Strain hardening and anisotropy in solid polymers · relation between molecular orientation and...

Strain hardening and anisotropy in solid polymers

Citation for published version (APA):Senden, D. J. A. (2013). Strain hardening and anisotropy in solid polymers. Technische Universiteit Eindhoven.https://doi.org/10.6100/IR747901

DOI:10.6100/IR747901

Document status and date:Published: 01/01/2013

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Strain Hardening and Anisotropyin Solid Polymers

Strain Hardening and Anisotropy in Solid Polymers by Dirk SendenTechnische Universiteit Eindhoven, 2013

A catalogue record is available from the Eindhoven University of Technology LibraryISBN: 978-90-386-3321-3

Reproduction: University Press Facilities, Eindhoven, The NetherlandsCover design: Dirk Senden

This research was supported by the Dutch Technology Foundation STW and thetechnology program of the Ministry of Economic Affairs, under grant number 07730.

Strain Hardening and Anisotropyin Solid Polymers

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan deTechnische Universiteit Eindhoven, op gezag van derector magnificus, prof.dr.ir. C.J. van Duijn, voor een

commissie aangewezen door het College voorPromoties in het openbaar te verdedigen

op maandag 18 februari 2013 om 16.00 uur

door

Dirk Jerome Andre Senden

geboren te Goirle

Dit proefschrift is goedgekeurd door de promotor:

prof.dr.ir. H.E.H. Meijer

Copromotoren:dr.ir. L.E. Govaertendr.ir. J.A.W. van Dommelen

Summary ix

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Processing-induced molecular orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Deformation and failure of oriented polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Scope of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5 Outline of the thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Strain hardening in oriented polymers 9

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Materials and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Influence of orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Modeling the Bauschinger effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4.1 Elastic strain hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4.2 Viscous strain hardening. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4.3 Elastic-viscous strain hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Aging and deformation kinetics of polycarbonate 33

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34


3.3 Aging and plastic deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.4 Aging and deformation kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.5 Physical interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

v

vi

4 Rate- and temperature-dependent strain hardening of polycarbonate 49

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.3 Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.3.1 Strain rate dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.3.2 Temperature dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.4 Constitutive modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.4.1 Kinematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.4.2 Total stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.4.3 Driving stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.4.4 Elastic strain hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.4.5 Viscous strain hardening. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.5 Model characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.6 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.6.1 Compression at different strain rates and temperatures . . . . . . . . . . . . . . 70

4.6.2 Aging dependence of the activation enthalpy. . . . . . . . . . . . . . . . . . . . . . . . . 71

4.6.3 Bauschinger effect in polycarbonate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5 Anisotropic yielding of injection molded polyethylene 81

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82


5.3 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.4 Modeling anisotropic yielding of oriented polymers . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.4.1 Yield function of Hill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.4.2 The Bauschinger effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.4.3 A combined approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.5 Constitutive modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.5.1 Kinematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.5.2 Kinematics of pre-deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

vii

5.5.3 Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.5.4 Anisotropic viscoplastic flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.5.5 Implementation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.6 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.6.1 Model characterization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.6.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.6.3 Results for yielding at constant strain rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.6.4 Results for yielding at a constant load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6 Conclusions and recommendations 107

6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.2.1 More detailed constitutive modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.2.2 Processing ⇒ structure ⇒ properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Samenvatting 113

Dankwoord 115

Curriculum vitae 117

List of publications 119

Mechanical properties of polymers strongly depend on the underlying microstructure. Forinstance, processing-induced molecular orientation may, in semi-crystalline polymers,lead to differences in lifetime up to a factor of 500 within a single injection moldedproduct. Furthermore, enormous differences between the mechanical performance intension and compression may arise as a result of an anisotropic distribution of themolecular orientation in a product. Considering that, due to their processing history,polymer products are rarely free of orientation, the importance of understanding therelation between molecular orientation and mechanical properties is evident. This thesismainly focuses on the development of modeling concepts that capture the effects ofmolecular orientation on the deformation kinetics of solid polymers.

For amorphous polymers, rubber-elastic theories are traditionally used to model theorienting network of entangled polymer chains, leading to reasonable descriptions ofthe experimentally observed strain hardening response. However, there are a numberof issues regarding such entropic strain hardening models that still need to be solved.For instance, experiments show that the strain hardening response changes with strainrate and has a negative temperature dependence, both of which cannot be explainedfrom an entropic origin. Another example is the large asymmetry between the yieldstress in tension and compression that is observed experimentally in oriented amorphouspolymers. Recently, it has been suggested that strain hardening in amorphous polymersprimarily has an intermolecular origin, which would imply a viscous stress contributionon the macroscopic scale. The physical origin is that deformation leads to locallyanisotropic chain conformations, which result in an intensification of activation barriersthat is accompanied by an increase in segmental relaxation times. It is shown thatthe issues mentioned are indeed solved by employing a combined elastic-viscous strainhardening description. The model proposed for the viscous strain hardening contributionconsists of an Eyring viscosity with parameters that evolve as a function of deformation.Incorporation of this modeling concept in the Eindhoven Glassy Polymer model andcharacterization using a set of uniaxial compression experiments on polycarbonate leadsto accurate, quantitative descriptions of the material’s strain hardening response for awide range of temperatures and strain rates. Additional support for the model proposedis found in the observation that it also quantitatively captures the mechanical responseof polycarbonate in cyclic loading conditions, involving tension and compression onoriented samples, without any additional parameter fitting.

In semi-crystalline polymers, frozen-in orientation in the amorphous domains is not the

ix

x Summary

only possible source of anisotropy; the presence of possibly oriented and intrinsicallyanisotropic crystalline domains can have a significant contribution as well. In themacroscopic mechanical response of oriented semi-crystalline polymers, the influenceof orientation is, therefore, twofold. On one hand, the frozen-in orientation in theamorphous phase leads to a large asymmetry between the mechanical response in tensionand compression and introduces a dependence on loading direction in the material’sresponse, albeit relatively weak. On the other hand, the preferred orientation ofthe crystalline lamella gives rise to a strong direction dependence in the mechanicalresponse, but does not enhance the differences between tension and compression.These aspects of the anisotropic deformation of oriented semi-crystalline polymers areassessed experimentally by measuring the tensile and compressive deformation kineticsof injection molded polyethylene at different loading angles with respect to the injectiondirection. For the first time, it is shown that a significant asymmetry between tension andcompression exists in a melt-oriented (semi-crystalline) polymer. Also from a modelingperspective, the different origins of anisotropy in oriented semi-crystalline polymers arediscussed; a model is formulated that combines the modeling concepts developed fororiented amorphous polymers with a model for anisotropic viscoplasticity. The modelproposed accurately describes the tensile and compressive yield kinetics of injectionmolded polyethylene, for loading both parallel and perpendicular to the main orientationdirection. Additionally, it captures the experimentally observed failure kinetics of theseoriented polyethylene samples when subjected to a constant tensile load in these loadingdirections.

The modeling concepts developed in this study provide a solid basis for future efforts tomodel failure of semi-crystalline polymers. An important open end is that the modelsproposed are essentially phenomenological, that is, the parameters only have a meaningon a macroscopic level and are not linked to a specific morphology. The next bigchallenge is to connect the macroscopic model parameters to microstructural parameters,such as crystallinity, lamellar thickness and orientation distribution. Ultimately, theseparameters can, in turn, be related to processing history in terms of temperature, pressureand flow history, given the details of the molecular weight distribution.

2 Introduction

1.1 Motivation

To many people, the subject of mechanical properties of polymers may seem only abstract.In reality, it is a topic that we all care about, perhaps without realizing it, and practicalproblems related to it are to be found everywhere. For instance, anyone who accidentallydrops his laptop, iPad, or phone on the floor anxiously checks whether cracks have formedin the body. Similarly, it is quite disappointing to find your expensive polaroid sunglassesbent or broken after you unintentionally sat down on them. Worries of an entirely differentorder are from people who have their knee or hip replaced by an artificial joint with aplastic bearing and wish for the new joint to last a life-time. Whatever the applicationis, polymer failure is unwanted in most cases, sometimes even dangerous, and the everincreasing use of polymers in structural applications calls for adequate solutions to thisproblem. Of course, one could resort to pragmatic, short-term solutions, such as shownin Figure 1.1, but to really improve our polymer products, a better understanding of theirmechanical properties is needed. In this respect, one should realize that for the mechanicalproperties of polymer products it does not only matter what polymer a product is madeof; it is also of critical importance how the product is made.

Figure 1.1: A pragmatic, short-term solution to polymer failure [1].

1.2 Processing-induced molecular orientation

This thesis focusses on the influence of processing-related parameters on mechanicalproperties rather than on the influence of polymer-specific details in the molecularstructure. Nevertheless, it is important to know that polymers consist of long, chain-like molecules that are made up of many small identical building blocks; hence the Greekorigin of their name πoλυς µερoς (= many parts). This chain-like nature of the polymer

Deformation and failure of oriented polymers 3

molecules is relevant because it has a tremendous influence on a polymer’s mechanicalresponse, both in the molten state and in the solid state.

Processing of polymer products usually involves large deformations to get the materialinto the desired shape. The ability of the long polymer molecules to respond to suchexternal deformations is dictated by their mobility, which strongly depends on the chaindirection relative to the deformation direction, and on temperature, time and stress(pressure). Due to these dependencies on external conditions, the conformation of thepolymer chains in the final product is heavily affected by, for instance, the specifictemperature and flow (deformation) history during processing. If little or no flow isapplied, e.g. in compression molding, and the polymer is cooled slowly from the melt, themechanical properties of the final product are isotropic, that is, the same in all directions.In the case of an amorphous polymer, this means that the molecules form an entanglednetwork of randomly coiled chains; in a semi-crystalline polymer, a spherulitic crystalstructure is formed, which is isotropic on a macroscopic scale. However, most processingtechniques used to make polymer products involve significant amounts of flow, either ofthe polymer melt, e.g. in injection molding, or of the solid polymer, e.g. in hot-drawing.Due to their chain-like structure, polymer molecules tend to become oriented in thedirection of flow and, upon rapid cooling of the material, this state of preferred orientationgets frozen-in in the final product. In the case of semi-crystalline polymers, the orientedmolecules direct the solidification process, leading to the formation of entirely different,highly oriented crystalline structures, such as stacked lamellae or shish-kebabs [2]. Animportant consequence of the molecular orientation that develops during processing isthat the mechanical properties of the final product become anisotropic, that is, direction-dependent. Such mechanical anisotropy is of prime interest in this thesis since it has atremendous influence on the deformation and failure of polymer products.

1.3 Deformation and failure of oriented polymers

An illustrative example of anisotropic mechanical properties, originating from molecularorientation, is shown in Figure 1.2. If a sheet of isotactic polypropylene is hot-drawnto six times its original length, the failure stress in the drawing direction becomes veryhigh, see Figure 1.2a, much higher than that of the isotropic material. Perpendicular tothe drawing direction, however, the material has comparatively very little strength, seeFigure 1.2b.

In the literature, a variety of similar examples is available that demonstrate the enormousinfluence of molecular orientation on deformation and failure of polymers. For instance,

4 Introduction

Figure 1.2: Anisotropic mechanical properties of hot-drawn polypropylene sheet.

anisotropy caused by processing-induced molecular orientation is observed in a widerange of polymers as a strong dependence of the tensile yield stress on the direction ofloading [3-11]. Additionally, the impact properties significantly improve with orientation,especially for semi-crystalline polymers [12, 13], but it is known that the failure modemay change from completely ductile to brittle fracture as a function of the deformationdirection. Last, it has been shown that the time-to-failure (life-span) in creep tests atconstant load may increase by decades due to molecular orientation [10, 11]. Analogousto the former two examples, anisotropy plays an essential role in the latter; differencesin time-to-failure of a factor 500 have been found within a single, injection moldedpolypropylene product, depending on the deformation direction [11].

1.4 Scope of the thesis

In short, molecular orientation that develops during processing offers opportunities fordesign engineers due to the improved mechanical properties in specific directions, butthe accompanying anisotropy also poses risks if it is not well understood. To reallytake advantage of molecular orientation in the design of new products, engineers needto have tools that enable them to predict the mechanical response of oriented polymers.The development of such tools is a great challenge and it requires efforts in differentareas. Morphological studies investigate how parameters like molecular orientation,

Outline of the thesis 5

crystallinity and crystal structure evolve during processing, as a function of the processingconditions, see e.g. [14]. This thesis does not consider these morphological details,but rather focuses on the consequences of orientation for the macroscopic deformationresponse of polymers. There is a wide variety of modeling approaches to study thedeformation of solid polymers, ranging from the molecular scale to the macroscopiccontinuum scale. Simulations on a microscopic level provide valuable insight in thephysics that underlie macroscopic deformation phenomena by investigating the relationbetween morphological features and macroscopic properties, see e.g. [15]. An importantdrawback for the practical application of such models in an engineering environment isthat they are inherently complex and computationally rather expensive. For this reason,this thesis focusses on the development of macroscopic, phenomenological modelingapproaches that capture deformation phenomena of oriented polymers without taking intoaccount any micromechanical detail. In a later stage, this macroscopic model will be fedwith the outcomes of the micromechanical and structural models.

A crucial concept in the development of such macroscopic, phenomenological constitu-tive models is the intrinsic deformation response of a polymer, that is, the true stressin response to a homogeneous deformation. Experimentally, a well-established methodto obtain this intrinsic deformation response is the uniaxial compression test, see Figure1.3a. In Figure 1.3b, a typical stress-strain curve for such a compression test is shown andthe main features of the response are pointed out. After an initial non-linear viscoelasticresponse, the material starts to deform plastically at the yield point, which is strongly rate-and temperature-dependent. The post-yield response often, but not always, exhibits a dropin stress with increasing deformation, called strain softening, before entering the strainhardening regime, in which the stress rises again. As the title of this thesis already givesaway, this strain hardening proves to be of key importance in the constitutive modelingoriented polymers.

1.5 Outline of the thesis

Chapters 2-4 concern the constitutive modeling of oriented amorphous polymers. InChapter 2, an experimental study shows that strain hardening plays a crucial role in themechanical response of oriented amorphous polymers. Moreover, it is demonstratedthat traditional constitutive models, which treat strain hardening as a purely elasticphenomenon, predict physically unrealistic behavior when deformation of orientedpolymers is concerned. Instead, an approach is proposed that treats strain hardening asa partly elastic, partly viscous phenomenon and modeling consequences are discussed.Chapter 3 actually is a bit of an intermezzo. Large plastic deformations do not only

6 Introduction

Figure 1.3: Intrinsic deformation response of a polymer, as measured in uniaxial compression.

affect the molecular orientation in a polymer, they also change its aging history. Theexperimental study in Chapter 3 shows that aging, and therefore also the erasure ofaging due to plastic deformation, changes the temperature dependence of the yield stress.Although such an effect is to be expected based on physical considerations, it has,until now, never been observed experimentally. In Chapter 4, an existing constitutivemodel, the Eindhoven Glassy Polymer model, is extended to incorporate the modelingapproaches developed in Chapters 2 and 3. The constitutive relation proposed accuratelycaptures the intrinsic deformation response of isotropic polycarbonate up to large strains,including the strain rate and temperature dependence of strain hardening. Additionally,both the tensile and compressive deformation responses of oriented polycarbonate areaccurately described.

Having established a constitutive model for oriented amorphous polymers, Chapter5 takes up the challenge of developing a model that describes yielding and ductilefailure of oriented semi-crystalline polymers. An experimental study is discussed, whichdemonstrates that similar macroscopic deformation phenomena are observed in injectionmolded semi-crystalline polyethylene, as in oriented amorphous polymers. However,the mechanical anisotropy is much more pronounced and partly has a different physicalorigin. Based on these physical considerations, a constitutive model is proposed thatquantitatively captures the deformation phenomena observed in the experiments. Finally,in Chapter 6, the main conclusions of the thesis are summarized and an outlook to futurechallenges is presented.

References 7

References

[1] Copyright: Zorilla. http://www.flickr.com/photos/barry b/76055201/sizes/m/in/photostream/.[2] A. Keller and M.J. Machin. Oriented crystallization in polymers. Journal of Macromolecular Science:

Part B: Physics. 1967, 1, 41-91.[3] A. Keller and J.G. Rider. On the tensile behaviour of oriented polyethylene. Journal of Materials

Science. 1966, 1, 389-398.[4] N. Brown, R.A. Duckett and I.M. Ward. The yield behaviour of oriented polyethylene terephthalate.

Philosophical Magazine. 1968, 18, 483-502.[5] C. Bridle, A. Buckley and J. Scanlan. Mechanical anisotropy of oriented polymers. Journal of

Materials Science. 1968, 3, 622-628.[6] J.G. Rider and E. Hargreaves. Yielding of oriented poly(vinyl chloride). Journal of Polymer Science:

Part A-2: Polymer Physics. 1969, 7, 829-844.[7] D. Shinozaki and G.W. Groves. The plastic deformation of oriented polypropylene: Tensile and

compressive yield criteria. Journal of Materials Science. 1973, 8, 71-78.[8] F.F. Rawson and J.G. Rider. A correlation of Young’s modulus with yield stress in oriented

poly(vinyl chloride). Polymer. 1974, 15, 107-110.[9] S.G. Burnay and G.W. Groves. Variation of yield stress with orientation in oriented polyethylene.

British Polymer Journal. 1978, 10, 30-34.[10] T.B. van Erp, C.T. Reynolds, T. Peijs, J.A.W. van Dommelen and L.E. Govaert. Prediction of

yield and long-term failure of oriented polypropylene: Kinetics and anisotropy. Journal of PolymerScience: Part B: Polymer Physics. 2009, 47, 2026-2035.

[11] T.B. van Erp, L.E. Govaert and G.W.M. Peters. Mechanical performance of injection-moldedpoly(propylene): Characterization and Modeling. Macromolecular Materials and Engineering. 2012,DOI, 10.1002/mame.201200116.

[12] Z. Bartczak, J. Morawiec and A. Galeski. Deformation of high-density polyethylene produced byrolling with side constraints. II. Mechanical properties of oriented bars. Journal of Applied PolymerScience. 2002, 86, 1405-1412.

[13] Z. Bartczak, J. Morawiec and A. Galeski. Structure and properties of isotactic polypropylene orientedby rolling with side constraints. Journal of Applied Polymer Science. 2002, 86, 1413-1425.

[14] T.B. van Erp. Structure development and mechanical performance of polypropylene. PhD Thesis.2012. Eindhoven University of Technology (the Netherlands).

[15] A. Sedighiamiri. Micromechanical modeling of the deformation kinetics of semicrystalline polymers.PhD Thesis. 2012. Eindhoven University of Technology (the Netherlands).

Abstract: The nature of strain hardening in glassy polymers is investigatedby studying the mechanical response of oriented polycarbonate. The yieldstress in tension increases strongly with pre-deformation, whereas it slightlydecreases in compression (the so-called Bauschinger effect). Moreover, orientedspecimens display increased strain hardening in tension, whereas this nearlyvanishes in compression. It is shown that these observations can be capturedby the introduction of a viscous contribution to strain hardening in terms of adeformation dependence of the flow stress. For polycarbonate, this originatesfrom a deformation-induced change of the Eyring rate constant, which causesthe room temperature yield kinetics to shift from the α to the (α + β)-regime.

9

10 Strain hardening in oriented polymers

2.1 Introduction

The post-yield stress-strain response of glassy polymers generally displays two character-istic phenomena: strain softening, the initial decrease of true stress with strain, and strainhardening, the subsequent upswing of the true stress-strain curve. Localization of strainis typically induced by intrinsic strain softening, whereas the evolution of this localizedplastic zone strongly depends on the stabilizing effect of strain hardening. In case ofinsufficient strain hardening, the material tends to form crazes; extremely localized zonesof plastic deformation that act as precursors for cracks and thus induce macroscopicallybrittle failure [1, 2]. 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 [3].

An important step in this direction was made by Haward and Thackray [4], whowere the first to envision strain hardening as an entropy-elastic contribution of theentangled molecular network. Their inspiration was found in the observation that plasticdeformation of a polymer glass is (almost) fully recovered by heating above the glasstransition temperature Tg [5-8], which gives evidence that the entangled molecularnetwork remains largely intact during plastic deformation. This concept was translatedinto a 1-D constitutive relation in which the post-yield stress is additively decomposedinto a viscous contribution, representing the stress-activated yield process, and a strainhardening contribution, representing the chain-orientation hardening, modeled with afinitely-extensible rubber-elastic spring [9]. The application of this stress decompositionformed a solid basis for the development of a number of 3-D constitutive models, startingwith the Boyce-Parks-Argon (BPA) model [10]. In its original form, the strain hardeningcontribution in the BPA model was captured by the ‘three-chain’ model of Wang and Guth[11, 12]; later this was replaced with the more realistic ‘eight-chain’ model [13]. Other 3-D models employing different hyperelastic strain hardening approaches followed, e.g. theOxford Glass Rubber (OGR) model [14, 15], incorporating the cross-link slip-link modelof Edwards and Vilgis [16], and the Eindhoven Glassy Polymer (EGP) model [17, 18],which uses a neo-Hookean model, equivalent to the application of the Gaussian networktheory of rubber elasticity. In this theory, the polymer strands between entanglementsnever reach a fully stretched conformation, and the elastic stress in uniaxial loading isrepresented by:

σ = Gr

(λ2 − λ−1

), (2.1)

where Gr is the strain hardening modulus and λ is the draw ratio. This expression proved

Introduction 11

successful from a phenomenological, descriptive point of view, since most amorphousand semi-crystalline polymers display this specific hardening response over a largedeformation range [19-26]. It should be noted that the responses of the eight-chain andthe Edwards-Vilgis model are indistinguishable from the neo-Hookean model at somedistance from the extensibility limit [26, 27], and that all are capable of capturing strainhardening responses in different loading geometries [26, 28, 29].

Although the constitutive models mentioned above enabled quantitative analysis oflocalization and failure in glassy polymers [18, 29-32], and revealed the crucial role of theintrinsic post-yield characteristics on macroscopic strain localization [33], there are manyarguments against a hyperelastic, entropic nature of strain hardening. The first argumentis related to the influence of the entanglement network density. In the Gaussian networktheory, the strain hardening modulus is expressed as:

Gr = NekBT, (2.2)

where Ne, kB and T represent the entanglement network density, Boltzmann’s constantand the absolute temperature, respectively. The proportionality between strain hardeningmodulus and network density, suggested by the Gaussian theory, was investigated byVan Melick et al. [34], who systematically altered the network density of polystyrene byblending with poly(2,6-dimethyl-1,4-phenylene-oxide) (PPO) as well as through cross-linking during polymerization (XPS). Although their results gave convincing evidence forthe hypothesized proportionality in these systems, it should not be concluded that networkdensity is the key parameter determining the magnitude of strain hardening. In Figure 2.1,the strain hardening moduli of various polymers are plotted versus their network densities.The values of the strain hardening moduli at room temperature are presented for: XPSand PS-PPO [34], polycarbonate (PC) [26], poly(methyl methacrylate) (PMMA) [35],polyoxymethylene (POM), polytetrafluoroethylene (PTFE), polyamide (PA) 6 and PA 66(all from [19]). The network densities in the melt were calculated from the results of Wu[36]. The scatter of the data in Figure 2.1 clearly demonstrates that network density cannot be regarded as the key parameter determining the magnitude of the strain hardeningmodulus.

Another intriguing argument against an entropic nature of strain hardening is found inthe experimental observation that strain hardening decreases with increasing temperature,whereas, according to Gaussian theory, it should increase [34, 37-39]. Initially, this nega-tive temperature dependence was interpreted in terms of a viscoelastic stress contributionoriginating from temperature-activated relaxation of the entanglement network through


Figure 2.1: 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 differentsources [19, 26, 36], for various polymers.

chain slip [34, 39], a view consistent with the observed molecular weight dependence ofstrain 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 Rolie-Poly conformational meltmodel [43] in an attempt to capture the effect of melt orientation on strain hardening. Inits current form, however, the model only captures chain orientation on the entanglementlength scale and it therefore under-predicts strain hardening at temperatures well belowTg, where it is dominated by sub-entanglement chain orientation.

Another, more promising, route appears to be the addition of a viscous contributionto strain hardening by introducing a deformation dependence in the flow stress. Thephysical picture is that plastic deformation induces chain orientation, leading to changesin inter-chain packing that result in an intensification of activation barriers [44, 45]. Thefirst to explore a viscous contribution to strain hardening were Wendtland et al. [46],who presented experimental evidence for a strain rate dependence of strain hardeningfor a selection of polymers [46, 47]. The data was successfully modeled by addinga deformation dependence to the Eyring flow term through a strain dependence of theactivation volume. This leads to a gradual increase of the strain rate dependence ofthe flow stress with deformation, which, in combination with a neo-Hookean strainhardening component, proved successful in describing uniaxial compression experimentsat different strain rates [46]. In a recent extension of their model, also the temperaturedependence of the post-yield response is described accurately for a number of polymers[48]. An alternative approach was suggested by Buckley [45], who proposed ananisotropic Eyring flow process. Also here, an additional hyperelastic term was requiredto obtain the level of strain hardening that is experimentally observed [45]. Additional

Materials and methods 13

experimental evidence for the existence of a viscous strain hardening contribution waspresented by Hoy and Robbins [49-51], who performed atomistic simulations of large-strain uniaxial compression of polymer glasses. Their results reproduce the importantexperimental observations on strain hardening, and suggest a major role for a deformationdependence of the flow stress and only a minor for the entropic back stress [49, 51].

The concept of a deformation dependence of the flow stress also appears consistent withthe aforementioned results of Van Melick et al. [34]. They showed that by plotting thestrain hardening moduli as a function of T−Tg, which changes with PS/PPO composition,the data for different PS/PPO blends collapse onto a single curve for temperatures farbelow Tg. Apparently, the value of Tg is of key importance in this range and the networkdensity does not play a significant role. This observation supports the existence of arelation between strain hardening and the flow stress; i.e. a viscous contribution to strainhardening.

The results discussed above lead to the conclusion that the strain hardening responseof glassy polymers consists of two separate components, one viscous and the otherelastic. From an experimental point of view, this constitutes a problem since thetest that is usually applied to study the intrinsic stress-strain response of polymers, auniaxial compression test, does not sufficiently discriminate between these components,as evidenced by the success of hyperelastic models in describing such experimental data.In order to characterize both components, the present study suggests an experimentalapproach that exploits the large difference between the compressive and tensile yieldstress that is observed in oriented polymers [52-56]. Based on shrinkage stress and yieldstress measurements on oriented PMMA, Botto et al. [57] were the first to propose thatthis Bauschinger effect [58] is related to a frozen-in network stress. To date, however,this relation has never been explored. In the present investigation, the Bauschinger effectin oriented polycarbonate is analyzed and its possible use for isolating the viscous andelastic contributions to strain hardening is evaluated.

2.2 Materials and methods

Axisymmetric tensile bars, with dimensions as shown in Figure 2.2, were machined fromcommercially available extruded polycarbonate (PC) rod (Lexan, Sabic).

Since strain localization (necking) subsequent to the yield point inhibits the character-ization of the large-strain post-yield response, the specimens were mechanically pre-conditioned by large-strain torsion. The samples were clamped, twisted over an angle


Figure 2.2: Axisymmetric tensile bar, dimensions are in millimeters.

of 990, and subsequently twisted back over the same angle. In this way, strain softeningis eliminated, resulting in homogeneous deformation of the specimen in a subsequenttensile test [17, 26].

To obtain specimens with different degrees of anisotropy, or pre-orientation, the pre-conditioned tensile bars were subjected to uniaxial tensile tests at a constant true strainrate of 10−4 s−1, up to true strain levels of 0, 0.15, 0.3, 0.45 and 0.6. These tests wereperformed on a Zwick Z010 tensile testing machine, equipped with an extensometer toaccurately measure the deformation. After reaching the desired pre-strain, the specimenwas unloaded to zero force at the same true strain rate and then used for either a tensileor a compression experiment. The different pre-strains that were applied led to differentamounts of residual plastic strain after unloading, as indicated in Table 2.1.

Table 2.1: Residual plastic strains after pre-orientation.

true pre-strain [-] residual plastic strain [-]0.15 0.12660.30 0.27440.45 0.41550.60 0.5542

Tensile experiments, at constant true strain rates ranging from 10−4 to 3 · 10−3 s−1, wereperformed on oriented specimens immediately after the pre-deformation.

For the uniaxial compression experiments, cylindrical specimens, with the diameter andheight both equal to 5 mm, were machined from the gauge sections of the pre-orientedtensile bars directly after applying the pre-orientation. Compression experiments at aconstant compressive true strain rate of 10−4 s−1 were performed on a servo-hydraulicMTS 831 Elastomer Testing System using two parallel, flat steel plates. In order toprevent any bulging of the sample, friction was reduced by applying a lubricating PTFEspray 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 testingequipment was measured and corrected for in a real-time feedback loop to ensure accuratestrain measurement and control.

Influence of orientation 15

All experiments, including the sample preparation, were performed at room temperature.From the recorded force and true strain signals, the true stress was calculated assumingisochoric deformation.

2.3 Influence of orientation

In this section, the effect of pre-orientation on the mechanical response of PC is exploredusing uniaxial compression and tensile tests at a single strain rate, 10−4 s−1. In Figure2.3a, the mechanical response of PC in uniaxial tension is given for various levels of pre-strain. It is evident that the influence of orientation is substantial, leading to a pronouncedincrease in both yield stress and strain hardening.

Figure 2.3: (a) Mechanical response of pre-oriented PC in uniaxial tension, the level of true pre-strain is indicated in the figure. (b) Same results, but the stress is plotted versus thetotal strain, i.e. including the residual plastic strain (see Table 2.1) in the sample.

When the stress is plotted as a function of the total strain, i.e. the strain measured in thetensile test plus the residual plastic strain in the specimen, the post-yield responses of allmeasurement curves collapse onto a single curve, see Figure 2.3b. This implies that, uponreloading, the pre-oriented specimens follow their regular path along the isotropic curve.The pre-oriented specimens show a hint of strain softening, although they are expected tobe rejuvenated. This is believed to originate from stress-accelerated physical aging thatoccurs during the unloading of the pre-strain.

In the case of uniaxial compression, the influence of orientation on the mechanicalresponse is entirely different, as illustrated in Figure 2.4a. The yield stress remainslargely unaffected, and the strain hardening modulus decreases with increasing pre-strain.As depicted in Figure 2.4b, the mechanical response of a pre-oriented specimen again


coincides with that of an isotropic specimen at large deformations, when the stress isplotted as a function of the total strain.

Figure 2.4: (a) Mechanical response of pre-oriented PC in uniaxial compression, the level oftrue pre-strain is indicated in the figure. (b) Same results, but the stress is plottedversus the total strain, i.e. including the residual plastic strain (see Table 2.1) in thesample.

To illustrate the different influences of orientation on the mechanical behavior in uniaxialtension and compression, these two responses are shown in a single graph for an isotropic(Figure 2.5a) and a pre-oriented (Figure 2.5b) specimen. In the isotropic case, the yieldstress in compression is slightly higher than in tension due to the influence of hydrostaticpressure [17, 59]. In the pre-oriented case, a strong Bauschinger effect is observed; theyield stress in tension is much higher than that in compression. Moreover, strongerstrain hardening is observed in tension than in compression. These effects completelyoverwhelm the relatively small hydrostatic pressure effect.

2.4 Modeling the Bauschinger effect

Traditionally, most modeling approaches for the description of the finite strain mechanicalresponse of polymers incorporate a rubber-elastic model for the strain hardening. Toevaluate whether such modeling approaches are able to capture the experimentallyobserved Bauschinger effect, simulations were performed using the Eindhoven GlassyPolymer (EGP) model [60, 61]. In these simulations, the imposed uniaxial deformationis cyclic: the specimen is first loaded in tension up to a certain pre-strain at a constantstrain rate and subsequently compressed back to its original length. The results arepresented in Figure 2.6a for four different pre-strain levels. For small pre-strains (0.15and 0.3) the model predicts a Bauschinger effect, i.e. the compressive yield stress issubstantially lower in magnitude than the momentary yield stress in tension at the point

Modeling the Bauschinger effect 17

Figure 2.5: The difference between the mechanical response of PC in uniaxial tension andcompression for (a) isotropic specimens, data taken from [26], and (b) specimensthat have been pre-oriented in tension up to a true strain of 0.6.

of load reversal. At higher pre-strains, however, no compressive yield stress is observedanymore. The reason for this is that the elastic strain hardening stress at such largepre-strains 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 modelpredicts that the amount of residual plastic strain (i.e. the residual strain after unloadingto zero stress) is identical for all pre-strains larger than 0.35. To give an impression of theactual behavior, experimental tension and compression curves are combined in a singleplot in Figure 2.6b. Please note that the compression experiments were not performedimmediately after unloading. Nevertheless, the discrepancy between the experimentalbehavior and the model predictions is tremendous.

Besides the EGP model, also other well-established models that describe the mechanicalbehavior of glassy polymers, such as the BPA model [10] and the OGR model [14, 62, 63],will predict this physically unrealistic behavior. In the following, a simple 1-D model isused to show that the problems are caused by the decomposition of the stress as it wasproposed by Haward and Thackray [4].

2.4.1 Elastic strain hardening

First, the traditional approach is used, incorporating a purely elastic description ofstrain hardening. In the present 1-D model, only the post-yield mechanical response isconsidered, assuming an additive decomposition of the stress σ into a viscous flow stressσflow and an elastic strain hardening stress σr, consistent with the work of Haward andThackray [4]:


.

Figure 2.6: Mechanical response of PC in cyclic uniaxial deformation: (a) as predicted by theEGP model, and (b) an impression of the actual behavior. The tensile pre-strainlevels are indicated in the graphs.

σ = σflow (ε) + σr (λ) . (2.3)

The viscous flow stress is a function of strain rate ε, which is described with an Eyringrelation [64]:

σflow (ε) =kBT

V ∗ sinh−1

(ε

ε0

), (2.4)

where the activation volume V ∗ and the rate constant ε0 are model parameters. Boltz-mann’s constant is denoted by kB and the absolute temperature T is 293 K in allsimulations. The elastic strain hardening stress is a function of the draw ratio λ, accordingto a neo-Hookean relation:

σr (λ) = Gr

(λ2 − λ−1

), (2.5)

where Gr represents the elastic strain hardening modulus. In Table 2.2, values of themodel parameters are given, which are representative for PC [18, 65]. Again, a cyclicdeformation path with a constant absolute true strain rate of 10−4 s−1 is used to evaluatethe response of the model.


Table 2.2: Model parameters for the simulations shown in Figure 2.7.

V ∗ [nm3] ε0 [s−1] Gr [MPa]3.0 10−20 26

In Figure 2.7a, the total stress response of the model is shown, as well as the individualcontributions of the viscous flow stress and elastic strain hardening stress. Dueto the absence of pre-yield elasticity in this simplified approach, the viscous stressinstantaneously attains its constant flow level, which changes sign when the deformationdirection is reversed. The elastic stress obeys a neo-Hookean relation, obviouslyfollowing the same curve during both the tensile and the compressive stage. Thetotal stress is simply an addition of these two components, revealing that the modelqualitatively predicts the same unrealistic response as the EGP model, with an apparent‘yield stress’ during unloading at positive stress levels.

The cause of these physically erroneous predictions is that the elastic contribution tothe stress dominates over the viscous contribution at high deformations, resulting in aresponse reminiscent of traditional kinematic strain hardening.

2.4.2 Viscous strain hardening

It was already argued in the introduction that strain hardening of polymers involvesa viscous contribution. To explore the implications of this, the 1-D model is firstreformulated such that the strain hardening is fully viscous. This can be achieved byadding the neo-Hookean dependence on deformation to the viscous stress:

σ = σflow (ε) +Gr

(λ2 − λ−1

)(2.6)

= σflow (ε)

(1 +

Gr

σflow (ε)

(λ2 − λ−1

)). (2.7)

This expression is generalized by introducing a parameter Cr that represents the mag-nitude of viscous strain hardening. Together with a substitution of Equation (2.4), thisyields:

σ =kBT

V ∗ sinh−1

(ε

ε0

)(1 + Cr

(λ2 − λ−1

)). (2.8)


In principle, this implies that the response of this model equals that of the model withpurely elastic strain hardening at a single strain rate, as long as the deformation directionis not reversed. At rates higher than this specific strain rate, stronger strain hardeningis predicted; at lower strain rates, the strain hardening response is weaker. Simulationresults using this model are depicted in Figure 2.7b, using Cr = 0.514. The overallresponse during the tensile part of the deformation is identical to that of the model withpurely elastic strain hardening, but it becomes completely different when the deformationdirection is reversed. The compressive yield stress is of equal magnitude, but opposite insign to the apparent tensile yield stress at the moment of load reversal, which is a responsereminiscent of traditional isotropic strain hardening.

The response of the purely viscous model also does not correspond with the experi-mentally observed influence of tensile pre-strain on the compressive yield stress, sincethe model predictions do not exhibit a Bauschinger effect. Moreover, a negative strainhardening modulus is predicted in the compressive phase of the test, while experimentsshow that the compressive strain hardening modulus of specimens that were pre-orientedin tension becomes small, but not negative under the influence of orientation, as shownin Figure 2.4.

2.4.3 Elastic-viscous strain hardening

Since neither the model with purely elastic strain hardening, nor the model with purelyviscous strain hardening is able to capture the experimentally observed Bauschingereffect, the applicability of a combination of these two approaches is now explored. Thisis done by combining Equations (2.5) and (2.8):

σ =kBT

V ∗ sinh−1

(ε

ε0

)(1 + ϕCr

(λ2 − λ−1

))+ (1− ϕ) Gr

(λ2 − λ−1

), (2.9)

where ϕ represents the amount of viscous strain hardening relative to the total amount ofstrain hardening. This idea is not new. In fact, there is a striking similarity with a classicalapproach to describe the Bauschinger effect in metals, using a combination of kinematicand isotropic hardening [66]. In some cases, a so-called Bauschinger ratio is defined [67],representing the ratio of kinematic to isotropic hardening, similar to the parameter ϕ usedhere. The response of the elastic-viscous model is shown in Figure 2.7c for an equaldistribution of the total strain hardening in an elastic and a viscous contribution (ϕ =

0.5). The overall response in the tensile part of the deformation is again identical tothat of the model with purely elastic strain hardening, but upon reversal of the loading


Figure 2.7: Simulated mechanical response in uniaxial deformation: (a) with purely elasticstrain hardening, (b) with purely viscous strain hardening, (c) with an equal amountof elastic and viscous strain hardening. The total stress (solid lines), as well as thecontributions of the viscous stress (dash-dotted lines) and the elastic strain hardeningstress (dashed lines) are plotted. (d) Total stress for different ratios of the elastic andviscous strain hardening contribution.

direction a pronounced Bauschinger effect is observed, with a yield stress equal to theinitial 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 andthe viscous strain hardening contribution, as illustrated in Figure 2.7d. An increase in theelastic contribution leads to a decrease in the compressive yield stress and an increase inthe compressive strain hardening modulus.

Deformation-induced changes in rate dependence

Now that it has been established that the model with combined elastic and viscous strainhardening enables a qualitative description of the Bauschinger effect, the nature of theviscous strain hardening contribution is further investigated. In the model discussed


above, this viscous contribution to strain hardening is introduced as a deformationdependence in the viscous flow stress, see Equation (2.9). Since this is modeled withan Eyring flow relation, it suggests that the viscous part of the strain hardening can beinterpreted either as a deformation dependence of the activation volume:

V ∗ (λ) =V ∗

f (λ), (2.10)

or as a deformation dependence of the rate constant:

ln (ε0 (λ)) = ln (ε0) f (λ) , (2.11)

where f(λ) = 1 + ϕCr (λ2 − λ−1) for the example discussed in the previous section.Whereas a deformation-dependent activation volume has already been suggested byseveral researchers [45, 46, 57, 68], a deformation-dependent rate constant has not.Both parameters have quite a different influence on the predicted yield kinetics, asschematically illustrated in Figure 2.8. A deformation dependence of the activationvolume V ∗ implies that the strain rate dependence changes with deformation, whereasa deformation dependence of the rate constant ε0 results in a shift along the logarithmicstrain rate axis. It is worth noting that the possibility that both parameters change withdeformation cannot be excluded at this point.

Figure 2.8: Schematic illustration of the effect of deformation on Eyring yield kinetics in caseof (a) a deformation-dependent activation volume, and (b) a deformation-dependentrate constant. The range of strain rates usually covered in mechanical experimentsis also indicated.


To examine the influence of pre-deformation on the yield kinetics of PC, the tensile yieldstress of oriented PC is presented as a function of the logarithm of strain rate in Figure 2.9a.The corresponding values for the activation volume, calculated from the slopes of thestraight lines that are formed by the data points, are plotted in Figure 2.9b. Remarkably,the activation volume already decreases strongly with small amounts of pre-strain andsubsequently remains essentially constant.

Figure 2.9: (a) Tensile yield kinetics of oriented PC. The various levels of true pre-strain areindicated in the figure; lines are a guide to the eye. (b) Corresponding activationvolumes as a function of pre-strain. Literature values [65] for the activation volumesassociated with the α and (α+ β)-processes of PC are indicated by the solid lines.

A possible explanation for this, almost stepwise, change in activation volume may befound in the fact that PC, as most polymers, exhibits so-called thermorheologicallycomplex behavior. This implies that, depending on the temperature and time scale ofthe experiment, multiple molecular processes may contribute to the mechanical response,each with its own characteristic activation volume and activation energy. At low tomoderate strain rates, the room temperature yield kinetics of isotropic PC are governed bythe α-process, which is associated with full main-chain segmental motion, i.e. the primaryglass transition. At higher strain rates, the contribution of the β-process becomes apparent,leading to a distinct change in slope of the strain rate dependence of the yield stress, seeFigure 2.10a. This contribution of the β-process is associated with partial mobility of themain chain, generally referred to as a secondary glass transition. Although several studiesindicate that phenyl ring motions play a role in this process [69-72], there is substantialexperimental evidence that these motions are restricted to the high temperature part ofthe observed relaxation [70, 72-74]. The main contribution to the β-process appears tooriginate from the mobility of the carbonate group [74-77].


Figure 2.10: (a) Tensile yield kinetics of isotropic PC at different temperatures, data taken from[65]. (b) Schematic illustration of the Ree-Eyring modeling approach.

As demonstrated by several researchers [65, 78-80], the deformation kinetics of a polymerthat exhibits a thermorheologically complex response is accurately described with theRee-Eyring [81] modification of the original Eyring theory [64]. Essentially, this theoryassumes that the two processes act independently and that their stress contributions areadditive, as illustrated in Figure 2.10b. For isothermal conditions, the yield stress isexpressed as:

σy (ε) =kBT

V ∗α

sinh−1

(ε

ε0,α

)+kBT

V ∗β

sinh−1

(ε

ε0,β

), (2.12)

where subscripts α and β indicate the process with which a parameter is associated.

As indicated in Figure 2.9b, there is a remarkable agreement between the changes inactivation volume upon deformation and the values reported in literature [65] for theactivation volumes of the α and (α+β)-processes in PC. This suggests that pre-orientationcauses a transition of the room temperature yield kinetics of PC from the α into the (α +

β)-regime, indicating that the contributions of these processes may have shifted alongthe strain rate axis, as illustrated in Figure 2.8b. To explore this transition, additionalmeasurements were performed using dog-bone shaped tensile bars with a gauge lengthof 20 mm. Given the limitations of the testing equipment, the use of small specimensenables the coverage of a much larger range of strain rates. Orientation was appliedto these specimens through a cold-rolling process. Results from these measurements, atpre-strains of 20% and 35%, are presented in Figure 2.11a, convincingly showing that pre-orientation causes a shift of the contributions of the α and β-processes along the strainrate axis. This implies that the activation volumes of both processes are independent of


the level of pre-orientation. The solid lines in the figure are fitted using Equation (2.12),with activation volumes as determined by Klompen et al. [65] and rate constants as listedin Table 2.3. Clearly, the measured yield kinetics are accurately described by assuminga deformation dependence of the rate constants of both processes. In Figure 2.11b, theexperimental data from Figure 2.9a is plotted once more, now accompanied by fits usingEquation (2.12). The activation volumes of the α and β-processes are constant and equalto those determined by Klompen et al. [65]. The rate constants for both processes arefitted separately for each level of pre-strain, but their values are not given here, sincethese cannot be uniquely determined from the data. Figure 2.11b demonstrates that theexperimental data on the large, axisymmetric tensile bars is also accurately described byEquation (2.12), assuming a deformation dependence of the rate constants. These resultsare perhaps somewhat unexpected, as the strain rate dependence of the strain hardeningbehavior of PC has previously been successfully modeled by assuming a deformation-dependent activation volume [46].

Figure 2.11: (a) Strain rate dependence of the tensile yield stress of small, cold-rolled PCspecimens; the thickness reduction achieved in the rolling process is indicatedin the figure. (b) Experimental data as in Figure 2.9a. In both figures, the solidlines are fits using Equation (2.12), assuming a deformation dependence of the rateconstants.

Table 2.3: Rate constants for the fits in Figure 2.11a.

20% 35%ε0,α = 1.4 · 10−18 s−1 ε0,α = 4.2 · 10−20 s−1

ε0,β = 3.8 · 10−3 s−1 ε0,β = 2.3 · 10−3 s−1


To demonstrate that the behavior observed in PC does not necessarily apply to allpolymers, reference is made to the experimental data from a recent study on theanisotropic yield behavior of isotactic polypropylene (iPP) by Van Erp et al. [82], whomeasured the tensile yield kinetics of hot-drawn tapes with different levels of orientation.Their results, depicted in Figure 2.12a, unambiguously show that the slope of the strainrate dependence of the yield stress increases continuously with increasing orientation. Thedashed lines in the figure are fits of the yield kinetics for each draw ratio using Equation(2.12) (omitting the β-process). Figure 2.12b shows the rate constants and activationvolumes as they are obtained in this fitting process as a function of pre-strain. The rateconstant is found, within experimental error, not to depend on pre-strain. It is thereforeconcluded that the influence of pre-strain on the yield kinetics of iPP is characterizedsolely by a deformation-dependent activation volume, which is shown in Figure 2.12c.Indeed, the solid lines in Figure 2.13 demonstrate that the yield kinetics are accuratelydescribed by Equation (2.12) (omitting the β-process) using the same rate constant (ε0 =3 · 10−10 s−1), but different activation volumes (see Table 2.4) for the various draw ratios.

Figure 2.12: (a) Tensile yield kinetics of hot-drawn iPP tapes, data taken from Van Erp et al.[82]. The draw ratio achieved in the drawing process is indicated in the figure;dashed lines are fits using Equation (2.12). Corresponding (b) rate constants and(c) activation volumes as a function of true pre-strain.

All in all it is evident that there is a marked difference between the influence of orientationon the tensile yield kinetics of PC and iPP. The former clearly show a deformationdependence of the rate constant, whereas the latter display a deformation-dependentactivation volume. The cause for this fundamental difference is presently unclear.

Conclusions 27

Figure 2.13: Experimental data as in Figure 2.12a. Solid lines are fits using Equation (2.12)with the same rate constant, but different activation volumes for the different drawratios.

Table 2.4: Activation volumes for the fits in Figure 2.13.

λ [-] V ∗ [nm3]1 2.072 1.304 0.416 0.21

2.5 Conclusions

Whereas the mechanical responses in uniaxial tension and compression are quite similarfor isotropic polymers, they increasingly deviate when the molecular chains in thematerial become oriented; a phenomenon which is referred to as the Bauschinger effect.Experiments on polycarbonate show a dramatic increase in both tensile yield stress andstrain hardening upon pre-orientation in tension, whereas the compressive yield stress andstrain hardening slightly decrease.

It was shown that the experimentally observed Bauschinger effect cannot be describedusing traditional modeling approaches for the mechanical response of polymers, whichassume that strain hardening is purely elastic. Simulations with a simple 1-D modeldemonstrate that such approaches fail to describe the observed Bauschinger effect since itresults in (extreme) kinematic strain hardening. A purely viscous strain hardening model


is also not suitable because this leads to an isotropic hardening response. The combinationof these two approaches yields a simple, but powerful model that is able to qualitativelycapture the Bauschinger effect.

An investigation of the influence of orientation on the tensile yield kinetics of poly-carbonate and isotactic polypropylene leads to the conclusion that the nature of theviscous contribution to strain hardening may differ between polymers. For polycarbonate,a deformation dependence of the rate constant is observed, which causes the roomtemperature yield kinetics of the material to shift from the α into the (α + β)-regime.On the other hand, isotactic polypropylene clearly exhibits a deformation-dependentactivation volume, which causes the yield kinetics to continuously change as the levelof orientation increases. The cause for this fundamental difference is presently unclear.

Experimental assessment of the Bauschinger effect and its influence on the yield kineticsof polymers proves to be a valuable tool in the correct characterization of both the elasticand the viscous contribution to strain hardening.

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Abstract: Considering the current view that physical aging of glasses results inan intensification of activation barriers to plastic deformation, it is surprising thatso far no influence of physical aging was found on the temperature dependenceof plastic deformation in polymeric glasses. The present study evaluates whysuch an influence has not been found and it is shown that detailed analysis ofa set of uniaxial compression data on polycarbonate at different strain rates andtemperatures leads to the conclusion that a significant influence indeed exists.As a consequence, the Eyring activation energy depends on the aging historyof the material. These experimental observations are rationalized in terms of asimple physical interpretation of the aging phenomenon.

33

34 Aging and deformation kinetics of polycarbonate

3.1 Introduction

Mechanical properties of polymeric glasses are known to change as a function of storagetime below the glass transition temperature Tg; this phenomenon is called physical aging(from now on simply referred to as aging) [1-3]. The aging phenomenon arises from thefact that glasses are generally not in a state of thermodynamic equilibrium at temperaturesbelow Tg. In time, this causes a structural relaxation towards equilibrium that leads toa decrease in thermodynamic variables such as specific volume and free energy [2, 3].Aging also has a pronounced influence on the mechanical response of glasses. Thisis illustrated in Figure 3.1, where typical compression curves for thermally quenchedand annealed (aged) polycarbonate (PC) are depicted. The upper yield stress, defined asthe first (local) maximum in the curve, increases with increasing aging time. At strainsbeyond the upper yield point, the stress decreases due to intrinsic strain softening andthe responses of quenched and annealed PC coincide in the lower yield point. In fact,the entire large-strain response of the material is unaffected by prior aging history [4-6].Because the upper yield stress increases with aging, while the lower yield stress does not,the amplitude of strain softening is directly related to the aging history of the material.To understand strain softening is important since it is the origin of strain localization and,upon exceeding a critical value, it eventually leads to brittle failure [7, 8].

Figure 3.1: Typical uniaxial compression curves of thermally quenched and annealed (aged) PC.

A more detailed understanding of the physics of aging may be obtained from atomisticsimulations [9-12]. Binary Lennard-Jones (L-J) glasses are popular in such studiesbecause the short relaxation times that characterize small-molecule glasses are compu-tationally more accessible than the long ones found in polymeric glasses. Similar topolymeric glasses, these L-J glasses show an increase in yield stress upon aging [10-12],while the post-yield flow stress remains unaffected by aging [10, 11]. Molecular dynamicssimulations on a coarse-grained bead-spring model have qualitatively reproduced the

Introduction 35

experimentally observed influence of aging on the mechanical response of polymericglasses in creep [13] and constant strain rate tests [14].

The physical interpretation of the aging mechanism is based on the notion that eachparticle is trapped in a potential well formed by its neighbors, from which it can escapethrough thermal activation [15]. Different arrangements of the particles in a systemcorrespond to different levels of the total potential energy, which gives rise to the well-known picture of the system as a point within an energy landscape in the multidimensionalconfiguration space [16]. Aging is regarded as thermally-activated hopping throughthis energy landscape, visiting progressively deeper energy minima as the aging timeincreases [10-12, 15]. Obviously, when a system resides in a deep energy minimumdue to aging, more energy is required to get it out than for a less aged system. In otherwords, aging increases the activation barriers to plastic deformation of a glass, which iswhy the yield stress increases, together with the density and the elastic modulus at smallstrains. The thermally-activated nature of aging is reflected in the drastic reduction of theaging time required to reach equilibrium upon an increase in temperature. Especially forpolymeric glasses this is noticeable because typical times required to reach equilibrium atmoderate temperatures are generally astronomical [2].

Macroscopically, aging has a pronounced influence on the yield kinetics of polymericglasses as measured in mechanical experiments as a function of strain rate and/ortemperature. Characteristic for the kinetics of PC is the linear dependence of the yieldstress on the logarithm of strain rate across a wide range of temperatures [17, 18]. This isusually modeled with Eyring’s flow theory [19]:

σ (ε, T ) =kBT

V ∗ sinh−1

(ε

ε0exp

(∆G

RT

)), (3.1)

which can be approximated by:

σ (ε, T ) ≈ kBT

V ∗ ln

(ε

12ε0

)+kB∆G

RV ∗ if σ (ε, T ) >>kBT

V ∗ . (3.2)

The experimental conditions are given by the strain rate ε and the temperature T ; theactivation volume V ∗, the rate constant ε0 and the activation energy ∆G are modelparameters; R and kB denote the gas constant and Boltzmann’s constant, respectively.With aging, the yield kinetics of PC as expressed in a yield stress versus logarithmof strain rate plot do not change slope, but merely shift to lower strain rates [6, 20].


This shift along the logarithmic strain rate axis is generally interpreted in terms of anaging-dependent rate constant, whereas the activation volume and activation energy areregarded to be aging-independent [6, 20, 21]. It is surprising that the activation energy isexperimentally observed not to depend on aging since this is in direct contradiction withthe physical picture of activation barriers that increase with aging. This apparent anomalywas experimentally investigated by Bauwens-Crowet and Bauwens [20], who concludedthat the changes observed in the activation energy of PC with aging were insignificantrelative to the uncertainty in their measurements. Indeed, an error analysis of their datareveals that the uncertainty in the activation energy is about 10%, or 25 kJ/mol, whilein a similar, more recent study activation energies of 279 kJ/mol and 286 kJ/mol werefound for unaged (unannealed) and aged (annealed) PC, respectively [22]. Consideringthese numbers, it is not surprising that a significant influence of aging on the activationenergy could not be detected. The present study aims to extend the experimental studyof Bauwens-Crowet and Bauwens [20] in two ways. First of all, the uncertainty in themodel parameters is minimized by assessing the yield kinetics across a wide range oftemperatures and strain rates. Secondly, an effort is made to maximize the difference inactivation energy between the unaged and aged state. The materials used in the studiesmentioned above were already aged quite severely (annealing at 120C for 46 hours [20]or 96 hours [22]), indicating that not much is gained by aging the material even more.However, it is shown in Figure 3.1 that although the difference in yield stress betweenannealed and quenched PC is limited, the difference between the upper and lower yieldstress of annealed PC is substantial. The fact that the lower yield stress is the samefor quenched and annealed PC suggests that it is independent of the aging history, and,perhaps, represents an unaged state of the material. However, this latter suggestion iscontroversial [23] and the relation between plastic deformation and the apparent de-agingis presently still debated.

The present study contributes to the aging/de-aging discussion by showing that, from amechanical point of view, the lower yield stress can indeed be regarded as a de-agedequivalent of the upper yield stress. Next, a detailed analysis of the upper and loweryield response of PC at various strain rates and temperatures is discussed. A significantinfluence of aging on the temperature dependence of yield is indeed found, consistentwith the physics discussed above. Constitutive models for glassy polymers need to takethis effect into account to correctly capture the temperature dependence of the yield andpost-yield deformation. Finally, the experimental observations are rationalized in termsof a physical interpretation of the aging phenomenon.



Tensile experiments were performed on a commercially available polycarbonate (Lexan,Sabic), which was supplied as 8 mm diameter extruded rod. The number-averaged and theweight-averaged molecular weight of this polycarbonate grade are 14.0 and 35.9 kg/mol,respectively [6]. From the extruded rod, axisymmetric dog-bone shaped tensile bars weremachined. Half of the specimens were mechanically pre-conditioned by twisting themto-and-fro over an angle of 990, which is a deformation well into the plastic regime.Subsequently, both the as-received and the pre-conditioned samples were annealed undervacuum at an annealing temperature of either 135C or 145C for a duration between 1and 2000 hours. The yield stress of the samples was measured in uniaxial tension at roomtemperature, at an engineering strain rate of 10−3 s−1. True stress values were calculatedassuming isochoric deformation of the samples.

3.3 Aging and plastic deformation

The mechanical responses of polymeric glasses with different ages become indistinguish-able at large strains [4-7, 24-27], see also Figure 3.1. This implies that plastic deformationbrings the material into a state that is independent of the prior aging history. At the sametime, application of plastic deformation lowers the yield stress [7, 28-31], as was elegantlyillustrated by G’Sell et al. [29], who measured the response of an aged PC specimen ina cyclic shear deformation experiment, shown in Figure 3.2. From A to B, the specimenyields at a shear strain of 0.08, after which a shear band is formed that grows through thesample until the deformation becomes fully homogeneous at a shear strain of 0.9. Next,the sample is unloaded and strained in the opposite direction, along the line indicated byC and D. Reloading of the sample along curve E clearly illustrates that the yield stressis substantially (∼20 MPa) lower than before the plastic deformation cycle, while theresponses coincide at shear strains above 0.9, as indicated by part F. The experimentaldata in Figures 3.1 and 3.2 suggest that plastic deformation erases the aging historyof a polymeric glass, making the material appear ‘younger’. The question now ariseswhether this is true ‘mechanical rejuvenation’, which would imply that the aging historyis erased in a reversible manner, without inducing any permanent (structural) changes inthe material.

Indications that mechanical pre-conditioning does induce permanent structural changeswere found by Aboulfaraj et al. [32], who measured the aging evolution of the shear yieldstress of an epoxy network. Figure 3.3 shows their data, comparing the aging-inducedevolution of samples that have been pre-conditioned through a plastic shear deformation


Figure 3.2: Mechanical response of a PC specimen subjected to a cyclic shear loading cycle.The loading path is in alphabetical order from A to F, as indicated in the plot.Experimental data taken from G’Sell et al. [29].

cycle with that of non-cycled (thermally quenched) samples. At long aging times, theyield stress evolution levels off because the material approaches its equilibrium level.Figure 3.3 clearly shows that although the evolution kinetics (slope) are similar, theyield stress levels of the non-cycled samples are significantly higher than those of thecycled samples, both in the evolution regime and the equilibrium regime. This means thatmechanical pre-conditioning appears to induce irreversible structural changes in theseepoxy networks, rather than to rejuvenate them in a reversible way. This observation doesnot agree with the work of Klompen et al. [6], who concluded from their experimentalobservations that mechanically pre-conditioned samples of PC age to the same yield stresslevels as thermally quenched samples, but this involved quite a substantial extrapolationof their data.

A physical understanding of the influence of large mechanical deformations on agedglasses may come from atomistic simulations. Mechanical rejuvenation in the sensethat the deformation responses of differently aged glasses become indistinguishable inthe large-strain regime has indeed been observed in atomistic simulations of both small-molecule glasses [10, 11, 33, 34] and polymeric glasses [13]. The physical interpretationof the rejuvenation phenomenon that arises from these studies is that plastic deformationforces an aged glass from its low-energy minimum to high-energy regions on the potentialenergy landscape [10, 34], thus suggesting an erasure of aging history. However, usingboth atomistic simulations [34] and simulations with a random energy landscape model[35], Lacks and co-workers showed that although the states of a plastically deformed glassresemble those of a less aged system, the regions of the energy landscape visited duringaging are different from those visited upon plastic deformation. A possible explanation forthis observation was given by Lee and Ediger [27], who performed direct measurements

Aging and plastic deformation 39

Figure 3.3: Influence of a plastic pre-deformation cycle on the aging-induced yield stressevolution of a DGEBA-PPO epoxy network; data taken from Aboulfaraj et al. [32].Lines are a guide to the eye.

of the molecular mobility of PMMA during creep and subsequent recovery. They tooobserved that the state of a plastically deformed glass clearly differs from that of aquenched glass immediately after deformation, but that these differences vanish aftera (short) period of strain recovery, apparently confirming the mechanical rejuvenationhypothesis.

To ascertain whether or not aging of PC changes due to mechanical pre-conditioning,an experimental study was conducted. The results are presented in Figure 3.4, showingthe evolution of the true yield stress of as-received and mechanically pre-conditioned PCthat results from annealing at 135C. Regardless of the prior thermo-mechanical history,after annealing for approximately 105 s or more the yield stresses of as-received andmechanically pre-conditioned PC coincide. This suggests that the pre-conditioning doesnot induce any permanent structural changes in PC, which is in contradiction with theobservations of Aboulfaraj et al. [32].

The data in Figure 3.4 can be a bit misleading because the initial yield stress differencebetween as-received and mechanically pre-conditioned PC appears small, which is due tothe high rate of aging at 135C. To properly demonstrate the extent to which mechanicalpre-conditioning reduces the yield stress, the results are combined with literature data [6]of mechanically pre-conditioned PC aged at room temperature, see Figure 3.5. Note thatthe horizontal axis of Figure 3.5 does not display annealing time because the annealingtimes for the different annealing temperatures, 23C and 135C, are not comparable dueto the different rates of aging at these temperatures. Instead, the effective aging time at areference temperature Tref = 23C is calculated according to [6]:


Figure 3.4: True yield stress values of as-received and mechanically pre-conditioned PC as afunction of annealing time at 135C. Lines are a guide to the eye.

teff (T (t)) =

∫ t

0

a−1T (T (t′))dt′, (3.3)

where teff is the effective aging time, t is the annealing time at a temperature T and aT isa temperature shift factor given by:

aT (T (t)) = exp

(∆Ha

R

(1

T (t)− 1

Tref

)). (3.4)

This time-temperature shift is governed by the activation enthalpy ∆Ha = 205 kJ/mol [6].

Figure 3.5 clearly shows that the reduction in yield stress upon mechanical pre-conditioningis dramatic, approximately 38 MPa. By way of comparison, the yield stress differencebetween the unaged (thermally quenched) and the aged PC in the study of Bauwens-Crowet and Bauwens [20], mentioned in the introduction, is about 8 MPa at the sametemperature and strain rate, see the dashed lines in Figure 3.5. Considering thissubstantial difference, it can be expected that aging-induced changes in activation barriersare significantly more pronounced when comparing the yield stresses of aged PC andmechanically pre-conditioned PC. Another difference between the mechanically pre-conditioned and the as-received samples is that annealing results in an increase in yieldstress at much shorter times for the former than it does for the latter. This observationis related to the fact that aging only increases the yield stress of a glassy polymer if theaging time exceeds the age that the material already has due to prior aging conditions,

Aging and plastic deformation 41

Figure 3.5: True yield stress values of as-received and mechanically pre-conditioned PC asa function of effective aging time at a reference temperature Tref = 23C. Theannealing data at 23C are taken from Klompen et al. [6]. Solid lines are modeldescriptions using Equation (3.5). The dashed lines indicate the yield stress levelsof the aged and unaged samples of Bauwens-Crowet and Bauwens [20].

including those experienced during the melt-processing of the samples [36, 37]. A simplemodel that defines the evolution of the yield stress as a function of the effective agingtime describes these kinetics [6]:

σy(T (t)) = σy,0 + c · log(teff (T (t)) + ta

t0

), (3.5)

where σy denotes the yield stress, σy,0 and c are model parameters, ta is the initial ageof the material and t0 = 1 s. Figure 3.5 shows that the model accurately describes theexperimental data, using σy,0 = 24.8 MPa and c = 3.55 MPa/decade. The initial agesof the as-received and the mechanically pre-conditioned material were found to be equalto 3 · 1014 s and 1 · 104 s, respectively. The model has been successfully applied tothe prediction of the yield stress, and the long-term ductile failure, of injection moldedPC, taking into account the temperature history experienced during processing [36, 38].These studies revealed the reason for the large discrepancy between the yield stress levelsof thermally quenched and mechanically pre-conditioned PC. The material already startsaging as soon as it is is cooled through its glass transition temperature and, therefore,experiences a substantial amount of aging during processing, which always involves finitecooling rates [37].

The last question that needs to be answered is whether the value of the equilibrium yieldstress of PC changes as a result of mechanical pre-conditioning. In the experiments de-scribed in Figure 3.4, equilibrium could not be reached within experimentally accessible


time scales. Figure 3.6 shows that at an annealing temperature of 145C, equilibriumis reached already after 104 s and it is obvious that the corresponding yield stresses ofboth the as-received and the mechanically pre-conditioned samples are indistinguishable.Also these observations are in contradiction with the observations of Aboulfaraj et al. [32].This discrepancy could be related to the fact that they studied a cross-linked system, incontrast to the amorphous thermoplast used in the present study.

Figure 3.6: True yield stress values of as-received and mechanically pre-conditioned PC as afunction of annealing time at 145C. Dashed lines are a guide to the eye.

In summary, the results presented confirm that large-strain mechanical pre-conditioningdoes not induce permanent structural changes in PC, but merely de-ages the polymer ina reversible manner. In that sense, the term ‘mechanical rejuvenation’ seems appropriateand will be used throughout the rest of this study. The fact that a highly aged state andthe mechanically rejuvenated state only differ in the extent of aging indicates that it ispossible to investigate the influence of aging on the temperature dependence of yield bycomparing these two states. This is the main objective of this study and is discussed inthe following sections.

3.4 Aging and deformation kinetics

Next, it is investigated how aging changes the temperature dependence of yield by probingthe deformation kinetics of PC across a wide range of strain rates and temperatures.Rather than comparing the upper yield stresses of aged and mechanically rejuvenatedsamples, this study focusses on the difference between the upper and the lower yieldstress as measured in a uniaxial compression test. Figure 3.1 shows that the lower yieldstress is equal for samples with different aging histories, indicating that it is representativefor the rejuvenated yield stress.

Aging and deformation kinetics 43

Figure 3.7: Uniaxial compression curves for PC at (a) different strain rates at a temperature of23C and (b) different temperatures at a strain rate of 10−3 s−1, data taken from [39]and [40], respectively. Upper and lower yield stresses corresponding to the data aredepicted in (c) and (d), in which lines represent model fits using Equation (3.1), withthe activation energy either aging-independent (dashed lines) or aging-dependent(solid lines).

Figures 3.7a and 3.7b show uniaxial compression curves for PC at different strain ratesand temperatures; the corresponding upper and lower yield stresses are plotted in Figures3.7c and 3.7d, respectively. The yield drop, defined as the difference between the upperand lower yield stress, is also plotted in these figures, it measures the strain softening.Both the upper and lower yield stresses show the linear dependence on the logarithm ofstrain rate and temperature reflected in Equation (3.2). Therefore, the data is fitted withthe Eyring model, see Equation (3.1). In accordance with earlier observations [6, 20],the data in Figure 3.7c shows that the activation volume does not significantly changewith aging because the slope of the linear dependence on the logarithm of strain rate isthe same for the upper and lower yield stress. Using a linear regression analysis on theupper and lower yield stresses plotted in Figure 3.7c, the average activation volume isdetermined: V ∗ = 3.35 nm3. Next, the activation energy is extracted from the upper yield


stress data in Figure 3.7d, leading to: ∆G = 342.6 kJ/mol. These parameters result inthe fits represented by the dashed lines, which give an accurate description of the upperyield data across the entire range of strain rates and temperatures. Considering the loweryield data, it is first assumed that the temperature dependence does not change with aging,which implies that the activation energy for the lower yield data is equal to that for theupper yield data. Fitting the rate constant to the lower yield stress at 23C and 10−3 s−1,this assumption leads to an accurate description of the strain rate dependence, as depictedby the dashed line in Figure 3.7c. However, the dashed line in Figure 3.7d shows that thedescription of the temperature dependence of the lower yield stress data is completely off,suggesting that the activation energy does significantly change with aging/rejuvenation.A separate linear regression analysis of the lower yield stress data in Figure 3.7d leads toa much lower value of the activation energy: ∆G = 291.3 kJ/mol. With this value, thefits represented by the solid lines are obtained. Note that the lower activation energy doesnot change the lower yield stress fit in Figure 3.7c, but only the value of the rate constantassociated with it. The final model parameters, corresponding with the fits representedby the solid lines, are listed in Table 3.1. An error analysis of the data reveals that theuncertainty in the activation energy values is about 7%, or 20 kJ/mol, slightly lower thanthe 25 kJ/mol found for the data of Bauwens-Crowet and Bauwens [20] discussed in theintroduction. Considering these numbers, it is concluded that the activation energy indeedsignificantly changes due to aging/rejuvenation.

Table 3.1: Model parameters for the fits represented by the solid lines in Figures 3.7c and 3.7d.

V ∗ [nm3] ∆G [kJ/mol] ε0 [s−1]upper yield - aged 3.35 342.6 1.4 · 1033

lower yield - rejuvenated 3.35 291.3 2.9 · 1029

3.5 Physical interpretation

The aim of this last section is to rationalize the observations regarding the influenceof aging on the activation energy in terms of the physical background of aging andrejuvenation. Each particle in a glass resides in a potential well, formed by its neighbors[15], introducing energy barriers that need to be overcome to dislodge the particle,allowing plastic deformation to occur. This is schematically illustrated in Figure 3.8a,where the barrier is represented as an enthalpy barrier ∆H , which is more appropriatesince materials are usually tested under pressure (or stress) control rather than under

Physical interpretation 45

volume control. There is no inherent activation barrier associated with the entropy ofa glass, as illustrated in Figure 3.8b.

Figure 3.8: Schematic illustration of the (a) enthalpy and (b) entropy of a glass in response to aplastic deformation event at constant temperature and pressure.

In contrast to the aged state, in the rejuvenated state there are no differences in theenthalpy or entropy of a glass before and after plastic deformation, but there is aninherent resistance to plastic deformation, an activation enthalpy ∆H0 that needs to beovercome, which corresponds with the rejuvenated yield stress. With aging, a glass visitsprogressively deeper minima within the energy landscape, consequently leading to anincrease in ∆H . At the same time, aging leads to a more densely packed configurationof the molecules in a glass. Such denser packing implies that the system has less possibleconformations available, which is why the entropy of a glass decreases with aging. For anaged glass, this phenomenon gives rise to an additional activation barrier ∆S that needsto be overcome before plastic deformation can occur. After plastic deformation the glassreturns to its rejuvenated state due to mechanical rejuvenation.

So far, it is still unclear how this physical understanding of aging-induced changes in∆H and ∆S relates to the experimentally observed changes in the Eyring parameters,discussed in the previous section. The main issue here is that the exact nature of theactivation energy ∆G is as yet unspecified. Of course, for a pragmatic application of theEyring flow theory, as done in the previous section, this does not really matter. Mostresearchers simply treat ∆G as a fit parameter that has dimensions J/mol, but Krausz andEyring [19] specifically state that ∆G represents an activation Gibb’s free energy thatgoverns the kinetics of plastic deformation. Using ∆G = ∆H − T∆S, Equation (3.1)can be rewritten:


σ (ε, T ) =kBT

V ∗ sinh−1

(ε

ε0exp

(∆H

RT

)exp

(−∆S

R

)). (3.6)

The change in entropy ∆S appears in this equation as a term that does not depend ontemperature. Therefore, it can be incorporated in the rate constant:

σ (ε, T ) =kBT

V ∗ sinh−1

(ε

ε♯0exp

(∆H

RT

)), (3.7)

with:

ε♯0 = ε0 exp

(∆S

R

). (3.8)

Comparing Equations (3.7) and (3.8) with Equation (3.1) reveals the relation between themodel parameters (∆G and ε0) and the physical quantities (∆H and ∆S). Referring tothe parameter values listed in Table 3.1 and the illustrations in Figure 3.8, it is concludedthat the experimentally observed increase in ∆G due to aging is related to the increasein activation enthalpy ∆H . Similarly, the aging-induced increase in ε0 is related to theincrease in activation entropy ∆S.

3.6 Conclusions

The first part of this study convincingly shows that large-strain mechanical pre-conditioningof aged polycarbonate truly rejuvenates the material in the sense that it reversiblyerases the aging-induced increase of the yield stress. This conclusion is of criticalimportance to address the influence of aging on the temperature-dependent deformationof polycarbonate. From a physical point of view, the existence of such an influence seemsevident because aging is known to increase activation barriers to plastic deformation. Acomparison of the temperature dependence of the yield stress of aged polycarbonate withthat of mechanically rejuvenated polycarbonate leads to the conclusion that the Eyringactivation energy indeed changes with aging and rejuvenation.

References 47

References

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[2] J.M. Hutchinson. Physical aging of polymers. Progress in Polymer Science. 1995, 20, 703-760.[3] G.B. McKenna. Glass formation and glassy behavior. In: Comprehensive Polymer Science: Vol.

2: Polymer Properties. C. Booth and C. Price, eds. Pergamon Press: Oxford, 1st Edition, 1989, p.311-362.

[4] G.A. Adam, A. Cross and R.N. Haward. The effect of thermal pretreatment on the mechanicalproperties of polycarbonate. Journal of Materials Science. 1975, 10, 1582-1590.

[5] O.A. Hasan and M.C. Boyce. Energy storage during inelastic deformation of glassy polymers.Polymer. 1993, 34, 5085-5092.



[8] H.G.H. van Melick, L.E. Govaert and H.E.H. Meijer. Localisation phenomena in glassy polymers:Influence of thermal and mechanical history. Polymer. 2003, 44, 3579-3591.

[9] W. Kob and J.-L. Barrat. Aging effects in a Lennard-Jones glass. Physical Review Letters. 1997, 78,4581-4584.

[10] M. Utz, P.G. Debenedetti and F.H. Stillinger. Atomistic simulation of aging and rejuvenation inglasses. Physical Review Letters. 2000, 84, 1471-1474.

[11] F. Varnik, L. Bocquet and J.-L. Barrat. A study of the static yield stress in a binary Lennard-Jonesglass. Journal of Chemical Physics. 2004, 120, 2788-2801.

[12] J. Rottler and M.O. Robbins. Unified description of aging and rate effects in yield of glassy solids.Physical Review Letters. 2005, 95, 225504.

[13] M. Warren and J. Rottler. Simulations of aging and plastic deformation in polymer glasses. PhysicalReview E. 2007, 76, 031802.

[14] A.Y.-H. Liu and J. Rottler. Aging under stress in polymer glasses. Soft Matter. 2010, 6, 4858-4862.[15] C. Monthus and J.-P. Bouchaud. Models of traps and glass phenomenology. Journal of Physics A:

Mathematical and General. 1996, 29, 3847-3869.[16] F.H. Stillinger. A topographic view of supercooled liquids and glass formation. Science. 1995, 267,

1935-1939.[17] R.E. Robertson. On the cold-drawing of plastics. Journal of Applied Polymer Science. 1963, 7, 443-

450.[18] C. Bauwens-Crowet, J.-C. Bauwens and G. Homes. Tensile yield-stress behavior of glassy polymers.

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Quantitative interpretation of the effect on yield stress and differential scanning calorimetrymeasurements. Polymer. 1982, 23, 1599-1604.



[22] C. Ho Huu and T. Vu-Khanh. Effects of physical aging on yielding kinetics of polycarbonate.Theoretical and Applied Fracture Mechanics. 2003, 40, 75-83.

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[25] H.G.H. van Melick, L.E. Govaert, B. Raas, W.J. Nauta and H.E.H. Meijer. Kinetics of ageing andre-embrittlement of mechanically rejuvenated polystyrene. Polymer. 2003, 44, 11711179.

[26] H.-N. Lee and M.D. Ediger. Interaction between physical aging, deformation, and segmental mobilityin poly(methyl methacrylate) glasses. The Journal of Chemical Physics. 2010, 133, 014901.

[27] H.-N. Lee and M.D. Ediger. Mechanical rejuvenation in poly(methyl methacrylate) glasses?Molecular mobility after deformation. Macromolecules. 2010, 43, 5863-5873.

[28] J.-C. Bauwens. A new approach to describe the tensile stress-strain curve of a glassy polymer.Journal of Materials Science. 1978, 13, 1443-1448.

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[32] M. Aboulfaraj, C. G’Sell, D. Mangelinck and G.B. McKenna. Physical aging of epoxy networks afterquenching and/or plastic cycling. Journal of Non-Crystalline Solids. 1994, 172-174, 615-621.

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Abstract: The modeling and quantification of a viscous contribution to strainhardening is discussed. Traditional strain hardening models, based on rubber-elasticity, show serious deviations from the experimentally observed large-strainresponse of glassy polymers. Therefore, to capture both the strain rate andtemperature dependence of strain hardening correctly, a part of the rubber-elastic strain hardening is replaced with a viscous contribution. This is realizedby introducing a deformation-dependent viscosity in the Eindhoven GlassyPolymer model. The performance of the proposed model is assessed usinga set of uniaxial compression tests on polycarbonate at different strain ratesand temperatures. An accurate description of all experimental data results and,moreover, the model quantitatively captures the Bauschinger effect observed inoriented polycarbonate.

49

50 Rate- and temperature-dependent strain hardening of polycarbonate

4.1 Introduction

Constitutive models that capture the mechanical response of glassy polymers up to largedeformations are applied in various types of studies. Typical examples are the modelingof solid-state forming processes, such as hydrostatic extrusion [1], hot-drawing [2, 3] andinjection stretch-blow molding [4], and the modeling of thin film and contact mechanics,such as indentation [5-9] and scratching [10, 11]. Other applications are found in polymercoated metals, such as the simulation of wall ironing processes [12, 13] and of polymerdelamination [14, 15]. Moreover, simulations demonstrate that the intrinsic large-strainresponse of polymers is of critical importance in the modeling of strain localization [16-18]. At large deformations, the stress response of glassy polymers is dominated bystrain hardening, the increase in stress with increase of strain as a result of orientingthe entanglement network [19]. Interestingly, the various constitutive models used in thestudies mentioned all employ rubber-elastic (entropic) strain hardening descriptions.

The use of entropic strain hardening models can be traced back to the seminal workof Haward and Thackray [20], who proposed an additive decomposition of the stressin a driving (flow) stress and an entropic strain hardening stress. Their work wasinspired by earlier observations [21-24] that plastic deformation in glassy polymersalmost fully recovers upon heating above the glass transition temperature, from whichit was concluded that the entanglement network of polymeric chains is preserved duringplastic deformation. The (scalar) Haward and Thackray model was later extended to3-D by Boyce, Parks and Argon [25], resulting in the well-known BPA model. Otherwell-established models describing glassy polymer deformation are the Oxford GlassRubber (OGR) model [26] and the Eindhoven Glassy Polymer (EGP) model [27, 28].Although all these models are based on the stress decomposition proposed by Haward andThackray, the details of the governing equations differ considerably. Regarding the strainhardening stress, various entropy-elastic models have been applied that were originallydeveloped to describe the mechanical response of elastomers. Popular models includethe neo-Hookean model [19], a number of 3-D network models [29-32] that employ theconcept of Langevin chain statistics [33] and the cross-link slip-link model derived byEdwards and Vilgis [34].

Although the constitutive models mentioned have proven to capture a number of aspectsof glassy polymer deformation, in recent years an increasing number of indicationsrevealed that strain hardening is a combined elastic-viscous phenomenon, rather than apurely elastic one, see for instance [35-38]. First, an entropic origin of strain hardeningsuggests that the corresponding modulus increases linearly with temperature, whereasa decrease is observed both in experiments [39-41] and in molecular simulations [42].

Introduction 51

Second, purely elastic models are unable to account for the strain rate dependencethat is observed [42, 43]. And, last, experiments show a strong dependence onhydrostatic pressure [44, 45], which is not captured by (incompressible) rubber-elasticstrain hardening models.

A viscous contribution to strain hardening is therefore suggested, and Chen and Schweizer[46] seek its physical origin in amplified activation barriers, as resulting from deformation-induced anisotropy in chain conformations and modifications in chain packing. In theirview, external deformation causes an increase in segmental relaxation times, whichleads to strain hardening stresses that are of intermolecular origin, as opposed to theintramolecular nature of stresses associated with an entropic strain hardening description[46]. Direct experimental evidence for this theory was presented by Lee et al. [47],who observed a deformation-induced decrease in the segmental mobility of poly(methylmethacrylate) during strain hardening, using an optical photobleaching method. Arelation between local anisotropy at plastic deformation sites and the viscous contributionto strain hardening was also advocated by Buckley [48], who argued that viscoplasticflow in amorphous polymers is intrinsically anisotropic on time scales shorter than theentanglement Rouse time because of the short length scales involved.

The concept of a viscous contribution to strain hardening has already been applied inthe constitutive modeling of glassy polymer deformation and a promising approach isto introduce a deformation dependence in the viscous flow stress [37, 43, 48-50]. Themodel proposed by Wendlandt et al. [43, 49] features an Eyring activation volume thatevolves with plastic deformation, resulting in an encouraging description of the rate- andtemperature-dependent strain hardening of a selection of polymers. Li and Buckley [50]followed another route, employing the idea of intrinsically anisotropic viscoplastic flow[48] and incorporating this in the OGR model to study its influence on strain localizationin polystyrene. Despite that the details of both modeling approaches differ considerably,an important similarity is that the viscous contribution to strain hardening expressesitself predominantly as a deformation-induced change in the strain rate dependence.Interestingly, recent experiments on polycarbonate showed that a change in strain ratedependence does not necessarily originate from a change in activation volume; it can alsoresult from a shift in the relative stress contributions from different relaxation processes[37].

A question that arises when discussing viscous strain hardening, is whether an entropiccontribution still plays a role, albeit weaker than previously assumed. Evidence for anaffirmative answer to this question was found in experimental [51] and simulation [42, 52]studies that investigated the relation between the strain hardening modulus, as obtained


from a rubber-elastic fit of the data, and the plastic flow stress. A linear relation wasfound, clearly evidencing that strain hardening does not have a purely entropic origin.However, extrapolation of the data to temperatures close to the glass transition, where theflow stress vanishes, leads to non-zero values of the strain hardening modulus, suggestingthat a (weak) entropic stress persists [52]. Additional indications that strain hardening isnot of a purely viscous nature are found in studies of the Bauschinger effect [53], which isthe asymmetry in the tensile and compressive yield stresses observed in oriented polymers[54-57]. The conclusions were twofold. On one hand, it was shown that all purely elasticstrain hardening models lead to erroneous descriptions of the Bauschinger effect and thata viscous contribution is required [37]. On the other hand, measurements of yield andshrinkage stresses of oriented poly(methyl methacrylate) made Botto et al. [58] concludethat the Bauschinger effect is related to a frozen-in network stress. These findings appearcontradictory at first sight, but it was shown that a weak rubber-elastic strain hardeningcontribution is indeed required to correctly model the Bauschinger effect [37].

The present study aims to quantitatively model the viscous contribution to strainhardening in polycarbonate (PC). The concept of a deformation-dependent flow stress[37, 43, 49, 50] is used to extend the EGP model, capturing all important experimentalobservations with a deformation-independent activation volume. The parameters in themodel proposed are obtained from the rate- and temperature-dependent strain hardeningresponse of PC as measured in uniaxial compression tests. Validation is done by assessingits performance in predicting the Bauschinger effect in PC.

4.2 Background

The deformation kinetics of amorphous polymers are characterized by a linear depen-dence of the yield stress on both the logarithm of strain rate and temperature. In manycases, these linear dependencies show a distinct change in slope, indicating that the yieldkinetics are governed by two molecular relaxation processes. This is illustrated in Figure4.1, where literature data [59] for the tensile yield stress of PC at different temperaturesand strain rates are presented. At low strain rates and high temperatures the yield kineticsare dominated by the contribution from the α relaxation process, which is associatedwith the full main-chain segmental relaxation that occurs at the primary glass transition.With increasing strain rate and/or decreasing temperature, the stress contribution of theβ relaxation process becomes significant and the yield kinetics show a distinct change inslope. This β relaxation process is related to a secondary glass transition observed around-100C in dynamic mechanical experiments at 1 Hz [60, 61].

Background 53

Figure 4.1: Tensile yield kinetics of PC. Symbols represent experimental data that were takenfrom [59]. Solid lines are fits using Equation (4.1) with the parameters from Table4.1. Dashed lines indicate the individual stress contributions from the α and βprocesses for the data measured at 22C.

The influence of temperature on the yield kinetics of PC is illustrated more clearly inFigure 4.2, where literature data [60] are presented for the tensile yield stress of PC ata single strain rate and across a wide range of temperatures, clearly demonstrating thecontribution from the β relaxation process.

Figure 4.2: Tensile yield stress of PC. Symbols represent experimental data that were taken from[60]. Solid lines are fits using Equation (4.1) with the parameters from Table 4.1.Dashed lines indicate the individual stress contributions from the α and β processes.

Deformation kinetics of polymers that are governed by two molecular relaxation processare accurately captured by the Ree-Eyring modification [62] of the original Eyring theory[63, 64]. A key assumption in this theory is that the stress contributions of the individualrelaxation processes (α and β) are additive:


σ (ε, T ) =∑x=α,β

kBT

V ∗x

sinh−1

(ε

ε0,xexp

(∆Hx

RT

)). (4.1)

The experimental conditions are given by the strain rate ε and the temperature T ; theactivation volume V ∗

x , the rate constant ε0,x and the activation enthalpy ∆Hx are modelparameters; kB and R denote Boltzmann’s constant and the gas constant, respectively.The solid lines in Figures 4.1 and 4.2 are fits with Equation (4.1), clearly demonstratingthat the model accurately captures the measured yield stresses. The individual stresscontributions of the α and the β-process, represented by the dashed lines, illustrate theadditive stress split proposed by the Eyring model. The parameter values used in thesefits are listed in Table 4.1; the differences between the rate constants associated with thedata in both figures are due to the different aging histories of the PC samples in both datasets.

Table 4.1: Ree-Eyring parameters for PC.

x V ∗x [nm3] ∆Hx [kJ/mol] ε0,x [s−1]

Figure 4.1: 6 · 1030

α 3.21 300 Figure 4.2: 6 · 1028

Figure 4.1: 8 · 1011

β 2.17 62 Figure 4.2: 1 · 1012

4.3 Phenomenology

As discussed in the introduction, the viscous contribution to strain hardening in polymerscan be interpreted in terms of a deformation dependence of the flow stress, or viscosity.Combined with the fact that this flow stress is well captured by the Eyring theory, asdiscussed in the previous section, this suggests a deformation dependence of one or moreof the Eyring parameters V ∗

x , ε0,x and ∆Hx, as defined Equation (4.1). A recent study[37] has confirmed this suggestion and also showed that for different polymers, differentEyring parameters depend on deformation. Direct experimental evidence was found that,in contrast to for instance polypropylene, for PC the activation volume does not changewith (pre-)deformation [37]. This conclusion is used in the present study, where the goalis to quantitatively model the viscous contribution to strain hardening.

Phenomenology 55

Essentially, there are two ways to directly investigate the influence of (pre-)deformationon the deformation kinetics of a polymer. The first is measuring the yield kinetics ofsamples that have been pre-deformed to various levels of strain, which involves a largeamount of experimental work [37]. A second, more promising method is to extractthe desired data directly from a set of uniaxial compression tests. This method, whichwas also used by Wendlandt et al. [43], is applied here because it provides much moreinformation from a smaller set of experiments.

4.3.1 Strain rate dependence

Figure 4.3a shows literature data [41] for the uniaxial compression of PC at differentstrain rates. The influence of deformation on the deformation kinetics, as shown inFigure 4.3b, is assessed by taking the stress at certain strain levels along the compressioncurves, which are indicated with the markers in Figure 4.3a. Note that only thelarge-strain regime is evaluated because at lower strains the flow stress is influencedby aging/rejuvenation effects, which cannot be distinguished from the influence ofdeformation-induced orientation. At the lower strain levels (|ε| < 0.6), the α-processdominates the kinetics and with increasing strain the onset of the β-process shifts to lowerstrain rates, marked by the distinct change in slope.

Figure 4.3: Uniaxial compression of PC at different strain rates and a temperature of 23C.(a) Compression curves, data taken from [9], with markers representing the stressat certain strain levels. (b) Deformation kinetics at these strain levels. Markersrepresent the measured stresses; solid lines are fits using Equation (4.1), with therate constants deformation-dependent; dashed lines represent stress contributionsfrom the α-process.

To further assess how deformation affects deformation kinetics, the experimental data inFigure 4.3b are fitted with Equation (4.1). For PC, it has been shown that the activationvolumes of both relaxation processes do not change with deformation [37]; the values


of the activation volumes are directly determined from the experimental data and listedin Table 4.2. The differences between the activation volume values listed in Tables 4.1and 4.2 result solely from the influence of the hydrostatic pressure, which is obviouslydifferent for tension and compression. Due to the isothermal character of the data set inFigure 4.3, it is not possible to distinguish the activation enthalpy from the rate constant.Therefore, the value of a combined parameter ε∗0,x is determined separately for each strainlevel, with:

ε∗0,x (T ) = ε0,x exp

(−∆Hx

RT

). (4.2)

This approach results in the model fits represented by the solid lines in Figure 4.3, leadingto the conclusion that Equation (4.1) accurately describes the experimental data when therate constant and/or the activation enthalpy are allowed to evolve with deformation. Inthe next section, an experimental data set at different temperatures is used to distinguishbetween the contributions of both parameters.

Table 4.2: Activation volumes used in the fits of Figures 4.3b, 4.4b and 4.5.

x V ∗x [nm3]

α 2.93β 1.98

4.3.2 Temperature dependence

Figure 4.4a shows literature data [41] for the uniaxial compression of PC at differenttemperatures. The influence of deformation on the deformation kinetics is again evaluatedby taking the stress at certain strain levels along the compression curves; Figure 4.4bshows these stresses as a function of temperature. Also in this data set, the transition fromthe α-regime to the (α+ β)-regime is clearly visible in the experimental data, marked bythe change in slope that appears at the higher strain levels (|ε| > 0.6).

Regarding the fitting of the experimental data in Figure 4.4b with Equation (4.1), it is firstassumed that the activation enthalpy does not depend on deformation. This assumptionresults in the model fits represented by the solid lines in Figure 4.4b, where the parametersare chosen such that the fits match the experimental data at 20C and that the values ofthe rate constant and the activation enthalpy combined match the values of ε∗0,x used in

Constitutive modeling 57

Figure 4.4: Uniaxial compression of PC at different temperatures and a true strain rate of 10−3

s−1. (a) Compression curves, data taken from [41], with markers representing thestress at certain strain levels. (b) Deformation kinetics at these strain levels. Markersrepresent the measured stresses; solid lines are fits using Equation (4.1), with the rateconstants deformation-dependent.

Figure 4.3b. Obviously, the temperature dependence of the deformation kinetics is notcaptured at all for the higher strain levels, which leads to the conclusion that not only therate constant, but also the activation enthalpy is deformation-dependent. Indeed, Figure4.5 clearly shows that the experimentally observed kinetics are accurately captured whenboth the rate constants and the activation enthalpies are determined at each individualstrain level. Also here, the values of both parameters at each strain level are such thatthey match the value of ε∗0,x used in Figure 4.3b at the corresponding strain level. Theobserved increase in the rate constants and the activation enthalpies with increasingdeformation leads to the conclusion that the effect of deformation-induced orientationon deformation kinetics is opposite to that of mechanical rejuvenation. It was recentlydemonstrated that the de-aging of a glassy polymer upon applying relatively small plasticstrains (in the strain softening regime) results in a decrease in both the rate constant andthe activation enthalpy [65]. However, this is not of influence to the results presentedhere since the analysis focusses on the large-strain regime, where the polymer is alreadyfully rejuvenated.

4.4 Constitutive modeling

The next step is to incorporate the phenomenological observations regarding the viscouscontribution to strain hardening, discussed in the previous section, in a 3-D constitutivemodel for glassy polymers: the Eindhoven Glassy Polymer (EGP) model [27, 28, 66-69].In this section, the governing equations of this model are briefly summarized.


Figure 4.5: Markers represent the same data as in Figure 4.4b; solid lines are fits using Equation(4.1), with both the rate constants and the activation enthalpies deformation-dependent; dashed lines represent stress contributions from the α-process.

4.4.1 Kinematics

The elastic and plastic parts of the total deformation are defined through a multiplicativedecomposition of the deformation gradient tensor F :

F = Fe · Fp, (4.3)

where the subscripts e and p refer to the elastic and plastic parts of the deformation,respectively. This decomposition implies that the velocity gradient tensor L is additivelysplit in an elastic and a plastic part:

L = F · F−1 (4.4)

= Fe · F−1e + Fe · Fp · F−1

p · F−1e (4.5)

= Le +Lp. (4.6)

The decomposition represented by Equation (4.3) is not unique because it does not specifythe amount of rotation associated with the elastic and the plastic parts of the deformation.This issue was elaborately discussed by Boyce et al. [70] and the solution used here is toassume that the plastic deformation is spin-free:


Lp = Dp +Ωp = Dp, (4.7)

where the plastic rate of deformation tensor Dp and the plastic spin tensor Ωp are thesymmetric and anti-symmetric parts of Lp, respectively. Furthermore, it is assumed thatplastic deformation is isochoric, implying that the volume ratio J only depends on theelastic deformation gradient tensor:

J = det(F ) = det(Fe). (4.8)

4.4.2 Total stress

The total Cauchy stress σ is defined in terms of its deviatoric and hydrostatic stresscomponents as:

σ = σh + σd = K (J − 1) I + σd. (4.9)

The superscripts h and d refer to the hydrostatic and deviatoric parts of the stress,respectively. The hydrostatic stress is straightforwardly described with a constant bulkmodulus K; the second-order unit tensor is denoted by I . The deviatoric stress isadditively split in an elasto-viscoplastic driving stress σs and an elastic strain hardeningstress σr:

σd = σs + σr. (4.10)

4.4.3 Driving stress

The additive split of the Eyring stress in cases where more than one relaxation processcontributes to the response, see Equation (4.1), is also applied in the definition of thedriving stress [59, 68, 69]. Furthermore, the stress contribution of each relaxation processis characterized not just by a single viscoelastic Maxwell mode, but by a spectrum ofrelaxation times and corresponding shear moduli for which the stress contributions areadditive [67, 71]:


σs = σα + σβ (4.11)

=n∑

j=1

σα,j +m∑k=1

σβ,k (4.12)

=n∑

j=1

Gα,jBdeα,j

+m∑k=1

Gβ,kBdeβ,k

. (4.13)

Subscripts α and β refer to the relaxation process; subscripts j and k denote the individualmodes associated with both processes; n and m are the total number of modes for eachprocess. The deviatoric part of the isochoric, elastic left Cauchy-Green deformation tensorof each mode is defined as:

Bdex,i

=(Fex,i · F T

ex,i

)d= J−2/3

(Fex,i · F T

ex,i

)d. (4.14)

The subscripts i and x indicate that a variable corresponds to a certain mode i that isassociated with process x and will be used as such in many of the following equations.The interrelation between the driving stress and the plastic rate of deformation is governedby a non-Newtonian flow rule:

Dpx,i =σx,i

2 ηx,i(T, τx, p, Sx), (4.15)

where the viscosity η is a scalar parameter that depends on temperature T , equivalentstress τx, pressure p and the state parameter Sx:

ηx,i = η0x,iτx/τ0x

sinh (τx/τ0x)exp

(∆Hx

RT

)exp

(µxp

τ0x

)exp (Sx) . (4.16)

Herein, η0x,i denotes the initial viscosity, τ0x is a characteristic stress and µx is the pressuredependence. In this viscosity definition, the stress and temperature dependencies arebased on the Eyring theory [27]; the dependencies on pressure and (thermodynamic) statewere added later [28]. The equivalent stress, characteristic stress and pressure are definedas:


τx =

√1

2σx : σx ; τ0x =

kBT

Vx; p = −1

3tr(σ). (4.17)

Note that the Eyring activation volume Vx is related to the shear equivalent stress and,therefore, has a different value than the activation volume V ∗

x that appears in Equation(4.1), which is implicitly related to a uniaxial equivalent stress. The description of theintrinsic strain softening that is observed in glassy polymers is incorporated in the stateparameter Sx, which is, therefore, a function of the equivalent plastic strain γp [28, 66]:

Sx(γp) = SaxRγx(γp). (4.18)

The initial value of the state parameter is denoted by Sax and represents the thermo-mechanical (aging) history of the material [66, 72, 73]. Intrinsic strain softening isdescribed with the function Rγx , which is a modified version of the Carreau-Yasudafunction [66]:

Rγx(γp) =

[1 + (r0x exp(γp))

r1x

1 + rr1x0x

] r2x−1

r1x

, (4.19)

where r0x , r1x and r2x are fit parameters. The values of these softening parameters aretaken the same for both relaxation processes since the experimental data do not providesufficient information to discriminate between them. The equivalent plastic strain iscalculated from the equivalent plastic strain rate of the mode with the highest initialviscosity because that mode marks the onset of macroscopic yielding [67]. Assumingthat this is the first α mode, the equivalent plastic strain rate ˙γp is defined as:

˙γp =τα,1ηα,1

where τα,1 =

√1

2σα,1 : σα,1. (4.20)

4.4.4 Elastic strain hardening

For modeling the elastic strain hardening stress, different rubber-elastic models are usedin the various constitutive models for glassy polymers. For instance, the original BPA


model [25] employs the three-chain model from Wang and Guth [30], but later the eight-chain model [31] was proposed instead. Studies on the EGP model have so far used aneo-Hookean description [28, 74]. In the present study, however, elastic strain hardeningis modeled with an Edwards-Vilgis model [34], providing a more accurate description ofthe non-linear strain hardening. This model has already been applied as a strain hardeningmodel in the OGR model [26, 75, 76]. In the absence of cross-links, the conformationalfree energy expression reads [34]:

Ac =Ne kB T

2

[(1 + ξr)(1− α2

r)

1− α2r

∑3k=1 λ

2k

3∑k=1

λ2k1 + ξr λ2k

. . .

+3∑

k=1

ln(1 + ξr λ

2k

)+ ln

(1− α2

r

3∑k=1

λ2k

)]. (4.21)

Herein, λk (k = 1,2,3) are the principal stretch ratios. The response of the model is shapedby the number-density of entanglements Ne, the mobility of entanglements (slip-links) ξrand a parameter αr that determines the limiting extensibility of the network. Note thatthis free energy expression is non-zero in the absence of deformation, which is peculiar,but it does not matter for the stress response predicted by the model. In tensor notation,Equation (4.21) is written as:

Ac =Gr

2

[(1 + ξr)(1− α2

r)

1− α2r tr(B)

tr(B ·

(I + ξr B

)−1). . .

+ ln(det(I + ξr B)

)+ ln

(1− α2

r tr(B))], (4.22)

where also the strain hardening modulus Gr = Ne kB T was introduced as a constant. Us-ing standard hyper-elasticity theory the elastic strain hardening stress can be determined:

σr =2

JF · ∂Ac

∂C· F T , (4.23)

where C denotes the right Cauchy-Green deformation tensor. The hardening stressaccording to the Edwards-Vilgis model is then given by:


σr =Gr

J

(B ·Z

)d, (4.24)

introducing the tensor Z as:

Z =α2r(1 + ξr)(1− α2

r)(1− α2

r tr(B))2 tr

(B ·

(I + ξr B

)−1)I

+(1 + ξr)(1− α2

r)

1− α2r tr(B)

((I + ξr B

)−1

− ξr

(I + ξr B

)−1

·(I + ξr B

)−1

· B)

+ξr

(I + ξr B

)−1

− α2r

1− α2r tr(B)

I. (4.25)

It is noteworthy that the Edwards-Vilgis expression for the elastic strain hardening stressreduces to the neo-Hookean expression (i.e. Z = I) in the case of αr = ξr = 0.

To demonstrate the differences between the Edwards-Vilgis strain hardening model andthe neo-Hookean one, the responses of the models are evaluated by simulating a uniaxialtensile test. Figure 4.6 shows literature data [37] for such a test, as well as the results ofsimulations with the single mode, single process EGP model, using the model parameterslisted in Table 4.3 and in the figure itself. The dashed line represents the model fit asobtained by Klompen et al. [66], underestimating the experimental data both at low andhigh strains. In the regime just past the yield point, the model predictions improve if theinitial viscosity is increased, but this is accompanied by a stronger underestimation ofthe stress at high strains, as illustrated by the dash-dotted line. In contrast, the solid lineshows that the Edwards-Vilgis strain hardening model does describe the experimentaldata accurately across the entire strain range, which is the reason that this model is usedin the present study. As it is known that the entanglement network of a glassy polymerremains intact during plastic deformation, the mobility of entanglements (slip-links) ξr ischosen zero in this study.

Table 4.3: Model parameters for the simulations in Figure 4.6.

K [MPa] G [MPa] V [nm3] ∆H [kJ/mol] µ [-] Sa [-] ξr [-]3000 750 5.32 289.8 0.08 0 0


Figure 4.6: Uniaxial tension of mechanically pre-conditioned PC at a true strain rate of 10−4 s−1.Markers represent experimental data, taken from [37]. Lines represent simulationswith the EGP model, using model parameters as listed in Table 4.3. Three differentmodel options for the elastic strain hardening stress are evaluated and the associatedparameters are listed in the figure.

4.4.5 Viscous strain hardening

Based on phenomenological considerations it was established in previous sectionsthat a viscous contribution to strain hardening expresses itself through deformationdependencies of both the Eyring rate constant and activation enthalpy. It is easy toconclude that in the case of the EGP model this implies that the viscosity becomesdeformation-dependent, but it is as yet unclear what scalar deformation function governsthis deformation dependence. This function should be an invariant function of theappropriate deformation tensors and it is derived by using a simplified single mode, singleprocess viscosity notation, but this does not influence the argument that is made. In short,a dependence on the invariant deformation function Ir(B) is introduced in the viscosity:

η(T, τ , p, S) ⇒ η(T, τ , p, S, Ir(B)). (4.26)

Note that, similar to the elastic contribution, the viscous contribution to strain hardeningis modeled as a function of the total deformation. By defining a temperature-, pressure-and state-dependent initial viscosity as:

η∗0(T, p, S) = η0 exp

(∆H

RT+µp

τ0+ S

), (4.27)


it is possible to rewrite Equation (4.16):

η (τ) =η∗0 τ

τ0

1

sinh(τ /τ0). (4.28)

From this expression, it becomes clear that there are two options for introducing adeformation dependence in the viscosity: in the pre-factor and in the argument of thehyperbolic sine function. Both quantities are directly proportional to the equivalent stressτ . Since the large-strain response is rather well approximated by Equation (4.24), asshown in Figure 4.6, the invariant deformation function for viscous strain hardeningshould result in a similar stress response. For the simple case of Z = I , Equation (4.24)reduces to a proportionality between the deviatoric parts of the Cauchy stress tensor andthe isochoric left Cauchy-Green deformation tensor: σd ∝ Bd. This is substituted in thedefinition of the equivalent stress (Equation (4.17)) to obtain:

τ ∝ Ir(B), (4.29)

where the invariant deformation function Ir(B), from now on referred to as equivalentstrain, is the following function of invariants of B:

Ir(B) =

√1

2Bd : Bd =

√1

3(I1(B))2 − I2(B), (4.30)

with:

I1(B) = tr(B) and I2(B) =1

2

(tr2(B)− tr(B · B)

). (4.31)

The deviatoric part of the isochoric left Cauchy-Green deformation tensor Bd is definedanalogous to Equation (4.14), albeit as a function of the total deformation gradient ratherthan the elastic.


4.5 Model characterization

The characterization of the EGP model is already well documented [66-69, 77] and forthe majority of the parameters it is clear how they can be extracted from experimentaldata. Therefore, this section focusses on the determination of the parameters associatedwith the viscous contribution to strain hardening. The first issue is to establish the correctdivision of the total strain hardening in its elastic and viscous contributions. In previouswork it was shown that an approximately equal distribution of elastic and viscous strainhardening is required to capture the Bauschinger effect that is observed in PC [37]. Inthe simulation represented by the solid line in Figure 4.6, only elastic strain hardening isincorporated. The final model parameters for elastic strain hardening, listed in Table 4.5,therefore feature a strain hardening modulus that is roughly half of the one used in thatsimulation.

Regarding the viscous contribution to strain hardening, the same approach is used aswas discussed in the ‘phenomenology’ section. In short, the deformation dependenceof the Eyring parameters is extracted from the strain rate and temperature dependenceof the stress at certain strain levels along a uniaxial compression curve. The maindifference is that, for characterization of the EGP model, the Eyring parameters shouldbe fitted to the experimental driving stress σs, which is obtained by subtracting the elasticstrain hardening contribution from the experimental stress σ. For the case of uniaxialcompression, this subtraction is expressed as [66, 67]:

σs(λ) = σ(λ)−√3√

3− µσr(λ), (4.32)

where λ is the uniaxial stretch. The (scalar) Edwards-Vilgis strain hardening stress σr isthe uniaxial stress that results from Equation (4.24) when it is evaluated for the case ofuniaxial isochoric deformation and taking into account that ξr = 0:

σr(λ) = Gr

[α2r (1− α2

r) (λ2 + 2λ−1)

(1− α2r (λ

2 + 2λ−1))2+

1− 2α2r

1− α2r (λ

2 + 2λ−1)

] (λ2 − λ−1

). (4.33)

Figure 4.7 shows the deformation dependence of the deformation kinetics based on theexperimental driving stress, comparable with Figures 4.3b and 4.5. At each strain level,the values of the initial viscosity and the activation enthalpy are uniquely determined bysimultaneously fitting the corresponding data points in Figures 4.7a and 4.7b. The solid

Model characterization 67

lines in Figure 4.7 demonstrate that also for the driving stress kinetics a good agreementis obtained between the model fits and the experimental data.

Figure 4.7: Driving stress as determined from uniaxial compression experiments. Markersrepresent experimental data, solid lines are fits using Equation (4.1) withdeformation-dependent initial viscosities and activation enthalpies. (a) Data fromFigure 4.3b. (b) Data from Figure 4.5. In both cases, the driving stress was obtainedby subtracting the elastic strain hardening contribution, see Equation (4.32).

The values of the activation enthalpy and the initial viscosity that are used to fit the data inFigure 4.7 at each strain level are plotted as a function of equivalent strain in Figures 4.8and 4.9, respectively. The equivalent strain is calculated according to Equation (4.30).Figures 4.8 and 4.9 clearly show that both model parameters change with increasingdeformation. The evolution of these model parameters is consistent with the physicalinterpretation of viscous strain hardening in terms of amplified activation barriers [46],as discussed in the introduction: the activation enthalpy increases and the decrease of theinitial viscosity implies that the activation entropy increases as well [64, 65].

The evolution of both the activation enthalpy and the initial viscosity as a function ofequivalent strain is incorporated in the EGP model using a set of simple functions thatdescribe the experimentally observed evolution:

∆Hα(Ir(B)) = ∆H0α + C1,α (Ir(B))2 (4.34)

∆Hβ(Ir(B)) = ∆H0β + C1,β Ir(B) (4.35)

η0α,i(Ir(B)) = η0α,i

exp(−C2,α (Ir(B))2

)(4.36)

η0β,i(Ir(B)) = η0β,i exp(−C2,β Ir(B)

)(4.37)


Figure 4.8: Activation enthalpy of the α and β-process versus equivalent strain Ir(B). Opensymbols represent the values obtained from the fits in Figure 4.7; closed symbolsrepresent the experimental values at the yield point. Solid lines are fits withEquations (4.34) and (4.35), using the parameters from Table 4.5.

Figure 4.9: Initial viscosity of the α and β-process versus equivalent strain Ir(B). Opensymbols represent the values obtained from the fits in Figure 4.7; closed symbolsrepresent the experimental values at the yield point. Solid lines are fits withEquations (4.36) and (4.37), using the parameters from Table 4.5.

The solid lines in Figures 4.8 and 4.9 represent the model fits with these functions,using the parameter values listed in Table 4.5. There is a good agreement between theexperimental values and the model description at large strains, but the filled markersindicate that, for the α-process, the experimental values for the activation enthalpy andthe initial viscosity at the yield point deviate considerably from the evolution function.This is related to the influence of aging on the activation enthalpy of PC [65], which isnot taken into account in the EGP model. Experimental activation enthalpy values at theyield point correspond to an aged PC, while the evolution function is characterized withexperimental stresses at large strains, where the material is fully rejuvenated. The issue isaddressed in detail in a later section.

Simulation results 69

As mentioned, the determination of the remaining model parameters is not discussed herein detail. It is, however, important to mention that a spectrum of 17 relaxation modes isused for the α-process, whereas a single mode is used for the β-process. Employinga spectrum of relaxation modes rather than a single mode provides a more accuratedescription of the pre-yield regime while the post-yield response remains unaffected[67, 71]. The relaxation spectrum for the α-process is identical to the one used by VanBreemen et al. [67], but it is scaled to get a higher total initial viscosity. The parametersassociated with the relaxation spectrum are listed in Table 4.4 and all others are listed inTable 4.5.

Table 4.4: Relaxation spectrum.

x, i Gx,i [MPa] η0x,i [MPa.s]α, 1 316.8 9.8221 · 10−28

α, 2 49.95 1.6277 · 10−29

α, 3 40.32 1.3798 · 10−30

α, 4 37.08 1.3283 · 10−31

α, 5 31.50 1.1880 · 10−32

α, 6 28.80 1.1412 · 10−33

α, 7 24.75 1.0290 · 10−34

α, 8 21.87 9.5415 · 10−36

α, 9 18.63 8.5592 · 10−37

α, 10 16.29 7.8577 · 10−38

α, 11 13.86 7.0625 · 10−39

α, 12 12.24 6.5481 · 10−40

α, 13 10.71 5.9400 · 10−41

α, 14 8.820 5.1449 · 10−42

α, 15 9.360 5.7529 · 10−43

α, 16 1.899 1.2254 · 10−44

α, 17 14.76 1.0009 · 10−44

β, 1 315.0 5.0000 · 10−11

4.6 Simulation results

All the simulations in this study are performed with the finite element software MSCMarc, using an implementation of the EGP model in the user subroutine HYPELA2. Thefinite element mesh consists of a single linear quadrilateral axisymmetric element anduniaxial loading conditions are applied.


Table 4.5: Final model parameters.

x ∆H0x [kJ/mol] C1,x [kJ/mol] η0x,i [MPa.s] C2,x [-] Gr [MPa] αr [-] ξr [-]α 240 80 1.0 · 10−27 24β 58 60 5.0 · 10−11 17 7 0.23 0

x Vx [nm3] µx [-] Sax [-] r0x [-] r1x [-] r2x [-] K [MPa]α 5.32 0.08 see text 0.965 50 -5β 5.06 0.08 0 0.965 50 -5 3000

4.6.1 Compression at different strain rates and temperatures

Now that the model has been fully characterized, finite element simulations are usedto assess the performance of the model in describing rate- and temperature-dependentstrain hardening of PC. For these simulations, the initial value of the state parameter isdetermined to be Saα = 19. Figure 4.10 shows that the uniaxial compression responseof PC at different strain rates is accurately captured across the entire strain range. Notethat the markers in Figure 4.10 represent the experimental compression curves depicted assolid lines in Figure 4.3a. By way of comparison, similar simulations with the EGP modelwithout the viscous contribution to strain hardening could only describe the experimentalcurves up to a compressive true strain of approximately 0.6 [66, 67], albeit with a neo-Hookean model for the elastic strain hardening, rather than the Edwards-Vilgis modelused here.

Figure 4.10: Uniaxial compression curves for PC at different strain rates. Markers represent thesame experimental data as in Figure 4.3a; solid lines represent model simulations.

At the same time, the proposed model accurately describes the large-strain uniaxialcompression response of PC at different temperatures. This is shown in Figure 4.11,


where the markers represent the experimental compression curves depicted as solid linesin Figure 4.4a. However, the temperature dependence of the yield point is not capturedwell; the experimental yield stress is highly overestimated at elevated temperatures. Thecause for this discrepancy was already noted in the discussion of Figure 4.8 and is thesubject of the next section.

Figure 4.11: Uniaxial compression curves for PC at different temperatures. Markers representthe same experimental data as in Figure 4.4a; solid lines represent modelsimulations.

4.6.2 Aging dependence of the activation enthalpy

The aging history of a glassy polymer has a pronounced influence on the yield stress, butnot on the large-strain response [66, 78, 79]. Over the years, this aging dependence ofthe yield stress has generally been interpreted in terms of an aging-dependent Eyring rateconstant (initial viscosity) [26, 66, 80], but recently it was shown that also the activationenthalpy exhibits a significant dependence on aging [65]. The poor description of thetemperature dependence at the yield point (see Figure 4.11) and the discrepancy betweenthe experimentally measured activation enthalpy at yield and the expected value based onthe large-strain response (see Figure 4.8) are both caused by the fact that the EGP modeldoes not take into account this aging dependence of the activation enthalpy. This canbe solved by adapting the expressions for the activation enthalpy (Equations (4.34) and(4.35)):

∆Hα(Ir(B), γp) = ∆H0α + C1,α (Ir(B))2 + SHαRγα(γp) (4.38)

∆Hβ(Ir(B), γp) = ∆H0β + C1,β Ir(B) + SHβRγβ(γp). (4.39)


Herein, SHx denotes the aging-dependent contribution to the activation enthalpy, which isequal to zero for a fully rejuvenated material. The softening function Rγx(γp) is identicalto the one defined in Equation (4.19). Due to the introduction of an aging-dependentactivation enthalpy, the role of the state parameter Sx (see Equation (4.18)) also changes.Essentially, the state parameter induces aging-related changes in the initial viscosity (seeEquation (4.16)), implying that they are related to changes in entropy [65]. As the entropyof a glassy polymer decreases with aging, Equation (4.18) is redefined:

Sx(γp) = −SSxRγx(γp). (4.40)

The new state parameter SSx embodies all aging-induced changes in entropy and its valueincreases with aging. All parameter values associated with the introduction of an aging-dependent activation enthalpy are listed in Table 4.6; the softening function parametersremain the same.

Table 4.6: Parameters for the influence of aging used in the simulations shown in Figure 4.12.

x SHx [kJ/mol] SSx [-]α 60 5.5β 0 0

Simulation results regarding the temperature-dependent uniaxial compression responseof PC are presented in Figure 4.12. The solid lines in the figure clearly demonstratethat incorporation of the influence of aging/rejuvenation on the activation enthalpy leadsto an accurate description of the experimental data across the entire strain range for alltemperatures measured.

4.6.3 Bauschinger effect in polycarbonate

Until now, this study focussed on the issue of modeling rate- and temperature-dependentstrain hardening of PC by incorporating a viscous contribution to strain hardening. This,however, also affects the EGP model predictions of the Bauschinger effect since, withouta viscous contribution, no model based on an additive decomposition of elastic andviscous stresses is able to capture this effect [37]. For this reason, an assessment ofthe capabilities of the present model to predict the Bauschinger effect in PC is a powerfulway to test how adequate the proposed model concepts actually are.


Figure 4.12: Uniaxial compression curves for PC at different temperatures. Markers representthe same experimental data as in Figure 4.11; solid lines represent modelsimulations, taking into account the influence of aging/rejuvenation on theactivation enthalpy.

One way to measure a Bauschinger effect is to use uniaxial deformation experimentsin which the deformation direction is reversed halfway through the test. A practicalprocedure is to first homogeneously extend a tensile bar, followed by a compressiontest performed on a sample that is machined from the oriented bar. Here, data of aprevious study [37] are used. An important feature of these particular experiments is thatthey were performed on mechanically pre-conditioned (rejuvenated) axisymmetric tensilebars to ensure homogeneous deformations during the tensile part of the test. Duringmechanical pre-conditioning (applying large-strain torsion over an angle of 990 to andfro), a small artefact is introduced since the axial force during torsion is not exactly zero,which induces some axial orientation in the tensile bar. This is demonstrated in Figure4.13, where uniaxial compression curves are shown for an as-received sample and amechanically pre-conditioned sample. It is known that the compression curves of sampleswith different pre-strains coincide in the large-strain regime if a strain-shift is appliedequal to the amount of pre-strain [37, 81]. Using this principle, the axial orientation inthe pre-conditioned sample is found to be equal to a tensile true strain of ε = 0.038, asillustrated in the figure.

The data from the Bauschinger effect experiments are presented in Figure 4.14 and arecorrected for the tensile pre-orientation that is present in the specimens as a result ofmechanical pre-conditioning. The dashed lines in the figure represent simulations usingthe EGP model without a viscous contribution to strain hardening and demonstrate thata purely elastic model for strain hardening results in dramatic errors in the prediction ofthe experimental data. The solid lines represent EGP model simulations that includethe viscous contribution to strain hardening, as proposed in this study; evidently, an


Figure 4.13: Uniaxial compression of PC at a true strain rate of 10−4 s−1. Curves are shown foran as-received sample (data taken from [9]) and for a mechanically pre-conditionedsample (data taken from [37]). The latter is shifted across the strain axis to accountfor the axial orientation that is induced during pre-conditioning.

accurate, quantitative prediction of the experimentally observed response is obtained. Itshould be stressed that no additional parameter fitting is involved in the prediction of theBauschinger effect presented in Figure 4.14; the only difference with the simulationsin the previous section is that here SSα = SHα = 0 because the Bauschinger effectexperiments were performed on mechanically pre-conditioned (rejuvenated) samples.Another interesting observation is that the tensile parts of the cyclic deformation pathsin Figure 4.14 are captured well by the new modeling approach, despite the fact thatthe model was only characterized using uniaxial compression data. This is an indicationthat the equivalent strain Ir(B), defined in Equation (4.30), is an adequate quantity fordescribing the viscous contribution to strain hardening. Upon reversal of the loadingdirection, the simulation for the highest pre-strain of 0.6 exhibits an unexpected yieldpeak, which is not observed in the experimental data. This minor effect is the combinedresult of the functional shapes of the viscous and elastic strain hardening contributions.

4.7 Conclusions

A detailed analysis of the uniaxial compression response of polycarbonate at variousstrain rates and temperatures reveals that the viscous contribution to strain hardeningmanifests itself as a deformation dependence of both the Eyring rate constant and thecorresponding activation enthalpy. In the range of experimental conditions probed inthis study, the deformation response of polycarbonate contains stress contributions fromtwo different relaxation processes, referred to as α and β. The Eyring parameters ofboth processes are observed to be deformation-dependent. In terms of the EGP model,

References 75

Figure 4.14: Cyclic uniaxial deformation of PC at a true strain rate of 10−4 s−1. Markersrepresent experimental data, taken from [37]; lines represent simulations without(dashed) and with (solid) viscous strain hardening.

this implies that the initial viscosities and activation enthalpies of both the α and theβ-process change as a result of deformation-induced orientation. Incorporation of thismodeling concept in the EGP model and characterization using uniaxial compression testsleads to an accurate description of the experimental data across the full range of strains,strain rates and temperatures considered. Additional support for the quality of the modelproposed is found in the observation that it also quantitatively captures the Bauschingereffect in oriented polycarbonate.

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Abstract: The anisotropic yielding of injection molded polyethylene isdiscussed, specifically focussing on the differences between the tensile andcompressive yield stress as a function of loading angle and strain rate. For thefirst time, it is demonstrated that a strong Bauschinger effect exists in polymersthat possess molecular orientation due to melt-processing. A macroscopicconstitutive model is proposed to capture the yielding phenomena observed inthe experiments. This model features two sources of anisotropy, the physicalsignificance of which is discussed: a frozen-in stress originating from theoriented elastic network, and an intrinsically anisotropic viscoplastic flow rulebased on the yield function of Hill, extended to incorporate the asymmetrybetween the tensile and compressive response. Model simulations demonstratethat the constitutive relation proposed accurately captures the important featuresof the experimental data.

81

82 Anisotropic yielding of injection molded polyethylene

5.1 Introduction

Mechanical properties of polymer products are the result of a complex interplay betweenmaterial-related parameters, such as the molecular architecture of the base material,and processing-related parameters, such as the flow and temperature conditions in themanufacturing process. The study specifically focusses on the effect of processing-induced molecular orientation on the mechanical response of solid polymers. Thischoice is motivated by the recognition that catastrophic failure of a polymer product isheavily affected by the degree of molecular orientation. More specifically, it is knownthat processing-induced orientation has a significant influence on both toughness [1-4] and time-to-failure (life span) [5-7]. Evidently, these two performance parametersare of key importance for the successful use of polymers in structural applications. Abetter understanding of the relation between processing-induced molecular orientationand failure will aid the development of predictive numerical tools, ultimately enablingdesign engineers to prevent premature failure of their polymer products.

For semi-crystalline polymers, the influence of processing-induced orientation is par-ticularly large due to their microstructure, which is a heterogeneous composition ofintrinsically anisotropic crystalline domains and randomly coiled amorphous regions.During solid-state forming processes, such as drawing [8], hydrostatic extrusion [9, 10]or rolling [2, 11], the polymer chains in the crystalline domains are oriented towards themain deformation direction; the chains in the amorphous domains are stretched along thissame direction. Similarly, the flow applied during melt-processing causes the polymermolecules to stretch and orient in the direction of flow, directly affecting the formation ofpolymer crystals upon cooling. This may lead to highly oriented crystalline morphologies,such as shish-kebab or stacked lamellae structures [12].

An important consequence of a preferred orientation of molecules in a specific directionis that it leads to a distinct anisotropy in mechanical properties, such as modulus oryield stress. For instance, increasing the loading angle, i.e. the angle between themain deformation direction and the main orientation direction, from 0 to 90 resultsin a significant drop of the tensile yield stress of polymer samples that were orientedby drawing [6, 13-18], plane-strain compression [19] or injection molding [7]. Thesimple, physical explanation of this phenomenon is that polymers become more difficultto deform when the molecules get closer to an aligned, stretched conformation. For failureof oriented polymers, this loading angle dependence is also of critical importance. In thecase of injection molding, substantially different failure modes may be found within asingle polymer product, covering the full range from brittle to ductile [7, 20]. Moreover,the ratio between the times-to-failure upon application of a constant load parallel and


perpendicular to the injection (flow) direction can be as large as 500 within a singleproduct of polypropylene [7].

Another phenomenon often found in oriented polymers is a pronounced asymmetrybetween the tensile and compressive responses that is not present in isotropic polymers.This type of asymmetry is often referred to as the Bauschinger effect [21], namedafter a phenomenon in metals that has quite similar phenomenology, but a differentphysical background. A classic example of such a response in hydrostatically extrudedpolypropylene was presented by Duckett et al. [22], who measured a tensile yield stressthat was about 8 times higher than the compressive one for an extrusion ratio of 5. Othermeasurements of the Bauschinger effect in polymers include tension and compressiontests on oriented samples [17-19, 23, 24], and simple shear tests on oriented poly(ethyleneterephthalate) at different loading angles [14, 15]. It is interesting to note that, so far, all ofthe experimental observations on the Bauschinger effect in polymers have been made onpolymers that received their molecular orientation through solid-state forming processes.Considering that the mechanical anisotropy of oriented polymers in terms of their loadingangle dependence is very similar for deformation-induced (solid-state processing) andflow-induced (melt-processing) molecular orientation [6, 7], one might expect that flow-induced orientation also gives rise to a Bauschinger effect. Until now, however, this hasnever been observed experimentally.

The goal of the present study is twofold. First, to establish that Bauschinger effects mayalso originate from flow-induced molecular orientation, the mechanical anisotropy ofinjection molded polyethylene (PE) plates is experimentally investigated by performinguniaxial tension and compression tests at various loading angles and strain rates. Second,a macroscopic constitutive model is developed that describes the influence of loadingangle on the yield kinetics of oriented PE, both in tension and compression. Previousmodeling efforts and the physics that underlie the model are elaborately discussed in theappropriate section. Finally, the performance of the model proposed in predicting thelong-term failure kinetics of injection molded PE in uniaxial tension at different loadingangles is evaluated.


All experiments in this study were performed on a grade of high-density polyethylene(PE) that is commercially known as Stamylan HD 8621 (Sabic Europe, Geleen, theNetherlands). This specific grade was selected because it was also used in an earlierstudy on the influence of processing-induced orientation on mechanical properties [20].


The number-averaged and weight-averaged molecular weights of this particular grade areequal to 7.0 and 210.0 kg/mol, respectively [20].

Square plates with dimensions of 70 x 70 x 1 mm were injection molded on an Arburg320S Allrounder 500-150 injection molding machine. The mold included a v-shapedrunner (5 mm thick) with a ramp, which causes the flow front of the polymer melt tobe uniform across the width of the plate. Figure 5.1a shows a top-view of the plate andrunner; the material is injected through the circular mark at the top of the runner, froman out-of-plane direction perpendicular to the plate. In the injection molding process, thepolymer was heated to a temperature of 250C before injecting it in the mold, which waskept at a temperature of 20C. A rather low injection flow rate was used, 10 cm3/s, to geta high degree of molecular orientation as a result of cooling during flow. The packingpressure was reduced stepwise from 600 to 200 bar in 8 s.

Two types of specimen were taken from these 1 mm thick plates. First, tensile bars witha gauge section of 54 x 5 x 1 mm were directly punched from the plates, at differentloading angles with respect to the main flow direction, ranging from 0 to 90. This isillustrated in Figure 5.1a, where the tensile bar cut at a loading angle of 0 is depicted inblack. Second, cubic compression samples of 1 x 1 x 1 mm were machined from the plates.Cubic samples were used instead of the regular cylindrical ones because the small size didnot allow for a turning lathe to be used to machine the samples. Moreover, as most of themolecular orientation in injection molded products is concentrated at the surface [20, 25],it was preferred to cut the samples in such a way that they span the entire thickness ofthe plate. The compression samples were made by first cutting a strip of 2 mm wide andthen carefully planing it to a smooth bar with a cross-sectional area of 1 mm2. At thispoint, the planed surfaces were marked black to be able to later identify the compressionsurfaces of the cubes. To cut cubes from this bar, it was first mounted in a custom-madejig and submerged in liquid nitrogen to prevent the formation of burrs. Next, the sampleswere cut from the center of the bar with a fine saw. Figure 5.1b illustrates how thesecompression samples were taken from the plate, although in reality the cubes are 2 timessmaller than depicted here, compared to the dimensions of the plate. Optical microscopypictures of two side walls of a single sample, which were used to determine the sampledimensions prior to the experiment, are shown in Figures 5.1c and 5.1d.

Both uniaxial tension and uniaxial compression experiments were performed on a ZwickZ010 tensile testing machine, equipped with a 1 kN load cell. In the tensile tests, constantnominal strain rates were applied, ranging from 10−4 to 10−2 s−1. In addition to theseconstant strain rate tests, a set of uniaxial tensile creep tests was done to measure thetime-to-failure for different constant loads. The compression tests were performed at

Experimental results 85

Figure 5.1: Schematic top-view of the injection molded plate and runner, illustrating how (a) thetensile bars and (b) the compression samples were taken from these plates. Sampleswith a loading angle of 0 are depicted in black. (c,d) Optical microscopy picturesof two side walls of a compression sample.

constant true strain rates, ranging from 10−4 to 10−2 s−1. In these tests, the sampleswere compressed between two parallel steel plates, the polished surfaces of which werelubricated with a PTFE spray (Griffon TF089). The finite stiffness of the test setupwas measured and corrected for in a real-time feedback loop to ensure accurate strainmeasurements. In all data presented in this study, true stresses were calculated byassuming that the polymer deforms isochorically.

5.3 Experimental results

In this section, only the results from the tension and compression tests at a constant strainrate of 10−3 s−1 are discussed, as these demonstrate, for the first time, the presenceof a pronounced Bauschinger effect in a flow-oriented polymer. The remainder of theexperimental data is presented in later sections, where it is used to characterize andvalidate the constitutive model proposed.

The influence of loading angle on the tensile response of injection molded PE is shown inFigure 5.2. As expected, the material has the highest yield stress when it is loaded parallel(0) to the main orientation direction. Increasing the loading angle towards 90 initiallyleads to a strong decrease in yield stress that is qualitatively very similar to responsesreported for solid-state oriented polymers like PE [13, 19] and others [6, 14-18]. Atloading angles above 60, the yield stress increases slightly again, which is typical fororiented PE [13, 19]. With the increase in loading angle, the failure mode also changes


from brittle to ductile. At loading angles below 20, the tensile bars fracture just after theonset of yielding, whereas at higher loading angles the samples show stable necking withextensive plastic deformation.

Figure 5.2: Engineering stress-strain curves for uniaxial tension of injection molded PE atvarious loading angles, as indicated in the graph. Crosses indicate fracture of thetensile bar, whereas arrows indicate that the sample did not yet fracture at the strainmaximum. Strain rate: 10−3 s−1.

Figure 5.3 shows the changes in the compressive response of injection molded PE as afunction of loading angle. Looking at the first yield point, marked by the diamonds, itis clear that the loading angle dependence is much less pronounced than in the case ofuniaxial tension, which is consistent with earlier observations on compression-orientedPE [19] and hot-drawn poly(vinyl chloride) [18]. With regard to the post-yield response,the loading angle appears to have a significant influence as well. At large loading angles,a distinct second yield point is visible, which is also observed in the uniaxial deformationof isotropic PE [26-28]. The curves at lower loading angles do not show this second yieldpoint and at 0 and 15 even a softening response is observed. However, the possibilitythat this is due to a geometric instability, e.g. a shear band that is initiated at the onsetof yielding, cannot be ruled out. One of the disadvantages of using cubic compressionsamples is that they lack the inherent suppression of such instabilities that cylindricalsamples have due to their shape.

The presence of a Bauschinger effect is characterized by an asymmetry between thetensile and compressive responses of an oriented polymer. Figure 5.4a, which comparesthe tensile and compressive responses of injection molded PE, clearly shows that thematerial exhibits a Bauschinger effect. When loaded parallel to the main orientationdirection, the tensile yield stress is 2.3 times higher than the compressive one, in theabsolute sense. This type of response has previously been observed for polymers oriented

Experimental results 87

Figure 5.3: True stress-strain curves for uniaxial compression of injection molded PE at variousloading angles, as indicated in the graph. Diamond-shaped markers indicate the firstyield points in the curves. Absolute true strain rate: 10−3 s−1.

Figure 5.4: Absolute true stress-strain curves of injection molded PE in tension (‘t’) andcompression (‘c’) at a loading angle of (a) 0 and (b) 90. (c) Loading-angledependence of the yield stress in tension and compression. Absolute strain rate:10−3 s−1.

through solid-state deformation [17-19, 22-24], but until now never for polymers thatreceived their molecular orientation during melt-processing. For the 90 loading angle,the absolute values of the yield stresses in tension and compression are identical, seeFigure 5.4b, indicating that the Bauschinger effect is loading-angle-dependent. Thisorientation dependence is demonstrated more clearly in Figure 5.4c, which shows tensileand compressive yield stresses across the full range of loading angles.

The experimental observations presented in this section form both the motivation anda guideline for the development of a constitutive model that captures the important


phenomena associated with yielding of oriented semi-crystalline polymers, which isdiscussed in the following sections.

5.4 Modeling anisotropic yielding of oriented polymers

Before elaborating on the constitutive equations that govern the model proposed, whichis the topic of the next section, some physical considerations are discussed along with areview of modeling concepts for oriented polymers that are currently available.

5.4.1 Yield function of Hill

The most well-known manifestation of anisotropy in oriented polymers is the widelyobserved variation of tensile yield stress as a function of loading angle. Arguably the mostestablished approach to describe the drop in tensile yield stress with increasing loadingangle is based on the anisotropic (orthotropic) yield function of Hill [29, 30], which canbe expressed as:

σ2y = σ2

h, (5.1)

where σy is the isotropic reference yield stress; the equivalent stress σh is defined as:

σ2h = F (σ22 − σ33)

2+G (σ33 − σ11)2+H (σ11 − σ22)

2+2Lσ223+2M σ2

13+2N σ212. (5.2)

The stress components σij (i, j = 1, 2, 3) are components of the Cauchy stress tensorwith respect to a material vector basis that coincides with the principal axes of orthotropy.Note that in the original Hill function, all terms are quadratic in stress, which impliesthat the model shows an identical response in tension and compression. Furthermore,the theory assumes that hydrostatic pressure does not affect yielding, which is why 6parameters suffice for the description of the anisotropy, rather than the 9 parametersthat are generally required, for instance, in orthotropic elasticity descriptions. These 6anisotropy parameters are a function of the yield stresses of the anisotropic material:

Modeling anisotropic yielding of oriented polymers 89

F =1

2

(1

R222

+1

R233

− 1

R211

); L =

3

2R223

;

G =1

2

(1

R211

+1

R233

− 1

R222

); M =

3

2R213

; (5.3)

H =1

2

(1

R211

+1

R222

− 1

R233

); N =

3

2R212

,

where the parameters Rij represent the ratio of the yield stress in the correspondingmaterial direction to the isotropic reference yield stress σy. Various studies havedemonstrated that the yield function of Hill provides an adequate description of theloading angle dependence of the tensile yield stress of oriented polymers at a singlestrain rate [6, 14-17, 31, 32]. However, very few studies have looked at the influenceof orientation on the strain rate dependence of yielding that is characteristic for polymers[6, 7, 33]. Van Erp et al. [6] developed a viscoplastic model, consisting of an associatedflow rule in combination with a Hill equivalent stress and Eyring rate dependence, thatsuccessfully captures the influences of initial draw ratio and loading angle on the tensileyield kinetics of hot-drawn polypropylene tapes. In addition, the model quantitativelydescribes the time-to-failure of these tapes upon application of a constant load. Althoughthis model may be successful in describing tensile yield kinetics, it cannot capture anasymmetry between tension and compression, such as shown in Figure 5.4. To resolvethis Bauschinger effect, other modeling approaches are needed.

5.4.2 The Bauschinger effect

To describe the Bauschinger effect, a model must account for differences between thetensile and compressive responses that arise as a result of molecular orientation in thepolymer. It has been experimentally established that the Bauschinger effect is stronglyrelated to a frozen-in internal stress in the polymer, which is of elastic origin since itmakes the polymer retract to its unoriented state at elevated temperatures [15, 34]. Thiselastic stress is typically attributed to the entropy-elastic response of the amorphousentanglement network. However, it was recently demonstrated that an elastic networkstress is not sufficient to capture the Bauschinger effect in oriented polymers, but that adeformation-induced evolution of the viscous flow stress is required as well [24, 35]. Theessence of the model proposed by Senden et al. [24, 35] is schematically illustrated inFigure 5.5, in which the solid line represents its post-yield stress response for a uniaxialtension-compression cycle. In the initially isotropic case, labeled ‘A’ in the figure,


the absolute yield stresses in tension and compression are identical. With increasingdeformation, the tensile yield stress increases, whereas the compressive yield stressremains unchanged, see labels ‘B’ and ‘C’, which is exactly the type of response observedin experiments. This model response is the combined result of the elastic and viscousstress contributions. The elastic network stress (dashed line) obviously loads and unloadsalong the same curve in this deformation cycle, leading to a kinematic hardening responsein the total stress. Furthermore, the viscous flow stress (dash-dotted line) displays adeformation-induced evolution, resulting in an isotropic hardening response. An equaldistribution of the total strain hardening between the elastic and viscous contributions isused, such that the total stress response in compression does not show any hardening.

Figure 5.5: Schematic post-yield stress response of the model developed by Senden et al. [24,35] in a tension-compression cycle. On the horizontal axis, λt represents the post-yield draw ratio, which equals 1 at the yield point. Lines represent the differentstress contributions; markers denote tensile and compressive yield points at threelevels of deformation.

The model illustrated in Figure 5.5 quantitatively captures the Bauschinger effect inoriented polycarbonate [35]. Additionally, it predicts a loading angle dependence as aresult of the frozen-in elastic network stress, but this is not sufficient to describe the strongloading angle dependence observed in oriented semi-crystalline polymers. The reason isthat two sources of anisotropy are present in oriented semi-crystalline polymers: a frozen-in elastic stress in the oriented amorphous phase and crystallographic texture. Therefore,based on this physical point of view, this model concept is combined with an anisotropicflow rule based on the yield function of Hill, as discussed in the previous paragraph.

Modeling anisotropic yielding of oriented polymers 91

5.4.3 A combined approach

For the combined approach, the aforementioned Hill-based anisotropic viscoplastic modelof Van Erp et al. [6] is taken as a starting point. Adding an elastic network contributionto the model is trivial and the key to incorporation of a deformation-induced evolutionof the viscous flow stress is already present in the original work, albeit not explicit. Theanisotropy in the model is governed by the Rij parameters (see Equation (5.3)) and ateach individual level of the initial orientation, these parameters were separately fitted.As expected, a systematic increase of the parameter values is observed as a functionof orientation. This is illustrated in Figure 5.6, where the values of R11, R22 and R12,obtained by Van Erp et al. [6] for hot-drawn polypropylene tapes, are plotted as a functionof draw ratio.

Figure 5.6: Deformation-induced evolution of the anisotropy parameters Rij , defined inEquation (5.3), fitted on data measured by Van Erp et al. on hot-drawnpolypropylene tapes. The initial draw ratio of the tapes is denoted with λ; linesare a guide to the eye.

Considering Figure 5.6, it appears that a deformation-induced evolution of the viscousflow stress is easily incorporated through the Rij parameters. However, in this casethe difficulty is not the modeling itself, but rather the subsequent characterization ofthe model parameters with experimental data, which is restricted by the availability ofdata at different levels of orientation, such as presented in Figure 5.6. To predict onlyanisotropic yielding and not evolution of yield stress, describing the evolution of the yieldparameters is not necessary. If this evolution is not taken into account, however, a tension-compression asymmetry is needed in the viscous flow stress, as explained in Figure 5.7.The effects of an orientation-induced evolution of the viscous flow stress are illustratedin Figure 5.7a, in which the solid lines schematically show, at an arbitrary amount oftensile pre-deformation (orientation), the tensile and compressive responses of a modelthat incorporates such an evolution. The grey lines in the figure represent the stress


evolutions starting from an isotropic material, compare with Figure 5.5. Starting in theorigin (0,0), the tensile curve shows a viscoelastic response until it reaches the yield point(square marker), after which a combined elastic and viscous strain hardening responseis observed. On the other hand, the compressive response shows a lower absolute yieldstress (diamond marker) and absence of strain hardening. The stress contribution fromthe elastic network (dashed line) has a non-zero value at λ = 1, which is the frozen-ininternal stress characteristic for oriented polymers. Of course, this internal network stressis balanced by the viscoelastic stress (dash-dotted line), which again coincides with thegrey evolution line at the onset of yielding.

Figure 5.7: Schematic model responses (black lines) in tension and compression at an arbitrarylevel of pre-orientation: (a) with and (b) without an orientation-induced evolutionof the viscous flow stress. Grey lines represent stress evolutions from the initiallyisotropic state, compare with Figure 5.5. The three different stress contributions areplotted; markers denote tensile and compressive yield points.

If one now discards the orientation-induced evolution of the viscous flow stress, whichis desirable for reasons mentioned earlier, the schematic picture shown in Figure 5.7b isobtained. The tensile and compressive yield stresses of the oriented polymer (square anddiamond markers, respectively) can still be captured, but this requires the incorporationof an inherent asymmetry between the tensile and compressive viscous flow stress. Thislatter conclusion is drawn directly from Figure 5.7b, which shows that |σv,t| > |σv,c|.Extensions of the Hill function that incorporate an asymmetry between tension andcompression are readily available. Perhaps the most promising of these is the yieldfunction of Caddell et al. [32], which is strikingly similar to a model for the brittle fractureof orthotropic materials that was proposed by Hoffman [36] a few years earlier. It isimportant to note that, when using the approach without an orientation-induced evolutionof the viscous flow stress, the strain hardening response following the yield point is notdescribed accurately anymore, compare the slopes of the total stress after the yield pointin Figures 5.7a and 5.7b. However, if one is primarily interested in describing yielding


of oriented polymers and the failure that is initiated by the onset of yielding, this is notnecessarily a problem. Therefore, the constitutive model developed in the present studyis based on this approach.

5.5 Constitutive modeling

This section deals with the detailed mathematical description of the constitutive modelproposed, which is composed of the components discussed in the previous section: anelasto-viscoplastic model that incorporates an anisotropic flow rule based on an extensionof the Hill function that accounts for an asymmetry in tension and compression combinedwith an elastic network stress.

5.5.1 Kinematics

Derivation of the model starts by defining an initial, isotropic configuration Ci and acurrent, deformed configuration Cc, which are linked through the deformation gradienttensor F . To distinguish between the elastic and plastic parts of the deformation, amultiplicative split of the deformation gradient is used [37]:

F = Fe · Fp. (5.4)

Herein, and in other variables that follow, subscripts e and p refer to the elastic andplastic parts, respectively. The multiplicative split of Equation (5.4) implies the existenceof a virtual, unstressed intermediate configuration C, but it does not uniquely define thisconfiguration since it does not specify the portion of the total rotation that is elastic, orplastic. It has been shown that the choice of the intermediate configuration does notinfluence model results [38]; here, it is assumed that plastic deformation does not involveany rotations:

Rp = I. (5.5)

The rotation tensors R, Re and Rp are defined through the polar decomposition of thecorresponding deformation gradients. Using Equation (5.4), the velocity gradient tensorL is written as:


L = F · F−1 = Fe · F−1e + Fe · Fp · F−1

p · F−1e , (5.6)

where the material time derivative of a variable is denoted by a dot above the cor-responding character and the inverse of a tensor is denoted by the superscript -1.From this equation, the plastic velocity gradient tensor associated with the intermediateconfiguration is defined:

Lp = Fp · F−1p = Dp + Ωp. (5.7)

Variables associated with the intermediate configuration C are marked with a circumflex.The plastic deformation rate tensor Dp and the plastic spin tensor Ωp are defined as thesymmetric and anti-symmetric parts of Lp, respectively. Last, it is assumed that plasticdeformation occurs isochorically, implying that the volume ratio J equals:

J = det(F ) = det(Fe). (5.8)

5.5.2 Kinematics of pre-deformation

The purpose of this study is to develop a constitutive model that describes yielding oforiented polymers and in the previous section it was discussed that an elastic networkstress is required for that. In an oriented polymer, this elastic network is pre-stretched,giving rise to a frozen-in internal stress, which indicates that pre-deformations mustbe accounted for in the constitutive model. The kinematic assumptions reflected inEquations (5.4), (5.5) and (5.8) also hold of the deformation gradient representing thepre-deformation F0:

F0 = F0e · F0p (5.9)

R0p = I (5.10)

J0 = det(F0e). (5.11)

Next, the elastic and plastic parts of the pre-deformation gradient defined as a part of thetotal deformation gradient, according to [39, 40]:


Fe = F1e · F0e (5.12)

Fp = F1p · F0p, (5.13)

where F1e and F1p are the elastic and plastic parts of the deformation from the pre-orientedstate to the current state.

5.5.3 Stress

The stress definition of the model is rather straightforward and starts by splitting theCauchy stress σ in a deviatoric and a hydrostatic part. The latter is simply described witha constant bulk modulus K:

σ = σh + σd = K (J − 1) I + σd. (5.14)

Superscripts h and d refer to the hydrostatic and deviatoric parts, respectively; the second-order unit tensor is denoted as I . Next, the deviatoric stress is additively split in a viscousdriving stress σs and an elastic hardening (network) stress σr [41, 42]:

σd = σs + σr. (5.15)

The driving stress component is modeled as a nonlinear, elasto-viscoplastic Maxwellelement, featuring a neo-Hookean spring that determines the stress:

σs =G

JBd

e . (5.16)

Herein, G denotes the shear modulus and Bde is the deviatoric part of the elastic, isochoric

left Cauchy-Green deformation tensor, defined as:

Bde = J−2/3Bd

e = J−2/3(Fe · F T

e

)d, (5.17)


where a superscript T denotes the transpose of a tensor. The elastic hardening stress isalso modeled with a neo-Hookean rubber-elastic model:

σr =Gr

JBd, (5.18)

where Gr represents the neo-Hookean hardening modulus and the calculation of Bd isanalogous to Equation (5.17).

5.5.4 Anisotropic viscoplastic flow

The next step is to define the anisotropic viscoplastic flow rule that determines the rate ofplastic deformation as a function of the stress state. An approach similar to the ones in [43]and [6] is used, separating the plastic deformation rate tensor in the (scalar) magnitude of(visco)plastic flow ξ and the direction of (visco)plastic flow N :

Dp = ξN . (5.19)

Note that the flow rule is evaluated in the intermediate configuration C. The direction ofplastic flow is given by an associated flow rule:

N =∂σ

∂σs

, (5.20)

where σs represents the driving stress tensor as defined in Equation (5.16), but pulled-back to the intermediate configuration C:

σs = F−1e · σs · F−T

e . (5.21)

The equivalent stress σ is based on the yield function of Hill [29, 30] and is essentiallythe same as that defined in Equation (5.2). However, a different definition is used here, interms of invariants of stress and orientation. In addition to that, an extension of the Hillfunction [32, 36] is used that is able to account for an asymmetry between the tensile andcompressive responses due to the addition of linear stress terms. Let n1, n2, n3 be anorthogonal vector basis that coincides with the principal axes of anisotropy in the material,


in the intermediate configuration. Then there are two independent orientation tensors thatdefine the state of anisotropy:

N1 = n1n1 (5.22)

N2 = n2n2. (5.23)

Using these two orientation tensors, the extended Hill equivalent stress can be written inan invariant formulation [44]:

σ =[α tr(σd

s · σds

)+ β tr2

(σd

s · N1

)+ γ tr

(σd

s · N1 · σds

)+ ζ tr2

(σd

s · N2

). . .

+χ tr(σd

s · N2 · σds

)+ ψ tr

(σd

s · N1

)tr(σd

s · N2

)]1/2. . .

+(κ1 − κ3) tr(σd

s · N1

)+ (κ2 − κ3) tr

(σd

s · N2

). (5.24)

The fit parameters α, β, γ, ζ , χ and ψ define the state of anisotropy analogous to theoriginal Hill function. Formally, these parameters are related to those defined in Equation(5.3) by:

α = L+M −N

β = F + 4G+H − 2M

γ = −2L+ 2N (5.25)

ζ = 4F +G+H − 2L

χ = −2M + 2N

ψ = 4F + 4G− 2H − 2L− 2M + 2N.

However, it is important to note that, due to the incorporation of the Hill function in thiselasto-viscoplastic framework, the direct connection between the anisotropy parametersRij (see Equation (5.3)) and the anisotropic yield stresses of the oriented polymer is lost.For this reason, the parameters α, β, γ, ζ , χ and ψ are simply used to define the state ofanisotropy, with α = 3

2and β = γ = ζ = χ = ψ = 0 in the case of an isotropic material.

Additionally, the parameters κ1, κ2 and κ3 govern the asymmetry between tension and


compression in the three principal anisotropy directions, with κ1 = κ2 = κ3 = 0 in theabsence of a Bauschinger effect.

So far, the magnitude of plastic flow ξ, as introduced in Equation (5.19), has not beendefined. If an equivalent plastic strain rate ˙εp is introduced such that it is plastic-workconjugated with the equivalent stress:

˙εp σ = σs : Dp, (5.26)

it turns out that this equivalent plastic strain rate equals the magnitude of plastic flow:

ξ = ˙εp. (5.27)

It is well-known that plastic flow in polymers is a stress-dependent process, whichstems from the stress-activated nature of the mobility of polymeric chains. Here,the phenomenological Eyring model [45, 46] is used, which has proven capable ofsuccessfully describing the strain rate dependence of yielding in polymers [47-51]:

˙εp(σ) = ε0 sinh

(σV ∗

kBT

). (5.28)

Herein, the rate constant ε0 and the activation volume V ∗ are fit parameters; kB and Tdenote Boltzmann’s constant and the absolute temperature, respectively.

5.5.5 Implementation

The constitutive model is implemented as a software routine that calculates the stressresponse for a certain deformation history given by F (or F0 and F1). More specifically,a numerical integration scheme based on Heun’s method is used to solve for the plasticright Cauchy-Green deformation tensor Cp:

Cpn+1 = Cpn + δt Cpn+1 , (5.29)

where the subscripts n and n + 1 refer to the current and the next increment in thecalculation; the incremental time step is denoted by δt. The material time derivative

Simulations 99

at increment n + 1, Cpn+1 , is calculated as the average of its value in increment n and afirst-order estimate for its value in increment n+ 1, Cpn∗:

Cpn+1 =1

2

(Cpn + Cpn∗

). (5.30)

The evolution equation for Cp that is required for these calculations, can be derived bymaking use of Equations (5.4) and (5.7):

Cp = F Tp · Fp (5.31)

Cp = 2F T · F−Te · Dp · F−1

e · F . (5.32)

5.6 Simulations

5.6.1 Model characterization

Before actually discussing the simulations of the yield kinetics of injection molded PE,it is important to provide insight in how the parameters of the constitutive model can bedetermined in a meaningful way. First, the elasticity parameters are simply fitted to theexperimental data. Then, without imposing any restrictions, it is assumed that the tensileyield kinetics measured at a loading angle of 90 represent the isotropic reference yieldkinetics, implying that the Eyring parameters ε0 and V ∗, see Equation (5.28), are uniquelydetermined. The experiments performed do not characterize the full anisotropic state ofthe material, since only the in-plane loading angle dependence is considered. Therefore,it is assumed that the out-of-plane yield resistance is equal to the bulk value, implyingthat the anisotropy parameters defined in Equation (5.25) are interrelated according to:

α =3

2− γ

2(5.33)

β = −2ζ (5.34)

χ = γ (5.35)

ψ = ζ + γ. (5.36)

The next step is to fit the value of ζ such that the slope of the predicted strain ratedependence of the yield stress for the 0 loading angle approximates the average of


the experimentally determined slopes for tension and compression at that loading angle.Inspired by the flow pattern in the mold during processing of the injection molded plates,the state of initial orientation in the plates is mimicked by assuming a (virtual) pre-deformation in the simulations:

F 0 =

λpre 0 0

0 1 0

0 0 1λpre

. (5.37)

The elastic and plastic parts of the pre-deformation gradient are calculated by solvingfor stress equilibrium in the (macroscopically stress-free) pre-deformed state. Two moreparameters can be eliminated from the fitting procedure in the present case. Due to theabsence of experimental data in the thickness direction of the plates, it is assumed thatκ3 = 0. Furthermore, the parameters Gr and λpre have a fully equivalent effect becausethe present study focuses on yielding, which is influenced by the frozen-in internal stress,rather than the elastic strain hardening response of the network. Therefore, either ofthe two parameters may be fixed at a particular value, in order to fit the other one to theexperimental data. Having characterized the Bauschinger effect for the 0 and 90 loadingdirection, the last step is to fit the value of γ to the yield stress data at intermediate loadingangles. The parameter set that is used in all simulations discussed, is presented in Table5.1.

Table 5.1: Model parameters for injection molded PE.

V ∗ [nm3] ε0 [s−1] G [MPa] K [MPa] Gr [MPa] λpre [-]2.1 9 · 10−10 200 3000 1 2.8

α [-] β [-] γ [-] ζ [-] χ [-] ψ [-] κ1 [-] κ2 [-] κ3 [-]-1.17 -1.12 5.34 0.56 5.34 5.90 0.24 0.12 0

5.6.2 Boundary conditions

The implementation of the constitutive model, discussed in the previous section, cal-culates the response of the model for a single material point, which has importantconsequences for the boundary conditions that are applied. Since all experiments inthis study are uniaxial deformation tests, the most obvious choice would be to applythe boundary conditions accordingly. In that case, however, the anisotropy in the

Simulations 101

material induces shear deformations that, in a single material point with uniaxial boundaryconditions, lead to a deformation response in both tension and compression that is notobserved in the experiments. Therefore, the deformation conditions in the experimentsare mimicked by choosing boundary conditions that suppress shear deformations. For theconstant strain rate tests, these are expressed as:

F 1 =

exp(εt) 0 x40 x1 x3x4 x3 x2

; σ =

σ1 σ2 0

σ2 0 0

0 0 0

, (5.38)

where σ1, σ2, x1, x2, x3, x4 are the unknown variables.

5.6.3 Results for yielding at constant strain rate

Figure 5.8 presents a comparison between experimental data and simulation results fortensile and compressive yielding of injection molded PE at various strain rates. Withinthe range of strain rates considered, the yield kinetics are accurately described by themodel, both for a loading angle of 0 and 90. Only for compression at a loading angle of0, the slope of the strain rate dependence is slightly underestimated.

Figure 5.8: Yield kinetics of injection molded PE in tension and compression at a loading angleof (a) 0 and (b) 90. Markers represent experimental data and solid lines aresimulation results.

Additionally, the performance of the proposed model is evaluated with regard to theloading angle dependence that the yield stress of injection molded PE displays. Figure5.9 compares the experimentally obtained yield stresses with those from the modelsimulations, showing a reasonable agreement for both tension and compression.


Figure 5.9: Loading-angle dependence of the tensile and compressive yield stress of injectionmolded PE at an absolute strain rate of 10−3 s−1. Markers represent experimentaldata and solid lines are simulation results.

5.6.4 Results for yielding at a constant load

In structural applications of polymers, it is more common that failure occurs as a resultof a load that is applied for a prolonged period of time, rather than the applicationof a certain deformation rate. Therefore, it is relevant to study the failure kinetics asa result of yielding upon the application of a constant load, both experimentally andnumerically. Due to practical limitations, these experiments were only performed inuniaxial tension. The results are presented in Figure 5.10, in which the period of timebetween load application and the onset of yielding, referred to as time-to-failure, is plottedfor a range of applied loads. In the experiments at 0 loading angle, the samples fractureddirectly after the onset of yielding, while the samples loaded in the 90 direction displayedstable necking. Of course, no necking was observed in the simulations, since calculationsare only made in a single material point. However, the simulations do show geometricsoftening, which results from the continuous decrease in cross-sectional area during a testand the corresponding increase in the true stress experienced by the bar. The solid line inFigure 5.10 shows that the experimental data is captured quite well using the parameterset that was characterized solely on the experimental data measured at constant strain rate.

5.7 Conclusions

The first part of this work discusses an extensive experimental characterization of thetensile and compressive deformation response of injection molded polyethylene. Inparticular, the influences of strain rate and loading angle with respect to the mainorientation direction are investigated. The prime conclusion that is drawn from these

Conclusions 103

Figure 5.10: Tensile failure kinetics of injection molded PE subjected to constant uniaxial loadsat loading angles of 0 and 90. Markers represent experimental data and solidlines are simulation results.

experiments is that a substantial Bauschinger effect exists, an asymmetry between thetensile and compressive yield stress that is not observed in isotropic polyethylene. Thisconclusion is particularly interesting because, until now, such Bauschinger effects haveonly been observed in polymers that possess molecular orientation due to solid-statedeformation.

In the second part, an elasto-viscoplastic constitutive model is proposed that capturesthe important phenomena observed in the experiments. Two sources of anisotropy arepresent in this model: a frozen-in elastic network stress due to the initial orientation in thematerial, and an anisotropic viscoplastic flow rule that is based on the yield functionof Hill, extended to account for an asymmetry between its tensile and compressiveresponse. It is demonstrated that the model proposed accurately captures the tensile andcompressive yield kinetics for the 0 and 90 loading direction; at intermediate loadingangles, a reasonable description is obtained as well. Additionally, simulations capture theexperimentally observed failure kinetics of injection molded polyethylene subjected to aconstant tensile load at loading angles of 0 and 90.

This study describes the effect of flow-induced orientation on the mechanical responseand failure of semi-crystalline polymers using a phenomenological, macroscopic con-stitutive model. Due to the limited availability of experimental data, a number ofeducated assumptions are made, which provides opportunities for future research. Forinstance, the model does not yet describe the evolution of the anisotropic yield stresseswith deformation, although a suggestion was made how this could be incorporated. Amore challenging extension of the model would be the incorporation of rotations ofthe principal axes of anisotropy that originate from plastic deformation. Regarding the


characterization of the model, it is interesting to study possible relations between themacroscopic modeling parameters and microstructural parameters, such as crystallinity,lamellar thickness and orientation distributions. Ultimately, these parameters can, in turn,be related to processing conditions in terms of temperature, pressure and flow history.

References

[1] B.S. Thakkar, L.J. Broutman and S. Kalpakjian. Impact strength of polymers 2: The effect of coldworking and residual stress in polycarbonates. Polymer Engineering and Science. 1980, 20, 756-762.

[2] Z. Bartczak, J. Morawiec and A. Galeski. Deformation of high-density polyethylene produced byrolling with side constraints. II. Mechanical properties of oriented bars. Journal of Applied PolymerScience. 2002, 86, 1405-1412.

[3] Z. Bartczak, J. Morawiec and A. Galeski. Structure and properties of isotactic polypropylene orientedby rolling with side constraints. Journal of Applied Polymer Science. 2002, 86, 1413-1425.

[4] J. Mohanraj, N. Chapleau, A. Ajji, R.A. Duckett and I.M. Ward. Fracture behavior of die-drawntoughened polypropylenes. Journal of Applied Polymer Science. 2003, 88, 1336-1345.

[5] D.H. Ender and R.D. Andrews. Cold drawing of glassy polystyrene under dead load. Journal ofApplied Physics. 1965, 36, 3057-3062.

[6] T.B. van Erp, C.T. Reynolds, T. Peijs, J.A.W. van Dommelen and L.E. Govaert. Prediction ofyield and long-term failure of oriented polypropylene: Kinetics and anisotropy. Journal of PolymerScience: Part B: Polymer Physics. 2009, 47, 2026-2035.

[7] T.B. van Erp, L.E. Govaert and G.W.M. Peters. Mechanical performance of injection-moldedpoly(propylene): Characterization and Modeling. Macromolecular Materials and Engineering. 2012,DOI, 10.1002/mame.201200116.

[8] J.M. Andrews and I.M. Ward. The cold-drawing of high density polyethylene. Journal of MaterialsScience. 1970, 5, 411-417.

[9] A.G. Gibson, I.M. Ward, B.N. Cole and B. Parsons. Hydrostatic extrusion of linear polyethylene.Journal of Materials Science - Letters. 1974, 9, 1193-1196.

[10] K. Nakayama and H. Kanetsuna. Hydrostatic extrusion of solid polymers. Part 5: Structure andmolecular orientation of extruded polyethylene. Journal of Materials Science. 1975, 10, 1105-1118.

[11] Z. Bartczak. Deformation of high-density polyethylene produced by rolling with side constraints. I.Orientation behavior. Journal of Applied Polymer Science. 2002, 86, 1396-1404.

[12] A. Keller and M.J. Machin. Oriented crystallization in polymers. Journal of Macromolecular Science:Part B: Physics. 1967, 1, 41-91.

[13] A. Keller and J.G. Rider. On the tensile behaviour of oriented polyethylene. Journal of MaterialsScience. 1966, 1, 389-398.


[15] C. Bridle, A. Buckley and J. Scanlan. Mechanical anisotropy of oriented polymers. Journal ofMaterials Science. 1968, 3, 622-628.

[16] J.G. Rider and E. Hargreaves. Yielding of oriented poly(vinyl chloride). Journal of Polymer Science:Part A-2: Polymer Physics. 1969, 7, 829-844.

References 105

[17] D. Shinozaki and G.W. Groves. The plastic deformation of oriented polypropylene: Tensile andcompressive yield criteria. Journal of Materials Science. 1973, 8, 71-78.


[19] S.G. Burnay and G.W. Groves. Variation of yield stress with orientation in oriented polyethylene.British Polymer Journal. 1978, 10, 30-34.

[20] B.A.G. Schrauwen, L.C.A. van Breemen, A.B. Spoelstra, L.E. Govaert, G.W.M. Peters andH.E.H. Meijer. Structure, deformation, and failure of flow-oriented semicrystalline polymers.Macromolecules. 2004, 37, 8618-8633.




[24] D.J.A. Senden, J.A.W. van Dommelen and L.E. Govaert. Strain hardening and its relation toBauschinger effects in oriented polymers. Journal of Polymer Science: Part B: Polymer Physics.2010, 48, 1483-1494.

[25] M.R. Kantz, H.D. Newman and F.H. Stigale. The skin-core morphology and structure-propertyrelationships in injection-molded polypropylene. Journal of Applied Polymer Science. 1972, 16,1249-1260.

[26] R. Popli and L. Mandelkern. Influence of structural and morphological factors on the mechanicalproperties of the polyethylenes. Journal of Polymer Science: Part B: Polymer Physics. 1987, 25,441-483.

[27] R. Seguela and F. Rietsch. Double yield point in polyethylene under tensile loading. Journal ofMaterials Science - Letters. 1990, 9, 46-47.

[28] N.W. Brooks, R.A. Duckett and I.M. Ward. Investigation into double yield points in polyethylene.Polymer. 1992, 33, 1872-1880.

[29] R. Hill. A theory of the yielding and plastic flow of anisotropic metals. Proceedings of the RoyalSociety of London, Series A. 1948, 193, 281-297.

[30] R. Hill. In: The Mathematical Theory of Plasticity. Oxford University Press, 1950, .[31] F.F. Rawson and J.G. Rider. Effects of internal stress on the yielding of oriented poly(vinyl chloride).

Journal of Polymer Science: Part C: Polymer Symposium. 1971, 33, 87-99.[32] R.M. Caddell, R.S. Raghava and A.G. Atkins. A yield criterion for anisotropic and pressure

dependent solids such as oriented polymers. Journal of Materials Science. 1973, 8, 1641-1646.[33] N.R. Karttunen and A.J. Lesser. Yield modeling of an aliphatic polyketone terpolymer in multiaxial

stress states. Polymer Engineering and Science. 2000, 40, 2317-2323.[34] P.A. Botto, R.A. Duckett and I.M. Ward. The yield and thermoelastic properties of oriented

poly(methyl methacrylate). Polymer. 1987, 28, 257-262.[35] D.J.A. Senden, S. Krop, J.A.W. van Dommelen and L.E. Govaert. Rate- and temperature-dependent

strain hardening of polycarbonate. Journal of Polymer Science: Part B: Polymer Physics. 2012, 50,1680-1693.

[36] O. Hoffman. The brittle strength of orthotropic materials. Journal of Composite Materials. 1967, 1,200-206.

[37] E.H. Lee. Elastic-plastic deformation at finite strains. Journal of Applied Mechanics. 1969, 36, 1-6.


[38] M.C. Boyce, G.G. Weber and D.M. Parks. On the kinematics of finite strain plasticity. Journal of theMechanics and Physics of Solids. 1989, 37, 647-665.

[39] M.C. Boyce, D.M. Parks and A.S. Argon. Plastic flow in oriented glassy polymers. InternationalJournal of Plasticity. 1989, 5, 593-615.

[40] E.M. Arruda, M.C. Boyce and H. Quintus-Bosz. Effects of initial anisotropy on the finite straindeformation behavior of glassy polymers. International Journal of Plasticity. 1993, 9, 783-811.



[43] J.J. Pereda, N. Aravas and J.L. Bassani. Finite deformations of anisotropic polymers. Mechanics ofMaterials. 1993, 15, 3-20.

[44] Y.F. Dafalias and M.M. Rashid. The effect of plastic spin on anisotropic material behavior.International Journal of Plasticity. 1989, 5, 227-246.


[46] A.S. Krausz and H. Eyring. Deformation Kinetics. John Wiley & Sons, Inc., 1975.[47] R.E. Robertson. On the cold-drawing of plastics. Journal of Applied Polymer Science. 1963, 7, 443-

450.[48] J.A. Roetling. Yield stress behaviour of polymethylmethacrylate. Polymer. 1965, 6, 311-317.[49] J.A. Roetling. Yield stress behaviour of isotactic polypropylene. Polymer. 1966, 7, 303-306.[50] C. Bauwens-Crowet, J.-C. Bauwens and G. Homes. Tensile yield-stress behavior of glassy polymers.

Journal of Polymer Science: Part A-2: Polymer Physics. 1969, 7, 735-742.[51] C. G’Sell and J.J. Jonas. Yield and transient effects during the plastic deformation of solid polymers.

Journal of Materials Science. 1981, 16, 1956-1974.

108 Conclusions and recommendations

6.1 Conclusions

This thesis primarily focuses on the development of modeling concepts that capture theeffects of molecular orientation on the deformation kinetics of solid polymers; this sectionsummarizes the main conclusions point by point.

1. Traditional constitutive models for amorphous polymers are unable to capture theBauschinger effect, i.e. the increasing asymmetry between the tensile and compressiveyield stress with increasing (tensile) orientation, that is observed in these materials. Theorigin of this deficiency is found in the purely elastic strain hardening description, whichleads to a macroscopically kinematic hardening response.

2. The Bauschinger effect in amorphous polymers is captured correctly if a part of thestrain hardening is attributed to a viscous contribution. This is achieved by introducinga deformation-dependent flow stress. A viscous contribution to strain hardening isconsistent with other experimental observations, such as the strain rate dependence and(negative) temperature dependence of the strain hardening modulus. Experimentally, theelastic and viscous contributions to strain hardening are separable through an evaluationof the Bauschinger effect, for instance by measuring the tensile and compressiveresponses of an oriented amorphous polymer.

3. In polycarbonate, this deformation-dependent flow stress clearly manifests itself as adeformation dependence of both the Eyring activation energy and the rate constant; withincreasing deformation (orientation), the room temperature yield kinetics shift from the αto the (α + β)-regime. In contrast, isotactic polypropylene displays a distinct change inEyring activation volume with increasing orientation.

4. Incorporation of the concept of partly viscous strain hardening in the Eindhoven GlassyPolymer model, and characterization using a set of uniaxial compression tests, leads to anaccurate description of the mechanical response of initially isotropic polycarbonate acrossthe full range of strains, strain rates and temperatures considered. Note that this includesthe strain rate and temperature dependence of strain hardening. Additionally, the modelproposed quantitatively captures the Bauschinger effect in oriented polycarbonate.

5. A drawback of the model is that it features additional parameters that need to bedetermined, requiring an extra set of experiments to uniquely extract the parametervalues. This added complexity is unnecessary for the evaluation of simple, monotonic

Conclusions 109

deformation of isotropic amorphous polymers in a limited range of strain rates and/ortemperatures.

6. To capture the full mechanical response of an amorphous polymer across a wide rangeof temperatures it is essential to recognize the influence of the material’s thermodynamicstate on the temperature dependence of yield. Although an aging-dependent activationenergy is expected from a physical point of view, this thesis shows, for the first time,evidence from macroscopic deformation experiments that the Eyring activation energyindeed changes due to physical aging.

7. In the analysis that led to the previous conclusion, use has been made of the notionthat the application of plastic deformation to an aged amorphous polymer results in truemechanical rejuvenation, i.e. the material de-ages in a reversible manner. This issue haslong been controversial, but this thesis convincingly shows that, from a mechanical pointof view, the mechanical pre-conditioning of polycarbonate only changes the age of thematerial, without inducing any permanent structural changes.

8. Frozen-in orientation in the amorphous phase of oriented semi-crystalline polymers leadsto a pronounced Bauschinger effect in these materials and this thesis shows, for the firsttime, that this phenomenon is also present when the molecular orientation originatesfrom (practical) melt-processing, rather than from (academic) solid-state deformation.Moreover, a strong anisotropy is visible in this Bauschinger effect: parallel to the mainorientation direction, the asymmetry between the yield stress in tension and compressionis large, but perpendicular to that it is completely absent.

9. In the deformation response of oriented semi-crystalline polymers, there are two sourcesof anisotropy. As mentioned, the frozen-in orientation in the amorphous phase resultsin a pronounced Bauschinger effect, at the same time introducing a relatively weakloading direction dependence in the mechanical response. Additionally, the preferredorientation of the crystalline lamellae gives rise to a strong loading direction dependencein the mechanical response, but does not enhance the differences between tension andcompression.

10. The constitutive model proposed, which incorporates these two sources of anisotropythrough a frozen-in elastic network stress and an intrinsically anisotropic viscoplasticflow rule, accurately describes the tensile and compressive yield kinetics of injectionmolded polyethylene, both parallel and perpendicular to the main orientation direction.


Additionally, the model captures the failure kinetics of injection molded polyethylenesubjected to a constant tensile load in these loading directions.

6.2 Recommendations

The modeling concepts developed in this study provide a solid basis for future effortsto model failure of semi-crystalline polymers. There are many issues to be resolvedand a number of subjects are worth studying; this section highlights the most prominentchallenges.

6.2.1 More detailed constitutive modeling

Regarding the macroscopic constitutive model developed in Chapter 5 to describeyielding of oriented semi-crystalline polymers, a number of refinements are worthconsidering, depending on the desired application of the model.

1. The anisotropic viscoplastic flow rule used in the model, which is essentially an associatedflow rule with a Hill equivalent stress, is successful in describing the relation between thestress state and the uniaxial strain rate as a function of loading angle for different semi-crystalline polymers. However, it has never been verified whether strain rates predictedby the model for the transverse directions are accurate. This issue is of great importancefor the application of the model in more complex loading geometries. A first step in thisdirection could be to monitor the local strain fields in a tensile bar during deformationtests, for instance using optical methods, such as digital image correlation.

2. Recently, a new viscoplastic flow rule has been developed for transversely isotropicmaterials [1], which is an interesting alternative for the Hill-based associated flow ruleused in Chapter 5. Both models give a similar description of the loading-angle-dependentyield kinetics of oriented semi-crystalline polymers, assuming that elastic deformationsremain small [1, 2]. Nevertheless, this new flow rule has the marked advantage that it isderived from a statistical-mechanics-based coarse graining analysis and, therefore, has amuch firmer physical basis than the purely phenomenological associated flow rule. It isinteresting to study the similarities and differences between both models in more detail,and to include the new flow rule in the experimental verification suggested in the previousrecommendation.

Recommendations 111

3. In its current form, the model describes ductile failure kinetics for a single state ofmolecular orientation (anisotropy). A natural extension of the model would be toincorporate an orientation dependence, such that the viscous strain hardening componentevolves with deformation, see Figure 5.6. The main challenge here is to characterizethis evolution function, which requires either a series of deformation experiments onsamples with different, well-defined levels of orientation, or a parameter study withan aptly equipped micromechanical model. Note that this extension only allows aunified description of yield for various initial orientations and not the evolution oforientation/anisotropy during a simulation with large plastic deformations.

4. If sufficiently large plastic deformations are applied to an initially anisotropic polymer,the state of orientation changes and the anisotropy directions rotate with respect to theirinitial configuration; this is currently not accounted for in the model. To correctly describethe intrinsic deformation response of oriented semi-crystalline polymers, rather than justthe yield and failure kinetics, such an evolution of the anisotropy directions needs tobe incorporated. Also with this model extension, a micromechanical model can provideimportant input for its development and characterization. Note that this is a far morecomplex addition to the model than the previous one and that it is not required if one isonly interested in the influence of initial orientation.

6.2.2 Processing ⇒ structure ⇒ properties

The PhD work that has resulted in this thesis was part of a larger research project, ofwhich the central objective was to predict catastrophic failure of semi-crystalline polymerproducts. In this research project, the approach adopted spans the entire life cycle of apolymer product, from manufacturing to final use. This approach was chosen becauseof the intimate relation between processing history, microstructure and mechanicalproperties of a product. Presently, the key elements required to achieve the objectiveare available. For instance, models have been developed that predict morphologydevelopment during flow, including crystallinity, crystal structure and orientation [3-5]. Additionally, it has been shown that, based on such information, micromechanicalmodeling quantitatively predicts macroscopic deformation responses for isotropic semi-crystalline polymers, including long-term creep failure of tensile bars [6]. The currentthesis addresses the aspect of macroscopic mechanical properties. Especially for practicalapplications, involving complex product geometries, the macroscopic constitutive modeldeveloped in this thesis is essential; the microscopic modeling approach is too complexand computationally too expensive to be used in an engineering environment.


5. Although the key elements are available, the entire processing ⇒ structure ⇒ propertiesproblem has not yet been tackled as a whole. Of course, in all these subareas there aremany issues that require attention; regarding the macroscopic constitutive modeling part,these issues were discussed in the previous paragraph. Nevertheless, perhaps the mostinteresting next challenge is not to focus on these issues per subarea, but to investigatehow well the different areas can already be connected. Regarding the macroscopicmodeling, the first step in this direction is to establish relations between macroscopicmodeling parameters and microstructural parameters, such as crystallinity, lamellarthickness and orientation distribution. In principle, it is possible to study such relationsexperimentally, but it is difficult to prepare samples with well-defined morphologies thatare suitable for macroscopic mechanical testing. A more promising approach is to use awell-characterized micromechanical model that incorporates the influence of morphology,making it quite straightforward to systematically study relations between macroscopicmodel parameters and micromechanical parameters. Ultimately, these microstructuralparameters can, in turn, be related to processing history in terms of temperature, pressureand flow history, given the details of the molecular weight distribution.

References

[1] M. Hutter and T.A. Tervoort. Statistical mechanics aspects of anisotropic elasto-viscoplasticdeformation. Submitted for publication.

[2] T.B. van Erp, C.T. Reynolds, T. Peijs, J.A.W. van Dommelen and L.E. Govaert. Prediction of yieldand long-term failure of oriented polypropylene: Kinetics and anisotropy. Journal of Polymer Science:Part B: Polymer Physics. 2009, 47, 2026-2035.

[3] R.J.A. Steenbakkers. Precursors and nuclei, the early stages of flow-induced crystallization. PhDThesis. 2009. Eindhoven University of Technology (the Netherlands).

[4] F.J.M.F. Custodio. Structure development and properties in advanced injection molding processes.PhD Thesis. 2009. Eindhoven University of Technology (the Netherlands).

[5] T.B. van Erp. Structure development and mechanical performance of polypropylene. PhD Thesis. 2012.Eindhoven University of Technology (the Netherlands).

[6] A. Sedighiamiri. Micromechanical modeling of the deformation kinetics of semicrystalline polymers.PhD Thesis. 2012. Eindhoven University of Technology (the Netherlands).

Mechanische eigenschappen van polymeren worden in belangrijke mate bepaald door deonderliggende microstructuur. Een voorbeeld hiervan is door stroming tijdens verwerkinggeınduceerde moleculaire orientatie; in het geval van semi-kristallijne polymeren kandit zelfs leiden tot een factor 500 verschil in levensduur binnen een enkel product.Ook kan een niet-willekeurige distributie van de moleculaire orientatie in een productresulteren in enorme verschillen in mechanische eigenschappen gemeten in trek of indruk. Het is daarom belangrijk het verband tussen moleculaire orientatie en mechanischeeigenschappen te begrijpen en te kwantificeren. Dit proefschrift richt zich op hetconstrueren van constitutieve modellen die de invloed van moleculaire orientatie op dedeformatiekinetiek van polymeren in vaste toestand kunnen beschrijven.

Bij amorfe polymeren wordt orientatie van het netwerk dat de polymeerketens vormenbeschreven met behulp van entropische, rubberelastische modellen. Deze modellenleveren weliswaar een redelijke beschrijving van de daadwerkelijke rekversteviging, maarer is nog een aantal fundamentele problemen op te lossen. Zo laten experimentenbijvoorbeeld zien dat rekversteviging een reksnelheidsafhankelijk proces is, met eennegatieve temperatuursafhankelijkheid; beide effecten zijn strijdig met een entropischebeschrijving. Ook de significante asymmetrie tussen vloeispanningen gemeten in trek-en drukproeven aan georienteerde amorfe polymeren wordt niet beschreven. Onlangs isgeopperd dat rekversteviging voornamelijk een intermoleculaire oorsprong heeft, hetgeenimpliceert dat rekversteviging macroscopisch gezien een gedeeltelijk viskeus proces is.De fysische verklaring is dat deformatie zorgt voor lokaal anisotrope ketenconformaties,en daardoor voor een verhoging van energiebarrieres en segmentele relaxatietijden.Bovengenoemde problemen kunnen worden verholpen door rekversteviging te beschrij-ven middels een deels elastische en deels viskeuze spanningsbijdrage. De viskeuzecomponent wordt gemodelleerd middels een Eyring vergelijking, waarvan de parametersevolueren als functie van de deformatie. Dit concept is geımplementeerd in hetzogenaamde Eindhoven Glassy Polymer model en vervolgens gekarakteriseerd via eenuitgebreide set drukproeven; het bleek mogelijk met deze aanpak de rekverstevigingvan polycarbonaat (als modelsysteem) nauwkeurig te beschrijven voor uiteenlopendereksnelheden en temperaturen. Daarnaast kan de mechanische respons van dit polymeerook onder cyclische belastingscondities kwantitatief worden beschreven. Dit is een extrabevestiging van de voorspellende kracht van het ontwikkelde model, aangezien trek- endrukbelastingen van georienteerd polycarbonaat juist bij cyclische belastingscondities eenbelangrijke rol spelen.

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114 Samenvatting

In semi-kristallijne polymeren is, afgezien van ingevroren orientatie in de amorfegebieden, ook de aanwezigheid van intrinsiek anisotrope kristallen een mogelijke bronvan aanzienlijke anisotropie. Macroscopisch gezien is de invloed van orientatie opde deformatie van semi-kristallijne polymeren dus tweeledig. Aan de ene kant leidtde ingevroren orientatie in het amorfe netwerk tot een sterke asymmetrie tussen devloeispanning in trek en in druk, terwijl dit effect slechts een lichte anisotropie inde mechanische respons oplevert. Aan de andere kant zorgt een niet-willekeurigeorientatie van de kristallen voor een sterke anisotropie in de mechanische respons,terwijl dit de asymmetrie tussen trek en druk niet vergroot. Deze aspecten van dedeformatie van georienteerde semi-kristallijne polymeren zijn experimenteel onderzochtdoor treken drukproeven te doen aan gespuitgiet polyetheen, belast onder verschillendehoeken ten opzichte van de spuitgietrichting. Deze proeven laten, voor het eerst,zien dat een aanzienlijke asymmetrie tussen de vloeispanning in trek en in drukook bestaat in polymeren die georienteerd zijn in de smelt. Door de aanpak vooramorfe polymeren te combineren met een anisotroop viscoplastisch model, wordenbeide eerder genoemde fysische bronnen van anisotropie meegenomen in het nieuwontwikkelde model. Er resulteert een goede beschrijving van de vloeikinetiek in treken druk van gespuitgiet polyetheen, zowel parallel als loodrecht op de belangrijksteorientatierichting. Daarnaast beschrijft het model ook kwantitatief de faalkinetiekvan georienteerd polyetheen, gemeten bij een constante trekspanning in dezelfde tweebelastingsrichtingen.

De modellen beschreven in dit proefschrift vormen een goede basis voor de verdereontwikkeling van constitutieve relaties die het falen van semi-kristallijne polymerenpogen te beschrijven en te kwantificeren. Een van belangrijkste onderwerpen voorverder onderzoek vloeit voort uit het feit dat de tot nu toe ontwikkelde modelleringenfenomenologisch van aard zijn. Daardoor hebben de modelparameters slechts betekenisop de macroscopische schaal en is er geen relatie met details van de onderliggendede morfologie. De volgende grote uitdaging is daarom het vinden van relaties tus-sen macroscopische en microstructurele parameters, zoals kristalliniteit, lameldikte enorientatiedistributie. Het ultieme doel hierbij is om, gegeven de molgewichtsverdelingvan het gekozen polymeer, deze microstructurele parameters op hun beurt weer tevoorspellen op basis van de specifieke verwerkingscondities, zoals de druk-, stromings-en temperatuursgeschiedenis tijdens spuitgieten.

Bijna vier jaar geleden ben ik de promotie-uitdaging aangegaan. Dat ik het in die tijdontzettend naar mijn zin gehad heb en nu een compleet proefschrift aflever, heb ik vooreen groot deel te danken aan mijn drie (co)promotoren. Han, ondanks dat jij in debegeleiding meestal niet op de voorgrond trad, was jouw memorabele, recht voor zijnrape feedback altijd erg verhelderend. Het was een onbeschrijflijk leuke, soms ronduitwonderlijke ervaring om een aantal jaar te mogen werken in jouw vakgroep, waar zo’ngoede sfeer hangt en zoveel mogelijk is. Hans, ik vond het erg fijn om met jou samente werken; jouw inzicht hielp mij om de lastigste problemen te kraken. Ook heb ik veelgehad aan jouw kritische en gedetailleerde feedback, waar ik altijd op kon rekenen. Leon,niet alleen ben jij degene die mij warm heeft gemaakt om te gaan promoveren, ook tijdensmijn promotie werkte jouw uitbundige enthousiasme voor de polymeermechanica vaakaanstekelijk (niemand is zo ‘blij met kunststof’ als jij). Los van het feit dat ik enorm veelvan jou geleerd heb, was het ook ontzettend gezellig.

Het kantoor waar ik het grootste deel van mijn promotietijd probeerde te werken, 4.22,zat precies tegenover de koffiekamer. Ik zeg probeerde, aangezien niet zelden dekoffiekamer een oase van rust leek vergeleken met de uitbundige zoete inval bij ons.Ondanks dat ik mij voor het schrijven van mijn proefschrift genoodzaakt zag om mijterug te trekken in rustigere oorden, ben ik oprecht dankbaar voor de onuitputtelijkebron van hilarische anekdotes en mooie herinneringen aan verhitte discussies, ontspannengesprekken, fruitmomenten, scherpe woordgrappen en andere humor. Ook terugdenkendaan de talloze borrels, barbecues, congressen en andere tripjes, kom ik tot dezelfdeconclusie: ik heb echt een schitterende groep collega’s om mij heen gehad.

Een bijzonder woord van dank is voor mijn familie en goede vrienden, de mensen die mijhet meest dierbaar zijn. Door hen voelde ik mij gesteund bij mijn promotie, maar vooralook bij andere, nog belangrijkere dingen in het leven. Een speciale plek hierin hebbennatuurlijk mijn ouders. Pap en mam, bedankt voor alle kansen die ik de afgelopen 29jaar van jullie gekregen heb en voor de support bij het verwezenlijken ervan. De laatstewoorden kunnen voor niemand anders zijn dan voor mijn fantastische vrouw. LieveStefanie, samen met jou de toekomst tegemoet... wat een prachtig avontuur!

DirkDecember 2012

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Dirk Senden was born on the 9th of February 1984 in Goirle, the Netherlands. In2002, he graduated from his secondary education at the Eckart College in Eindhoven.After spending a year working and traveling to broaden his horizon, he enrolled as astudent in the department of Mechanical Engineering at the Eindhoven University ofTechnology, where he received his Bachelor’s degree in 2006. During his Master inthe Polymer Technology group, chaired by prof.dr.ir. H.E.H. Meijer, he performed aninternship at the ETH Zurich, Switzerland, in the Polymer Technology group of prof.dr.P. Smith. In the beginning of 2009, he received his Master’s degree on the thesisentitled ‘Towards a macroscopic model for the finite-strain mechanical response of semi-crystalline polymers’, under the supervision of dr.ir. L.E. Govaert and dr.ir. J.A.W. vanDommelen.

In April 2009, he took a position in the Polymer Technology group as a PhD student,which has resulted in the present thesis. During his PhD he successfully completed thepostgraduate course Register Polymer Science of the National Dutch Research SchoolPTN (Polymeer Technologie Nederland) and obtained the title of Registered PolymerScientist in January 2012. The course consists of the following modules: A - PolymerChemistry, B - Polymer Physics, C - Polymer Properties, D - Polymer Rheology and E -Polymer Processing.

117

This thesis has resulted in the following publications:

⇒ D.J.A. Senden, J.A.W. van Dommelen and L.E. Govaert. Strain hardening and itsrelation to Bauschinger effects in oriented polymers. Journal of Polymer Science: PartB: Polymer Physics. 2010, 48, 1483-1494.

⇒ D.J.A. Senden, J.A.W. van Dommelen and L.E. Govaert. Physical aging anddeformation kinetics of polycarbonate. Journal of Polymer Science: Part B: PolymerPhysics. 2012, 50, 1589-1596.

⇒ D.J.A. Senden, S. Krop, J.A.W. van Dommelen and L.E. Govaert. Rate- andtemperature-dependent strain hardening of polycarbonate. Journal of Polymer Science:Part B: Polymer Physics. 2012, 50, 1680-1693.

⇒ D.J.A. Senden, G.W.M. Peters, L.E. Govaert and J.A.W. van Dommelen. Anisotropicyielding of injection molded polyethylene: Experiments and modeling. Submitted forpublication.

Additionally, the author contributed to a few publications outside the scope of this thesis:

⇒ D.J.A. Senden, T.A.P. Engels, S.H.M. Sontjens and L.E. Govaert. The effect ofphysical aging on the embrittlement of steam-sterilized polycarbonate. Journal ofMaterials Science. 2012, 47, 6043-6046.

⇒ L.C.A. van Breemen, T.A.P. Engels, E.T.J. Klompen, D.J.A. Senden and L.E. Govaert.Rate- and temperature-dependent strain softening in solid polymers. Journal of PolymerScience: Part B: Polymer Physics. 2012, 50, 1757-1771.

⇒ M. Hutter, D.J.A. Senden and T.A. Tervoort. Comment on the associated flow rulebased on the yield criterion of Hill. Submitted for publication.

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