strain hardening in polymer glasses phenomenology and modeling · this compressible leonov model...

Strain hardening in polymer glassesphenomenology and modeling

Esther WeltevredenMT 09.08

Section Polymer TechnologyDepartment of Computational and Experimental MechanicsFaculty of Mechanical EngineeringEindhoven University of Technology

prof. dr. ir. H.E.H. Meijerdr. ir. L.E. Govaert

Eindhoven, March 26, 2009

Abstract

Although the complete reversibility of plastic deformation by heating above the glass-transition temperature suggest an elastic nature of strain hardening, the combination ofits strain rate dependence and it anomalous temperature dependence seem to indicate thatpart of the strain hardening response has a viscous origin.In this study an attempt is made to separate these contributions by studying the mechan-ical response of oriented polycarbonate in uniaxial extension and compression. The yieldstress in extension is observed to increase with pre-deformation, whereas it decreases incompression (the so-called Bauschinger effect). Moreover, the oriented specimens displaya strongly increased strain hardening in extension, whereas it nearly diminishes in com-pression.A comparison of these experimental results with simulations, that assume a fully elasticnature of strain hardening (Eindhoven-Glassy-Polymer model) also show deviations, point-ing to a more complex nature of strain hardening. Alternatively, it is demonstrated thatthese observations can be qualitatively captured by introduction of a viscous contributionto strain hardening in terms of a plastic strain-dependence of the activation volume.

Contents

1 Introduction 2

2 Experimental 52.1 Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Deformation glassy polymers: Phenomenology 73.1 Intrinsic stress-strain response of isotropic polycarbonate . . . . . . . . . . 73.2 Deformation of oriented polycarbonate . . . . . . . . . . . . . . . . . . . . 8

4 Deformation of oriented glassy polymers: Modeling 164.1 Simulation by EGP-model . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2 New modeling approach to strain hardening . . . . . . . . . . . . . . . . . 194.3 Validation of the elastic-viscous representation . . . . . . . . . . . . . . . . 234.4 Influence of strain hardening representation on activation volume . . . . . 24

5 Physical background 265.1 Molecular relaxation mechanisms . . . . . . . . . . . . . . . . . . . . . . . 265.2 Influence of orientation on deformation kinetics . . . . . . . . . . . . . . . 28

6 Conclusion and recommendations 30

A Constitutive modeling 35

Chapter 1

Introduction

The post-yield response of polymer glasses generally displays two characteristic phenom-ena: 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 typicallyinduced by intrinsic strain softening whereas the evolution of this localized plastic zonestrongly depends on the stabilizing effect of strain hardening. In the case of insufficientstrain hardening, the material will be inclined to deform plastically by crazing, extremelylocalized zones of deformation that act as precursors for cracks and thus induce macroscop-ically brittle failure [1]. As the latter applies to most polymer glasses, it is evident that afundamental understanding of the origin of strain hardening is essential in the moleculardesign of novel, ductile polymer systems [2].A general representation of the intrinsic deformation behavior of glassy polymers can beseen in figure 1.1, described by the solid line. The first step in this direction was theobservation that plastic deformation of a polymer glass can be almost fully recovered byheating above the glass transition temperature [3]. This finding suggests that the entangledmolecular network remains largely intact during plastic deformation, and inspired Hawardand Thackray [4] to propose a decomposition of the stress at large deformations into twoseparate contributions: a viscous component usually referred to as the flow stress, and arubber-elastic strain hardening component originating from the entangled molecular net-work. The dotted lines in figure 1.1 represent the decomposition of the stress by Hawardand Thackray.

3

Figure 1.1: Schematic representation of the intrinsic deformation behavior of an amorphous polymer. The

solid line represents the total response, which consists of a intermolecular interaction part and a molecular

network part.

Boyce et al. [5] extended the proposition by Haward and Thackray into a three-dimensionalformulation, called the BPA-model. Later this model was refined by several authors [6, 7]resulting in the eight and the full-chain model. More recently the Oxford-model was pro-posed by Buckley [8]. However, the applicability of this model is limited to moderatetemperature and time. Another addition to the Haward approach was derived by Tervoortet al. by means of incorporating a Neo-Hookean description of the strain hardening behav-ior in a Leonov model. This was done using a Neo-Hookean spring and an Eyring dashpot[9]. This compressible Leonov model was further extended by Klompen et al. [10] and vanBreemen et al. [11, 12] leading to the so-called Eindhoven-Glassy-Polymer (EGP) model.Although the above Haward approaches have proven successful from a phenomenological,descriptive point of view [13], there are many arguments against a rubber-elastic, entropicnature of strain hardening. The most puzzling anomalies are the well-known experimentalobservations that the strain hardening intensity is orders of magnitude larger than whatcan be expected from the network density determined in the melt [9, 14], and that strain-hardening decreases with increasing temperature, whereas an entropic stress contributionwould be expected to increase [14, 15, 16]. Moreover, strain hardening was also shown tobe rate-dependent, which contradicts a purely elastic origin [17].Initially, this rate- and temperature-depend strain hardening was interpreted is in terms ofa viscoelastic stress contribution originating from temperature-activated relaxation of theentanglement network [15, 17]. Unfortunately, however, this approach proves less success-ful from a descriptive point of view [18]. Another, more promising route is the additionof a viscous contribution to strain hardening by introducing a deformation-dependencein the flow stress. Examples are recent models that employ the Eyring model equippedwith a strain-dependent activation volume [17], or anisotropic activation volume [19]. In

4

these approaches, a quantitative description was only obtained by including a neo-Hookeanelastic component. Attempts to fit the stress-strain curves solely with a viscous contribu-tion employing a non-constant activation volume were not successful [17, 19]. A strongindication for the existence of this elastic component of the strain hardening response isfound in the unusually pronounced Bauschinger effect [20] in oriented polymer materials,typically observed in uniaxial deformation as a strong increase in tensile yield stress at theexpense of the compressive yield stress [21, 22]. Based on shrinkage stress and yield stressmeasurements on oriented PMMA, Botto, Duckett and Ward [23] were the first to proposethat this Bauschinger effect is related to a frozen-in network stress. To date, however, thisrelation has never been explored. In the present investigation the Bauschinger effect inorient polycarbonate (PC) is analyzed and its possible use for isolating the viscous andelastic contributions to strain hardening is evaluated.To observe the Bauschinger effect, experiments on oriented polycarbonate extruded rodsin uniaxial compression and tension have been conducted. The strain hardening responseof these experiments has been compared with the response given by the material modeldeveloped in the Eindhoven group, the EGP-model. There was found that a discrepancyexists between the model and the experiments. This deviation and results found in liter-ature, lead to the believe that the current constitutive model might not yet be completeand adjustments have to be done.First the experimental setup will be described in Chapter 2, followed by some backgroundon the subject of orientation and the phenomenology of the Bauschinger effect in Chapter3. Hereto the results of experiments on rejuvenated extruded polycarbonate rod are used.The decomposition by Haward and Thackray and the corresponding constitutive model isfurther explored by means of simulations in Chapter 4. After evaluation of the experimentsand validation of the constitutive model, a explanation for the observations is proposed inChapter 5. Finally, overall conclusions are drawn and recommendations for future researchare given in Chapter 6.

Chapter 2

Experimental

2.1 Material

All experiments are performed on an extrusion grade ofLexan (Mn = 14.0 kg/mol, Mw =35.9 kg/mol) supplied as an extruded rod with a nominaldiameter of 10 mm. Having a gauge diameter of 8 mm and a gauge length of 50 mm. Thegeometry of the sample is shown in figure 2.1. Since the viscoelastic and plastic deformation

Figure 2.1: Geometry of the un-oriented extruded polycarbonate rod used in the experiments.

are strongly influenced by the thermal history [10, 24], all the specimens are mechanicallyrejuvenated by torsion to prevent localization and ensure a homogene deformation. Heretothe samples are twisted to-and-fro over a twist angle of 720◦ [10]. After this mechanicaltreatment the samples were pre-strained. To accomplish this, a strain is applied to therejuvenated extruded rod by means of tension, where the strain evolution was monitoredby an extensometer. The pre-strained samples are then used for the uniaxial tension andcompression experiments. For tension no further preparation is needed, for compressionhowever, cylindrical samples (Ø 6 mm × 6 mm) are machined from the deformed samples.The geometry of the compression sample is depicted in figure 2.2.

2.2 Method 6

Figure 2.2: Geometry of compression sample taken from the oriented extruded polycarbonate rod.

2.2 Method

All experiments are performed at an ambient temperature of 23℃, using samples pre-strained in tension.Uniaxial extension test are performed on a Zwick Z010 universal tensile tester at DSM.The mechanically rejuvenated samples are extended to a predefined pre-strain; εp = 0.3,εp = 0.45 and εp = 0.6, using true strain controle. A constant true strain rate ofε̇p = 3 × 10−3 s−1 is used for pre-straining. After pre-straining, the sample is unloadedwith a rate of ε̇ = 3 × 10−3 s−1 and loaded again, loading is done till fracture occurs.The reloading is done using different constant strain rates, ranging from ε̇ = 1× 10−3 s−1

to ε̇ = 3 × 10−2 s−1. In the assumption of incompressible material behavior, engineeringstresses were converted to true stresses. An extensometer was employed to monitor theevolution of the strain during the test.Uni-axial compression tests are performed on a servo-hydraulic MTS 831 Elastomer TestingSystem. The specimens are cylindrical shaped and compressed between two flat, paral-lel steel plates at constant true strain rate control, ranging from ε̇ = 3 × 10−4 s−1 toε̇ = 3 × 10−2 s−1. A correction has been made for the finite stiffness of the compressionsetup. Friction between the sample and the compression stamps is reduced by applyingteflon tape (3M 5480, PTFE skived film tape) on the ends of the sample and using a mix-ture of soap and water (1 : 1) for lubrication. During the compression tests no bulging orbuckling of the sample was observed, meaning friction is sufficiently reduced. True stresseswere calculated in the assumption of incompressibility.

Chapter 3

Deformation glassy polymers:

Phenomenology

3.1 Intrinsic stress-strain response of isotropic poly-

carbonate

Figure 3.1 (left) presents the intrinsic stress-strain response of a glassy polymer in uniaxialcompression, under different strain rates ranging from 10−4 to 10−2. In these experiments,the stress is observed to increase steadily until a maximum is reached, the yield stress.At this point main-chain segmental motion sets in due to the applied stress, resulting in aplastic rate of deformation, which exactly matches the one applied. After the yield point thematerial displays a decrease in the true stress with increasing deformation, the so-calledsoftening. At larger deformations the stress increases again, showing strain hardening.Which is the subject of this study. Although the exact origin of strain hardening is unknownat the moment, it is clear that it is related to the orientating of the moleculair network.The typical stress-strain curve of polycarbonate in figure 3.1 (left) shows a deformation rateor strain rate dependency for glassy polymers. The responses are similar to the intrinsicdeformation behavior, albeit that the stress for a given strain increases due to strain rate.This is due to the fact that increasing the strain rate requires a higher molecular mobilityto obtain the balance between the plastic strain rate and the applied strain rate again.Consequently, leading to a higher yield stress and shifting the entire curve upwards. Thestrain rate dependence of the yield stress is presented in figure 3.1 (right), displaying thetypical linear increase with the logarithmic of the strain rate. This interdependency can

3.2 Deformation of oriented polycarbonate 8

0 0.1 0.2 0.3 0.4 0.50

10

20

30

40

50

60

70

true strain [−]

com

p. tr

ue s

tres

s [M

Pa]

10−2 s−1

10−3 s−1

10−4 s−1

10−4

10−3

10−2

0

10

20

30

40

50

60

70

strain rate [s−1]

com

p. y

ield

str

ess

[MP

a]

Figure 3.1: Left: True stress versus true strain in uniaxial compression. Right: Yield stress versus strain

rate, symbols represent experimental data solid line a fit using the Eyring relation.

be easily described using the so-called Eyring relation 3.1.

σY =k T

V ∗ sinh−1

(ε̇

ε̇0

)(3.1)

Where σY is the yield stress, k is the Boltzmann constant with a value of 1.38065e−23 JK−1,T is the absolute temperature, V ∗ is the activation volume, ε̇ is the strain rate and ε̇0 is apre-exponential factor.Another important feature of the material behavior, is the fact that the yield stress is pres-sure dependent. A depiction is show in figure 3.2 (left), showing the impressive influenceof pressure on the yield stress, e.g. an increase of the pressure by 1000 MPa results inalmost a quadruplicating of the yield stress. For this purpose uniaxial tensile tests areperformed under superimposed pressure. To emphasize the pressure dependence of poly-carbonate figure 3.2 (right) is shown. Visualizing the effect of the pressure dependence onthe stress response for un-oriented polycarbonate in different loading directions, resultingin a distinct difference between tension and compression, i.e. higher value for the stress incompression than the one conducted in tension.

3.2 Deformation of oriented polycarbonate

As mentioned in Chapter 1, the Bauschinger effect is studied, in an attempt to quan-tify the strain hardening phenomena, using polycarbonate as a reference material. TheBauschinger effect [20] can be described as direction-dependent yield behavior for orientedpolymers, where there is a significant increase of the yield stress in tension to the expenseof the compressive yield stress [21, 22]. To investigate the Bauschinger effect in orientedpolycarbonate, experiments in uniaxial compression and tension were performed. To ori-ent the polycarbonate rod an excitation, as shown in figure 3.3 (left) was applied to the


0 250 500 750 10000

50

100

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200

250

hydrostatic pressure [MPa]

true

yie

ld s

tres

s [M

Pa]

data Christiansen et al.model prediction

0 0.5 1 1.5 2 2.5 30

50

100

150

compression

tension|true

str

ess|

[MP

a]

|λ 2 − λ−1|

Figure 3.2: Left: True yield stress versus applied hydrostatic pressure. Symbols represent experimental

data by Christiansen et al [25] at a strain rate of 1.7×10−4 s−1 and solid line the prediction by the Eyring

model [10]. Right: True stress versus true strain for rejuvenated polycarbonate, constant strain rate of

ε̇ = 1× 10−2 s−1 [9].

material. The material was loaded up to the required strain value, followed by unloadingto a force of 0 N. For different orientations this lead to the pre-straining responses as statedin figure 3.3 (right). Showing the deformation and the unloading of the sample, leadingto a plastic strain, which throughout the thesis is referred to as the pre-strain. However,

0 100 200 300 4000

0.3

0.45

0.6

time [s]

true

str

ain

[−]

Figure 3.3: Left: Excitation using a constant strain rate of ε̇p = 3×10−3 s−1, to achieve pre-strain. Right:

Pre-straining response using a constant strain rate of ε̇p = 3× 10−3 s−1.

due to viscoelastic recovery, this pre-strain will not have the exact value as stated in theprevious section, it is a target value. Throughout the thesis the target value will be usedinstead of the residual value for the pre-strain. The residual pre-strain values are statedin table 3.1. To ensure the amount of deformation resulting in plastic strain, a recoveryexperiment has been performed. For this purpose rejuvenated extruded polycarbonaterods were pre-strained, after which the retardation was monitored. The result of which


Target pre-strain value [−] Residual pre-strain value [−]

0.3 0.27

0.45 0.42

0.6 0.55

Table 3.1: Pre-strain values

can be seen in in figure 3.4. Since measurements have been performed on a large timescale,

101

102

103

104

105

106

107

108

0.4

0.42

0.44

0.46

0.48

0.5

εp = 0.5

1hour 1day 1month

time [s]

resi

dual

true

str

ain

[−]

101

102

103

104

105

106

107

108

0.6

0.62

0.64

0.66

0.68

0.7

εp = 0.7

1hour 1day 1month

time [s]

resi

dual

true

str

ain

[−]

Figure 3.4: Left: Time dependent recovery of a polycarbonate rod with a pre-strain of εp = 0.5. Right:

Time dependent recovery of a polycarbonate rod with a pre-strain of εp = 0.7.

it is clear that the recovery will be no more than 6% of the applied strain. Hence, theresidual pre-strain value will not deviate enough from the target pre-strain value to havea significant influence on the response after pre-straining.To prevent recovery, the excitation as shown in figure 3.5 (left) is used for the uniaxial ten-sile tests. An example of a tensile test on a pre-oriented sample, with the pre-orientationpart included, is presented in figure 3.5 (right). First the sample is loaded using a constantstrain rate of ε̇p = 3× 10−3 s−1 to achieve a plastic deformation, the pre-strain. Once thetarget value for the strain is reached, the sample is unloaded using the same strain rate,until the required applied force reaches zero. At this point the sample is loaded again untilfracture occurs. Using the same strain rate for pre-straining as well as for the second load-ing (the actual experiment) leads to the response shown in figure 3.5 (right). Throughoutthe further part of the thesis, only the actual experiment is depicted. Therefore the totalstrain is corrected by subtracting the residual plastic strain. The correction is illustratedin figure 3.6, where the different stress-strain responses in tension are given for differentpre-strains, but with the same strain rate. In figure 3.6 (left) it is clear that the strainhardening responses of the oriented material coincide with the one of the isotropic material,having their yield stress on this curve as well. It is obvious that due to orientation the


0 100 200 300 4000

0.3

0.45

0.6

0.8

pre−orientation

F = 0 N

experiment

time [s]

true

str

ain

[−]

Figure 3.5: Left: Excitation using a constant strain rate of ε̇p = 3 × 10−3 s−1 to achieve pre-strain and

a constant strain rate of ε̇ = 3 × 10−3 s−1 for the transient loading. Right: Stress-strain response for

polycarbonate in transient loading, εp = 0.45, ε̇p = 3× 10−3 s−1, ε̇ = 3× 10−3 s−1.

0 0.3 0.45 0.6 10

50

100

150

200

true strain [−]

true

str

ess

[MP

a]

εp = 0

εp = 0.3

εp = 0.45

εp = 0.6

Figure 3.6: Left: Effect of pre-strain on the material response in tension at a constant strain rate of

ε̇ = 1× 10−3 s−1. Right: Influence of pre-strain on the stress-strain response using a constant strain rate

of ε̇ = 1× 10−3 s−1 in tension.

yield stress and the strain hardening increase significant.Results of uniaxial compression tests on samples pre-strained in tension are presented infigure 3.7. Again it is observed that the compressive stress-strain curve of the orientedsamples coincide with the test of an isotropic sample. Albeit that the strain hardening up-swing is not that pronounced. Therefore, when looking at figure 3.7, showing compressivetrue stress as a function of strain, the complete opposite from figure 3.6 can be seen. Thereis hardly no influence of orientation visible, the change in yield stress seems abundant andthe strain hardening modulus even shows a slight decrease.


−1 0 0.3 0.45 0.6−150

−100

−50

0

50

true strain [−]

true

str

ess

[MP

a]

εp = 0

εp = 0.45

εp = 0.6

0 1 20

50

100

150

|λ 2 − λ−1|

com

p. tr

ue s

tres

s [M

Pa]

εp = 0.45

εp = 0.6

Figure 3.7: Left: Effect of pre-strain on the material response in compression at a constant strain rate of

ε̇ = 1× 10−3 s−1. Right: Influence of pre-strain on the stress-strain response using a constant strain rate

of ε̇ = 1× 10−3 s−1 in compression.

To show the Bauschinger effect in oriented polycarbonate more clearly, a direct comparisonbetween the response in tension and compression is presented in figure 3.8. The responsesfor tension and compression for a pre-strain of εp = 0.45 and εp = 0.6, and a strain rate ofε̇ = 1×10−3 s−1 are depicted in figure 3.8 (left) and figure 3.8 (right), respectively. One can

Figure 3.8: Left: Influence of pre-strain on the stress-strain response using a constant strain rate of

ε̇ = 1× 10−3 s−1 and a pre-strain of εp = 0.45. Right: Influence of pre-strain on the stress-strain response

using a constant strain rate of ε̇ = 1× 10−3 s−1 and a pre-strain of εp = 0.6.

see that for oriented polycarbonate the direction of deformation has a significant influenceon the global material response. The yield stress and strain hardening modulus are stronglyincreased in uniaxial tension, where as the compressive response is hardly influenced. Thepressure dependence of polymers was previously seen in figure 3.2 (left), showing that forisotropic polycarbonate the compressive stress is higher than the tensile stress. Comparingthis to figure 3.8, the complete opposite is visible, i.e. the tensile stress by far exceeds the


compressive stress. Hence, stating the impact of orientation. To further elucidate the effectof pre-strain, the yield stress and strain hardening modulus are depicted for different pre-strains in figure 3.9. Here it is clear that the yield stress in tension increases significantly

Figure 3.9: Left: Influence of pre-strain on the yield stress in tension and compression using a constant

strain rate of ε̇ = 1 × 10−3 s−1. Symbols represent experimental data and dotted line a guide to the

eye. Right: Influence of pre-strain on the strain hardening modulus using a constant strain rate of

ε̇ = 1× 10−3 s−1. Symbols represent experimental data and dotted line a guide to the eye.

due to orientation, where as in compression the yield stress even slightly decreases. Pleasenote: in the isotropic case the compressive yield stress is always higher than the yieldstress in extension, clearly pre-straining treatment changes this. In figure 3.9 (right) theinfluence of the pre-orientation on the strain hardening modulus is even more pronounced,for tension the hardening modulus increases rapidly and in compression it shows a strongdecrease. So far only the influence of orientation on the material response is elaborated.However, the strain rate dependency is still pronounced, although the effect of orientationcan be seen over the entire range, as is depicted in figure 3.10.


10−4

10−3

10−2

10−1

25

45

65

85

105

strain rate [s−1]

yiel

d st

ress

[MP

a]

εp = 0.3

εp = 0.45

εp = 0.6

10−4

10−3

10−2

10−1

25

45

65

85

105

strain rate [s−1]

yiel

d st

ress

[MP

a]

εp = 0.45

εp = 0.6

Figure 3.10: Left: Influence of pre-strain on the yield stress in tension. Right: Influence of pre-strain on

the yield stress in compression. Symbols represent experimental data and solid lines are fits.

As to be expected when a comparison is made with the un-oriented rejuvenated material,the yield stress increases due to increasing strain rate. This increase is clear for tensionas well as for compression. To describe this interdependency, the Eyring relation can beapplied. Therefore this relation can be used to determine the stress dependency by com-puting the activation volume using equation 3.1. The activation volume with pre-strainare depicted in figure 3.11. It is clear that pre-deformation induces a significant change

0 0.3 0.45 0.60

1

2

3

4

pre strain [−]

activ

atio

n vo

lum

e [n

m3 ]

tension

compression

Figure 3.11: Activation volume versus pre-strain. Un-oriented material values derived by Klompen [10]

in the activation volume. This deformation induced change of activation volume impliesthat the strain hardening of polycarbonate displays a significant rate-dependence. Thisis alternatively demonstrated in figure 3.12, where the compressive stress-strain responseof PC is presented for a range of strain rates. There should be mentioned that the dataacquired for small strains is not reliable and are therefore represented by a dotted line. Infigure 3.12 (left) the divergence of the curves at higher deformation, indicate an increaseof the rate dependence. The corresponding values of the activation volume are shown in


Figure 3.12: Left: Compression data achieved by van Breemen [12], Lexan 141R at 23℃. Right: Influence

of strain on the activation volume. Line represents data achieved by van Breemen in a continues way [12],

symbols data achieved in a intermitted way.

figure 3.12 (right) as a function of applied strain. A gradual decrease of the activationvolume is observed with increasing deformation. These results compare reasonably withthe results of figure 3.11. Although the results on pre-oriented samples appear to displaya significantly stronger decrease. To summarize the observations stated in this chapter;

In tension:? the strain hardening responses of the pre-strained samples all coincide with theisotropic material.? the yield stress increases due to strain rate.? the yield stress increases due to pre-strain.? the strain hardening modulus increases due to pre-strain.? the activation volume decreases due to pre-strain.

In compression:? the strain hardening responses of the pre-strained samples all coincide with theisotropic material.? the yield stress increases due to strain rate.? the yield stress hardly changes due to pre-strain.? the strain hardening modulus decreases due to pre-strain.? the activation volume decreases due to pre-strain.

Chapter 4

Deformation of oriented glassy

polymers: Modeling

Many papers have been written trying to explain and model the phenomenon known asstrain hardening. Most of these papers propose a constitutive strain hardening model basedon an elastic contribution of the orientating of the molecular network. The first ones toapply such approach to strain hardening, were Haward and Thackray [4]. They propose adecomposition of the stress, consisting of a part due to the intermolecular interactions anda part related to the orientation of the molecular network, the so-called viscous and therubber-elastic component respectively. This can be represented mechanically by a Maxwellelement, i.e. a spring and dashpot combination, parallel to a rubber-elastic spring. Thiswas only a one-dimensional equation and many three-dimensional models following thesame approach followed, one of them being the Eindhoven-Glassy-Polymer (EGP) model.A brief description concerning the EGP-model can be found in appendix A. Although arubber-elastic approach looks very convincing from a phenomenological point of view, thereare some arguments that take the edge of a fully elastic description of the strain hardening.For one, it was shown that the strain hardening modulus tends to decrease with increasingtemperature. This does not stroke with the rubber-elastic spring assumption, since themodulus of a true entropic spring would increase with temperature [14]. Even more so, thestrain hardening modulus is orders of magnitude larger than what is to be expected fromthe network density in the melt [13, 9]. Furthermore the strain hardening displays a strainrate dependency [17].These observations can be explained by a (partial) viscous origin of the strain hardening.This led to the believe that the current EGP-model might not yet be complete. To verifythe assumption that an additional contribution in EGP-model is needed to properly cap-ture the strain hardening, simulations are done.In the previous chapter a strong Bauschinger effect in PC was found, in this section a

4.1 Simulation by EGP-model 17

investigation is conducted in how this can help in identifying the elastic and viscous com-ponents in strain hardening. Firstly the influence of orientation is checked on the EGP-model, which employs a fully elastic hardening. The simulations are done using the finiteelement software package MSC.Marc/Mentat (version 2005r3), for the implementation ofthe EGP-model the subroutine HYPELA2 is used [11].

4.1 Simulation by EGP-model

For a rough estimation, the EGP-model using one axisymetric element in multimode [12]is used. This element is subjected to different loadcases in series, representing the pre-straining, the unloading and the transient or reversed loading. The parameters are statedas in the experiments, of which the results can be found in chapter 2. The modeling pa-rameters are of polycarbonate, stated in table A.1 and table A.2, the pre-strains used areεp = 0.3, εp = 0.45 and εp = 0.6, the pre-straining strain rate is ε̇p = 3× 10−3 s−1.In figure 4.1 the pre-straining responses are shown. Here the experimental pre-strain re-sponse is compared to the response achieved by simulations using the EGP-model. Striking

0 0.3 0.45 0.60

50

100

strain rate [s−1]

true

str

ess

[MP

a]

sim.

ε = 0.3

ε = 0.45

ε = 0.6

Figure 4.1: Pre-straining response. Left: Experimental data. Right: Simulation data.

is the discrepancy between the residual strain for the experiment and the simulations. Inthe experiment, the sample is unloaded upon reaching the target value for the strain. Thesample unloads, showing little recovery, as shown in figure 4.1 (left). When this trajectoryis compared to the simulation in figure 4.1 (right), there is a pronounced recovery visiblein the simulation upon unloading. Even more so, it appears that if a pre-strain larger than0.3 is applied, the simulation output value for the residual strain is independent of theapplied strain.Finally the pre-strained element is loaded using a strain rate of ε̇ = 1 × 10−3 s−1. Fortension the element was first loaded up to the desired pre-strain, unloaded and loadedagain to ε = 0.8. The results for the different pre-strains for the tension simulations areshown in figure 4.3, stating εp = 0.3 and εp = 0.45 respectively. Remarkable is the small

4.1 Simulation by EGP-model 18

0 0.3 0.45 0.60

50

100

150

true strain [−]

true

str

ess

[MP

a]

0 0.3 0.45 0.60

50

100

150

true strain [−]

true

str

ess

[MP

a]

Figure 4.2: Left: Simulation of cyclic transient loading, εp = 0.3. Right: Simulation of cyclic transient

loading, εp = 0.45.

hysteresis in figure 4.3 (left) and the extreme hysteresis shown in figure 4.3 (right). Again,this figure shows that the residual strain is independent of the applied strain, adding tothis the observation that different loads lead to the same response. From these simulationsit is clear that the influence of the rubber-elastic spring is fairly exaggerated.For the compression samples an extra variable is present, namely the time it took to makecompression samples after pre-straining in tension, at first the waiting time is kept con-stant for all the simulations. The results for the different pre-strains for the compressionsimulations are shown in figure 4.3 (left). It appears that different loads lead to exactly thesame response. Remarkable is, that if the waiting time (tw) is varied in the compressionsimulation, the yield stress in compression becomes time dependent, see figure 4.3 (right).If immediately after unloading a compressive force is applied to the sample, no compressive

0 0.3 0.45 0.6−50

0

50

100

true strain [−]

true

str

ess

[MP

a]

εp = 0.3

εp = 0.45

εp = 0.6

Figure 4.3: Left: Simulation of cyclic reversed loading. Right: Influence of waiting time in simulation.

yield stress is observed. Actually in that case, the yield stress has already been passed inthe unloading path. If now the sample is allowed to recover before reloading, a yield stressin compression is visible. However, this yield stress will depend on the amount of recovery,

4.2 New modeling approach to strain hardening 19

making it a time dependent yield point. There should be noted, that these observationsonly hold for large strains, i.e. ε > 0.3. The rate of retardation is made visible in figure4.4. In this figure the residual strain is plotted versus the logarithmic time, showing the

102

104

106

108

1010

0

0.1

0.2

0.3

0.4

0.5

loading

unloading

retardation

logaritmic time [s]

true

str

ain

[−]

Figure 4.4: Retardation according to the EGP-model for a pre-strain of εp = 0.45.

amount of time it takes, according to the EGP-model, to let all the strain relaxate away.This simulation shows that it takes about 318 years for the plastic strain to minimize.This and the observations mentioned before require an extensive study on the modelingapproach. Recently adjustments have been conducted by Senden [26], nevertheless theyfall outside the scope of this thesis.

4.2 New modeling approach to strain hardening

In previously mentioned models the stress response is decomposed in a viscous part and arubber-elastic part, hereby the strain hardening contribution is represented by a rubber-elastic spring. However, it seems that different loads lead to the same response and that theresidual strain is independent of the applied strain. To emphasize the modeling questionraised in chapter 1, three different strain hardening modeling concepts are considered.

Fully elastic approach to strain hardening

First the currently used representation, a fully elastic modeling approach, for the hardeningresponse is stated. Hereby the yield stress and equation 4.1 [10], which is a one-dimensionalrepresentation of the EGP-model for post yield only, are used to describe the process.

σ = σY + Gr ∗(λ2 − λ−1

)(4.1)


In which σ is the stress, Gr is the hardening modulus, λ is the draw ratio and σY is theyield stress given by the Eyring flow rule as stated in equation 4.2 [10].

σY =k T

V ∗ sinh−1

(ε̇

ε̇0

)(4.2)

Where k is the Boltzmann constant, T is the absolute temperature, V ∗ is the activationvolume, ε̇ is the strain rate and ε̇0 is the rate constant. Combining the equations above givesthe qualitative description stated in figure 4.5. As can be seen, a fully elastic description of

Figure 4.5: Qualitative description of elastic representation of strain hardening in cyclic loading.

the strain hardening leads to a ”kinematic” hardening response. However, for large strainvalues, upon unloading this results in a positive value for the compressive yield stress,which does not comply with the response found in experiments and therefore the fullyelastic representation is not a correct modeling approach.

Viscous approach to strain hardening

Since a fully elastic description of the strain hardening does not capture the real strainhardening response accurately, as shown by simulations using the EGP-model, a fullyviscous description is considered. This suggestion was earlier done by Hoy and Robbins


[27], they show by means of simulation that the strain hardening scales with the yield stress.To a first approximation the assumption is made that the flow process hardens similarlyto the elastic representation. For the viscous description equation 4.3 is proposed.

σ = σY ∗(1 + Cr ∗

(λ2 − λ−1

))(4.3)

In which Cr has a constant value and σY is determined by the Eyring flow rule as statedin equation 4.2. Hereto the equation can be rewritten.

σ =k T

V ∗ sinh−1

(ε̇

ε̇0

)∗ (

1 + Cr ∗(λ2 − λ−1

))(4.4)

A qualitative description of the stated equation is given in figure 4.6. A fully viscous

Figure 4.6: Qualitative description of viscous representation of strain hardening in cyclic loading.

representation of the strain hardening leads to an ”isotropic” hardening response. Whenloading in extension, this model will give a response similar to the elastic approach. Butjust as for the elastic description, the response in figure 4.6 does not completely look like theexperimental response. There is a negative value for the compressive yield stress, howeverthe stress decreases during compression. This does not stroke with the observations madein experiments, in which the stress has a more or less constant value. Therefore the strainhardening response shown in figure 4.6 can be called pseudo isotropic.Equation 4.3 is equivalent to a strain dependent activation volume.

V ∗ = V ∗0 ∗

1

1 + Cr ∗ (λ2 − λ−1)(4.5)

Herein V ∗0 is the initial value, for rejuvenated un-oriented material, of the activation volume

and Cr is a constant determined by the yield stress and the hardening modulus. Thepossibility of a strain dependent activation volume was studied before by Wendland et al.[17] and Buckley [19]. Where they state a modeling proposition capable of capturing thepronounced strain hardening, due to orientation.


Combined elastic-viscous strain hardening

It was shown that a completely elastic or a completely viscous representation of the strainhardening, does not capture the real response accurately. However, both the descriptionscapture a portion of the response very well, since there is a strain rate dependency andthe Bauschinger effect is visible. Therefore a elastic-viscous combination might capturethe entire strain hardening response. A combination of equation 4.1, equation 4.4 andequation 4.5 is proposed in equation 4.6.

σ =k T

V ∗ sinh−1

(ε̇

ε̇0

)+ Gre ∗

(λ2 − λ−1

)(4.6)

Because of different contributions to the total strain hardening response, Gr, it can berepresented by Gr = Gre + Grv , where the subscripts denote the elastic and viscous con-tribution, respectively. Therefore the constant Cr is given by Cr = Grv

σY. In figure 4.7 the

combination of the pseudo kinematic hardening response and pseudo isotropic hardeningresponse, with an equal elastic versus viscous contribution, can be seen. This qualitative

Figure 4.7: Combining elastic and viscous representation of strain hardening in cyclic loading, ratio 50:50.

description looks like the expected trend [22, 28] and is capable of describing an extremeBauschinger effect. Most promising at the moment therefore, seems to be to representthe strain hardening respons by means of a viscous dashpot parallel to the rubber-elastic

4.3 Validation of the elastic-viscous representation 23

spring. However the ratio of the different contributions still has to be determined. In figure4.8 the influence of a change in contribution can be seen.

0

|λ 2 − λ−1|

true

str

ess

[MP

a]

0

|λ 2 − λ−1|

true

str

ess

[MP

a]

0

|λ 2 − λ−1|

true

str

ess

[MP

a]

Figure 4.8: Combining elastic and viscous representation of strain hardening in cyclic loading. Left: Ratio

70:30. Center: Ratio 50:50 Right: Ratio 30:70.

4.3 Validation of the elastic-viscous representation

Previously a description of the strain hardening by means of an elastic-viscous combinationwas proposed. Before being able to properly implement and validate this adjustment, theratio of the elastic versus viscous contribution has to be determined. Hereto the constantC is used. This constant consist of a ratio between the yield stress and the hardeningmodulus, since equation 4.3 is used for the determination. Using this ratio and Gr, theproportion elastic versus viscous strain hardening can be determined. A comparison be-tween the viscoelastic representation for the strain hardening and the experimental resultsis conducted in figure 4.9. The parameters used to conduct the fit are stated in table 4.1.Besides some small deviations the simulation response in figure 4.9 is consistent with theglobal trend observed in the experiments. Therefore it seems advisable to represent thestrain hardening in a viscoelastic matter. A derivation of the three-dimensional represen-

σY [MPa] Gre [MPa] Grv [MPa] Cr [−]

19.7 15 15 0.76

Table 4.1: Used fitting parameters in figure 4.9 for polycarbonate

tation of the viscoelastic strain hardening was conducted by Senden [26].

4.4 Influence of strain hardening representation on activation volume 24

Figure 4.9: Combining elastic and viscous representation of strain hardening in cyclic loading. Combination

experiment and simulation, for εp = 0.45, ε̇ = 1× 10−3s−1.

4.4 Influence of strain hardening representation on

activation volume

In figure 3.11, the activation volume is stated as being a function of the pre-strain. Usingthe Eyring flow rule, stated in equation 3.1 and the representation of the stress as statedin equation 4.3, the activation volume is described in equation 4.7 as a function of strain.

V ∗ = V ∗0 ∗

1

1 + Cr ∗ (λ2 − λ−1)(4.7)

In the equation V ∗0 is the initial value of the activation volume and Cr is a constant deter-

mined by fitting equation 4.7 to the experimental values and using the activation volumesof εp = 0 in tension and compression as a value for V ∗

0 . In figure 4.10 the experimen-tally determined activation volumes and the values, determined by using equation 4.7 areplotted versus |λ2 − λ−1|.

4.4 Influence of strain hardening representation on activation volume 25

0 1 2 3 40

1

2

3

4

|λ 2 − λ−1|

activ

atio

n vo

lum

e [n

m3 ]

tension

compression

Figure 4.10: Activation volume as a function of pre-strain. Symbols represent experimental data and line

a fit using equation 4.7. Un-oriented material values derived by Klompen [10]

Apparently equation 4.7 gives a fairly good description of the pre-strain dependence of theactivation volume for polycarbonate. The used parameters are stated in table 4.2, withthe remark that this parameter set is fitted on the yield stress and strain.

V ∗0 [nm3] Cr [−]

3.11 0.757

Table 4.2: Parameters for polycarbonate, obtained from fitting the activation volume in figure 4.10 with

equation 4.7

Chapter 5

Physical background

In figure 3.11 it was show that there is an influence of orientation on the activation volume.Using equation 4.7 and figure 4.10, it was stated that the dependency can be capturedwith one parameter. This observation raises new questions concerning the influence oforientation on the material response.

5.1 Molecular relaxation mechanisms

The mechanical response of a polymer glass is strongly determined by the mobility of thepolymer chains and the level of activation by stress and temperature. In the glassy statethe polymer chains are ”frozen-in”, their segmental mobility is extremely low and largechanges in conformation are impossible. Under the influence of stress, however, the mo-bility strongly increases, and, at the yield point, the chains have obtained a state of fullsegmental mobility. this degree of mobility is comparable to that in the rubbery state, withthe main difference that it is induced by a stress increase, not by a temperature increase.At the yield stress the segmental mobility allows the chains to change their conformation,which leads to a plastic flow rate that exactly balances the one applied. With respect to thedeformation kinetics it is important to realize that there are several molecular process thatcontribute to the stress regarded to deform a polymer. The most important process is theglass-transition, full main-chain segmental motion, general referred to as the α-transition.However, one can also observe a stress contribution of other molecular motions, the mostwell-known being the secondary glass-transition, or β-transition, which implies the onset ofmobility of a side-chain, or a part of the main-chain. The resulting conformational changesare small and have a low activation energy, implying that they are thermally active withinthe glassy state.In the mechanical response the secondary transition is clearly visible when monitoring theloss angle as a function of temperature, an example of such a dynamic mechanical analyses

5.1 Molecular relaxation mechanisms 27

is presented in figure 5.1 (left). As observed in figure 5.1 (left), for polycarbonate the

Figure 5.1: Left: Loss angle versus temperature, comparison between polycarbonate and poly(methyl

methacrylate) [24]. Right: Crankshaft motion of a segment of a molecular chain [29] and structural

formula of polycarbonate.

β-transition sets in at -100℃, giving rise to a fall in modulus of approximately 1 GPa anda peak in loss angle. At 150℃, full segmental mobility is achieved; the glass-transition.For poly(methyl methacrylate) (PMMA), however, the activation of the β-mobility clearlyrequires a higher temperature, i.e. 25℃. The β-process also contributes to the yield stress,as depicted in figure 5.2 (right).For PMMA the β-transition is linked to side-chain mobility, where in the case of poly-carbonate it is generally assigned to crankshaft motion of a part of the main-chain. Thecrankshaft motion mechanism is schematically depicted in figure 5.1 (right), in which thebonds 1 and 7 remain collinear, while the intermediary groups move as crankshafts. There-fore this is a local motion of the segment of the main chain. In figure 5.1 (right) the struc-tural formula if PC is depicted, clarifying that crankshaft rotation will be around the theO-C bonds of the phenylene-rings.The β-transition also manifests itself in the rate dependence of the yield stress [30]. Thisis schematically represented in figure 5.2 (left). The total response of the polymer can bewell described by placing two Eyring flow processes in parallel [10]. At low strain ratesthe α-process dominates, at higher strain rates the β-transition starts to contribute. Withdecreasing temperature, the β-transition will shift to lower frequencies, as demonstratedin figure 5.2 (right) for polycarbonate. Since the α- and the β-process, have differenttemperature dependencies, time-temperature superposition only applies to the individualcontributions.

5.2 Influence of orientation on deformation kinetics 28

Figure 5.2: Rate dependence of the yield stress. Left: Schematic representation of α and β contribution

to the yield stress. Right: Tensile yield stress versus strain rate of polycarbonate for various temperatures

versus strain rate [24].

5.2 Influence of orientation on deformation kinetics

In figure 3.10 it was clearly shown that there is an increase in strain rate dependence. Thisincrease is schematically depicted in figure 5.3 (left). Analogous to Wendlandt et al. [17]it is proposed that the increase depends on deformation. The deformation dependencyof the strain rate dependence can be interpreted as a deformation dependent activationvolume, shown in figure 5.3 (right). This interpretation is in accordance with current viewsin literature [17, 19, 31]. They show that chain mobility decreases, due to chain orienta-tion. This increase in viscous resistance against deformation is generally interpreted as a

Figure 5.3: Left: Schematic depiction of stress as a function of strain rate. Right: Schematic depiction of

activation volume as a function of pre-strain.

deformation-induced restriction on the mobility of an orientating chain [32, 33].To further elucidate the possible origin of the observed deformation dependence, the acti-vation volumes obtained from oriented samples are compared with those of the α and α+β

5.2 Influence of orientation on deformation kinetics 29

regime for isotropic PC [10], a visualization is given in figure 5.4. Remarkably, it appears

0 0.3 0.45 0.60

1

2

3

4

pre strain [−]

activ

atio

n vo

lum

e [n

m3 ]

V*α

V*α + β

tension

compression

Figure 5.4: Left: Activation volume as a function of pre-strain. Un-oriented material values derived

by Klompen [10]. Right: Stress as a function of strain rate, visualization of the apparent shift of the

β-relaxation.

that upon deformation, the activation volume almost equals the α+β regime. This suggestthe possibility of an alternative interpretation, schematically depicted in figure 5.4. Hereit is assumed that the only β-transition shifts to lower strain rate, upon deformation. Thisobservation is analogous to the decrease in mobility under constant stress observed by Itoet al. [32]. That the β-transition in polycarbonate is orientation dependent is also shownby Arrese-Igor et al. [33], they show that deformation of the phenylene-ring in polycar-bonate leads to less chain mobility. Therefore a higher stress level is needed to achieve thesame deformation as for the un-oriented polycarbonate, making it easier for the secondaryrelaxation mechanism to come into play. So far it is assumed that only the activationvolume changes continuously due to orientation, however, the trend shown in figure 5.4(left) can also implicate a discontinues change. Such a mechanism would require a slightlydifferent approach where not the activation volume, V ∗ but the pre-exponential factor,ε̇0, is deformation dependent. Since the experimental data at this moment only covers alimited range of strain rates, it is inconclusive which of the two actually occur.

Chapter 6

Conclusion and recommendations

Because of unclarities concerning the kinetics and phenomenology of oriented material aswell as the orientation process itself, the Bauschinger effect in polycarbonate has beenexaminated by means of experiments. It was found that oriented polycarbonate exhibitsa large Bauschinger effect. The main focus was on the strain hardening respons, sinceprevious research has raised questions concerning the mechanical representation of strainhardening in glassy polymers.So far the hardening response has been represented by a rubber-elastic spring, followingthe stress decomposition by Haward and Thackray and the fact that plastic deformationis reversed once the material is brought above the glass transition temperature. However,the presence of a negative temperature dependency of the hardening modulus, and the factthat the strain hardening modulus is strain rate dependent, suggest a viscous contribution.Even more so, it was found that for amorphous polymers the hardening modulus is ordersof magnitude larger than what is to be expected from the network density in the melt.Therefore the strain hardening can consists of an elastic as well as a viscous contribution.To validate this hypotheses, the EGP-model was used to conduct simulations of theBauschinger effect present in polycarbonate, of which the outcome was compared withexperimental results. The result was not consistent, a significant discrepancy exists be-tween the experimental responses and the responses obtained by the EGP-model. Themodel states, in contrast to experimental observation, that different pre-strain loads leadto the same response and that the residual strain is independent of the applied strain. Tobe able to describe the strain hardening response more accurately, an approach is pro-posed that combines elastic and viscous contributions to strain hardening. Future workconcerning the strain hardening, should focus on validation of the adjusted representationfor different strain rates, temperatures and materials. Besides this, a closer look should betaken at the different relaxation mechanisms. Since at room temperature and moderatestrain rates, a β-transition appears due to pre-strain. Therefore making the stress additiondue to the β-transition depended on the applied pre-strain. However, due to a limited

31

range of strain rates of the experimental data it is inconclusive whether the activation vol-ume or the pre-exponential factor is deformation dependent. Nevertheless, further researchand validation is required concerning this topic.

Bibliography

[1] H.E.H. Meijer and L.E. Govaert, Mechanical performance of polymer systems: therelation between structure and properties, Prog. Polym. Sci., 30, 915−938, 2005

[2] E.J. Kramer, Open questions in the physics of deformation of polymer glasses, J. Pol.Sci. Part B: Pol. Phys., 43, 3369−3371, 2005

[3] R.N. Haward, , Trans. Faraday Soc., 38, 394-403, 1942

[4] R.N. Haward and G. Thackray, The use of a mathematical model to describe isother-mal stress-strain curves in glassy themoplastics, Proc. R. Soc. London A, 302,453−472, 1968

[5] M.C. Boyce, D.M. Parks and A.S. Argon, Large inelastic deformation of glassy poly-mers. part I: rate dependent constitutive model, Mech. Mater., 7, 15−33, 1988

[6] E.M. Arruda and M.C. Boyce, A three-dimensional constitutive model for the largestrech behavior of rubber elastic materials, J. Mech. Phys. Solids, 41, 389−412, 1993

[7] P.D. Wu and E. van der Giessen, The use of a mathematical model to describeisothermal stress-strain curves in glassy themoplastics, J. Mech. Phys. Solids, 41(3),472−756, 1993

[8] C.P. Buckley and D.C. Jones, Glass-rubber constitutive model for amorphous poly-mers near the glass transition, Polymer, 36(17) 3301−3312 , 1995

[9] T.A. Tervoort and L.E. Govaert, Strain-hardening behavior of polycarbonate in theglassy state, J. Rheol., 44(6), 1263−1277, 2000

[10] E.T.J. Klompen, Mechanical properties of solid polymers, constitutive modelling oflong and short term behaviour, PhD Thesis, University of Technology Eindhoven,2005

[11] L.C.A. van Breemen, Implementation and validation of a 3D model describing glassypolymer behavior, Master Thesis, University of Technology Eindhoven, 2004

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[12] L.C.A. van Breemen, E.T.J. Klompen, L.E. Govaert and H.E.H. Meijer, Constitutivemodeling of polymer glasses: a multi-mode approach, J. Mech. Phys. Solids, in press,2009

[13] R.N. Haward, Strain hardening of thermoplastics, Macromolecules, 26(22),5860−5869, 1993

[14] H.G.H. van Melick, L.E. Govaert and H.E.H. Meijer, On the origin of strain harden-ing, Polymer, 44, 2493−2502, 2003

[15] L.E. Govaert and T.A. Tervoort, , J. Polym. Sci. Part B: Polym. Phys., 42, 2041-2049, 2004

[16] E.M. Arruda, M.C. Boyce and R. Jayachandran, Effects of strain rate, temperatureand thermomechanical coupling on the finite strain deformation of glassy polymers,R. Mech. Mater., 19, 193-212, 1995

[17] M. Wendlandt, T.A. Tervoort and U.W. Suter Non-linear, rate-dependent strain-hardening behavior of polymer glasses, Polymer, 46, 11786−11797, 2005

[18] D.S.A. de Focatiis and C.P. Buckley, Constitutive modelling of large deformations inoriented polymer glasses: bridging the gap between molecularly aware melt rheologyand solid-state deformation, Proceedings DYFP2009, 289-292, 2009

[19] C.P. Buckley, Viscoplastic flow in solid polymers as an intrinsically anisotropicEyring process, Proceedings DYFP2006, 57−60, 2006

[20] J. Bauschinger, Der Zivilingenieur, 27, 289-348, 1881

[21] R.M. Caddell and A.R. Woodliff, Macroscopic yielding of oriented polymers, J. Mater.Sci., 12, 2028-2036, 1977

[22] C. M. Sargent and D. M. Shinozaki, Large Strain Cyclic Deformation of DuctilePolymers, Mater. Sci. Eng., 43, 125−134, 1980

[23] P.A. Botto, R.A. Duckett and I.M. Ward, The yield and thermoelastic properties oforiented poly(methyl metacrylate), Polymer, 28, 257262, 1987

[24] T.A.P. Engels, Predicting performance of glassy polymers. Evolution of the thermody-namic state during processing and service life, PhD Thesis, University of TechnologyEindhoven, 2008

[25] A.W. Christiansen, E. Bear and S.V. Radcliffe, The mechanical behaviour of polymersunder high pressure, Phil. Mag., 24, 451−467, 1971

[26] D.J.A. Senden, Towards a macroscopic model for the finite-strain mechanical responseof semi-crystalline polymers, Master Thesis, University of Technology Eindhoven,2009

BIBLIOGRAPHY 34

[27] R.S. Hoy and M.O. Robbins, Strain hardening in polymer glasses: limitations ofnetwork models, Phys. Rev. Lett., 99, 117801, 2007

[28] C. G’Sell, H. El Bari, J. Perez, J.Y. Cavaille and G.P. Johari, Effect of plastic de-formation on the microstructure and properties of amorphous polycarbonate, Mater.Sci. Eng., A110, 223−229, 1989

[29] D.K. Das-Gupta, Polyethylene: Structure, morphology, moleculair motion and di-electric behavior, IEEE Elec. Ins. Mag., 10(3), 0883−7554, 1994

[30] C. Bauwens-Crowet and J.C. Bauwens, Effect of thermal history on the tensile yieldstress of polycarbonate in the β transition range, Polymer, 24, 921924, 1983

[31] K. Chen and K.S. Schweizer, Suppressed segmental relaxation as the origin of strainhardening in polymer glasses, Phys. Rev. Lett., 102, 038301, 2009

[32] E. Ito, K. Sawamura and S. Saito, Effects of drawing on molecular motions in poly-carbonate, Col. Polym. Sci., 253, 480−484, 1975

[33] S. Arrese-Igor, O. Mitxelena, A. Arbe, A. Alegria, J. Colmenero and B. Frick, Effectof stretching on the sub-Tg phenylene-ring dynamics of polycarbonate by neutronscattering, Phys. Rev. E, 78, 021801, 2008

Appendix A

Constitutive modeling

A three-dimensional constitutive model is used for the description of large strain, time-dependent mechanical behavior of polymers. In the model a distinction is made between thecontribution of secondary interactions between polymer chains and the entangled polymernetwork, which govern the strain hardening. Based on the work of Haward and Thackray,the total Cauchy stress σ can be decomposed in a driving stress σs and a hardening stressσr.

σ = σs + σr (A.1)

The hardening stress, which is physically interpreted as a rubber-elastic contribution of theorienting entangled network, is mathematically described using a Neo-Hookean relation.

σr = GrB̃d (A.2)

Where Gr is the strain hardening modulus and B̃d the deviatoric part of the isochoric leftChaucy-Green deformation tensor.The driving stress, attributed to intermolecular interactions, is split into a deviatoric stressσd

s and a hydrostatic stress σhs . Since a multimode modeling approach is conducted, the

deviatoric part is modeled as a combination of n, parallel linked, Maxwell elements.

σs = σhs + σd

s = κ (J − 1) I +n∑

i=1

GiB̃de,i (A.3)

Here κ is the bulk modulus, J the volume change ratio, I the unity tensor, G the shearmodulus and B̃d

e the elastic part of the isochoric left Cauchy-Green strain tensor. Thesubscript i refers to a specific mode, i = [1, 2, 3, ..., n]. The evolution of J and B̃d

e is givenby the the following equations:

J̇ = J tr(D) (A.4)

36

˙̃Be,i =(L̃−Dp,i

)· B̃e,i + B̃e,i ·

(L̃c −Dp,i

)(A.5)

the plastic deformation rate tensors Dp,i are related to the driving stresses σds,i by a non-

Newtonian flow rule with modified Eyring equations.

Dp,i =σd

s,i

2ηi (τ̄ , p, Sa)(A.6)

Where p, the hydrostatic pressure, and τ̄ , the total equivalent stress, depend on the totalstress according to:

p = −1

3tr (σ) (A.7)

τ̄ =

√1

2σd

s : σds (A.8)

The viscosities are described by an Eyring flow rule, which has been extended to takepressure dependence and intrinsic strain softening into account.

ηi = η0,i,rej

τ̄τ0

sinh(

τ̄τ0

)

︸︷︷︸I

exp

(µp

τ0

)

︸︷︷︸II

exp (S)︸︷︷︸III

(A.9)

The zero-viscosities, η0,i,rej, are defined according to the rejuvenated state. The first partof equation A.9 captures the deformation kinetics. For low values of the equivalent stress,τ̄ < τ , this part equals unity and it decreases exponentially with increasing stress. PartII expresses the pressure dependency governed by the parameter µ, while the third partcaptures the dependency of the viscosity on the thermodynamic history, expressed in thestate parameter S. Where S is related to the equivalent plastic strain (γp) according to:

S (γ̄p) = Sa Rγ (γ̄p) where S ∈ [0, Sa] (A.10)

The initial thermodynamic state of the material is uniquely defined by the state parameterSa. If the material is in its rejuvenated state, Sa has a value of zero. The softening functionR(γp) describes the strain softening process, i.e. the erasure of thermal history with theonset of plastic deformation. The softening function can be normalized as a function ofthe equivalent plastic strain (γp) using a modified Careau-Yassuda relation:

R (γ̄p) =(1 + (r0 exp (γ̄p))

r1)(r2−1)

r1

(1 + rr10 )

(r2−1)r1

where R (γ̄p) ∈< 0, 1] (A.11)

and r0, r1 and r2 are fitting parameters. To summarize: the yield stress increases from itsrejuvenated reference state with increase of Sa and the momentary stress decreases on theonset of plastic deformation finally back to its rejuvenated state [12].

37

The parameters for polycarbonate have been determined in a previous study [24] and aregiven in table A.1

parameter value dimension

Hardening modulus, Gr 26 [MPa]

Bulk modulus, κ 3750 [MPa]

Characteristic stress, τ0 0.7 [MPa]

Initial value for softening variable, Sa 0 [-]

Parameter describing pressure dependence, µ 0.08 [-]

Fitting parameter, r0 0.9657 [-]

Fitting parameter, r1 50 [-]

Fitting parameter, r2 -3.0 [-]

Table A.1: Input parameters for polycarbonate.

38

mode η0,i,rej [MPa · s] Gi [MPa]

1 2.38 · 1011 3.47 · 102

2 5.62 · 109 4.79 · 101

3 7.87 · 108 3.92 · 101

4 1.43 · 108 4.14 · 101

5 2.07 · 107 3.51 · 101

6 3.21 · 106 3.18 · 101

7 4.40 · 105 2.55 · 101

8 9.76 · 104 3.30 · 101

9 6.47 · 103 1.28 · 101

10 4.46 · 103 5.15 · 101

11 5.53 · 101 5.56 · 101

12 4.56 · 10−1 3.99 · 101

13 4.33 · 10−3 3.29 · 101

14 3.80 · 10−4 2.52 · 100

15 2.81 · 10−7 1.62 · 101

16 8.37 · 10−10 4.21 · 100

Table A.2: Modeling values for polycarbonate

strain hardening in polymer glasses phenomenology and modeling · this compressible leonov model...

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