYSICS DOI 10.1002/POLB
that all are capable of capturing the strain hardening response
in different loading geometries.26, 28, 29
Although the constitutive models mentioned above enabled
quantitative analysis of localization and failure in glassy polymers18, 29–32 and revealed the crucial role of the intrinsic
postyield characteristics on macroscopic strain localization,33
there are many arguments against a hyperelastic, entropic
nature of strain hardening. The first argument is related to the
influence of the entanglement network density. In the Gaussian network theory, the strain hardening modulus may be
expressed as:
Gr = nkT,
(2)
where n, k, and T represent the network density (number of chains per unit volume in the network), Boltzmann’s
constant, and the absolute temperature, respectively. The proportionality between strain hardening modulus and network
density, suggested by the Gaussian theory, was investigated
by Van Melick et al.,34 who systematically altered the network
density of polystyrene by blending with poly(2,6-dimethyl1,4-phenylene-oxide) (PPO) and through crosslinking during
polymerization (XPS). Although their results gave convincing
evidence for the hypothesized proportionality in these systems, it should not be concluded that network density is the
key parameter determining the magnitude of strain hardening.
This is shown in Figure 1, where the strain hardening modulus
is plotted versus the network density for a range of polymers.
The values of the strain hardening moduli at room temperature are presented for the following: XPS and PS-PPO,34 PC,26
PMMA,35 POM, PTFE, PA6, and PA66 (all from ref. 19). The network densities in the melt were calculated from the results of
Wu.36 The scatter of the data in Figure 1 clearly demonstrates
that network density cannot be regarded as the key parameter
determining the magnitude of the strain hardening modulus.
Another intriguing argument against an entropic nature of
strain hardening is found in the experimental observation
that strain hardening decreases with increasing temperature, whereas, according to Gaussian theory, it would be
expected to increase.34, 37–39 Initially, this negative temperature dependence was interpreted in terms of a viscoelastic
stress contribution originating from temperature–activated
relaxation of the entanglement network through chain slip,34, 39
a view consistent with the observed molecular weight dependence of strain hardening.39 The idea was elegantly put to the
test by De Focatiis et al.,40–42 who combined the OGR model
with the well-known RoliePoly conformational melt model43
in an attempt to capture the effect of melt orientation on strain
hardening. However, in its current form, the model only captures chain orientation on the entanglement length scale, and
it, therefore, underpredicts strain hardening at temperatures
well below Tg , where it is dominated by sub-entanglement
chain orientation.
Another, more promising, route seems to be the addition of
a viscous contribution to strain hardening by introducing a
deformation dependence in the flow stress. The physical picture is that plastic deformation induces chain orientation,
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FIGURE 1 The relation between strain hardening modulus and
entanglement network density. Open symbols denote the data
for polystyrene systems from Van Melick et al.,34 see text. Closed
symbols represent a collection of data obtained from different
sources,19, 26, 36 for various polymers.
leading to changes in interchain packing that result in an
intensification of activation barriers.44, 45 The first to explore a
viscous contribution to strain hardening were Wendtland et al.,
who presented experimental evidence for a strain-rate dependence of strain hardening for a selection of polymers.46, 47
The data were successfully modeled by adding a deformation dependence to the Eyring flow term through a strain
dependence of the activation volume. This leads to a gradual increase of the strain-rate dependence of the flow stress
with deformation, which, in combination with a neo-Hookean
strain hardening component, proved successful in describing
uniaxial compression experiments at different strain rates.46
In a recent extension of their model, also the temperature
dependence of the postyield response is described accurately
for a number of polymers.48 An alternative approach was
suggested by Buckley,45 who proposed an anisotropic Eyring
flow process. Also herein, an additional hyperelastic term was
required to obtain the level of strain hardening that is experimentally observed.45 Additional experimental evidence for the
existence of a viscous strain hardening contribution was presented by Hoy and Robbins,49–51 who performed atomistic
simulations of large strain uniaxial compression of polymer
glasses. Their results reproduce the important experimental
observations on strain hardening and suggest a major role for
a deformation dependence of the flow stress and only a minor
role for the entropic back stress.49, 51
The concept of a deformation dependence of the flow stress
also seems consistent with the results of Van Melick et al.34
They showed that by plotting the strain hardening moduli
as a function of T − Tg , that is, the distance to Tg , which
changes with PS/PPO composition, the data for different
PS/PPO blends collapse onto a single curve for temperatures
far below Tg . Apparently, the value of Tg is of key importance
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in this range, and the network density does not play a significant role. This observation supports the existence of a relation
between strain hardening and the flow stress, that is, a viscous
contribution to strain hardening.
The results discussed above lead to the conclusion that the
strain hardening response of glassy polymers consists of two
separate components, one viscous and the other elastic. From
an experimental point of view, this constitutes a problem
because the test that is usually applied to study the intrinsic stress–strain response of polymers, a uniaxial compression
test, does not sufficiently discriminate between these components, as evidenced by the success of hyperelastic models in
describing such experimental data. To characterize both components, this study suggests an experimental approach that
exploits the large difference between the compressive and tensile yield stress that is observed in oriented polymers.52–56
Based on shrinkage stress and yield stress measurements on
oriented PMMA, Botto et al.,57 were the first to propose that
this Bauschinger effect is related to a frozen-in network stress.
To date, however, this relation has never been explored. In this
investigation, the Bauschinger effect in oriented polycarbonate is analyzed and its possible use for isolating the viscous
and elastic contributions to strain hardening is evaluated.
EXPERIMENTAL
Axisymmetric tensile bars, with dimensions as shown
in Figure 2, were machined from commercially available
extruded polycarbonate (PC) rod (Lexan, Sabic).
Because strain localization (necking) subsequent to the yield
point inhibits the characterization of the large strain postyield
response, the specimens were mechanically preconditioned by
large strain torsion. The samples were clamped, twisted over
an angle of 990◦ , and subsequently twisted back over the same
angle. In this way, strain softening is eliminated, resulting in
homogeneous deformation of the specimen in a subsequent
tensile test.17, 26
To obtain specimens with different degrees of anisotropy, or
preorientation, the preconditioned tensile bars were subjected
to uniaxial tensile tests at a constant true strain rate of 10−4
s−1 , up to predefined true strain levels of 0, 0.15, 0.3, 0.45,
and 0.6. These tests were performed on a Zwick Z010 tensile
testing machine, equipped with an extensometer to accurately
measure the deformation. After reaching the desired prestrain,
the specimen was unloaded to zero force at the same true
strain rate and then used for either a tensile or a compression
experiment. The different prestrains that were applied led to
different amounts of residual plastic strain after unloading, as
indicated in Table 1.
FIGURE 2 Axisymmetric
millimeters.
tensile
bar,
dimensions
are
in
TABLE 1 Residual Plastic Strains After Pre-Orientation
True Pre-Strain
Residual Plastic Strain
0.15
0.1266
0.30
0.2744
0.45
0.4155
0.60
0.5542
Tensile experiments, at constant true strain rates ranging
from 10−4 to 3 × 10−3 s−1 , were performed on oriented
specimens immediately after the predeformation. For the uniaxial compression experiments, cylindrical specimens, with
the diameter and height both equal to 5 mm, were machined
from the gauge sections of the preoriented tensile bars directly
after applying the preorientation. Compression experiments
at a constant compressive true strain rate of 10−4 s−1 were
performed on a servo-hydraulic MTS 831 Elastomer Testing
System using two parallel, flat steel plates. To prevent any
bulging of the sample, friction was reduced by applying a
lubricating PTFE spray to the polished steel plates. Moreover,
a layer of PTFE skived tape (3M 5480) was placed between
the sample and the lubricated plates. The stiffness of the testing equipment was measured and corrected for in a real-time
feedback loop to ensure accurate strain measurement and
control.
All experiments, including the sample preparation, were performed at room temperature. From the recorded force and
true strain signals, the true stress was calculated assuming
isochoric deformation.
INFLUENCE OF ORIENTATION
In this section, the effect of preorientation on the mechanical response of PC is explored using uniaxial compression and
tensile tests at a single strain rate, 10−4 s−1 . In Figure 3(a),
the mechanical response of PC in uniaxial tension is given for
various levels of prestrain. It is evident that the influence of
orientation is substantial, leading to a pronounced increase in
both yield stress and strain hardening.
When the stress is plotted as a function of the total strain,
that is, the strain measured in the tensile test plus the residual plastic strain in the specimen, the postyield responses
of all measurement curves collapse onto a single curve, see
Figure 3(b). This implies that, upon reloading, the preoriented specimens follow their regular path along the isotropic
curve. The preoriented specimens show a hint of strain softening, although they are expected to be rejuvenated. This is
believed to originate from stress-accelerated physical ageing
that occurs during the unloading of the prestrain.
In the case of uniaxial compression, the influence of orientation on the mechanical response is entirely different, as
illustrated in Figure 4(a). The yield stress remains largely
unaffected, and the strain hardening modulus decreases with
increasing prestrain. As depicted in Figure 4(b), the mechanical response of a preoriented specimen again coincides with
that of an isotropic specimen at large deformations, when the
stress is plotted as a function of the total strain.
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capture the experimentally observed Bauschinger effect, simulations were performed using the EGP model.59, 60 In these
simulations, the imposed uniaxial deformation is cyclic: the
specimen is first loaded in tension up to a certain prestrain
at a constant strain rate and subsequently compressed back
to its original length. The results are presented in Figure 6(a)
for four different prestrain levels. For small prestrains (0.15
and 0.3), the model predicts a Bauschinger effect, that is,
the compressive yield stress is substantially lower in magnitude than the momentary yield stress in tension at the
point of load reversal. At higher prestrains, however, no compressive yield stress is observed anymore. The reason for
this is that the elastic strain hardening stress at such large
prestrains is sufficiently high to induce plastic deformation
during the unloading phase, leading to an apparent “yield
point” at a positive stress level. As a result, the model predicts
that the amount of residual plastic strain (i.e., the residual
FIGURE 3 (a) Mechanical response of preoriented PC in uniaxial
tension, the level of true pre-strain is indicated in the figure. (b)
Same results, but the stress is plotted versus the total strain, that
is, including the residual plastic strain (cf. Table 1) in the sample.
To illustrate the different influences of orientation on the
mechanical behavior in uniaxial tension and compression,
these two responses are shown in a single graph for an
isotropic [Fig. 5(a)] and a preoriented [Fig. 5(b)] specimen. In
the isotropic case, the yield stress in compression is slightly
higher than in tension because of the influence of hydrostatic
pressure.17, 58 In the preoriented case, a strong Bauschinger
effect is observed; the yield stress in tension is much higher
than that in compression. Moreover, stronger strain hardening
is observed in tension than in compression. These effects completely overwhelm the relatively small hydrostatic pressure
effect.
MODELING THE BAUSCHINGER EFFECT
Traditionally, most modeling approaches for the description
of the finite strain mechanical behavior of polymers incorporate a rubber-elastic model for the strain hardening response.
To evaluate whether such modeling approaches are able to
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FIGURE 4 (a) Mechanical response of preoriented PC in uniaxial
compression, the level of true prestrain is indicated in the figure.
(b) Same results, but the stress is plotted versus the total strain,
that is, including the residual plastic strain (cf. Table 1) in the
sample.
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ARTICLE
model used here, only the postyield mechanical behavior is
considered, assuming an additive decomposition of the stress
into a viscous flow stress flow and an elastic strain hardening
stress r , consistent with the work of Haward and Thackray:4
= flow (˙) + r ().
(3)
The viscous flow stress is a function of strain rate ˙ , which is
described with an Eyring relation:63
˙
kT
,
(4)
flow (˙) = ∗ sinh−1
V
˙0
where the activation volume V ∗ and the rate constant ˙0 are
model parameters. Boltzmann’s constant is denoted by k, and
the absolute temperature T is 293 K in all simulations. The
elastic strain hardening stress is a function of the draw ratio
, according to a neo-Hookean relation:
r () = Gr (2 − −1 ),
(5)
FIGURE 5 The difference between the mechanical response of
PC in uniaxial tension and compression for (a) isotropic specimens, data taken from ref. 26, and (b) specimens that have been
preoriented in tension up to a true strain of 0.6.
strain after unloading to zero stress) is identical for all prestrains >0.35. To give an impression of the actual behavior,
uniaxial tension and compression curves are combined in a
single plot, see Figure 6(b). Please note that the compression
experiments were not performed immediately after unloading. Nevertheless, the discrepancy between the experimental
behavior and the model predictions becomes frighteningly
clear.
Besides the EGP model, also other well-established models
that describe the mechanical behavior of glassy polymers, such
as the BPA model10 and the OGR model,14, 61, 62 will predict this
physically unrealistic behavior. In the following, a simple 1-D
model is used to show that the problems are caused by the
decomposition of the stress as it was proposed by Haward and
Thackray.4
Elastic Strain Hardening
First, the traditional approach is used, incorporating a purely
elastic description of the strain hardening behavior. In the 1-D
FIGURE 6 Mechanical response of PC in cyclic uniaxial deformation: (a) as predicted by the EGP model, and (b) an impression
of the actual behavior, see text. For clarity, the tensile prestrain
levels are indicated in the graphs.
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TABLE 2 Model Parameters.
V ∗ (nm3 )
˙0 (s−1 )
Gr (MPa)
3.0
10−20
26
where Gr represents the elastic strain hardening modulus. In
Table 2, values of the model parameters are given, which are
representative for polycarbonate.18, 64 Again, a cyclic deformation path with a constant absolute true strain rate of 10−4 s−1
is used to evaluate the response of the model.
In Figure 7(a), the total stress response of the model is
shown, as well as the individual contributions of the viscous
flow stress and elastic strain hardening stress. Because of
the absence of preyield elasticity in this simplified approach,
the viscous stress instantaneously attains its constant flow
level, which changes sign when the deformation direction is
reversed. The elastic stress obeys a neo-Hookean relation,
obviously following the same curve during both the tensile
and compressive stages. The total stress is simply an addition
of these two components, revealing that the model qualitatively predicts the same unrealistic response as the EGP model,
with an apparent “yield stress” on unloading at positive stress
levels.
The cause of these physically erroneous predictions is that
the elastic contribution to the stress dominates over the viscous contribution at high deformations, resulting in a response
reminiscent of traditional kinematic strain hardening.
Viscous Strain Hardening
It was already argued in the introduction that the strain hardening behavior of polymers contains a viscous contribution. To
explore the implications of this, the 1-D model is first reformulated such that the strain hardening is fully viscous. This
can be achieved by adding the neo-Hookean dependence on
deformation to the viscous stress:
= flow (˙) + Gr (2 − −1 ),
Gr
= flow (˙) 1 +
(2 − −1 ) .
flow (˙)
(6)
(7)
FIGURE 7 Simulated mechanical response in uniaxial deformation: (a) with purely elastic strain hardening, (b) with purely viscous
strain hardening, and (c) with an equal amount of elastic and viscous strain hardening. The total stress (black), as well as the contributions of the viscous stress (red) and the elastic strain hardening stress (green) are plotted. (d) Total stress for different ratios of the
elastic and viscous strain hardening contribution.
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This expression is generalized by introducing a parameter Cr ,
which represents the magnitude of viscous strain hardening.
Together with a substitution of eq 4, this yields:
˙ kT
1 + Cr (2 − −1 ) .
(8)
= ∗ sinh−1
V
˙0
In principle, this implies that the response of this model equals
that of the model with purely elastic strain hardening at a
single strain rate, as long as the deformation direction is not
reversed. At rates higher than this specific strain rate, stronger
strain hardening is predicted, and at lower strain rates, the
strain hardening response is weaker. Simulation results using
this model are depicted in Figure 7(b), using Cr = 0.514. The
overall response during the tensile part of the deformation is
identical to that of the model with purely elastic strain hardening, but the behavior becomes completely different when
the deformation direction is reversed. The compressive yield
stress is of equal magnitude, but opposite in sign to the apparent tensile yield stress at the moment of load reversal; a
response reminiscent of traditional isotropic strain hardening.
The response of the purely viscous model also does not correspond with the experimentally observed influence of tensile
prestrain on the compressive yield stress, because the model
predictions do not exhibit a Bauschinger effect. Moreover, a
negative strain hardening modulus is predicted in the compressive phase of the test, whereas experiments show that
the compressive strain hardening modulus of specimens that
were preoriented in tension becomes small but not negative
under the influence of orientation, compare Figure 4.
Elastic-Viscous Strain Hardening
Because neither the model with purely elastic strain hardening
nor the model with purely viscous strain hardening is able to
capture the experimentally observed Bauschinger effect, the
applicability of a combination of these two approaches is now
explored. This is done by combining eqs 5 and 8:
=
kT
sinh−1
V∗
˙
˙0
1 + Cr 2 − −1
+ (1 − ) Gr (2 − −1 ),
(9)
where represents the amount of viscous strain hardening
relative to the total amount of strain hardening. This idea is
not new. In fact, there is a striking similarity with a classical
approach to describe the Bauschinger effect in metals, using a
combination of kinematic and isotropic hardening.65 In some
cases, a so-called Bauschinger ratio is defined,66 representing
the ratio of kinematic to isotropic hardening, similar to the
parameter used here. The response of this model is shown
in Figure 7(c) for an equal distribution of the total strain hardening in an elastic and a viscous contribution, that is, = 0.5.
The overall response in the tensile part of the deformation is
again identical to that of the model with purely elastic strain
hardening, but upon reversal of the loading direction, a pronounced Bauschinger effect is observed, with a yield stress
equal to the initial yield stress of the unoriented material and
a complete absence of strain hardening. The response of the
model may be changed by altering the ratio between the elastic and the viscous strain hardening contribution, as illustrated
in Figure 7(d). An increase in the elastic contribution leads to
a decrease in the compressive yield stress and an increase in
the compressive strain hardening.
DEFORMATION-INDUCED CHANGES IN
STRAIN-RATE DEPENDENCE
Now that it is established that the model with combined elastic
and viscous strain hardening enables a qualitative description of the Bauschinger effect, the nature of the viscous strain
hardening contribution is further investigated. In the model
discussed above, this viscous contribution to strain hardening is introduced as a deformation dependence in the viscous
flow stress. Because this is modeled with an Eyring flow relation, it suggests that the viscous part of the strain hardening
can be interpreted either as a deformation dependence of the
activation volume:
V ∗ () =
V∗
,
f ()
(10)
or as a deformation dependence of the rate constant:
ln(˙0 ()) = ln(˙0 ) f (),
(11)
where f () = 1 + Cr (2 − −1 ) for the example discussed
in the previous section. Although a deformation-dependent
activation volume has already been suggested by several
researchers,45, 46, 57, 67 a deformation-dependent rate constant
has not. Both parameters have quite a different influence on
the predicted yield kinetics, that is, the strain-rate dependence
of the yield stress, as schematically illustrated in Figure 8.
A deformation dependence of the activation volume V ∗ implies
that the strain-rate dependence changes with deformation,
whereas a deformation dependence of the rate constant ˙0
results in a shift along the strain rate axis. It is worth noting that the possibility that both parameters change with
deformation cannot be excluded at this point.
To examine the influence of predeformation on the yield kinetics of PC, the tensile yield stress of oriented PC is presented
as a function of the logarithm of the strain rate in Figure 9(a).
The corresponding values for the activation volume V ∗ , calculated from the slope of the straight lines that are formed by the
data points, are plotted in Figure 9(b). Remarkably, the activation volume already decreases strongly with small amounts
of prestrain and subsequently remains essentially constant.
A possible explanation for this, almost stepwise, change in
activation volume may be found in the fact that polycarbonate,
as most polymers, exhibits so-called thermorheologically complex behavior. This implies that, depending on the temperature
and time scale of the experiment, more than one molecular process may contribute to the mechanical response, each
with their own characteristic activation volume and activation
energy. At low to moderate strain rates, the room temperature
yield kinetics of isotropic PC are governed by the -process,
which is associated with full main chain segmental motion,
that is, the primary glass transition. At higher strain rates,
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independently and that their stress contributions are additive, as illustrated in Figure 10(b). For isothermal conditions,
the yield stress of the model can be expressed as:
kT
˙
kT
˙
+ ∗ sinh−1
,
(12)
y (˙) = ∗ sinh−1
V
˙0,
V
˙0,
where subscripts and indicate the process with which a
parameter is associated.
As indicated in Figure 9(b), there is a remarkable agreement
between the changes in activation volume upon deformation
and the values reported in literature64 for the activation volumes of the and ( + )-processes in PC. This suggests that
preorientation causes a transition of the room temperature
yield kinetics of PC from the into the ( + ) regime, indicating that the contributions of these processes may have
shifted along the strain rate axis, compare Figure 8(b). To
FIGURE 8 Schematic illustration of the effect of deformation on
the yield kinetics in case of (a) a deformation-dependent activation volume, and (b) a deformation-dependent rate constant.
The range of strain rates that is usually covered with mechanical
experiments is also indicated in the figure.
the contribution of the -process becomes apparent, leading
to a distinct change in slope of the strain-rate dependence of
the yield stress, see Figure 10(a). This contribution of the process is associated with partial mobility of the main chain,
generally referred to as a secondary glass transition. Although
several studies indicate that phenyl ring motions play a role
in this process,68–71 there is substantial experimental evidence
that these motions are restricted to the high temperature part
of the observed relaxation behavior.69, 71–73 The main contribution to the -process seems to originate from the mobility
of the carbonate group.73–76
As demonstrated by several researchers,64, 77–79 the deformation kinetics of a polymer that exhibits thermorheologically complex behavior can be accurately described using
the Ree-Eyring80 modification of the original Eyring theory.63
Essentially, this theory assumes that the two processes act
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FIGURE 9 (a) Strain-rate dependence of the tensile yield stress
of oriented PC, the various levels of true prestrain are indicated
in the figure. Lines are a guide to the eye. (b) Activation volume
as a function of prestrain. Literature values64 for the activation
volume of the and ( + )-processes in PC are indicated by the
solid lines.
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ARTICLE
explore this transition, additional measurements were performed using dog-bone-shaped tensile bars with a gauge
length of 20 mm. Given the limitations of the testing equipment, the use of small specimens enables the coverage of a
much larger range of strain rates. Orientation was applied
to these specimens through a cold rolling process. Results
from these measurements, at prestrains of 20% and 35%, are
presented in Figure 11(a), which convincingly shows that preorientation causes a shift of the contributions of the and
-processes along the strain rate axis. This implies that the
activation volumes of both processes are independent of the
level of preorientation. The solid lines in the figure are fitted using eq 12, with activation volumes as determined by
Klompen et al.64 and rate constants as listed in Table 3. It can
be concluded that the measured yield kinetics are accurately
described by assuming a deformation dependence of the rate
constants of both processes. In Figure 11(b), the experimental
data from Figure 9(a) is plotted once more, now accompanied
by fits using eq 12. The activation volumes of the and
FIGURE 11 (a) Strain-rate dependence of the tensile yield stress
of small, cold-rolled PC specimens. The thickness reduction
achieved in the rolling process is indicated in the figure. (b) Experimental data as in Figure 9(a). In both figures, the solid lines are
fits using eq 12, assuming a deformation dependence of the rate
constants.
-processes are constant and equal to those determined by
Klompen et al.64 The rate constants for both processes are
fitted separately for each level of prestrain, but their values
are not given here, since these cannot be uniquely determined from the data. Clearly, the experimental data on the
large, axisymmetric tensile bars is also accurately described
by eq 12, assuming a deformation dependence of the rate
constants. These results are perhaps somewhat unexpected,
because the strain-rate dependence of the strain hardening
TABLE 3 Rate Constants for the Fits in Figure 11(a)
FIGURE 10 (a) Strain rate dependence of the tensile yield stress
of isotropic PC at different temperatures, data taken from ref. 64.
(b) Schematic illustration of the Ree-Eyring modeling approach.
20%
35%
˙0, = 1.4 × 10−18 s−1
˙0, = 4.2 × 10−20 s−1
˙0, = 3.8 × 10−3 s−1
˙0, = 2.3 × 10−3 s−1
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JOURNAL OF POLYMER SCIENCE: PART B: POLYMER PHYSICS DOI 10.1002/POLB
behavior of PC has previously been successfully modeled by
assuming a deformation-dependent activation volume.46
To demonstrate that the behavior observed in PC does not
necessarily apply to all polymers, reference is made to the
experimental data from a recent study on the anisotropic yield
behavior of isotactic polypropylene (iPP) by Van Erp et al.,81
who measured the tensile yield kinetics of hot drawn tapes
with different levels of orientation. Their results, depicted
in Figure 12(a), unambiguously show that the slope of the
strain-rate dependence of the yield stress increases continuously with increasing orientation. The dashed lines in the
figure are fits of the yield kinetics for each draw ratio using
eq 12 (omitting the -process). Figure 12(b) shows the rate
constants and activation volumes obtained in this fitting process, as a function of prestrain. The rate constant is found,
within experimental error, to remain constant with increasing prestrain. It is, therefore, concluded that the influence
FIGURE 13 Experimental data as in Figure 12(a). Solid lines are
fits using eq 12 (omitting the -process) with the same rate
constant, but different activation volumes for the different draw
ratios.
of prestrain on the yield kinetics of iPP is characterized
solely by a deformation-dependent activation volume, which
is also shown in Figure 12(b). Indeed, the solid lines in Figure 13 demonstrate that the yield behavior can be accurately
described by eq 12 (omitting the -process) using the same
rate constant (˙0 = 3 × 10−10 s−1 ), but different activation
volumes (cf. Table 4) for the various draw ratios.
All in all, it is evident that there is a marked difference between
the influence of orientation on the tensile yield kinetics for
PC and that for iPP. The former clearly shows a deformation
dependence of the rate constant, whereas the latter displays a
deformation-dependent activation volume. The cause for this
fundamental difference is presently unclear.
CONCLUSIONS
Although the mechanical responses in uniaxial tension and
compression are quite similar for isotropic polymers, they
increasingly deviate when the molecular chains in the material become oriented; a phenomenon which is referred to as
the Bauschinger effect. Experiments on polycarbonate show a
dramatic increase in tensile yield stress and strain hardening
upon preorientation in tension, whereas the compressive yield
stress and strain hardening slightly decrease.
TABLE 4 Activation Volumes for the Fits in Figure 13
FIGURE 12 (a) Strain-rate dependence of the tensile yield stress
of iPP tapes, data taken from Van Erp et al.81 The draw ratio
achieved in the hot drawing process is indicated in the figure.
Dashed lines are fits using eq 12 (omitting the -process), see
text. (b) Rate constants (top part) and activation volumes (bottom
part) as a function of true prestrain.
1492
V ∗ (nm3 )
1
2.07
2
1.30
4
0.41
6
0.21
INTERSCIENCE.WILEY.COM/JOURNAL/JPOLB
ARTICLE
It was shown that the experimentally observed Bauschinger
effect cannot be described using traditional modeling
approaches for the mechanical behavior of polymers, which
assume that the strain hardening response is purely elastic. Simulations with a simple 1-D model demonstrate that
such approaches fail to describe the observed Bauschinger
effect, because it results in (extreme) kinematic strain hardening behavior. A purely viscous strain hardening model is also
not suitable, because this leads to isotropic hardening behavior. The combination of these two approaches yields a simple
but powerful model that is able to qualitatively capture the
Bauschinger effect.
An investigation of the influence of orientation on the tensile yield kinetics of polycarbonate and isotactic polypropylene
leads to the conclusion that the nature of the viscous contribution to the strain hardening behavior may differ between
polymers. For polycarbonate, a deformation dependence of
the rate constant is observed, which causes the room temperature yield kinetics of the material to shift from the into
the ( + ) regime. On the other hand, isotactic polypropylene
clearly exhibits a deformation-dependent activation volume,
which causes the yield kinetics to continuously change as the
level of orientation increases. The cause for this fundamental
difference is presently unclear.
Experimental assessment of the Bauschinger effect and its
influence on the yield kinetics of a polymer proves to be a
valuable tool in the correct characterization of both the elastic
and the viscous contribution to the strain hardening behavior.
This research was supported by the Dutch Technology Foundation STW, applied science division of NWO, and the Technology
Program of the Ministry of Economic Affairs (under grant
number 07730).
REFERENCES AND NOTES
1 Boyce, M. C.; Haward, R. N. The Physics of Glassy Polymers,
2nd ed.; Chapman & Hall: London, 1997; chapter 5, pp 213–293.
13 Arruda, E. M.; Boyce, M. C. J Mech Phys Solids 1993, 41,
389–412.
14 Buckley, C. P.; Jones, D. C.; Polymer 1995, 36, 3301–3312.
15 Li, H. X.; Buckley, C. P. Int J Solids Struct 2009, 46, 1607–1623.
16 Edwards, S. F.; Vilgis, Th. Polymer 1986, 27, 483–492.
17 Govaert, L. E.; Timmermans, P. H. M.; Brekelmans, W. A. M.
J Eng Mater Technol 2000, 122, 177–185.
18 Klompen, E. T. J.; Engels, T. A. P.; Govaert, L. E.; Meijer, H. E. H.
Macromolecules 2005, 38, 6997–7008.
19 Haward, R. N.; Macromolecules 1993, 26, 5860–5869.
20 G’Sell, C.; Jonas, J. J. J Mater Sci 1981, 16, 1956–1974.
21 G’Sell, C.; Boni, S.; Shrivastava, S. J Mater Sci 1983, 18,
903–918.
22 Cross, A.; Haward, R. N.; Polymer 1978, 19, 677–682.
23 Mills, P. J.; Hay, J. N.; Haward, R. N. J Mater Sci 1985, 20,
501–507.
24 Haward, R. N. Polymer 1987, 28, 1485–1488.
25 Haward, R. N. J Polym Sci: Part B: Polym Phys 1995, 33,
1481–1494.
26 Tervoort, T. A.; Govaert, L. E.; J Rheol 2000, 44, 1263–1277.
27 Sweeney, J.; Comput Theoret Polym Sci 1999, 9, 27–33.
28 Arruda, E. M.; Boyce, M. C. Int J Plast 1993, 9, 697–720.
29 Boyce, M. C.; Arruda, E. M.; Jayachandran, R. Polym Eng Sci
1994, 34, 716–725.
30 Boyce, M. C.; Arruda, E. M.; Polym Eng Sci 1990, 30, 1288–
1298.
31 Wu, P. D.; Van Der Giessen, E. Int J Plast 1995, 11, 211–235.
32 Van Melick, H. G. H.; Bressers, O. F. J. T.; Den Toonder, J. M. J.;
Govaert, L. E.; Meijer, H. E. H. Polymer 2003, 44, 2481–2491.
33 Meijer, H. E. H.; Govaert, L. E. Macromol Chem Phys 2003, 204,
274–288.
34 Van Melick, H. G. H.; Govaert, L. E.; Meijer, H. E. H. Polymer
2003, 44, 2493–2502.
2 Meijer, H. E. H.; Govaert, L. E. Prog Polym Sci 2005, 30, 915–938.
35 Van Breemen, L. C. A.; Engels, T. A. P.; Pelletier, C. G. N.;
Govaert, L. E.; Den Toonder, J. M. J. Philos Mag 2009, 89, 677–696.
3 Kramer, E. J. J Polym Sci: Part B: Polym Phys 2005, 43,
3369–3371.
36 Wu, S.; J Polym Sci Part B: Polym Phys 1989, 27, 723–741.
4 Haward, R. N.; Thackray, G. Proc R Soc London Ser A 1968, 302,
453–472.
5 Gurevich, G. I.; Kobeko, P. P. Rubber Chem Technol 1940, 13,
904–917.
6 Haward, R. N.; Trans Faraday Soc 1942, 38, 394–403.
7 Hoff, E. A. W. J Appl Chem 1952, 2, 441–448.
8 Haward, R. N.; Murphy, B. M.; White, E. F. T. J Polym Sci: Part
A-2: Polym Phys 1971, 9, 801–814.
9 Kuhn, W.; Grün, F. Kolloid Z 1942, 101, 248–271.
10 Boyce, M. C.; Parks, D. M.; Argon, A. S. Mech Mat 1988, 7,
15–33.
11 James, H. M.; Guth, E. J Chem Phys 1943, 11, 455–481.
12 Wang, M. C.; Guth, E. J Chem Phys 1952, 20, 1144–1157.
37 Engels, T. A. P.; Govaert, L. E.; Meijer, H. E. H. Macromol Mater
Eng, in press.
38 Arruda, E. M.; Boyce, M. C.; Jayachandran, R. Mech Mater
1995, 19, 193–212.
39 Govaert, L. E.; Tervoort, T. A. J Polym Sci Part B: Polym Phys
2004, 42, 2041–2049.
40 De Focatiis, D. S. A.; Buckley, C. P. In Book of Abstracts: 14th
International Conference on Deformation, Yield and Fracture of
Polymers, Kerkrade, The Netherlands, MaTe, Eindhoven, April
6–9, 2009; pp 289–292.
41 De Focatiis, D. S. A.; Embery, J.; Buckley, C. P. In AIP Conference Proceedings, American Institute of Physics, 2008; Vol. 1027,
pp 1327–1329.
42 De Focatiis, D. S. A.; Embery, J.; Buckley, C. P. J Polym Sci
Part B: Polym Phys 2010, 48, 1449–1463.
STRAIN HARDENING IN ORIENTED POLYMERS, SENDEN, VAN DOMMELEN, AND GOVAERT
1493
JOURNAL OF POLYMER SCIENCE: PART B: POLYMER PHYSICS DOI 10.1002/POLB
43 Likhtman, A. E.; Graham, R. S. J Non-Newtonian Fluid Mech
2003, 114, 1–12.
61 Buckley, C. P.; Dooling, P. J.; Harding, J.; Ruiz, C. J Mech Phys
Solids 2004, 52, 2355–2377.
44 Chen, K.; Schweizer, K. S. Phys Rev Lett 2009, 102, 038301.
62 Wu, J. J.; Buckley, C. P. J Polym Sci Part B: Polym Phys, 2004,
42, 2027–2040.
45 Buckley, C. P. In Book of Abstracts: 13th International Conference on Deformation, Yield and Fracture of Polymers, Kerkrade,
The Netherlands, MaTe, Eindhoven, April 10–13, 2006; pp 57–60.
46 Wendlandt, M.; Tervoort, T. A.; Suter, U. W. Polymer 2005, 46,
11786–11797.
47 Govaert, L. E.; Engels, T. A. P.; Wendlandt, M.; Tervoort,
T. A.; Suter, U. W. J Polym Sci Part B: Polym Phys 2008, 46,
2475–2481.
48 Wendlandt, M.; Tervoort, T. A.; Suter, U. W. J Polym Sci Part
B: Polym Phys 2010, 48, 1464–1472.
49 Hoy, R. S.; Robbins, M. O. Phys Rev E 2008, 77, 031801.
50 Hoy, R. S.; Robbins, M. O. J Polym Sci Part B: Polym Phys
2006, 44, 3487–3500.
51 Robbins, M. O.; Hoy, R. S. J Polym Sci Part B: Polym Phys
2009, 47, 1406–1411.
52 Brown, N.; Duckett, R. A.; Ward, I. M. Philos Mag 1968, 18,
483–502.
53 Duckett, R. A.; Ward, I. M.; Zihlif, A. M. J Mater Sci - Lett 1972,
7, 480–482.
63 Eyring, H.; J Chem Phys 1936, 4, 283–291.
64 Klompen, E. T. J.; Govaert, L. E. Mech Time-Depend Mater
1999, 3, 49–69.
65 Sowerby, R.; Uko, D. K.; Tomita, Y. Mater Sci Eng 1979, 41,
43–58.
66 Cardoso, R. P. R.; Yoon, J. W. Int J Plast 2009, 25, 1684–1710.
67 Coates, P. D.; Ward, I. M. J Mater Sci 1978, 13, 1957–1970.
68 Yee, A. F.; Smith, S. A. Macromolecules 1981, 14, 54–64.
69 Wehrle, M.; Hellmann, G. P.; Spiess, H. W. Colloid Polym Sci
1987, 265, 815–822.
70 Hansen, M. T.; Boeffel, C.; Spiess, H. W. Colloid Polym Sci
1993, 271, 446–453.
71 Monnerie, L.; Laupretre, F.; Halary, J. L. Adv Polym Sci 2005,
187, 35–213.
72 Kim, C. K.; Aguilar-Vega, M.; Paul, D. R. J Polym Sci Part B:
Polym Phys 1992, 30, 1131–1142.
54 Rawson, F. F.; Rider, J. G. Polymer 1973, 15, 107–110.
73 Wimberger-Friedl, R.; Schoo, H. F. M. Macromolecules 1996,
29, 8871–8874.
55 Raghava, R. S.; Caddell, R. M. Int J Mech Sci 1974, 16, 789–799.
74 Boyer, R. F.; Polym Eng Sci 1968, 8, 161–185.
56 Sargent, C. M.; Shinozaki, D. M. Mater Sci Eng 1980, 43,
125–134.
75 Heijboer, J.; Int J Polym Mater 1977, 6, 11–37.
57 Botto, P. A.; Duckett, R. A.; Ward, I. M. Polymer 1987, 28,
257–262.
58 Rabinowitz, S.; Ward, I. M.; Parry, J. S. C. J Mater Sci 1970, 5,
29–39.
59 Tervoort, T. A.; Klompen, E. T. J.; Govaert, L. E. J Rheol 1996,
40, 779–797.
60 Van Breemen, L. C. A.; Klompen, E. T. J.; Govaert, L. E.;
Meijer, H. E. H. Contact Mechanics in Glassy Polymers, Ph.D.
Thesis, Eindhoven University of Technology, Chapter 2, 2009.
http://www.mate.tue.nl/mate/pdfs/10677.pdf.
1494
76 Starkweather, H. W.; Avakian, P. Macromolecules 1989, 22,
4060–4062.
77 Roetling, J. A.; Polymer 1965, 6, 311–317.
78 Roetling, J. A.; Polymer 1966, 7, 303–306.
79 Bauwens-Crowet, C.; Bauwens, J. C.; Homès, G. J Mater Sci
1972, 7, 176–183.
80 Ree, T.; Eyring, H. J Appl Phys 1955, 26, 793–800.
81 Van Erp, T. B.; Reynolds, C. T.; Peijs, T.; Van Dommelen,
J. A. W.; Govaert, L. E. J Polym Sci Part B: Polym Phys 2009,
47, 2026–2035.
INTERSCIENCE.WILEY.COM/JOURNAL/JPOLB