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Prediction of long-term performance of load-bearingthermoplasticsCitation for published version (APA):Kanters, M. J. W. (2015). Prediction of long-term performance of load-bearing thermoplastics. Eindhoven:Technische Universiteit Eindhoven.

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Prediction of Long-Term Performance ofLoad-Bearing Thermoplastics

Marc Kanters

Prediction of Long-Term Performance of Load-Bearing Thermoplastics

by Marc J.W. Kanters, Technische Universiteit Eindhoven, 2015.

A catalogue record is available from the Eindhoven University of Technology Library

ISBN: 978-90-386-3896-6

This thesis was prepared with the LATEX 2ε documentation system.

Reproduction: Gildeprint Drukkerijen

Cover: Kevin Rhoe (art direction & design), Hen Metsemakers (photos).

Illustration: Birefringence by stress that surrounds the edge of a bended polycarbonate plate

under load, visualized with crossed polarisers.

This work has been financially supported by DSM Ahead, Geleen.

Prediction of Long-Term Performance ofLoad-Bearing Thermoplastics

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag

van de rector magnificus prof.dr.ir. F.P.T. Baaijens, voor een commissie aangewezen door het

College voor Promoties, in het openbaar te verdedigen op donderdag 3 september 2015 om

16:00 uur

door

Marc Johannes Wilhelmus Kanters

geboren te Weert

Dit proefschrift is goedgekeurd door de promotoren en de samenstelling van de promotiecom-

missie is als volgt:

voorzitter:

promotor:

co-promotoren:

leden:

adviseur:

prof.dr. L.P.H. de Goey

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

dr.ir. L.E. Govaert

dr.ir. T.A.P. Engels

prof.dr. A.J. Lesser (University of Massachusetts Amherst)

Univ.-Prof.Dipl.-Ing.Dr.mont. G. Pinter (Montanuniversitat Leoben)

prof.dr.ir. M.G.D. Geers

Jan Stolk PhD (DSM Ahead)

Contents

Summary v

1 Introduction 1

1.1 An example: degradable polymer implants . . . . . . . . . . . . . . . . . . . . 2

1.2 Failure modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Scope and outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 6

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

2 A new protocol for accelerated screening of long-term plasticity-controlled fail-

ure of polyethylene pipe grades 9

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

2.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Time-to-failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.2 Characterisation of plastic flow kinetics . . . . . . . . . . . . . . . . . . 13

2.2.3 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.1 Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.2 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.3 Mechanical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.4 Hydrostatic pressure testing . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.5 Influence of processing . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4.1 Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4.2 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

i

Contents

2.4.3 Time-to-failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.4 Extrapolation to obtain long-term predictions . . . . . . . . . . . . . . . 24

2.4.5 Characterisation protocol . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.5.1 Different PE100’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.5.2 Activation energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.5.3 Performance modification . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.7 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Appendix 2A: Combined viscosity approach . . . . . . . . . . . . . . . . . . . . . . . 32

Appendix 2B: Certification data PE100 pipe grades . . . . . . . . . . . . . . . . . . . 34

3 Prediction of plasticity-controlled failure in glassy polymers in static and cyclic

fatigue: interaction with physical ageing 35

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.1 Materials and sample preparation . . . . . . . . . . . . . . . . . . . . . 37

3.2.2 Mechanical testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.3 Thermo-mechanical treatments . . . . . . . . . . . . . . . . . . . . . . 38

3.3 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3.1 Physical ageing and mechanical rejuvenation . . . . . . . . . . . . . . . 38

3.3.2 Deformation kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3.3 Ageing kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3.4 Plasticity-controlled failure . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.4.1 Characterisation of the ageing kinetics . . . . . . . . . . . . . . . . . . 44

3.4.2 Cyclic loading conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.4.3 Model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.4.4 Lifetime predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54


References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Appendix 3A: Derivation of the shift factors . . . . . . . . . . . . . . . . . . . . . . 58

Appendix 3B: Expression for the evolution of the yield stress . . . . . . . . . . . . . . 59

4 Direct comparison of the compliance method with optical tracking of fatigue

crack propagation in polymers 61

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

ii

Contents

4.3 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65



4.3.4 Camera data acquisition and processing . . . . . . . . . . . . . . . . . . 67

4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.4.1 The influence of load ratio, R, and temperature . . . . . . . . . . . . . 68

4.4.2 The influence of load . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.4.3 Variations in initial crack length . . . . . . . . . . . . . . . . . . . . . . 70

4.4.4 Confirmation: a HDPE pipe grade . . . . . . . . . . . . . . . . . . . . . 71

4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.5.1 Changes in compliance . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.5.2 Crack propagation rates . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5 Competition between plasticity-controlled and crack-growth controlled failure

in static and cyclic fatigue of polymer systems 81

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

5.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.2.1 Crack-growth controlled failure . . . . . . . . . . . . . . . . . . . . . . 83


5.2.3 Distinction between failure mechanisms . . . . . . . . . . . . . . . . . . 90

5.2.4 Characterisation and distinction . . . . . . . . . . . . . . . . . . . . . . 91

5.3 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.3.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92


5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97


References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6 Integral approach of crack-growth in static and cyclic fatigue in a short-fibre

reinforced polymer; a route to accelerated testing 103

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106


6.2.2 Crack-growth controlled failure . . . . . . . . . . . . . . . . . . . . . . 107

6.3 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

iii

Contents

6.3.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110


6.3.3 Mechanical testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.4.1 Influence of frequency on load ratio dependence . . . . . . . . . . . . . 111

6.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.5.1 Phenomenological description . . . . . . . . . . . . . . . . . . . . . . . 114

6.5.2 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121


References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Appendix 6A: Damage based approach . . . . . . . . . . . . . . . . . . . . . . . . . 126

Appendix 6B: Estimation of the initial flaw size . . . . . . . . . . . . . . . . . . . . . 128

7 Conclusions and recommendations 131

7.1 Main conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Samenvatting 135

Dankwoord 139

Curriculum Vitae 141

List of publications 143

iv

Summary

As a result of their low density and high specific strength, polymers are increasingly employed

in load-bearing applications, usually combined with demanding environmental conditions. The

most important problem encountered in these applications is that all polymers eventually display

time-dependent failure; i.e. it is not the question whether failure will occur, but rather on what

time scale. In order to prevent premature failure in service, it is therefore of the utmost impor-

tance to be able to predict the long-term performance.

From efforts in estimating the lifetime via product testing, it is known that three distinct stages

with different failure processes can be recognized: Region I: plasticity-controlled failure, or ductile

failure. Region II: failure caused by slow crack growth, better known as brittle failure. Region

III: failure caused by molecular degradation, but, given the chemical nature of this process, it is

excluded from this investigation, that specifically focuses on stress activated phenomena.

Current options to estimate the product’s lifetime are time- and material consuming, which ren-

ders it impractical for development and ranking of new materials. Therefore this thesis aims at

the development of test methods which enable to access the long-term properties via short-term

measurements, without the necessity of large amounts of material. Eventually these methods are

validated on long-term failure data. The chapters in this thesis can be divided into two parts: one

focussing on plasticity-controlled failure (chapters 2 and 3) and one focussing on crack-growth

controlled failure (chapters 4-6).

In Chapter 2, an approach is provided which is able to predict plasticity-controlled failure, in-

cluding materials that display multiple deformation mechanisms (multiprocess). This method is

applied on a polyethylene pipe grade and subsequently validated on long-term certification data.

It is proven that long-term plasticity-controlled failure, can indeed be assessed via this route,

within the order of weeks.

v

Summary

In Chapter 3, time-to-failure is studied for an extensive range of temperatures and loading

conditions. The experiments clearly evidenced the existence of an apparent fatigue limit, which

is no more than an increase in resistance against deformation due to physical ageing during the

test. Remarkably, its development appears to proceed much faster under dynamic loading condi-

tions. However, from the evolution of the yield stress in time, for a broad range of temperatures

and loads, both static and dynamic, we learned that there is no significant enhancement under

dynamic loading. It is shown that for large applied stresses the acceleration by stress is only

limited, likely because mechanical rejuvenation starts to retard, or even reverse the effects of

ageing, and it is the rate of mechanical rejuvenation is lower during cyclic fatigue.

For measuring fatigue crack propagation, a well-established method is the compliance method.

Here, the change in stiffness of the test sample, due to an increase in crack length, is used

to translate the crack opening displacement into a crack length. In Chapter 4, the compli-

ance method is compared with direct optical tracking of the crack tip. From these experiments

we learned that the non-linear and viscoelastic behaviour of polymers proves to cause a strong

loading condition- and time dependency of the calibration curves and, as a result, no unique

relation can be found for crack length as function of dynamic compliance. The deviations be-

tween calibration curves appears to be related to stress enhanced physical ageing during the test.

Therefore, the compliance method yields acceptable results for large amplitude/high frequency

measurements (thus short measuring times), but determination of the crack length via optical

tracking prevails.

In Chapter 5, both failure mechanisms, accumulation of plastic strain and crack-growth, are

systematically discussed, and the influence of cyclic fatigue loading on each is investigated. This

shows that when increasing the load amplitude, with equal load maxima, (i) plasticity-controlled

failure is postponed by a decreasing rate of strain accumulation, and (ii) crack-growth controlled

failure is significantly enhanced by accelerated crack propagation. Therefore, the distinction

between plasticity- and crack growth-controlled failure can be made by comparing a polymer’s

lifetime under static loading with that under cyclic fatigue loading. This method of distinction is

demonstrated on a multitude of engineering polymers, including glass-fibre reinforced variants.

Chapter 6 studies an approach for fast assessment of slow crack propagation via cyclic fatigue

on glass-fibre reinforced smooth bars. By varying load ratio and frequency, it became clear that

the number of cycles-to-failure is only independent of frequency for large(r) load amplitudes,

and therefore the amplitude dependency of the time-to-failure varies with frequency. By sepa-

rating the total crack propagation rate into two contributions, a static and a cyclic component,

the time-to-failure for different load amplitudes and frequencies can be accurately be described.

Although the procedure is still rather time and material consuming, we showed that long-term

crack growth controlled failure under a static load can be estimated via fatigue experiments.

vi

CHAPTER 1

Introduction

In daily life one encounters a vast amount of applications that involve synthetic polymers, also

known as plastics. Their versatility enables contributions to transportation, safety, security,

health, shelter, communication, entertainment and innovations.1 Many applications are taken

for granted, like protective packaging of food or the pipes that transport drinking water, and the

material’s performance may not seem very exciting. However, sometimes properties and long-

term performance of polymers becomes clearly relevant, e.g. when they are applied in primary

structures in airplanes. Boeing’s 787 Dreamliner nowadays consists for 50% in weight (80% in

volume) out of polymers,2,3 in the form of advanced composites. They offer weight savings of

20%, compared to conventional aluminium designs,4 and therefore contribute to tremendous fuel

savings during the lifetime of the aircraft.

The continuously growing demand for polymers for more than 50 years, has led to a global pro-

duction in 2013 of an estimated 229 million tonnes, and is expected to continue to increase even

further for the next few years.5 Properties and performance of polymers improved over the years

and applications are becoming more and more demanding. Polymers are consequently increas-

ingly employed in load-bearing applications, often combined with rather extreme environmental

conditions, like high temperatures and humidities. The most important problem encountered in

these load-bearing applications is that all polymers eventually display time-dependent failure; it

is not the question whether failure will occur, but rather on what time-scale. Hence, in order to

prevent premature failure, it is of the utmost importance to be able to predict long-term failure

that inevitably limits the performance of an application.

1

1 Introduction

1.1 An example: degradable polymer implants6–9

A tangible example illustrating challenges one can encounter when applying polymers in load-

bearing configurations is a study that investigates the suitability of degradable polymers for spinal

implants. The primary function of skeletal tissues is mechanical support. However, when skeletal

tissues fail due to trauma or disease, fixations are required to reposition the structures and to

create a proper mechanical environment for functional healing. Usually, metal implants are used

and are quite successful, but have their drawbacks, since they are permanent, they eclipse the

fusion zone on radiological imaging, and they cause delayed union due to shielding of the stress

over the fusion area. From both a clinical and biomechanical point of view, removal of the

support is desired once healing is achieved. This has motivated the development of degradable

polymer implants, with the advantage that they do not interfere with most imaging techniques,

plus they degrade over time, and thus eliminate the necessity of retrieval surgery.

Polylactides, like poly(L-lactic acid) (PLLA) appear attractive candidates, since they are relatively

strong and have excellent biocompatibility. However, as the skeleton can be subject to large

amplitudes of dynamic loading, the mechanical strength of degradable polymers is a concern,

since they usually have limited strength (as compared to metals), which is known to decrease

upon degradation.

0 1 2 3 4 50

1

2

3

4

5

6

7

displacement [mm]

load

[kN

]

strength

a0 10 20 30 40

0

1

2

3

4

5

6

7

time [weeks]

load

max

imum

[kN

]

yield strengthvertebral segment

3.5 kN

b

Figure 1.1: a) Load-displacement curve for a dry cage at a loading rate of 10−3 mm/s at 23◦C and a photo of

a cage. Its strength is defined as the maximum force before collapse. b) Real-time degradation study of PLLA

cages at 39◦C, showing the decrease of the residual strength as function of time, measured at a velocity of 1.3

mm/min (0.022 mm/s) at 23◦C.

To investigate the suitability of PLLA spinal cages as resorbable implants, cages (10 x 18 x 10

mm3) were produced and tested, before implanting them into the spine of a goat for in-vivo

studies. As Figure 1.1a shows, the short-term strength of such a cage, defined as the maximum

load measured in a constant rate experiment, is approximately 5.9 kN, which is well above the

strength of 3.5 kN of a goat’s vertebrae. Figure 1.1b shows that PLLA indeed degrades in time,

but that the strength of a cage remains higher than that of a goat lumbar spine segment for a

2

1.2. Failure modes

period of at least 30 weeks, or seven months, which is longer than the typical period required

for fusion. Nevertheless, when implanted in actual goats, all cages showed plastic deformation

and micro-cracks already after a follow-up of only three months (see Figure 1.2).

Figure 1.2: PLLA cage after three months follow-up. Histology (left) shows micro-cracks after only three months

and micro-MRI (right) confirms these cracks and also shows some plastic deformation of the cage.

The issue here is that one should have recognized that the mechanical behaviour of polymers

is strongly load and time-dependent. Figure 1.3a shows that the cage strength is strongly

influenced by temperature, humidity, but also by loading speed. Decreasing the velocity by

a factor 10 decreases the cage strength with around 1 kN. Increasing the temperature to 37◦C

(body temperature) additionally decreases the strength by 1.5 kN, and wetting causes a decrease

by another 0.5 kN, for all loading velocities. As indicated with the solid marker, the cage

was designed such that its initial strength at room temperature for standard test conditions

(1.3 mm/min) was 7.1 kN. However, at lower loads the cages slowly deform in time and the

deformation does not remain zero. As a consequence, the cage can actually bear far less. As

can be seen in Figure 1.3b, at 37◦C under a load of 4.5 kN, the cages collapse already after

two to five minutes loading, and under a load equal to the strength of a goat lumbar vertebral

segment (3.5 kN) the cages fail after only one to three hours. Due to the decrease in strength,

wet samples are expected to perform even worse, and are predicted to collapse under loading

of 50% of the short-term strength in less than one hour, and under 25% of the cage strength

the lifetime is approximately one month. Note that the time-scales of these experiments are too

short for actual degradation. Therefore the time-dependent behaviour of polylactide is solely

due to its intrinsic properties, and not caused by the fact that it is (bio)degradable. The main

conclusion is that, unlike with metals, knowledge of a polymer’s instant strength is insufficient to

predict its applicability under load over long times, and time-dependent processes lead to failure

even at loads far below the short-term strength.

1.2 Failure modes

Typically, service lifetimes of load-bearing polymer applications are in the order of decades, and

therefore real time loading to estimate their lifetime is not an option. Despite it is imperative

3

1 Introduction

10−3

10−2

10−1

0

2

4

6

8

displacement rate [mm/s]

load

max

imum

[kN

]

23°C dry37°C dry37°C wet

7.1 kN

1.3

mm/min

kN

a10

210

310

410

510

610

710

80

1

2

3

4

5

6

time−to−failure [s]

appl

ied

load

[kN

]

37°C drypred. 37°C wet

hour day month

3 months2.0 kN

1.5 kN

b

Figure 1.3: a) Maximum load of as function of the loading speed, for dry PLLA cages measured at 23◦C,

37◦C, and wet PLLA cages at 37◦C. The closed marker indicates the cage strength used as design criterion. b)

Time-to-failure for dry PLLA cages at 37◦C, loaded at various compressive forces, far below the instantaneous

compressive strength. The gray line indicates the predicted performance of a wet PLLA cage.

to be able to predict the long-term properties and performance. From efforts in developing

predictive methods, and work on pressurized polyethylene pipes in particular, it is known that

three failure mechanisms are present that restrict the lifetime of polymers, see Figure 1.4: I)

”ductile failure”, caused by accumulation of plastic strain, II) ”brittle failure”, caused by slow

crack propagation, and III) brittle failure caused by molecular degradation.10–13

I) ductiletearing

II) brittlefracture

III) chemicaldegradation

Figure 1.4: Schematic representation of typical time-to-failure behaviour of plastic pipes subjected to a constant

internal pressure, with illustration of the three failure modes that are associated with each region.

In the ductile failure region (region I), the applied stress induces accumulation of plastic defor-

mation in time. In most cases, but not as a rule,14 this leads to failure that is accompanied with

large (local) plastic deformation (e.g. bulging of pipes, see Figure 1.4), followed by a ductile

tearing process.15,16 In region II, precursors of cracks are assumed to grow in time until one of

them becomes unstable or has reached a length that causes functional problems in the specific

4

1.2. Failure modes

application (e.g. leakage once the crack has breached the pipe wall).12,17,18 The failure mode is

therefore usually referred to as ”brittle”. In region III, molecular degradation (chemical aging)

leads to disintegration of the material. The failure mode is also brittle, essentially stress inde-

pendent and strongly influenced by stabilizers and molecular weight (Mn).11,19,20 In principle, all

these processes act simultaneously, until one of the three initiates catastrophic failure. However,

as stabilisation techniques improved over the years, region III shifted towards such long failure

times that it is no longer regarded as the lifetimes’ limiting factor,21 and therefore this thesis

focusses on the stress-induced mechanisms in region I and II.

A well-established approach for the characterisation of these failure mechanisms and to predict

the long-term performance for certification of pipe materials is performing creep-rupture tests

on pipe segments. To do so, pipe segments are subjected to various constant pressures up to

failure at several temperatures, according to ISO 1167,22 and the time-to-failure is extrapolated

via linear regression models, according to ISO 9080.23 Via time-temperature superposition this

standardized method can be employed to estimate the stress level that yields a 50-year lifespan

at room temperature, entitled the minimal required strength (MRS), or long-term hydrostatic

strength (LTHS). This enables ranking of different grades, e.g. when the pipe is made from

polyethylene and the MRS is over 8 MPa (80 bar), the grade is called a PE80, and when the

MRS is over 10 MPa (100 bar), the grade is ranked as a PE100.10 The method takes approxi-

mately 1.5 years to be experimentally completed.

a

0.1−0.1

−0.27−0.36−0.43

−0.65−1

R = −1.6

10

cycleslife:103

104

105

106

107

extrapolated

b

Figure 1.5: Stress range versus cycles (S-N curves) (a) and a Goodman diagram (b) for a [0/± 30]3S car-

bon/epoxy laminate. Markers represent measurements, gray lines are added as guide to the eye, and the solid

lines in (b) are lines for constant load ratio. Reproduced from Ramani et al.24

For automotive applications the practice is very different, since actual loading conditions usu-

ally contain a pronounced dynamic component.25,26 Design criteria are based on the number of

cycles-to-failure for a certain load (found in so-called S-N curves, as presented in Figure 1.5a) at

specific temperatures and load ratios, or R-values (σmin/σmax), that are considered typical for

the application. Since the R-value can vary, the accommodation to the mean stress sensitivity is

characterised by measuring the fatigue life for a wide range of test conditions and combine this

5

1 Introduction

in a Constant Fatigue Life (CFL) diagram,27 likely better known as a Goodman diagram,28 as

shown in Figure 1.5b. Such a presentation offers identification of the safe stress region for the

cyclic loading condition with a certain load ratio to guarantee that the composite does not fail

before a specified number of cycles. However, with the large number of mean loads and load

ratios, such a protocol quickly leads to large experimental programs.

Even though these procedures are proven to work well, both are extremely time-consuming and

require a large amount of material (to provide the pipe segments and fatigue samples), which

renders them impractical for fast and flexible material selection and optimization. Furthermore,

since the resistance against crack growth has significantly improved over the years, current gen-

eration pipe grades no longer display region II failure during the certification tests within 1.5

years, indicating that acceleration by temperature is no longer sufficient, and other means have

to be addressed to access this failure mechanism. Additionally, the methods only offer insight

in the performance under loading with constant variables (load ratio and frequency are fixed),

which is hardly ever the case in actual real-life applications.

Therefore, methods are required that can predict long-term failure of load-bearing plastics for

each failure mechanism on the basis of short-term testing, preferably for both static and cyclic

loading.

1.3 Scope and outline of the thesis

This thesis aims at qualifying and quantifying the mechanisms that lead to failure in loaded

polymers, to identify the different mechanisms, and develop methods that enable both access

and prediction of the long-term properties, based on short-term measurements only. Chapters

2 and 3 focus on plasticity-controlled failure, region I. Chapters 4 to 6 focus on crack-growth

controlled failure, region II.

Chapter 2 provides an approach that allows within a few weeks prediction of the long-term

plasticity-controlled failure. The method is validated on long-term data. Chapter 3 investigates

the interaction of progressive ageing with plasticity-controlled failure in static and cyclic fatigue.

Predictions are made to estimate the resulting ”endurance limit” for both. In Chapter 4 two

methods to measure crack propagation (rates) are compared, the compliance method and di-

rect optical tracking, enabling proper characterisation. Chapter 5 investigates the mechanisms

leading to failure in each region, and the influence of fatigue loading on each failure mechanism

separately. This enables the identification and characterisation of each mechanism. Chapter 6

addresses the prediction of long-term crack-growth controlled failure, done via characterisation

of the lifetime in cyclic fatigue for various load ratios (amplitudes) and frequencies. Results are

captured in a phenomenological framework. Finally, at the end of the thesis in Chapter 7, the

main conclusions are summarized, together with some recommendations for future research.

6

References

References

[1] SPI, the plastics industry trade association.

[2] Teresko, J. “Boeing 787: A Matter of Materials – Special Report: Anatomy of a Supply Chain”. Industry-

Week, 2007.

[3] “787 Dreamliner Program Fact Sheet”. http://www.boeing.com/commercial/787/#/overview. Re-

trieved: 24-6-2015.

[4] Hale, J. “Boeing 787, from the Ground Up”. AERO, 2006. pp. 27–23.

[5] Plastics Europe.

[6] Smit, T.H., Engels, T.A.P., Wuisman, P.I.J.M., and Govaert, L.E. “Time-dependent mechanical strength

of 70/30 poly(L,DL-lactide): Shedding light on the premature failure of degradable spinal cages”. Spine,

2008. 33, 14–18.

[7] Govaert, L.E., Engels, T.A.P., Sontjens, S.H.M., and Smit, T.H. Time-dependent failure in load-bearing

polymers. A potential hazard in structural applications of polylactides. Nova Science Publishers, Inc., 2009.

[8] Smit, T.H., Engels, T.A.P., Sontjens, S.H.M., and Govaert, L.E. “Time-dependent failure in load-bearing

polymers: A potential hazard in structural applications of polylactides”. Journal of Materials Science:

Materials in Medicine, 2010. 21, 871–878.

[9] Engels, T.A.P., Sontjens, S.H.M., Smit, T.H., and Govaert, L.E. “Time-dependent failure of amorphous

polylactides in static loading conditions”. Journal of Materials Science: Materials in Medicine, 2010. 21,

89–97.

[10] Andersson, U. “Which factors control the lifetime of plastic pipes and how the lifetime can be extrapolated.”

In: “Proceedings of Plastic Pipe XI”, 2001 .

[11] Gedde, U.W., Viebke, J., Leijstrom, H., and Ifwarson, M. “Long-term properties of hot-water polyolefin

pipes - A review”. Polymer Engineering & Science, 1994. 34, 1773–1787.

[12] Lang, R.W., Stern, A., and Doerner, G. “Applicability and limitations of current lifetime prediction models

for thermoplastics pipes under internal pressure”. Die Angewandte Makromolekulare Chemie, 1997. 247,

131–145.

[13] Chudnovsky, A., Zhou, Z., Zhang, H., and Sehanobish, K. “Lifetime assessment of engineering thermoplas-

tics”. International Journal of Engineering Science, 2012. 59, 108–139.

[14] Crissman, J.M. and McKenna, G.B. “Relating creep and creep rupture in PMMA using a reduced variable

approach”. Journal of Polymer Science Part B: Polymer Physics, 1987. 25, 1667–1677.

[15] Erdogan, F. “Ductile fracture theories for pressurised pipes and containers”. International Journal of

Pressure Vessels and Piping, 1976. 4, 253–283.

[16] Gotham, K. “Long-term strength of thermoplastics: the ductile-brittle transition in static fatigue”. Plastics

and Polymers, 1972. 40, 59–64.

[17] Gray, A., Mallinson, J.N., and Price, J.B. “Fracture Behavior of Polyethylene Pipes”. Plastics and Rubber

Processing and Applications, 1981. 1, 51–53.

[18] Frank, A., Freimann, W., Pinter, G., and Lang, R.W. “A fracture mechanics concept for the accelerated

characterization of creep crack growth in PE-HD pipe grades”. Engineering Fracture Mechanics, 2009. 76,

2780–2787.

[19] Hussain, I., Hamid, S.H., and Khan, J.H. “Polyvinyl chloride pipe degradation studies in natural environ-

ments”. Journal of Vinyl and Additive Technology, 1995. 1, 137–141.

[20] Burn, S. Long-term Performance Prediction for PVC Pipes. AWWA Research Foundation, 2005.

[21] Schulte, U. “A vision becomes true: 50 years of pipes made from High Density Polyethylene”. In: “Pro-

ceedings of Plastic Pipes XIII, Washington”, 2006 .

[22] “ISO 1167 Plastics pipes for the transport of fluids - Determination of the resistance to internal pressure”.

7

http://www.boeing.com/commercial/787/#/overview

References

[23] “ISO 9080 Plastic piping and ducting systems - Determination of the long-term hydrostatic strength of

thermoplastics materials in pipe form by extrapolation”.

[24] Ramani, S. and Williams, D. “Notched and unnotched fatigue behavior of angle-ply graphite/epoxy com-

posites”. Fatigue of filamentary composite materials, ASTM STP, 1977. 636, 27–46.

[25] Sonsino, C.M. and Moosbrugger, E. “Fatigue design of highly loaded short-glass-fibre reinforced polyamide

parts in engine compartments”. International Journal of Fatigue, 2008. 30, 1279–1288.

[26] Bernasconi, A., Davoli, P., and Armanni, C. “Fatigue strength of a clutch pedal made of reprocessed short

glass fibre reinforced polyamide”. International Journal of Fatigue, 2010. 32, 100–107.

[27] Kawai, M. “Fatigue life prediction of composite materials under constant amplitude loading”. In: A.P.

Vassilopoulos (editor), “Fatigue Life Prediction of Composites and Composite Structures”, Woodhead Pub-

lishing Series in Composites Science and Engineering, chap. 6, pp. 177–219. Woodhead Publishing, 2010.

[28] Goodman, J. Mechanics applied to engineering. No. v. 1 in Mechanics Applied to Engineering. Longmans,

Green, and Co., 1899.

8

CHAPTER 2

A new protocol for accelerated screening of

long-term plasticity-controlled failure of

polyethylene pipe grades

Abstract

In this study, a new experimental protocol to evaluate long-term, plasticity-controlled failure

using short-term testing is validated for a high-density polyethylene (PE100) pipe grade. In the

protocol, the strain rate dependence of the yield stress is determined using uniaxial tensile tests

at various temperatures. Complementary uniaxial compression tests are performed to determine

the influence of hydrostatic stress. The plastic flow kinetics is subsequently captured using a

Ree-Eyring modification of the pressure-modified Eyring flow equation. In combination with the

hypothesis that failure occurs at a critical amount of accumulated plastic strain, a versatile tool

to predict time-to-failure is obtained.

Reproduced from: M.J.W. Kanters, K. Remerie, and L.E. Govaert. Submitted 9

2 Accelerated screening of long-term plasticity-controlled failure

2.1 Introduction

As a result of their low density and high specific strength, polymers are increasingly employed in

load-bearing applications. The environmental conditions are usually demanding, with elevated

temperatures up to 140◦C (under the hood), often combined with high humidities (hydroblocks),

while the loading conditions that are generally assumed static, usually contain a pronounced

dynamic component.1,2 The most important problem encountered in load-bearing applications

is, however, that all polymers eventually display time-dependent failure; it is not the question

whether failure will occur, but rather on what time-scale. In order to prevent premature failure,

it is therefore of the utmost importance to be able to predict the long-term performance.

The application of polyethylene in pressurised pipe systems in potable water-, domestic water-

and natural gas supply networks, which started in the early 50’s,3,4 was a strong driving force

in the development of testing methodologies to estimate the hoop stress allowable for a lifetime

of 50 years. From these efforts, it became clear that three distinct regions with different failure

processes can be recognized:5–8 I) ductile failure, II) brittle fracture, and III) degradation con-

trolled failure, as illustrated in Figure 2.1.

I) ductiletearing

II) brittlefracture




In the ductile failure region (region I), the applied stress induces accumulation of plastic de-

formation in time. In most cases, but not as a rule,9 this leads to failure that is accompanied

with large local plastic deformation (e.g. bulging of pipes, see Figure 2.1), followed by a ductile

tearing process.10,11 In region II, precursors of cracks are assumed to grow in time until one of

them becomes unstable or has reached a length that causes functional problems in the specific

application (e.g. leakage once the crack has breached the pipe wall).7,12,13 The failure mode is

therefore usually referred to as ”brittle”. In region III, molecular degradation (chemical aging)

leads to disintegration of the material. The failure mode is also brittle, essentially stress inde-

pendent and strongly influenced by stabilizers and molecular weight (Mn).6,14,15 In essence, all

these processes act simultaneously, until one of the three initiates catastrophic failure. However,

10

2.1. Introduction

as stabilisation techniques improved over the years, region III shifted towards such long failure

times that it is no longer regarded as the limiting factor for pipe materials.3

It is important to note that region I failure does not necessarily have to manifest itself in large,

voluminous plastic deformation before failure, and in some cases the localization of plastic strain

is extreme and local crazing may lead to failure.16,17 In these cases, failure appears to be brittle

because of the small macroscopic deformations, whereas its origin is related to local accumu-

lation of plastic strain. Hence, it is better to distinguish between ”plasticity-controlled” and

”crack-growth controlled” failure, rather than between ”ductile” and ”brittle” failure.

A well-established approach for the characterisation of the long-term performance and certifi-

cation of pipe materials is performing creep-rupture tests on pipe segments. To do so, pipe

segments are subjected to various pressures up to failure at several temperatures, according to

ISO 1167,18 and the time-to-failure is extrapolated via linear regression models, according to

ISO 9080.19 Via time-temperature superposition, this standardized method of analysis can be

employed to estimate the stress level that yields a 50-year lifespan at room temperature, entitled

the minimal required strength (MRS), or long-term hydrostatic strength (LTHS). This enables

ranking of different grades, e.g. when the MRS is over 8 MPa (80 bar), it is called a PE80, and

when the MRS is over 10 MPa (100 bar), the grade is ranked as a PE100.5 Unfortunately this

procedure requires a large amount of material (to provide pipe segments), and to experimentally

complete the method takes approximately 1.5 years. This renders it rather impractical for flex-

ible material selection and optimization and, therefore, methods are required that can predict

long-term failure in each region on the basis of short-term testing. Not necessarily to completely

replace the standardised and accepted certification test, but rather to estimate or predict its

outcome on beforehand.

Approaches to predict failure in region II are often based on Linear Elastic Fracture Mechanics,

which enables lifetime predictions by combining the crack propagation rate with an initial flaw

size and the critical crack length.12,20,21 Since the resistance against crack growth of polyethy-

lene grades has significantly improved over the years, current generation pipe grades no longer

display region II failure during the certification tests within 1.5 years. Therefore, in this work

the focus is on plasticity-controlled failure (region I). A characterisation method is presented to

predict long-term performance, using short-term experiments only, which enables description of

the long-term behaviour within the order of weeks, including the contributions of multiple molec-

ular deformation processes. Its accuracy and validity is checked by applying the new method

to a PE100 pipe grade and by comparing the extrapolation with long-term failure data of pipe

certification tests.

11


2.2 Background

2.2.1 Time-to-failure

Subjection to a constant load causes solid polymers to deform in time (creep) since, similar to

temperature, stress induces mobility that allows the material to flow. However, since deforma-

tion cannot be indefinite, eventually failure results. Figure 2.2a shows the creep response of

polycarbonate under constant load. After an initial elastic response, a region is found where

the strain rate decreases in time (primary creep), followed by a region where the strain rate

remains (approximately) constant, εpl, (secondary creep), to arrive at long loading times in a

region where the strain rate gradually increases due to intrinsic- or geometric softening (tertiary

creep). Eventually this leads to plastic strain localisation and failure. As illustrated in Figure

2.2b, the polymer’s response strongly depends on the load applied: an increase in stress and/or

an increase in temperature results in shorter times-to-failure.

a

PC

primarycreep

secondarycreep

tertiarycreep

tf

εpl

εf

b

PC

σ↑,T ↑

Figure 2.2: a) Evolution of strain in time of polycarbonate in uniaxial extension under a constant stress. b)

Strain versus time for increasing stresses and temperatures.

It has been observed9,22 that in creep rupture the time-to-failure, tf , multiplied by the strain

rate at failure, εf , is constant for different applied stresses, σ, or:

εf (σ) · tf (σ) = C ortf (σ1)

tf (σ2)=εf (σ2)

εf (σ1)(2.1)

Following the observation by Mindel and Brown that the stress dependence of flow is independent

of strain,23 it can be shown that, under a static load, the ratio between the strain rate at failure

is equal to the ratio of the plastic flow rates during secondary creep, εpl, for different applied

stresses, which means:

εpl (σ) · tf (σ) = C (2.2)

The validity of this equation is demonstrated in Figure 2.3a which shows for four different

polymers the constant plastic flow rate during secondary creep for each applied load versus the

12

2.2. Background

corresponding time-to-failure. Indeed a linear relation with slope -1 in a double logarithmic plot

is found.

As demonstrated in Figure 2.3b, the constant C can be regarded as a critical strain, εcr, which

equals the accumulated plastic strain for a material subjected to the plastic flow rate, εpl, for its

entire lifetime up to failure. This phenomenological measure enables quantitative prediction of

the time-to-failure under a constant load, using the stress- and temperature-dependence of the

plastic flow rate, via:

tf (σ, T ) =εcr

εpl (σ, T )(2.3)

Note that this critical strain is smaller than the actual strain at failure as in reality the strain

rate gradually increases. For polymer glasses its value is in the order of 1-10%.

101

102

103

104

105

10−6

10−5

10−4

10−3

10−2

10−1

−1


plas

tic fl

ow r

ate

[s−

1 ]

PCPMMAPPPE a b

PC

εcr εcr

tf

εpl

Figure 2.3: a) Plastic flow rate during secondary creep rate versus time-to-failure for four different polymers:

polycarbonate (PC), poly(methyl methacrylate) (PMMA), polypropylene (PP), and polyethylene (PE). Markers

are measurements, the dotted lines are added as guide to the eye. b) Illustration of the critical strain for

polycarbonate (PC), for a low and a high applied load.

2.2.2 Characterisation of plastic flow kinetics

Although a creep test is easy to perform, it is rather difficult to estimate how much time is

required to reach failure, since a too high load results in immediate failure, and a too low

load in very long testing times. This makes it rather impractical to determine the stress and

temperature dependence of the plastic flow rate. A much easier test, from a logistic point of

view, is a constant rate experiment where the time up to a certain strain is fixed.

The stress-strain response in a constant rate experiment, as shown in Figure 2.4a, is based on

stress-enhanced molecular mobility. In the initial stage of the loading, where the stress is still

low, chain mobility is negligible and the modulus is determined by the intermolecular interactions

between chains. When the stress increases, changes in chain conformation start to contribute to

13


a

PC ε↑,T ↓

10−9

10−7

10−5

10−3

10−1

101

30

40

50

60

70

80

22.8°C

40°C

60°C

80°C

strain rate [s−1]

appl

ied

stre

ss [M

Pa]

applied strain rateapplied stress b

PC

Figure 2.4: a) Stress versus strain for increasing strain rates for polycarbonate (PC). b) Plastic flow rate versus

the stress at yield (open markers) or the applied stress (solid markers) for polycarbonate (PC), reproduced from

Bauwens-Crowet et al.24 Markers are measurements, the lines are added as guide to the eye.

the deformation (plastic deformation). Upon further straining, the mobility continues to increase

with increasing stress, until it exactly matches the strain rate applied, which is at the yield point.

In other words, the stress at yield induces a state of mobility resulting in a steady state of plastic

flow equal to the rate applied. So to be able to strain a material at a higher rate, a higher stress

is required to induce a higher mobility. The magnitude of this plastic strain rate does not only

depend on the stress, but also on the temperature. The latter implies, as first demonstrated by

Bauwens-Crowet et al.,24 that the steady state reached at the yield point in a constant strain

rate experiment is identical to the steady state reached in secondary creep (see Figure 2.4b)

and, therefore, we can use the stress- and temperature dependence measured in well-defined,

short-term constant strain rate experiments to describe the kinetics of plastic flow.

2.2.3 Modelling

The kinetics of plastic flow are described using Eyring’s activated flow theory.25 To obtain a

description independent of the loading geometry, the Von Mises stress or equivalent tensile

stress, σ, and equivalent strain rate, ˙ε, are used; also the influence of hydrostatic pressure, p, is

taken into account. The pressure-modified Eyring flow relation, as first proposed by Ward,26 is

used to describe the stress and temperature dependence of the equivalent plastic flow rate:

˙εpl (σ, T ) = ε0︸︷︷︸I

exp

(−∆U

RT

)︸︷︷︸

II

sinh

(σV ∗

kT

)︸︷︷︸

III

exp

(−µpV

∗

kT

)︸︷︷︸

IV

(2.4)

Part (I) of Equation 2.4 is a rate factor, ε0. The exponential term in part (II) covers the tem-

perature dependence, part (III) takes care of the stress dependency of the material, and part

(IV) captures the effect of hydrostatic pressure. V ∗ is the activation volume, ∆U the activation

energy, µ the pressure dependence, R the universal gas constant, k the Boltzmann’s constant

14

2.3. Experimental

and T the absolute temperature. In most cases only the parameter ε0 depends on the thermody-

namic state of the material (age, crystallinity). The definitions for the equivalent (plastic) strain

rate, ˙ε, equivalent stress, σ, and hydrostatic pressure, p, are given in Table 2.1, and show that

the equivalent strain rate and stress are equal to the strain rate and stress measured in uniaxial

tension and compression.

Definition Tens. Comp. Shear

˙ε =√

23

√(ε11 − ε22)2 + (ε22 − ε33)2 + (ε33 − ε11)2 + 6 (ε2

12 + ε223 + ε2

13) ε ε γ√3

σ =√

22

√(σ11 − σ22)2 + (σ22 − σ33)2 + (σ33 − σ11)2 + 6 (σ2

12 + σ223 + σ2

13) σ σ√

3τ

p = −13

(σ11 + σ22 + σ33) −1

3σ

1

3σ 0

Table 2.1: Definitions of the equivalent Von Mises plastic strain rate, ˙εpl, stress, σ, and hydrostatic pressure,

p, expressed in components of the deformation and stress tensor, respectively. And the explicit expressions for

tension, compression and shear.

The Eyring based flow rule, both with and without the pressure modification, in combination with

the critical strain concept, has successfully been applied to predict time-to-failure of polycarbon-

ate, poly(vinyl chloride),27 poly(lactic acid),28 (oriented) polypropylene,29,30 and on plasticity-

controlled failure in fatigue for various wave types, frequencies and amplitudes, for both glassy

and semi-crystalline polymers.31 In the present study, the validity of the approach is checked on

long-term pressurized pipes made of polyethylene.

2.3 Experimental

2.3.1 Material

The material used in this study was a bimodal high density polyethylene (PE100) pipe grade,

kindly provided by SABIC Europe. This grade was selected on availability of raw material and

certification data for validation of the long-term extrapolations of the ductile failure descriptions.

2.3.2 Sample preparation

Sheet material of various thickness are compression moulded from the PE100 pipe grade. To do

so, a mould is placed in a hot-press (set at 230◦C) and the force is gradually increased (to 100

kN) before keeping it constant for 3 minutes.

Three different cooling rates are used. The lowest rate is about 0.5◦C/min, obtained by turning

the hot press off after compression moulding with the mould still in the machine and let it cool

overnight to room temperature (hot press). An intermediate rate of approximately 5◦C/min is

achieved by taking the mould and compression plates from the hot press and allow it to cool at

15


ambient air (ambient air). The highest applied cooling rate is 50◦C/min and obtained by cooling

the samples using the cold press which is kept at 20◦C (cold press). For uniaxial tension tests,

dog-bone shaped samples (ISO 527 Type 1BA) are prepared from the compression moulded

plates, either by punching (for thicknesses <1.5 mm) or by milling (2-4 mm). For uniaxial

compression tests, cylindrical samples (Ø6 mm×6 mm) are machined from ambient cooled, 20

mm thick compression moulded plates.

2.3.3 Mechanical tests

Uniaxial tensile and compression tests are performed using Zwick Universal Testing Machines,

equipped with 10 kN load-cells. All measurements above room temperature are performed on

a machine equipped with a temperature chamber. To characterise the deformation kinetics,

uniaxial tensile tests are performed, at least in duplicates, at strain rates ranging from 10−5 s−1

up to 10−1 s−1. Before starting the measurement, a pre-load of 0.1 MPa is applied at a speed of

1 mm/min. The test is stopped after the yield point has been reached and is clearly noticeable.

Creep measurements are performed for a wide range of applied stresses and temperatures, and

chosen in such way that the measurement times are not exceeding 3·105 s. The stress is applied

within 10 seconds and subsequently kept constant until failure. The time-to-failure is corrected

for the load application time and is regarded to be the time when the creep rate reaches a

maximum (during neck formation), as found in a so called Sherby-Dorn plot,32 in which the

strain rate is plotted versus the strain. From this analysis, it becomes clear that this point

roughly coincides with a macroscopic strain of 0.8.

Uniaxial compression tests are performed at room temperature under true strain control, at

constant true strain rates of 10−5 − 10−1 s−1, between two parallel, flat steel plates. To obtain

the true deformation of the sample, the applied deformation is corrected during the test for the

stiffness of the experimental setup. Friction between samples and plates is reduced by attaching

adhesive PTFE tape (3M 5480, PTFE skived film tape) on the samples ends. The contact area

between steel and tape is lubricated using PTFE spray (Griffon TF89). During the test, no

bulging of the sample is observed, indicating that the friction is sufficiently reduced.

2.3.4 Hydrostatic pressure testing

The hydrostatic pressure tests are performed in accordance with ISO 1167 and the data are

extrapolated according to ISO 9080 by Exova Nykoping Polymer, which classifies the material

as a PE100. ISO 1167 states that two different geometries can be used, so-called type A and

type B, see Figure 2.5. For both types, end caps are mounted on the pipe segments to allow

sealing and connection of the pressurizing equipment. For the type B geometry, the end caps

are connected to one another via a metal rod and, therefore, the applied internal pressure only

results in a hoop stress acting on the specimen.

For the type A geometry, the fitting that seals the end is connected only to the test piece, hence

16

2.3. Experimental

type A

inlet inlet

type B

Figure 2.5: Schematic representation of the geometries used for hydrostatic pressure testing of pipe segments;

type A and type B.

transmitting the hydrostatic end trust to the test piece, resulting in biaxial loading due to the

resulting longitudinal stress component. Type A is applied for the long-term failure data used

for the long-term validation. The stress components for both types (thin walled pressure vessels)

and the resulting equivalent stress and hydrostatic pressure are presented in Table 2.2, expressed

in terms of the hoop stress, σh, resulting from the internal pressure, pi:

σh =(D − 2t) pi

2t(2.5)

where D and t are the outer diameter and thickness of the pipe segment, respectively.

σ11 σ22 σ33 σ p

Type A: σh12σh 0

√3

2σh −1

2σh

Type B: σh 0 0 σh −13σh

Table 2.2: Components of the stress tensor and the resulting equivalent stress and hydrostatic pressure for

thin-walled pressure vessels according to geometry A and B.

2.3.5 Influence of processing

The certification tests are performed on extruded pipes, which requires approximately 50 kg

of material to produce. Compression moulding is much less material consuming (250 gr) and

therefore more practical for testing (especially in the case of experimental grades). To be able

to compare the results of compression moulded samples to those of pressurised pipes, the poly-

mer should have experienced the same temperature (and deformation) history as those in the

processing of the pipes. During the extrusion process, pipes are slowly cooled at a rate of ap-

proximately 15◦C/min and this exact rate cannot be achieved with the available compression

mould setup. Therefore, the three different possible cooling rates (hot press, ambient air, and

cold press), are compared for the 1.5 mm samples, and, as can be concluded from Figure 2.6a,

the yield stress increases with decreasing cooling rate, while the rate dependence (the slope of

the line) remains constant. A similar observation was previously reported on polycarbonate (PC)

and poly(vinyl chloride) (PVC).27 Additionally, it is clear that only a small difference in yield

17


stress exists between samples cooled by turning the machine off and samples cooled in ambient

air. The slowest cooling method is therefore excluded from further tests.

10−4

10−3

10−2

10−1

0

5

10

15

20

25

30

strain rate [s−1]

yiel

d st

ress

[MP

a]

cold pressambient airhot press a

10−4

10−3

10−2

10−1

0

5

10

15

20

25

30

ambient air

cold press

strain rate [s−1]yi

eld

stre

ss [M

Pa]

1.5 mm3 mm4 mm b

Figure 2.6: a) Influence of cooling rate on the yield stress for 1.5 mm samples of PE100. b) Influence of sample

thickness on the yield stress for fast 50◦C/min (closed symbols) and slow 5◦C/min (open symbols) cooling.

Markers represent data, lines are a guide for the eye.

Since the cooling rate influences the yield stress it is expected that, due to a cooling gradient

in thickness, the yield stress also increases with increasing thickness. However, as Figure 2.6b

shows, this effect is not very large and can only be observed with the cold press cooled sam-

ples. When cooled in ambient air, there is no significant difference in yield stress for different

thicknesses. Therefore, cooling in ambient air was chosen as the standard procedure for sample

preparation, since its rate of approximately 5◦C/min is the closest to the cooling rate of the

pipes and at this rate the effect of sample thickness is negligible.

2.4 Results

2.4.1 Phenomenology

Figure 2.7a shows the stress strain response of the PE100 pipe grade at three different tempera-

tures over a range of strain rates. All curves show an initial elastic response, non-linear behaviour

up to the yield point, and a subsequent decrease in stress (due to geometrical softening). The

strain at yield increases for increasing temperature and decreasing strain rates. As this figure

also shows, the overall stress increases with increasing strain rate and decreasing temperature.

The increase in yield stress with increasing strain rate is smaller at the higher temperatures.

This becomes more apparent in Figure 2.7b, that plots the yield data from Figure 2.7a (as well

as some additional data) versus the strain rate applied. A clear change in strain rate depen-

dence is noticeable for the higher temperatures and lower strain rates and the slope of yield

stress versus strain rate changes. Such a response has already been reported for various polymer

18

2.4. Results

systems,33–36 and is generally interpreted as the result of an additional molecular deformation

process contributing to the stress.

a

ε↑,T ↓

10−5

10−4

10−3

10−2

10−1

0

5

10

15

20

25

30

strain rate [s−1]yi

eld

stre

ss [M

Pa]

23°C50°C65°C80°C

b

Figure 2.7: a) Tensile response at different temperatures and strain rates (10−4-3·10−3s−1). The markers

represent the yield point. b) Yield stresses versus strain rate applied for several temperatures, lines are added as

a guide to the eye.

2.4.2 Modelling

A successful way to model such a multi-process response was proposed by Ree and Eyring in the

50’s;37 later successfully applied to PMMA,33 PEMA,38 iPP,39 PVC and PC.34 The Ree-Eyring

modification is based on the assumption that both molecular processes act in parallel; i.e. the

stress contributions are additive. The rate dependence of the system is therefore captured by

summation of the two individual processes, both described by an Eyring-process, each having its

own activation energy, ∆Ux, activation volume, V ∗x , and rate factor, ε0,x, where x = I, II:

σ( ˙εpl, T ) = σI( ˙εpl, T ) + σII( ˙εpl, T )

=kT

V ∗Isinh−1

(˙εplε0,I

exp

(∆UI

RT

))+ ...

...+kT

V ∗IIsinh−1

(˙εplε0,II

exp

(∆UII

RT

))+ µp (2.6)

Note that the hydrostatic pressure, p, is the total hydrostatic pressure; this implies that the

influence of hydrostatic pressure is regarded to be identical for both processes.

The decomposition of the yield response at 65◦C into its two components is presented in Figure

2.8a. In polyethylene, the high temperature, low strain rate process (process I), was proposed to

be related to intralamellar deformation due to crystal slip (screw dislocations),41,42 while process

II is the so-called α-transition, related to interlamellar deformation, which finds its origin in the

migration of Gauche defects along the crystalline stem, see Figure 2.8b. Each defect passing

along the chain leads to a 180◦ twist and a displacement of half a unit cell. The resulting ”chain

19


diffusion” initiates subsequently relaxation within the interlamellar amorphous region.43 From

Dynamical Mechanical Thermal Analysis (DMTA) studies,36 we know that the relaxation time

of the α-relaxation is about 1 second at 80◦, i.e. the α-transition temperature. This implies

that it can be anticipated that above 80◦C, and/or in sufficiently slow tests, this process will no

longer contribute to the yield stress. Considering the above, it seems likely that process II in

Figure 2.8a, is the α-relaxation mechanism.

To determine the parameters of Equation 2.6 to describe both processes, the influence of the

hydrostatic pressure has to be determined. Since the effect of hydrostatic pressure is likely to

depend on crystallinity,44 its value cannot be derived from literature and has to be determined

on the same grade, processed under the same conditions. Ideally, the pressure dependence µ is

determined by applying a superimposed hydrostatic pressure during a tensile test and measuring

its influence on the yield stress.35,45,46 In the present study we employ instead a combination

of both uniaxial compression and tensile tests. As summarized in Table 2.1, the expressions for

the equivalent strain rate and equivalent stress are equal for uniaxial deformation in each case,

but the hydrostatic pressure terms have opposite signs. In uniaxial tension we can regard an

increase in hydrostatic pressure as an enhancement of effective load applied, or an increase in

mobility, while in uniaxial compression the hydrostatic pressure increases the resistance against

deformation. This implies that the difference between the yield stress in compression and that

in tensile can be used to estimate the pressure dependency, µ.

Uniaxial compression experiments are performed under several (true) strain rates at room tem-

perature (see Figure 2.9a). The strain rate dependence of the stress remains constant with

increasing strain, with an exception for the highest strain rate, where the stress decreases slightly

at larger strains due to viscous heating. Furthermore, in the true stress-true strain response

two yield points can be distinguished: one at smaller strains (≈ 0.1) and the other at larger

a

at 65◦C

I+II

I

II

I

II I II

Intralamellar Interlamellar

b

Figure 2.8: a) Yield stresses versus strain rate applied at 65◦C, and the two contributions in the Ree-Eyring

description separated. b) Schematic representation of the mechanisms governing the deformation in the different

regions, inspired by Pepels et al.40

20

2.4. Results

a

ε↑

10−5

10−4

10−3

10−2

10−1

0

10

20

30

40

strain rate [s−1]

eng.

yie

ld s

tres

s [M

Pa]

23°C50°C65°C80°C

compression

tensile

b

Figure 2.9: a) True stress versus true strain in uniaxial compression at room temperature, for constant true

strain rates ranging from 10−5 − 10−1 s−1 and b) the engineering yield stress versus strain rate applied for both

uniaxial compression (closed markers) and tension (open markers).

strains (≈ 0.4). The deformation mechanism of the first yield point is associated with diffuse

shear within the lamellae and the second yield point is related to a process of heterogeneous

slip resulting in break-up of the lamellae.47–50 Since the mechanism of the first yield point is

similar to that observed during yielding in uniaxial tension,48,49 the rate dependence of this first

yield point will be used to determine the pressure dependency. Although this first yield point is

clearly recognized at high strain rates, where it appears to occur at a constant strain of 0.097,

it is less easy to be defined at lower rates. Therefore the first yield point was taken at a strain

level of 0.097 for all strain rates. The engineering stresses, corresponding to that strain at yield,

are combined with the engineering yield stresses from the tensile experiments in Figure 2.9b,

all as function of strain rate. Note that at yield the difference between engineering strain rate

and true strain rate is negligible. When substituting the equivalent terms and the expression for

the hydrostatic pressure (from Table 2.1) for compression, σc, and tension, σt, into Equation

2.6, only the part capturing the hydrostatic pressure, µp, varies for equal strain rate applied and

temperature. Therefore, via the difference in yield stresses obtained from both experiments the

pressure dependency term, µ, is obtained:

µ = 3 · σc − σtσc + σt

(2.7)

This results in a value of 0.415 for the pressure dependency, µ, for this particular PE100, and the

corresponding Eyring parameters for the stress and temperature dependency of the two mech-

anisms are presented in Table 2.3. Figure 2.9b shows that using these parameters enables an

accurate description of the rate dependence of both the (first) yield stress in compression and

the yield stress in tension, and, as shown for the latter, over a wide range of strain rates and

temperatures.

The value of 900 kJ/mol for the activation energy of process I is pretty large and even sub-

stantially larger than the energy required to break a covalent C-C bond (284-368 kJ/mol51);

21


Material x V ∗x [nm3] ∆Ux [kJ/mol] ε0,x [s−1] εcr [-] µ [-]

HDPEI 33.9 900 2·10107

0.247 0.415II 3.17 141 4.6·1018

Table 2.3: Pressure modified Ree-Eyring-parameters and the average critical strain for HDPE obtained from the

plastic flow rates in Figure 2.9b and time-to-failure data in Figure 2.12a.

this issue will be addressed extensively in the discussion below. In the Ree-Eyring approach, it

is less straightforward to determine the plastic flow rate as function of the load applied, since

the total stress is distributed over the two mechanisms. Despite, it can be estimated using: (1)

˙εpl,I = ˙εpl,II and (2) σ = σI + σII . This set of equations cannot be solved analytically, but a

solution can be achieved numerically using simple optimization methods to find a constant strain

rate at which (2) is met. An approximative approach, which does give an analytical solution, is

provided in Appendix A.

2.4.3 Time-to-failure

Figure 2.10a shows that larger applied stress and/or a higher temperature results in larger plas-

tic flow rates, with subsequently shorter times-to-failure. The moment of failure is taken at a

constant strain of 0.8 (but using the maximum strain rate to determine failure, would result in

practically the same time-to-failure). The evolution of the strain rate with strain clearly shows

the three distinctive regions: primary creep, secondary creep, and tertiary creep up to failure, see

Figure 2.10b.

a

at 65◦C

σ↑,T ↑

εf

0 0.2 0.4 0.6 0.8 1

10−5

10−4

10−3

10−2

10−1

strain [−]

stra

in r

ate

[s−

1 ]

b

at 65◦C

εf

εf

Figure 2.10: a) Creep curves for several loads applied at 65◦C; the open markers indicate the time taken as

time-to-failure. b) Sherby-Dorn plots of the curves in Figure 2.10a. Solid markers represent the plateau creep

rate and open markers the rate and strain at failure.

This so-called Sherby-Dorn plot32 shows that the strain at which the steady state during sec-

22

2.4. Results

ondary creep, ˙εpl, is reached (solid markers), shifts towards larger strains for decreasing loads.

Remarkably if, similar to Figure 2.3a, this steady state plastic flow rate, ˙εpl, is plotted versus

the corresponding time-to-failure, tf , we observe on a double logarithmic scale a slope -1, or

˙εpl · tf = εcr, see Figure 2.11a. Therefore, the time-to-failure can still be calculated with Equa-

tion 2.3, using the steady state plastic flow rate, ˙εpl, and a single critical strain value, εcr, as

illustrated in Figure 2.11b:

tf (σ, T ) =εcr

˙εpl (σ, T )(2.8)

a b

at 65◦C

εcr

tf

˙εpl

Figure 2.11: a) Plastic flow rate during secondary creep rate versus time-to-failure; for the black line the average

critical strain is used; for the gray dashed lines the temperature dependent critical strain as illustrated in the inset

is used; markers are the individual critical strains and the dashed black line indicates the average. b) Illustration

of the critical strain.

Figure 2.12a demonstrates that the kinetics of the plastic flow rate during secondary creep

(solid markers) exactly matches the strain rate and temperature dependence of the yield stress

measured in short-term constant strain rate experiments (open markers). The lines prove that

modelling using the Ree-Eyring modification, Equation 2.6, can indeed be successfully applied

to accurately describe the kinetics of this PE100 pipe grade. Time-to-failure predictions by

Equation 2.8, as shown in Figure 2.12b, illustrate that the Ree-Eyring model combined with the

critical strain gives an accurate description of the measured time-to-failure, using the parameters

as presented in Table 2.3. Note that the lines in both figures have the same absolute slope, or

stress dependency, but with opposite sign. The inset in Figure 2.11a indicates that the critical

strain increases with increasing temperature from 0.18 to 0.38. However, using this temperature

dependent critical strain yields only a small improvement in time-to-failure prediction (dashed

lines in Figure 2.12b) and, therefore, the choice is made to use a single (average) critical strain

for all temperatures, as presented in Table 2.3.

23


10−6

10−5

10−4

10−3

10−2

10−1

0

5

10

15

20

25

appl

ied

stre

ss [M

Pa]

plastic flow rate [s−1]

80°C

65°C50°C

23°C

a

tensilecreep

101

102

103

104

105

106

0

5

10

15

20

25


appl

ied

stre

ss [M

Pa]

23°C50°C65°C80°C

b

Figure 2.12: a) Plastic flow rate versus applied stress; from constant load experiments (creep) and constant

strain rates (tensile). Markers represent data, lines are descriptions using Equation 2.6. b) Time-to-failure as

function of applied stress for different temperatures. Markers represent data, lines are descriptions using Equation

2.8 with a constant critical strain (solid lines) and using a temperature dependent critical strain (dashed lines),

as plotted with the markers in the inset of Figure 2.11a.

2.4.4 Extrapolation to obtain long-term predictions

The main reason to use in this study a PE pipe grade, is the availability of long-term certification

data. The pipe stress rupture data of the long-term certification according to ISO 9080 has to

be obtained in agreement with ISO 1167 and, according to the latter, two geometry types can be

used: type A (with longitudinal stress component) and type B (uniaxial loading). Experimentally

in testing this PE100 pipe grade type A was used. Via the expressions for the hydrostatic pressure

and equivalent stress, as presented in Table 2.2, the plastic flow rate can be determined for the

two loading conditions via Equation 2.6, and the time-to-failure can be estimated using the

critical strain definition in Equation 2.8.

Extrapolation using the model describes the certification data rather exact. The accuracy to

predict data at all temperatures illustrates that both stress and temperature dependence are

described correctly. Surprisingly, the differences in the predictions for restricted and unrestricted

pipes are rather small, which is related to the rather large value of µ. Contact with SABIC

confirmed that the difference in lifetime under a certain pressure between type A and type B

specimens is usually very small and even negligible within experimental error. Finally, from the

modelling, it is clear that the long-term behaviour is determined solely by the high temperature

process I (intralamellar slip).

2.4.5 Characterisation protocol

Clearly lifetime extrapolations based on proper constitutive modelling fits the certification data

very well, providing an effective prediction tool. To enable these extrapolations for different

grades of polymer, requires constant rate experiments in uniaxial tension for a wide range of

24

2.4. Results

101

102

103

104

105

106

107

108

0

5

10

15

20

25

20°C

60°C80°C


appl

ied

hoop

str

ess

[MP

a]

Type AType B a

104

105

106

107

108

109

0

5

10

15

20°C

60°C

80°C

95°C


appl

ied

hoop

str

ess

[MP

a]

Type AType B b

Figure 2.13: a) Certification data of the supplied PE100 pipe grade at three different temperatures. b) Certifi-

cation data of 16 PE pipe-grades, 15 obtained from Exova’s website,52 at four different temperatures. Markers

represent certification data, lines are predictions using the equivalent stress and hydrostatic pressure terms, as

presented in Table 2.2, in Equations 2.8 and 2.6 and the parameters from Table 2.3.

strain rates and temperatures. Since yielding occurs at strains smaller than 50%, the strain rate

dependence (10−5 − 10−1 s−1) at each temperature can be measured within 48 hours. For the

translation towards creep, only few creep experiments are needed to estimate the critical strain.

To be able to make a quantitative prediction for different loading geometries, also compression

experiments have to be performed, but still the total testing time is at maximum in the order

of two weeks. This yields a simple characterisation protocol, which enables prediction of the

long-term behaviour within two weeks using a single tensile machine only, via:

1. Constant rate experiments: Perform tensile tests at several strain rates and tempera-

tures to find the temperature- and rate dependence of the yield stress, i.e. the stress- and

temperature dependence of the plastic flow rate.

2. Constant stress experiments: Perform creep tests to determine the critical strain. Start

by applying a stress equal to the yield stress at 10−3 s−1, which typically results in time-

to-failures of approx. 100 seconds, and start decreasing the stress based on the kinetics

from the rate experiments (remember, similar absolute slopes, but with opposite signs).

3. Hydrostatic pressure dependence: Perform constant rate experiments under a super-

imposed hydrostatic pressure or, if not available, on a different loading geometry (e.g.

compression, see Equation 2.7).

25


2.5 Discussion

2.5.1 Different PE100’s

In Figure 2.13b, predictions are also compared with the data of 15 other PE100’s, as specified

in Appendix B.52 As expected, since all pipes are ranked as a PE100, all data overlap and show

the same trend within a certain experimental error. One of these grades was also tested at 95◦C,

and since the description of the data at this temperature fits rather well, this shows that the

prediction is also valid for a larger temperature range.

Surprisingly however, the results at higher temperatures include data with failure times ranging

from about 3·104 to 5·108 seconds, whereas the data at 20◦C only include data at failure times

starting from approximately 3·105 seconds. This is remarkable, since here process II should be

noticeable according to the model. The reason for this missing data is that, according to ISO

9080, it is allowed to omit failure points at times below 1.000 hours (approximately 106 seconds)

at temperatures equal to or less than 40◦C, to exclude a so-called ”elbow effect”.5 These data

points would strongly influence the extrapolated value at 20◦C from the linear regression model.

The effect is the largest for the unrestricted geometry (type A). The necessity of this exercise

shows that the regression model cannot correctly describe the actual material behaviour with

two mechanisms, and the phenomenon appears to be not well understood, since data is simply

excluded to improve their prediction.

2.5.2 Activation energy

Now we return to the large temperature dependence observed for deformation mechanism I,

which corresponds to a high activation energy of 900 kJ/mol. This appears an unrealistic high

value, since the activation energy required for chain scission (breaking of a covalent C-C bond) is

only 284-368 kJ/mol,51 suggesting that chain scission is the actual failure mechanism at elevated

temperatures. However, it has been shown that (considerable) molecular degradation only occurs

in failure region III and not in region I.6 To confirm this, and exclude chain scission as cause of

failure, Size Exclusion Chromatography (SEC) was performed on as-received samples, as well as

on samples loaded for various times at 80◦C (up to failure after 12.5 hours and interrupted after

7 hours loading); a temperature and time-scale where only mechanism I is active. The results

are presented in Table 2.4, and show that there is no significant difference between the three

which excludes chain scission as a major contributor to deformation.

To further investigate this issue, activation energies are determined for all the different PE100’s

used in Figure 2.13b. The data coincide and, therefore, show the same large temperature

dependence, and consequently all grades have this same high activation energy in mechanism I

with values between 1250 and 1366 kJ/mol.

The deformation at high temperatures proceeds through crystal slip, facilitated by nucleation

and propagation of dislocations and/or defects. The change in the mobility of a defect with

26

2.5. Discussion

as-received loaded failed

Mn 9.500 9.600 9.600

Mw 260.000 290.000 260.000

Mz 1300.000 1500.000 1200.000

Table 2.4: a) The molecular masses in g·mol−1 from Size Exclusion Chromatography (SEC) measurements for

a sample as-received, and a sample loaded for 7 hours at 80◦C and a sample which loaded until failure after 12.5

hours at 80◦C. Kindly provided by SABIC.

temperature is captured using an Arrhenius relation. However, since the defect density also

increases with temperature,53–55 the overall mobility increases much stronger. It is therefore

hypothesised that the high activation energy observed is related to the collective effect of both

thermal activation of the defect mobility and a temperature dependency of defect density.

2.5.3 Performance modification

Figure 2.13b illustrates that the long-term data of all PE100 pipe grades coincide. However,

as already illustrated in Figure 2.6, processing has a significant influence on the performance of

the material and the yield stress can strongly be influenced by e.g. the cooling rate. In Figure

2.6 samples were measured at room temperature and a strain rate range where contributions of

both molecular mechanisms I and II are active, while the long-term behaviour, Figure 2.13, is

determined solely by mechanism I. Since the two processes have a different molecular origin, it

might be possible that each process is influenced differently by e.g. cooling rates. To further

investigate this, also the fast cooled samples are tested at a larger range of temperatures and

strain rates and compared with samples with a lower cooling rate, see Figure 2.14a. Similar

to the results in Figure 2.6, the yield stress decreases significantly with increasing cooling rate

but, remarkably enough, this difference is only observed in the region where molecular process

II is active, since process I seems unaffected. This is also reflected in the time-to-failure data in

Figure 2.14b. An accurate fit on the data of the fast cooled samples can be obtained by a mere

change of the rate constant of the second process, ε0,II . The sensitivity of process II to cooling

rate might also explain the overestimation of the actual lifetime in Figure 2.13 at 20◦C and short

failure times. The difference in yield stress for process II is about 4 MPa between slow- and

fast cooled samples, which is substantial (about 20% at room temperature). Such a decrease

can be expected to have a significant effect on impact properties, and, with pipes failures mainly

caused through third-party damage (thus impact loads),56 and impact properties could improve

by quenching while the long-term performance remains unchanged.

27


10−5

10−4

10−3

10−2

10−1

0

5

10

15

20

25

30

strain rate [s−1]

yiel

d st

ress

[MP

a]

23°C65°C

ambient cooledcold press a

101

102

103

104

105

106

0

5

10

15

20

25


appl

ied

stre

ss [M

Pa]

23°C65°C

b

Figure 2.14: Ambient cooled and cold press cooled samples: a) the rate dependence and b) the time-to-failure.

2.6 Conclusions

In the present study, we demonstrated that the long-term failure of PE100 pipes, as measured

during hydrostatic pressure testing using a procedure that takes approximately 1.5 years, can be

evaluated quantitatively using a novel experimental protocol which takes approximately 2 weeks

on a single tensile testing machine.

The plastic flow rate, including multiple molecular deformation processes, can be described using

the modified Ree-Eyring approach and the yield stress from constant strain rate experiments can

be used to predict the magnitude of the plastic flow rate during secondary creep at different

stresses and temperatures. In combination with a single critical strain a good description of

the time-to-failure data is found. The influence of the hydrostatic pressure is estimated by

comparing the yield stress in uniaxial tension with the yield stress in uniaxial compression. From

these combined results, long-term predictions are made for the two common test geometries, type

A and type B, which shows that the long-term predictions are in excellent agreement with the

certification data. Due to the rather large pressure dependency, the differences between the two

geometries are rather small. Furthermore, the modelling reveals that the long-term performance

is determined by the high temperature process only.

By varying the cooling rate, only process II proves to be affected. Therefore, the two molecular

processes can be influenced separately, and as an example, by quenching during fabrication the

long-term performance remains unchanged whereas the short-term impact properties improve.

2.7 Acknowledgements

The authors would like to thank Dr. L. Havermans and Dr. M. Boerakker from SABIC Europe

for providing the PE100 grade, long-term certification data, and especially for the stimulating

discussions.

28

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[44] Parry, E.J. and Tabor, D. “Effect of hydrostatic pressure and temperature on the mechanical loss properties

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scattering during polyethylene deformation - II. Cold drawing of linear polyethylene”. Polymer, 1998. 39,

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scattering during polyethylene deformation 3. Compression of polyethylene”. Polymer, 1998. 39, 781–792.

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31

http://polymer.exova.com/wp-content/uploads/2014/09/


Appendix 2A: Combined viscosity approach

The rate dependence of polymers shows Newtonian-like behaviour; therefore the stress, σ, and

strain rate, ˙ε, are related through the viscosity, η:

σ = η ˙ε (A.1)

The stress determining the deformation should be used, σ′, and, when equivalent values are used

to define the Eyring parameters, this equals the applied equivalent stress σ minus the contribution

of the hydrostatic pressure, µp:

σ′ = σ − µp, or e.g. σ′ = σ ·(

1 +1

3µ

)for uniaxial tension (A.2)

The strain rate can be described using the Eyring-formulation.

εi (T, σ′) =

1

A0,i (T )sinh

(σ′

σ0,i (T )

)with i = I, II (A.3)

Where

σ0,i (T ) =kT

V ∗iand A0,i (T ) =

1

ε0,i

exp

(∆UiRT

)with i = I, II (A.4)

and R is the universal gas constant, k the Boltzmann constant, T the absolute temperature. V ∗iis the activation volume, ∆Ui the activation energy and ε0,i the rate factor of each process.

Rearranging this equation gives the expression for the stress:

σ′i (T, ε) = σ0,i (T ) sinh−1 (A0,i (T ) · εi) with i = I, II (A.5)

The viscosity of each process can be expressed as function of stress and temperature by substi-

tuting the expression for the strain rate in Equation A.1:

ηi (T, σ′i) =

σ′iε (T, σ′)

= A0,i (T )σ0,i (T )σ′i/σ0,i (T )

sinh (σ′i/σ0,i (T ))with i = I, II (A.6)

If we now distinguish between two stress regions: a region where only process I contributes (I)

and a region where both I and II contribute (I+II), where the stress is expressed via:

σ′I+II (T, ε) = σ0,I+II (T ) sinh−1 (A0,I+II (T ) · ε) (A.7)

and because the stress in the I+II-region is actually the sum of the stresses of the two processes,

the activation volume, activation energy and pre-exponential factor corresponding to this region

can be expressed in terms of the parameters of both processes, by using sinh−1 (x) ≈ ln (2x) for

x� 1.

32

Appendix 2A

The total viscosity, ηtot, is the sum of both viscosities, ηI and ηI+II, or:

ηtot (T, σ′) = ηI (T, σ′) + ηI+II (T, σ′)

= A0,I (T )σ0,I (T )σ′/σ0,I (T )

sinh (σ′/σ0,I (T ))+ ...

A0,I+II (T )σ0,I+II (T )σ′/σ0,I+II (T )

sinh (σ′/σ0,I+II (T ))(A.8)

with

σ0,I+II (T ) = σ0,I (T ) + σ0,II (T ) (A.9)

and

A0,I+II (T ) =1

2exp

(σ0,I (T ) ln (2A0,I (T )) + σ0,II (T ) ln (2A0,II (T ))

σ0,I (T ) + σ0,II (T )

)(A.10)

By using Equation A.1 in combination with a critical strain, this expression of the total viscosity

enables direct calculation of the time-to-failure as function of applied stress:

tf (T, σ) =εcr

ε (T, σ′)=εcr · ηtot (T, σ′)

σ′=εcr · ηtot (T, σ − µp)

σ − µp(A.11)

To be able to express the parameters in region (I+II) in terms of the two processes, the hyperbolic

sine is simplified via the approximation that sinh−1 (x) ≈ ln (2x). As a result, a small error is

found, which can only be observed in the region where the second process starts becoming

noticeable. As shown in Figure A.1, with the viscosity approach the transition between the

mechanisms is rather sharp compared to the approach without simplifications, as used in this

chapter. Nonetheless, this method provides a useful tool to directly calculate and estimate the

time-to-failure for an applied stress and temperature.

33


101

102

103

104

105

106

0

5

10

15

20

25


appl

ied

stre

ss [M

Pa]

23°C50°C65°C80°C

Figure A.1: Difference between the description of the time-to-failure using the viscosity approach (Equation

A.11) (- -) and the description without simplifications (Equations 2.8 and 2.6) (-).

Appendix 2B: Certification data PE100 pipe grades

Manufacturer Grade

SINOPEC Beijing Yanshan Company YanSan HDPE 7600 MBL

The DOW Chemical Company DGDA-2490BK GL

Formosa Plastics Corporation TAISOX 8001BL

Braskem S.A. GP 100 BK

Tosoh Corporation NIPOLON HARD 6600 BLUE

PetroChina Dushanzi Petrochemical Company TUB121N3000 Black

LyondellBasell Industries HOSTALEN CRP 100 RESIST CR BLACK

Reliance Industries Limited PE PIPE COMPOUND RELENE 46GP003 B

LyondellBasell Industries HOSTALEN CRP 100 BLACK

LyondellBasell Industries HOSTALEN CRP 100 BLUE

Reliance Industries Limited Relene 46GP003 O

LyondellBasell Industries HOSTALEN CRP 100 Orange-Yellow

Asahi Kasei Chemicals Corporation SUNTEC - HD B781

PT. Chandra Asri Petrochemical ASRENE SP4808 Natural + CB MB

SCG Performance Chemicals Co., Ltd. EL-LENE H1000PC

Table B.1: PE100 pipe grades corresponding to the certification data used in Figure 2.13b, obtained from the

Exova plastic pipes website.52

34

CHAPTER 3

Prediction of plasticity-controlled failure in

glassy polymers in static and cyclic fatigue:

interaction with physical ageing

Abstract

The deformation and ageing kinetics of polyphenylsulfone (PPSU) are extensively studied to

predict plasticity-controlled failure and how it is effected by progressive ageing. The deformation

kinetics are accurately captured using Eyring’s flow theory. It is shown that activation volume and

energy for deformation are independent of the thermodynamic state of the material. Physical

ageing is accelerated by both temperature and stress and, for temperatures well below Tg,

acceleration by temperature can be described with an Arrhenius expression. For sufficiently low

stresses, acceleration by stress is accurately described with an Eyring formulation, which breaks

down for larger stresses and long time-scales, where mechanical rejuvenation starts to retard, or

even reverse, the effects of ageing. It is shown that the acceleration by stress is determined by the

stress history that the material experienced, and therefore ageing occurs at a lower rate under a

cyclic load than that under static load (with equal maxima). The deformation and ageing kinetics

obtained, offer accurate predictions for the time-to-failure under cyclic loading conditions, and,

once a limit of maximum acceleration (due to mechanical rejuvenation) is introduced, also the

lifetime under static fatigue is predicted accurately.

Reproduced from: M.J.W. Kanters, T.A.P. Engels, and L.E. Govaert. In preparation 35

3 Prediction of plasticity-controlled failure: interaction with physical ageing

3.1 Introduction

Polymers are increasingly employed in demanding, load-bearing applications. With service life-

times typically in the order of decades, it is essential to be able to predict the long-term properties

and performance. To do so, one has to acknowledge that in polymers several mechanisms are

active that eventually lead to failure:1–4 I) accumulation of plastic strain, II) slow crack growth,

and III) chemical degradation. In the present study, the focus will be on the first mechanism:

failure due to accumulation of plastic strain, i.e. plasticity-controlled failure.

Similar to an increase in temperature, the application of stress results in an increase in molecular

mobility within a glassy polymer.5,6 This mobility results in plastic flow,7 that eventually leads

to failure. Since polymer glasses are typically not in thermodynamic equilibrium, material prop-

erties, such as the density, the elastic modulus, the yield stress, and the hardness, evolve in time,

known as physical ageing.8–10 The rate of this structural relaxation process strongly depends on

loading conditions (temperature and stress11–14), and, under the right circumstances, affect the

performance during service life;15–17 a phenomenon known as progressive ageing.8

Due to physical ageing the resistance against plastic deformation increases18–21 and, therefore,

when subjected to a load of similar magnitude, annealed samples show a longer time-to-failure

compared to non-annealed samples. When a quenched (i.e. relatively young sample) is subjected

to loads and temperatures such that progressive ageing can be witnessed within the experimental

time-scale, its properties slowly evolve during the experiment, leading to an apparent ”endurance

limit”.20–24 Implementation of ageing kinetics into a constitutive model,14 offers a framework to

accurately capture lifetimes during static fatigue, including progressive ageing. However, when

the same model is used to predict lifetimes under cyclic fatigue,25 progressive ageing proves to

be significantly underestimated. It appears that the evolution of the yield stress proceeds much

faster in cyclic fatigue loading, even though during cyclic loading the stress is significantly lower

for most of the time compared to static loading with the same maximum. To fully understand

and predict long-term plasticity-controlled failure in both static and cyclic fatigue, this deviation

has to be understood.

The characterisation of the ageing kinetics by Klompen et al.,14 also used by Janssen et al.,25 was

based on measuring the yield stress after several annealing treatments (both thermal and thermo-

mechanical). A constant strain rate and constant temperature was used, based on the assumption

that the deformation kinetics are independent of the thermodynamic state. Acceleration by stress

was measured using two constant loads only at a single elevated temperature. Failure at room

temperature was predicted assuming that the expression used for stress-acceleration is valid for

all stresses and temperatures. The characterisation of acceleration under cyclic loading25 was

limited, and performed at a temperature different from that of the static loading.

In this work we will extend all experimental conditions to measure and validate performance under

static and cyclic loading conditions. The deformation and ageing kinetics of a high performance

glassy polymer, polyphenylsulfone (PPSU), are studied after different annealing treatments to

36

3.2. Experimental

investigate whether characterisation via a single constant strain rate is allowed. Master curves

of the yield stress versus effective annealing time (created by time-temperature superposition)

are obtained at different temperatures, and the acceleration by stress is investigated at multi-

ple temperatures for a wide range of stresses, for both static and cyclic loading, enabling the

investigation of time-stress superposition. Subsequently, the resulting ageing kinetics are used

to predict time-to-failure (including progressive ageing) during both static and cyclic fatigue as

function of the applied (maximum) load and temperature. An engineering approach is used,

which is validated using fatigue measurements.

3.2 Experimental

3.2.1 Materials and sample preparation

The materials used in this study are polycarbonate (for comparison with previous work) and

mainly polyphenylsulfone. Polycarbonate (PC) is provided by SABIC Innovative Plastics, Bergen

op Zoom (LEXAN™ 121R resin). Polyphenylsulfone (PPSU), with a glass transition, Tg, of

220◦C, is provided by Solvay Specialty Polymers (Radel® R-5000). The material is obtained as

granulate, from which tensile bars are injection moulded according to ASTM D638 Type I test

specimen specifications, at a mould temperature of 150◦C.

3.2.2 Mechanical testing

Uniaxial tensile tests are performed on a Zwick Z010 Testing Machine, equipped with a 10 kN

load-cell and temperature chamber. Deformation kinetics are studied by measuring the strain

rate and temperature dependence of the yield stress, at least in duplicates, at strain rates ranging

from 10−5 s−1 up to 10−1 s−1 and temperatures ranging from 23◦C to 125◦C. To characterise

ageing kinetics, the evolution of the yield stress is determined (at a strain rate of 10−3 s−1) after

each annealing treatment at (measurement) temperatures ranging from 23◦C to 150◦C.

Static fatigue experiments are performed in constant force loading for a wide range of engineering

stresses; the stress is applied in 10 seconds and subsequently kept constant until failure. The

time-to-failure is corrected for the load application time. Cyclic fatigue experiments are performed

on a servo-hydraulic MTS Testing System, equipped with a 25 kN load cell, applying a sinusoidal

load up to failure. The load amplitude is varied via the load ratio, R (defined as σmin/σmax),

which is either 0.1, 0.55 or 1 (static fatigue). During each test, the load maximum and load

ratio are kept constant. At 23, 75, and 125◦C, the frequency applied is 1 Hz, at 100◦C it is 0.3

Hz.

37


3.2.3 Thermo-mechanical treatments

Annealing of samples is performed for different periods of time, ta, in an air circulated oven at

various temperatures, Ta (100-200◦C). After predefined times, the samples are removed from

the oven and allowed to cool to room temperature, followed by a day rest before testing. Some

samples are also subjected to a combined thermal and mechanical history, a schematic represen-

tation is provided in Figure 3.1: In the first, load controlled region, constant or sinusoidal loads

are applied with magnitude σa for various periods of time, ta, at different temperatures, Ta. This

is followed by unloading to a load of σrest = 0.1 MPa during rest period, trest, of 5 minutes.

After this rest period, a tensile test is performed in the strain controlled region to determine the

yield stress.

tapp ta trest

ε

σy

σa

σrest

load controlledstrain

controlled

Figure 3.1: Schematic representation of the loading applied in time during the thermo-mechanical treatments

and the characterisation after.

3.3 Background

3.3.1 Physical ageing and mechanical rejuvenation

Since polymer glasses are typically not in a state of thermodynamic equilibrium at temperatures

below Tg, they will strive towards equilibrium and mechanical properties will gradually evolve in

time,. This process is called physical ageing, or structural relaxation.8–10 Figure 3.2 schematically

displays specific volume versus temperature and at temperatures below Tg the specific volume

gradually decreases in time. The rate at which this proceeds is enhanced by temperature and

stress.11–14 Physical ageing has an effect on the mechanical response of polymer glasses. As

illustrated in Figure 3.2b both the modulus and the yield stress increase with age, indicating an

increase in resistance against deformation,26–28 accompanied by a more pronounced softening

after yield that causes brittleness. Continuing deformation after yield make the differences

38

3.3. Background

between the curves to disappear at a strain of 0.2. The strain response for strains larger than 0.2

is independent of the prior ageing history since all its influence is erased. This process is called

mechanical ’rejuvenation’.14,29

a

ageing

T

Tg

0 0.2 0.4 0.6 0.80

20

40

60

80

100

true strain [−]

true

str

ess

[MP

a] annealed

quenched b

Figure 3.2: a) Specific volume versus temperature; below Tg the densification effect slows down; at temperature

T it proceeds in time during physical ageing. b) Intrinsic behaviour of polycarbonate (PC) in uniaxial compression,

for quenched and annealed samples.26

Stress causes plastic flow,7 caused by an increase in molecular mobility.5,6 Figure 3.3a displays

the evolution of strain of polycarbonate (PC) in time applying a constant load of 50 MPa at

room temperature. Although disguised by the logarithmic time axis, the strain rate initially

decreases (primary creep) until it reaches a steady state, where the strain rate remains more

or less constant (secondary creep) to finally end up when geometrical and/or intrinsic softening

occurs that results in an increase in strain rate until failure occurs (tertiary creep). Creep curves

are intermitted by unloading at tun and the residual strain is measured. Next a constant rate

experiment is performed to measure the yield stress. As shown in Figure 3.3b, it is observed that

its magnitude initially remains constant, but decreases with increasing time-under-load. The

larger the residual strain, the smaller the resulting yield stress. This indicates the occurrence of

mechanical rejuvenation.

The stress-induced mobility enables plastic flow, that on its turn causes mechanical rejuvenation.

However, stress also promotes physical ageing. Therefore, upon application of stress, there is a

competition between physical ageing (that causes an increase in resistance against deformation)14

and mechanical rejuvenation (that results in a decrease in resistance against deformation)20.

Apparently, the temperature and stress applied for the data in Figure 3.3 were not sufficiently

high to display physical ageing before mechanical rejuvenation.

39


101

102

103

104

105

0

0.02

0.04

0.06

0.08

0.1

time [s]

stra

in [−

]

a

PC tf

tun

tre

PC tf

εresσy

b

Figure 3.3: PC: a) Evolution of strain in time of polycarbonate in uniaxial extension under a constant stress, up

to failure (black line). Unloading is done at different times, tun, which is followed by constant rate experiment

to measure the yield stress. Markers indicate the moment of unloading, tun, (solid) and reloading, tre, (open).

b) The corresponding yield stresses, σy, (circles) and residual strain, εres, (diamonds) measured slightly before

reloading, εres, versus time-under-load. Markers represent measurements, the line is the time of failure.

3.3.2 Deformation kinetics

The influence of stress and temperature on the plastic flow rate can be described using Eyring’s

activated flow theory:30

εpl (σ, T ) = ε0︸︷︷︸I

exp

(−∆UdRT

)︸︷︷︸

II

sinh

(σV ∗dkT

)︸︷︷︸

III

(3.1)

Part (I) of Equation 3.1 is a rate factor, ε0. The exponential term in part II covers the temperature

dependence and part III the stress dependency of the material, where σ is the yield stress, V ∗d the

activation volume, ∆Ud the activation energy, R the universal gas constant, k the Boltzmann’s

constant and T the absolute temperature.

At yield, the stress-induced plastic flow rate in the material exactly matches the experimentally

applied rate. Therefore can the yield stress as function of applied strain rate, ε, be expressed as:

σy (ε, T ) =kT

V ∗dsinh−1

(ε

ε0

exp

(∆UdRT

))(3.2)

Using sinh−1 (x) ≈ ln (2x) for x � 1, the ratio of the yield stress to temperature is expressed

by:

σyT

=k

V ∗dsinh−1

(ε

ε0

exp

(∆UdRT

))(3.3)

=k

V ∗dln (ε) +

k

V ∗dln

(2

ε0

)+

∆Udk

RV ∗d· 1

T(3.4)

which illustrates that its dependence on strain rate is defined by k/V ∗d , and its dependence on

the reciprocal of temperature by ∆Udk/RV∗d .

40

3.3. Background

10−4

10−3

10−2

0.1

0.15

0.2

0.25

0.3

strain rate [s−1]

yiel

d st

ress

/T [M

Pa/

K]

a

23◦C

75◦C

125◦C

2.4 2.6 2.8 3 3.2 3.4

x 10−3

0.1

0.15

0.2

0.25

0.3

1/T [1/K]

yiel

d st

ress

/T [M

Pa/

K]

30hrs at 200°C30hrs at 187°C30hrs at 175°C30hrs at 150°Cas received

b10−3s−1

Figure 3.4: PPSU: The ratio of yield stress to temperature versus strain rate applied for several temperatures

(a), illustrating a constant activation volume V ∗d , and versus the reciprocal temperature for a strain rate of

10−3 s−1 (b), proving a constant activation energy ∆Ud, for as-received samples and several anneal treatments.

Markers represent measurements, lines model fits according to Equation 3.3 and the parameters presented in

Table 3.3, only varying the rate factors.

Figure 3.4 shows the strain rate and temperature dependence of the ratio of the yield stress to

temperature for PPSU. As Figure 3.4a displays, the strain rate dependence of this ratio is the

same for as-received samples and samples that are annealed. This proves that the activation

volume for deformation, V ∗d , is, to a good approximation, independent of the thermodynamic

state of the material. Furthermore, as Figure 3.4b shows, also the temperature dependence of

this ratio is the same for as-received samples and samples that are annealed. This, in combina-

tion with the constant activation volume, implies that also the activation energy for deformation,

∆U∗d , is independent of the thermodynamic state. Hence, the only variable dependent on the

age of the material is the rate factor, ε0. As shown in Figure 3.5, the rate- and temperature

dependence of the as-received (AR) and annealed samples (AN) are accurately described using

Equation 3.2, the parameters in Table 3.3, and ε0,ar = 2.11 ·1023 s−1 and ε0,an = 1.35 ·1019 s−1,

respectively. Because only the rate factor is subject to change, ageing kinetics can successfully

be characterised by the evolution of the yield stress in time measured at a single constant strain

rate only.

These observations regarding the activation volume and energy are in full agreement with ob-

servations reported in literature,14,19 albeit that there are also references that actually report an

increase in activation volume and energy for deformation upon ageing. Senden et al.31 showed

on polycarbonate, by following Krausz and Eyring32 stating ∆Ud represents an activation Gibb’s

free energy, that an ageing induced increase in activation energy is related to the increase of the

activation enthalpy with age.1 The increase in activation energy was approximately 10% from

fully rejuvenated to severely aged materials. It is likely that such an increase is not observed

1And similarly, the ageing-induced increase in the rate factor, ε0, is related to the increase in activation

entropy, linking the increase in ε0 to the entropy activation barrier.

41


10−5

10−4

10−3

10−2

10−1

0

20

40

60

80

100

strain rate [s−1]

yiel

d st

ress

[MP

a]

23°C75°C125°C a

as-received

10−5

10−4

10−3

10−2

10−1

0

20

40

60

80

100

strain rate [s−1]

yiel

d st

ress

[MP

a]

23°C75°C125°C

annealed

b

Figure 3.5: PPSU: Yield stress versus strain rate applied at different temperatures for a) as-received samples

and b) annealed samples (96 hours at 200◦C). Markers represent measurements, lines model fits according to

Equation 3.2 and the parameters presented in Table 3.3.

here, simply because the range of thermodynamic states that are probed is not as large as theirs,

and therefore the activation energy for deformation can be assumed constant within the range

of thermodynamic states probed in this work.

3.3.3 Ageing kinetics

Since the rate factor ε0 is the only variable depending on age, the influence of the thermodynamic

state on the deformation kinetics can be included by modifying the rate factor:

ε0 (t) = ε0,rej exp (−Sa (teff )) (3.5)

where ε0,rej is the rate factor for unaged material, and Sa a state parameter that uniquely

determines the thermodynamic state. Sa displays a logarithmic evolution in effective ageing

time:33

Sa (t) = c0 + c1 · ln(teff + ta

t0

)(3.6)

where t0 = 1 s, c0 and c1 are constants, ta is the initial age that determines the onset of ageing,

and teff is the effective ageing time defined as:

teff =

t∫0

1

aT (T ) aσ (σ, T )dt (3.7)

with aT and aσ the shift factors capturing the influence of temperature and stress, under the

assumption that time-temperature and time-stress superpositions are valid. The expressions of

42

3.3. Background

the shift functions are of Arrhenius type and Eyring type, respectively14,25,34 (see Appendix A):

aT (T ) =T

Trefexp

(∆UaR

(1

T− 1

Tref

))(3.8)

aσ (σ, T ) =

σV ∗akT

sinh

(σV ∗akT

) (3.9)

where V ∗a and ∆Ua are the activation volume and activation energy for ageing, and Tref the

reference temperature at which the ageing kinetics are obtained.

The evolution of the yield stress and the plastic flow rate in time follow from substitution of

Equations 3.5 and 3.6 into Equations 3.1 and 3.2. However, since it is practically impossible to

obtain unaged material (ta = 0), it is difficult to determine the rate factor for the rejuvenated

state, ε0,rej. Consequently, no unique solution can be obtained for the parameters in the product

ε0,rej exp(−c0), that is found by combining Equations 3.5 and 3.6. Instead, as shown in previous

section, the rate factor for the as-received material ε0,ar, with t = 0 and thus teff = 0, can easily

be determined. Therefore, this issue of too many unknowns can be circumvented by replacing

that product by a reference rate factor, ε0,ref , chosen such that:

ε0,ar = ε0,ref · exp

(−c1 · ln

(tat0

))(3.10)

This results in an expression for the plastic flow rate as function of the effective ageing time:

εpl (σ, T, teff ) = ε0,ref exp

(−c1 · ln

(teff + ta

t0

))exp

(−∆U

RT

)sinh

(σV ∗

kT

)(3.11)

and for the evolution of the yield stress for a constant strain rate:

σy (T, ε, teff ) =kT

V ∗sinh−1

(ε

ε0,ref

exp

(c1 · ln

(teff + ta

t0

))exp

(∆U

RT

))(3.12)

An expression to directly obtain the evolution of the yield stress in time and the corresponding

ageing kinetics is provided in Appendix B.

Once both deformation and ageing kinetics are fully characterised, this framework allows evalua-

tion of the material’s thermodynamic state in time, and subsequently that of the corresponding

deformation kinetics.

3.3.4 Plasticity-controlled failure

Stress-induced mobility results in plastic flow; plastic deformation in time (creep) is steadily

accumulated. Accumulation cannot be indefinite and eventually failure is observed. This fail-

ure is called plasticity-controlled failure. The moment of failure can be estimated, to a good

43


approximation, by introducing a critical value of the accumulated plastic strain that triggers fail-

ure.21,27,35–41 Therefore can the time-to-failure be calculated by integration in time of the plastic

flow rate during secondary creep until the total accumulated strain exceeds this critical value:

εpl(t) =

t′∫0

εpl (σ, T, t′) dt′ with failure once εpl = εcr (3.13)

where εpl is the plastic strain, εpl the plastic strain rate for the load and temperature applied,

and εcr the critical plastic strain at failure.

As demonstrated by Bauwens-Crowet et al.,35 the steady state reached at yield in a constant

strain rate experiment is identical to that reached during secondary creep in a constant stress

experiment. This implies that the deformation kinetics of the yield stress, and its evolution

in time, can be used directly to describe the evolution of the plastic flow rate in time and,

subsequently, the accumulation of the plastic strain. In combination with the critical strain

in Equation 3.13, this allows prediction of plasticity-controlled failure, even for time-dependent

loading conditions.36

3.4 Results

3.4.1 Characterisation of the ageing kinetics

Effect of temperature

Figure 3.6 displays the evolution of the ratio of yield stress to temperature versus the annealing

time for several annealing temperatures, Ta, and the master curves constructed thereof by time-

temperature superposition (TTS). The increase in yield stress is more pronounced for higher

annealing temperatures.18,19 The data can manually be shifted to a single master curve using

the same set of shift factors for each measurement temperature, see Figure 3.7, leading to the

master curves presented in Figure 3.6 for each temperature. As Figure 3.7 shows, for annealing

temperatures at or below the reference temperature of 150◦C, the manually determined shift

factors can accurately be described by an Arrhenius relation, using an activation energy of 150

kJ/mol. However, for higher annealing temperatures, the activation by temperature deviates

from the Arrhenius relation; the activation energy appears to increase, in agreement with the

increase in activation energy reported by Senden et al.31 At high temperatures, the ageing rate is

sufficiently high that the age is (almost) instantaneously higher, which may lead to the observed

increase in activation energy when annealing is performed at high temperatures.

Note that when ∆Ua is determined using the shift factors at high annealing temperatures only,

as is usually done, its value will be overestimated and, as a result, the onset of physical ageing

is underestimated for temperatures below the reference temperature.

44

3.4. Results

23◦C

75◦C

100◦C

125◦C

TTS

Figure 3.6: The evolution of the ratio of yield stress to temperature versus the annealing time for several

annealing temperatures and the master curves constructed thereof, measured at different temperatures. Markers

represent measurements, lines model descriptions according to Equation 3.12 and the parameters presented in

Table 3.3. The data measured at 100◦C on the left are presented in gray for clarity purposes.

∆UaR

Figure 3.7: Arrhenius plot of the temperature dependence of the shift factor aT (Ta) for a reference annealing

temperature of 150◦C, corresponding to the master curves in Figure 3.6. Markers represent the experimentally

obtained shift factors, the dashed line a fit using Equation 3.8 in combination with a activation energy, ∆Ua, of

150 kJ/mol.

Effect of stress

Figure 3.8 displays the evolution of the ratio of yield stress to temperature versus the annealing

time for several applied stresses, σa. The stresses used for each temperature are presented in

45


Table 3.1. The range of stresses and annealing times that successfully can be applied during

these thermo-mechanical treatments is limited, since too low loads do not lead to an increase

in yield stress within reasonable time-scales, while too large loads (for too long times) result in

failure at relatively short times, already before the sample can be unloaded and a yield stress

can be measured. However, from yield stresses obtained as function of the annealing time,

a master curve can be created for each temperature by performing time-stress superposition

(TSS), manually shifting the data to that of the minimum stress applied for that temperature.

The resulting shift factors, aσ, are presented in Figure 3.9. For all temperatures, the manually

determined shift factors decrease with increasing stress, but only up to a certain extent. For

larger stresses, the shift factor reaches a plateau value, and it even increases for larger stresses.

This is most likely caused by the fact that for these high loads tertiary creep is reached during the

mechanical treatment, which leads to a decrease in yield stress due to mechanical rejuvenation

and, consequently, to an apparent smaller shift. Note that these deviations already occur when

the system is still far from actual failure. Either way, this shows that time-stress superposition

is only valid up to a (temperature dependent) maximum load.

The shift factors to shift the data to every reference stress, σref , can be calculated via:

aσ,σref (σ, T ) =aσ (σ, T )

aσ (σref , T )(3.14)

Equation 3.14 enables to describe the shift factors, obtained by shifting data towards the min-

imum applied load for each temperature. Figure 3.9 indicates that the stress dependency of

the shift factors obtained is, for low stresses, accurately described by an Eyring type function,

Equation 3.9 using an activation volume for ageing of 1.55 nm3. The shift factors for a reference

stress of 0 MPa can be calculated as well, as shown in Figure 3.9b. The results clearly display

the different minima in shift factors per temperature. Time-stress superposition of the ratio of

yield stress to temperature versus the effective annealing time, using calculated shift factors by

Equation 3.9, results in the master curves for 0 MPa shown in Figure 3.8. Since the influence of

mechanical rejuvenation is not taken into account, the shift factors for the higher stresses applied

are underestimated and the data is shifted to too long times. Data for which the expected shift

is significantly overestimated, presented in gray, is excluded for the determination of the model

parameters.

Temperature

75◦C 30 35 40 45 47.5 50 - - - -

100◦C 20 25 30 35 40 42.5 - - - -

125◦C 15 20 25 30 35 36 37 38 39 40

150◦C 5 10 15 20 25 30 32.5 - - -

Table 3.1: Stresses applied (MPa) corresponding to each marker in Figure 3.8 per measurement temperature.

46

3.4. Results

75◦C

100◦C

125◦C

150◦C

TSSσa,min

TSS

0MPa

Figure 3.8: The evolution of the ratio of yield stress to temperature versus the annealing time, measured at

different temperatures, for several stresses applied (left) and the master curves constructed thereof, first shifted

manually with the minimum stress applied as a reference (middle), and subsequent with the stress activation

according to Equation 3.9 in combination with an activation volume, V ∗a , of 1.5 nm3 (right). Markers represent

measurements, each type corresponding to a stress applied as presented in Table 3.1, lines model descriptions

according to Equation 3.12 and the parameters presented in Table 3.3. The gray markers indicate data from high

stresses applied, for which the stress activation is overestimated due to rejuvenation during the test (see Figure

3.9)

0 0.05 0.1 0.15

10−2

100

102

104

applied stress/T [MPa/K]

a σ [−]

75°C100°C125°C150°C

a

σref = σa,min

0 0.05 0.1 0.1510

−6

10−4

10−2

100


a σ [−]

75°C100°C125°C150°C

σref = 0MPa

b

Figure 3.9: Stress dependence of the shift factor aσ (σa) a) for the minimum stress applied as a reference and b)

for 0 MPa as a reference, corresponding to the master curves in Figure 3.8. Markers represent the experimentally

obtained shift factors, the lines descriptions using Equation 3.9 in combination with an activation volume, V ∗a ,

of 1.5 nm3.

47


3.4.2 Cyclic loading conditions

Instead of applying a constant load, thermo-mechanical treatments are also performed by applying

a cyclic wave form for several loading times. Observations by Klompen et al.20 and Janssen et

al.,25 indicate that ageing is more pronounced when a cyclic load is applied, compared to static

loading. This would suggest that, due to the cyclic nature of the loading, the enhancement by

stress under a cyclic load depends on frequency. Figure 3.10 displays the increase in yield stress

for a cyclic load with various frequencies, and a load maximum of 20 MPa and R = 0.1 at 125◦C,

versus the cycles under load (a) and the time under load (b), and shows that the evolution of the

yield stress is independent of frequency applied and solely determined by the time under load.

102

103

104

105

106

47

48

49

50

51

52

cycles under load [−]

yiel

d st

ress

[MP

a]

0.1Hz1Hz10Hz

a

20MPa,R = 0.1

102

103

104

105

106

47

48

49

50

51

52

time under load [s]

yiel

d st

ress

[MP

a]

0.1Hz1Hz10Hz

b

Figure 3.10: a) Yield stress versus both cycles under load (a) and time under load (b) for a cyclic load with

R = 0.1 and a maximum stress applied of 20 MPa and several frequencies, measured at 125◦C. Markers represent

measurements, the line in b) descriptions according to Equation 3.12.

From this perspective it seems reasonable to hypothesise that the activation by stress after a

certain loading period is simply determined by the stress history that the material experiences in

that period, or for a time-dependent load signal with constant load maximum, σa, load ratio, R,

and temperature:20,21,25

teff =

t∫0

1

aσ (σ(t, R, σa), T )dt =

t

aσ (R, σa, T )(3.15)

Although Equation 3.15 no longer gives an, easy, expression, the acceleration in time by aσ (R, σa, T )

can easily be calculated by numerical integration. Figure 3.11a shows that it is expected that

the activation is less for a cyclic load than for a static load with equal maximum, simply because

during cyclic loading the stresses are lower for the majority of the time.

To investigate the validity of this hypothesis, the evolution of yield stress is studied for a si-

nusoidal load with several maximum applied loads and two load ratios at 125◦C and 150◦C, as

presented in Figure 3.12, that displays the ratio of yield stress to temperature versus the anneal-

ing time. The maximum stresses used for each temperature are presented in Table 3.2. Also in

48

3.4. Results

this case, the range of stresses and annealing times that successfully can be applied is limited,

due to the same reasons as for constant loading. However, for cyclic loading an additional cause

can be premature failure due to fatigue crack propagation, since an oscillating load enhances the

crack propagation rate (as extensively elaborated on in Chapter 5). Similar to the results from

static loading, a master curve is created for each temperature by manually performing time-stress

superposition, shifting the data of the maximum applied stresses that yielded sufficient data, to

that of the minimum (maximum) stress applied for that temperature, and subsequent, with shift

factors calculated according to Equations 3.14 and 3.15, to a reference stress of 0 MPa. The

shift factors that result from the manual shift are presented in Figure 3.11b which shows that

the manual shifts for cyclic loading are indeed smaller (and therefore give larger aσ’s) than the

ones for a static load with the same maximum load. Furthermore, the activation by stress under

a cyclic load is accurately described for both R = 0.1 and R = 0.55 by Equation 3.15, within a

reasonable approximate. This validates the hypothesis that activation by stress is simply deter-

mined by the stress history experienced by the material.

Remarkably, one can see that for certain (maximum) stresses, that for static loading (R = 1)

result in an underestimation of the shift factor by Equation 3.9 (due to mechanical rejuvena-

tion), the corresponding shift factors for cyclic loading are still described accurately, although for

R = 0.55 also deviations are starting to become noticeable for larger applied maximum loads. For

high maximum stresses at R = 0.1, the shift factors could not be accurately determined, because

yield stresses could only be obtained for single annealing times. However, as the shifts of the

data towards a reference stress of 0 MPa show, using the calculated shifts, they are accurately

described even for larger stresses and deviations between the master curves and the data are

only minor compared to those of static loading. Note that for these stresses and time-scales also

crack-growth could become apparent within the time scale of the thermo-mechanical treatment,

a

amplitude

0 0.02 0.04 0.06 0.08 0.1 0.1210

−4

10−3

10−2

10−1

100

maximum stress/T [MPa/K]

a σ [−]

R=0.1R=0.55R=1

σref = 0MPa

125◦C

150◦C

b

Figure 3.11: a) Illustration of the dependency of aσ on R-value. b) Dependence of the shift factor aσ (σa) on

maximum stress applied, for several R-values at 125◦C (open markers) and 150◦C (solid markers), corresponding

to the master curves in Figure 3.12. Markers represent the experimentally obtained shift factors, the lines

descriptions using Equation 3.15, with an activation volume, V ∗a , of 1.5 nm3.

49


likely affecting the magnitude of the measured yield stress, and thus the shift factors, implying

a different cause of the deviations than mechanical rejuvenation.

125◦C

125◦C

150◦C

TSSσa,min

TSS

0MPa

Figure 3.12: The evolution of the ratio of yield stress to temperature versus the annealing time, measured at

different temperatures, for a cyclic load with R = 0.1 (open markers) and R = 0.55 (solid markers) and several

maximum stresses applied (left) and the master curves constructed thereof, first shifted manually with the lowest

maximum stress applied as a reference (middle), and subsequent with the stress activation according to Equation

3.15 in combination with an activation volume, V ∗a , of 1.5 nm3 (right). Markers represent measurements, each

type corresponding to a stress applied as presented in Table 3.15, lines model descriptions according to Equation

3.12 and the parameters presented in Table 3.3.

Temperature

125◦C 10 20 30 35 40 45

150◦C 5 10 15 20 25 -

Table 3.2: Maximum stresses applied (MPa) corresponding to each marker in Figure 3.12 per measurement

temperature.

3.4.3 Model parameters

The parameters for PPSU to describe both the deformation and the ageing kinetics, are presented

in Table 3.3. These parameters offer accurate description of both the deformation kinetics (see

Figures 3.4 and 3.5) and the ageing kinetics at different temperatures (Figures 3.6, 3.8, 3.10, and

50

3.4. Results

3.12). The description of the master curves for 0 MPa (Figure 3.8) at the different temperatures

prove that the activation energy for ageing correctly describes the onset of physical ageing at

these temperatures and that (for stresses where time-stress superposition is valid) the evolution

of the yield stress from stress annealing is identical to that from temperature annealing.

One might argue whether the dependence of the master curves on the measurement temperature

(Figure 3.6), that is captured by ∆Ud, is accurately described, since the yield stresses after

annealing at high temperatures are captured well for the high measurement temperatures, but

are underestimated for low measurement temperatures. However, as Figure 3.4b displays, the

activation energy for deformation is clearly independent of thermodynamic state within this

range, and the initial yield stresses are described accurately for all temperatures. A possible

explanation could be that the stress history experienced by the material during the uniaxial

tensile test causes deviations, since at the higher measurement temperatures, the acceleration by

stress might be sufficiently high to cause a minor increase in yield stress even during the constant

strain rate experiment, as also can be observed for the lower strain rates at higher measurement

temperatures in Figure 3.5, while at low measurement temperatures the onset of ageing simply

occurs at too long times.

deformation kinetics ageing kinetics

ε0,ref [s−1] ∆Ud [kJ/mol] V ∗d [nm3] εcr c1 [-] ta [s] ∆Ua [kJ/mol] V ∗a [nm3]

8.8 · 1027 295.8 3.35 0.018 1.05 104.4 150 1.55

Table 3.3: Model parameters to describe the deformation and ageing kinetics of PPSU.

3.4.4 Lifetime predictions

The deformation and ageing kinetics are used to calculate the accumulation of plastic strain

in time, via Equation 3.13, and predict the time-to-failure. Two intrinsic phenomena are of

interest here: i) due to a decrease in rate of plastic strain accumulation, the time-to-failure for

cyclic loading is longer compared to static loading with equal load maxima, ii) under constant

(maximum) load applied, the plastic flow rate decreases in time due to the increase in resistance

against deformation caused by progressive ageing, leading to an increase in lifetime.

Figure 3.13 shows the measured lifetime under static loading conditions compared with predic-

tions without (gray lines) and with progressive ageing (coloured lines) for static (a) and cyclic

loading (b), using a critical strain of 0.018. As, this figure shows, for equal maximum load,

the lifetime is largest for cyclic loading. Also crack-growth controlled failure can be recognized

under cyclic loading, as indicated with the gray solid markers in Figure 3.13b, causing failure

at shorter times. Describing this is beyond the scope of this chapter, and will be dealt with in

Chapters 5 and 6. Regarding the description of plasticity-controlled failure, one can recognize

51


101

102

103

104

105

106

30

40

50

60

70

80

90


appl

ied

stre

ss [M

Pa]

23°C75°C100°C125°C

a

R = 1

101

102

103

104

105

106

30

40

50

60

70

80

90


appl

ied

(max

.) s

tres

s [M

Pa]

23°C75°C100°C125°C

R = 0.1

b

Figure 3.13: a) Stress applied versus time-to-failure at several temperatures for static loading (a) and cyclic

loading with R = 0.1. Markers represent measurements, lines predictions according to Equations 3.13 and 3.11,

either without (gray) and with ageing (coloured). Solid gray markers indicate crack-growth controlled failure.

that the predictions without ageing, for both static as cyclic loading, yield a good approximation

for short failure times and low temperatures, but start to deviate when lower stress levels are

applied, where times-to-failure increase due to progressive ageing. The descriptions with pro-

gressive ageing, however, overestimate the lifetime for static loading conditions for all stresses

and temperatures applied, while the time-to-failure under cyclic loading is accurately described

for all maximum stresses and temperatures. This overestimation for static loading conditions

indicates that the activation by stress is too large. Indeed, as discussed earlier (see Figure 3.9),

for static loading the experimentally obtained shift factors already indicated that activation by

stress is limited, since it reaches a plateau value due to mechanical rejuvenation. And because

this is not included in the present description, the activation is significantly overestimated for

large applied stresses. As indicated by the ranges in Figure 3.14, for all temperatures, the stresses

applied relevant here (i.e. resulting in lifetimes between 10 and 106 seconds according to the

predictions) are all exceeding the stresses that resulted in the minimal values for aσ by far (≥ 10

MPa), explaining the overestimation of the time-to-failure. In contrast, the activation by stress

under cyclic loading conditions is reasonably described, even for stresses in the same order of

magnitude as the experimentally relevant stress,2 explaining why also the lifetime under cyclic

loading is described more accurately. This is likely related to the lower rate of plastic strain

accumulation under cyclic loading compared to that under static loading, and therefore no, or

less, mechanical rejuvenation during thermo-mechanical treatment.

To investigate this in more detail, the experimentally obtained minimum shift factors are used

as a limit for maximum stress activation for each temperature, as indicated in Figure 3.15a, and

used to predict the lifetime under static loading conditions. As can be seen in Figure 3.15b, this

results in an accurate description of the times-to-failure under static loading for all temperatures.

2For cyclic loading at 125◦C, the stress activation under cyclic load is measured up to 45 MPa, while the

range for maximum stresses applied that results in lifetimes of 10− 106 s is 45-53 MPa

52

3.4. Results

a

estimated

Figure 3.14: Applied stresses versus measurement temperature. Markers indicate the stresses applied that

resulted in the maximum activation by stress, lines the range of stresses that results in lifetimes of 10− 106 s.

0 0.05 0.1 0.1510

−6

10−4

10−2

100


a σ [−]

75°C100°C125°C150°C a

σref = 0MPa

101

102

103

104

105

106

30

40

50

60

70

80

90


appl

ied

stre

ss [M

Pa]

23°C75°C100°C125°C

R = 1

b

Figure 3.15: a) Shift factor aσ versus applied stress, where the lines indicate the limit in acceleration by stress

as used for each temperature. b) Stress applied versus time-to-failure for static fatigue. Markers represent

measurements, lines predictions according to Equations 3.13 and 3.11, either without (gray) and with ageing

(coloured). To predict the lifetime in static fatigue a limit in acceleration by stress is used as displayed in (a).

The gray markers in (b) indicate samples that failed due to crack-growth.

This illustrates that proper description of progressive ageing should include the effect of mechan-

ical rejuvenation, which would naturally result in accurate descriptions of the activation by stress

for both static and cyclic loading. The suitable way to do so would be by constitutive modelling

and, since the complete history of temperature and stress has to be taken into account, the

ageing kinetics should be determined while simulating the entire testing protocol (e.g. thermo-

mechanical treatment, unloading, relaxation, constant rate experiment). The implementation of

such a constitutive model lies beyond the scope of this thesis, and is topic of future work, but we

are convinced that the data and observations presented in this work provide a good basis for the

53


determination and validation of such a constitutive model that can capture progressive ageing.

Please note that both Klompen et al.20 and Janssen et al.25 already used constitutive modelling

for their predictions of progressive ageing in static and cyclic fatigue and here the same frame-

work to describe physical ageing is used. The discrepancies between their results, that initiated

this work, likely originate from an overestimation of the activation energy for ageing, since it

was determined mainly on high temperature data, as illustrated in Figure 3.16a. This has led to

an overestimation of the time at which ageing starts to become apparent (Figure 3.16b), and

therefore is the activation by stress at room temperature under a cyclic load not sufficient.

a

PC, Tref = 80◦C pred.

actual

predicted

actual

b

Figure 3.16: a) Shift factor aT versus annealing temperature for PC, manually determined for the data presented

by Klompen14 (open) and data on rejuvenated samples (solid). The line indicates the prediction according to

the activation energy used by Klompen, resulting in a mismatch between the actual activation by temperature

and the predicted as indicated with the arrow. b) Illustration of the evolution of the yield stress and the effect

of the mismatch in predicted (dashed) and the actual activation (solid).

3.5 Conclusions

The aim of this work is to predict the effect of progressive ageing on plasticity-controlled failure

of glassy polymers. The deformation kinetics and ageing kinetics of polyphenylsulfone (PPSU)

are studied in great extent. It is shown that both the activation volume and energy for defor-

mation are independent of the state of the material, and the only parameter subject to change

is the rate factor ε0, justifying evaluation of the ageing kinetics via the yield stress evolution

obtained with a single constant strain rate.

Physical ageing can be significantly accelerated by both temperature and stress. For relatively

low temperatures, the acceleration by temperature is accurately described by an Arrhenius ex-

pression, while for higher temperatures the activation energy for ageing appears to increase,

related to an increase in activation enthalpy. For low applied stresses, the acceleration by stress

is accurately described by an Eyring type formulation, while for large stresses and long time-

scales the acceleration by stress is overestimated. Here, tertiary creep is reached during the

54

3.6. Acknowledgements

thermo-mechanical treatment , causing mechanical rejuvenation and a decrease in yield stress.

Furthermore, it is shown that the acceleration by stress is determined by the stress history that

the material experienced, and therefore ageing occurs at a lower rate under a cyclic load than

that under static load (with equal maxima).

The deformation and ageing kinetics obtained offer accurate predictions for the time-to-failure

under cyclic loading conditions, but overestimate the lifetime under static fatigue. However, once

the experimentally observed limit for acceleration by stress is taken as maximum acceleration,

also the lifetime under static fatigue is predicted accurately.

The effect of mechanical rejuvenation has to be taken into account to properly capture the

effect of progressive ageing on the lifetime of glassy polymers. This can be done by constitutive

modelling, and is topic of future work.


The authors would like to thank Rene le Clercq and Stijn Arntz for their efforts and contributions

within the experimental work.

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[15] Bouda, V., Zilvar, V., and Staverman, A.J. “Effect of cylcic loading on polymers in a glassy state”. Journal

of Polymer Science, Polymer Physics Edition, 1976. 14, 2313–2323.

[16] Szocs, F. and Klimova, M. “Fatigue-effect on free radical decay in irradiated polymethyl methacrylate”.

European Polymer Journal, 1996. 32, 1087–1089.

[17] Szocs, F., Klimova, M., and Bartoa, J. “An ESR study of the influence of fatigue on the decay of free

radicals in gamma irradiated polycarbonate”. Polymer Degradation and Stability, 1997. 55, 233–235.

[18] Golden, J.H., Hammant, B.L., and Hazell, E.A. “The effect of thermal pretreatment on the strength of

polycarbonate”. Journal of Applied Polymer Science, 1967. 11, 1571–1579.

[19] Bauwens-Crowet, C. and Bauwens, J.C. “Annealing of polycarbonate below the glass transition: quantitative

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1982. 23, 1599–1604.

[20] Klompen, E.T.J., Engels, T.A.P., van Breemen, L.C.A., Schreurs, P.J.G., Govaert, L.E., and Meijer, H.E.H.

“Quantitative prediction of long-term failure of polycarbonate”. Macromolecules, 2005. 38, 7009–7017.

[21] Visser, H.A., Bor, T.C., Wolters, M., Wismans, J.G.F., and Govaert, L.E. “Lifetime assessment of load-

bearing polymer glasses: The influence of physical ageing”. Macromolecular Materials and Engineering,

2010. 295, 1066–1081.

[22] Ender, D.H. and Andrews, R.D. “Cold drawing of glassy polystyrene under dead load”. Journal of Applied

Physics, 1965. 36, 3057–3062.

[23] Matz, D.J., Guldemond, W.G., and Cooper, S.L. “Delayed yielding in glassy polymers”. J Polym Sci Part

A-2 Polym Phys, 1972. 10, 1917–1930.

[24] Gotham, K.V. and Turner, S. “Procedures for the evaluation of the long term strength of plastics and some

results for polyvinyl chloride”. Polymer Engineering & Science, 1973. 13, 113–119.

[25] Janssen, R.P.M., De Kanter, D., Govaert, L.E., and Meijer, H.E.H. “Fatigue life predictions for glassy

polymers: A constitutive approach”. Macromolecules, 2008. 41, 2520–2530.

[26] Govaert, L.E., Engels, T.A.P., Sontjens, S.H.M., and Smit, T.H. Time-dependent failure in load-bearing

polymers. A potential hazard in structural applications of polylactides. Nova Science Publishers, Inc., 2009.

[27] Engels, T.A.P., Sontjens, S.H.M., Smit, T.H., and Govaert, L.E. “Time-dependent failure of amorphous


89–97.

[28] Smit, T.H., Engels, T.A.P., Sontjens, S.H.M., and Govaert, L.E. “Time-dependent failure in load-bearing

polymers: A potential hazard in structural applications of polylactides”. Journal of Materials Science:

Materials in Medicine, 2010. 21, 871–878.

[29] Meijer, H.E.H. and Govaert, L.E. “Mechanical performance of polymer systems: The relation between

structure and properties”. Progress in Polymer Science (Oxford), 2005. 30, 915–938.



[31] Senden, D.J.A., Van Dommelen, J.A.W., and Govaert, L.E. “Physical aging and deformation kinetics of

polycarbonate”. Journal of Polymer Science, Part B: Polymer Physics, 2012. 50, 1589–1596.

[32] Krausz, A.S. and Eyring, H. Deformation Kinetics. Wiley, New York, 1975.

[33] Govaert, L.E., Engels, T.A.P., Klompen, E.T.J., Peters, G.W.M., and Meijer, H.E.H. “Processing-induced

properties in glassy polymers: Development of the yield stress in PC”. International Polymer Processing,

2005. 20, 170–177.

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[34] Tervoort, T.A., Klompen, E.T.J., and Govaert, L.E. “A multi-mode approach to finite, three-dimensional,

nonlinear viscoelastic behavior of polymer glasses”. Journal of Rheology, 1996. 40, 779–797.





2008. 41, 2531–2540.

[37] Engels, T.A.P., Schrauwen, B.A.G., Govaert, L.E., and Meijer, H.E.H. “Improvement of the Long-Term

Performance of Impact-Modified Polycarbonate by Selected Heat Treatments”. Macromolecular Materials

and Engineering, 2009. 294, 114–121.






Engineering, 2010. 295, 637–651.



2012. 50, 1438–1451.

[41] van Erp, T.B., Govaert, L.E., and Peters, G.W.M. “Mechanical Performance of Injection-Molded

Poly(propylene): Characterization and Modeling”. Macromolecular Materials and Engineering, 2013. 298,

348–358.

57


Appendix 3A: Derivation of the shift factors

The expressions for shift factors capturing the activation by temperature and stress of the de-

formation can be derived by substituting the Eyring expression in Equation 3.1 in the expression

for the total viscosity, η, of the system:

η (σ, T ) =σ

ε (σ, T )(A.1)

=σ

ε0 · exp

(−∆UdRT

)sinh

(σV ∗dkT

) (A.2)

=kT

ε0V ∗dexp

(∆UdRT

) σV ∗dkT

sinh

(σV ∗dkT

) (A.3)

=kTrefε0V ∗d

exp

(∆UdRTref

)· T

Trefexp

(∆UdR

(1

T− 1

Tref

))·

σV ∗dkT

sinh

(σV ∗dkT

) (A.4)

= η0,Tref · aT (T ) · aσ (σ, T ) (A.5)

with

η0,Tref =kTrefε0V ∗d

exp

(∆UdRTref

)(A.6)

aT (T ) =T

Trefexp

(∆UdR

(1

T− 1

Tref

))(A.7)

aσ (σ, T ) =

σV ∗dkT

sinh

(σV ∗dkT

) (A.8)

where R is the universal gas constant, k the Boltzmann constant, T the absolute temperature,

Tref the absolute temperature at which the reference viscosity is obtained, V ∗d the activation

volume, ∆Ud the activation energy, and ε0 the rate factor. This set of equations gives an

expression for the reference viscosity, η0,Tref , the shift factor for the acceleration by temperature,

aT (T ), and the shift factor for the acceleration by stress, aσ (σ, T ). The latter two, presented

in Equation A.7 and A.8, also hold for the acceleration of physical ageing in time simply by

replacing the activation energy and volume for deformation, ∆Ud and V ∗d respectively, with

those for ageing; V ∗a and ∆Ua.

58

Appendix 3B

Appendix 3B: Expression for the evolution of the yield stress

An expression for the evolution of the yield stress, σy, in time can easily be derived from substi-

tution of the time dependent rate factor in Equation 3.5 into the Eyring expression in Equation

3.2:

σy (t, ε, T ) =kT

V ∗dsinh−1

(ε

ε0,rej

exp

(∆UdRT

exp (Sa (teff ))

))(B.1)

=kT

V ∗dsinh−1

(ε

ε0,rej

exp

(∆UdRT

)exp

(c0 + c1 ln

(teff + ta

t0

)))(B.2)

By using sinh−1 (x) ≈ ln (2x) for x� 1, this yields:

σy (t, ε, T ) =kT

V ∗dln

(ε

ε0,rej exp(c0)exp

(∆UdRT

))+kTc1

V ∗dln

(teff + ta

t0

)(B.3)

= σy,0 (ε, T ) + c · ln(teff + ta

t0

)(B.4)

where R is the universal gas constant, k the Boltzmann constant, T the absolute temperature,

V ∗d the activation volume, ∆Ud the activation energy, and ε0,rej the rate factor of the unaged

material, Sa the state parameter, with t0 = 1 s, c0 and c1 are constants, ta is the initial age,

and teff is the effective ageing time, provided in Equation 3.7.

This illustrates that the yield stress is determined by σy,0, that denotes the yield stress at a

certain strain rate and temperature of the reference situation (here as-received), and a term that

increases with time. The slope of the yield stress versus the logarithm of the effective time is

determined by c/ln (10), from which the value of c1 can directly be derived.

59

CHAPTER 4

Direct comparison of the compliance method

with optical tracking of fatigue crack

propagation in polymers

Abstract

The compliance method is based on simple force-displacement data and is successfully applied to

determine fatigue crack propagation in linear elastic, isotropic materials like metals and ceramics.

Here, we investigate its potential use in non-linear, time dependent materials like polymers, by

comparing its results with those of direct optical tracking experiments. The non-linear and vis-

coelastic behaviour of polymers proves to cause a strong loading condition- and time dependency

of the calibration curves and, as a result, no unique relation can be found for crack length as

function of dynamic compliance. Normalization of the dynamic compliance, using an apparent

modulus, slightly reduces the difference, but this still does not yield a unique functional descrip-

tion, since the deviations between calibration curves appear to be related to stress enhanced

physical ageing during the experiment. Determination of the crack length via optical tracking

prevails. When impractical and when therefore the compliance method is used instead, results

should be taken with care.

Reproduced from: M.J.W. Kanters, J. Stolk, and L.E. Govaert. Polymer Testing (accepted) 61

4 Direct comparison of the compliance method with optical tracking

4.1 Introduction

With polymers increasingly employed in load-bearing engineering applications, it is imperative

to be able to estimate the product’s lifetime under design specific loading conditions. More-

over, it is essential to understand the mechanisms causing failure of the material. It is known

that regarding long-term failure of polymers, several mechanisms are operative that compete to

eventually cause failure. Three failure regions can be discerned: I) ”ductile failure”, caused by

accumulation of plastic strain, II) ”brittle failure”, caused by slow crack propagation, and III)

brittle failure caused by molecular degradation.1–3 In the present study, the focus will be on the

mechanism active in region II: failure due to crack propagation.

In region II, precursors of cracks are assumed to grow in time until the crack becomes unstable

or reaches a length that causes functional problems in a specific application (e.g. leakage of a

pipe once a crack has breached the pipe wall). Therefore, the lifetime of a product is basically

determined by two quantities: the initial flaw size, which may originate from processing (voids,

impurities, etc.) and/or handling (scratches), and the crack propagation rate, which strongly

depends on loading and environmental conditions.3–8 To evaluate the crack propagation kinetics

experimentally, a variety of methods can be used to monitor the crack length in time; some

are specific to the type of material. Methods used are based on direct (optical) observations,

on changes in mechanical response of the specimen,9,10 on use of surface gages11,12 or on the

specimens electrical characteristics.13,14 A well-established method, based on changes in the me-

chanical response, is the so-called compliance method.15 The basis of this method is that the

stiffness of the sample decreases with increasing crack length. For an isotropic, linear elastic

material, the crack tip opening displacement for a certain load can be related to the crack

length inside the material.10 The compliance method relates the crack opening displacement

to two easily measurable quantities: applied load and specimen deflection. It uses a calibra-

tion curve to relate crack length to compliance. Therefore, are crack lengths easily obtained,

even for situations where direct crack length measurements are difficult (e.g. in environmental

chambers). There is no need for complex data acquisition. The method can be performed on

multiple machines without the need for a multitude of (expensive) set-ups, like cameras, and

computers. The compliance method is therefore popular, well accepted, and widely applied to

obtain the fracture toughness of metals16–20 and ceramics.9,21 Inspired by these excellent results,

the method was adopted to determine fracture characteristics also for other materials that are

not linear elastic and isotropic, such as bone,22–28 and polymers, including reinforced29,30 and

non-reinforced thermoplastics,31 among which ultra-high molecular weight polyethylene32,33 and

polyethylene pipe grades.34–37

It is not trivial to apply a method developed for linear elastic behaviour on non-linear, time-

dependent materials, where the (apparent) modulus strongly depends on loading conditions and

loading time. Modifications of the method were therefore proposed; e.g. the use of an (av-

eraged) apparent modulus rather than the elastic modulus,25,26,28 or by separating the total

62

4.2. Background

displacement in a geometry function and deformation function, via the so-called single speci-

men normalization method.38,39 These modifications, the latter in particular, were reported to

enable accurate fracture toughness measurements on polymers.30,33,40–44 However, these studies

focussed on determination of the (static) fracture resistance (J − R-curves), via experiments

usually in displacement control. In other words, the time-scale hardly differs for different experi-

ments, and samples are loaded for rather short time-scales before fracture. In contrast, at present

is the characterisation of (fatigue) crack propagation rates has been forced towards significantly

longer testing times, due to increasing resistance against crack propagation of currently available

materials.4,45 The time-scale of an experiment is, therefore, now directly related to the crack

propagation rate, which is known to be influenced by test conditions3,5–8 such as the magnitude

of the load applied, the load amplitude, the frequency, and the temperature, but also at speci-

men level by details such as the initial crack size. Due to the non-linearity and time dependency

of polymers, the question rises whether a method such as the compliance method is actually

applicable in these cases.

In the present study, fatigue crack propagation is measured for various polymers under sev-

eral test conditions via optical tracking using a camera set-up. Simultaneous acquisition of both

displacement and loads, used to calculate the corresponding compliances, a large amount of cal-

ibration curves result for a wide range of testing conditions, allows a direct comparison between

the crack length obtained via the compliance method and via direct optical observations. To

evaluate the accuracy of the compliance method, results from (separate) investigations on poly-

carbonate and nylon are combined using different testing conditions and time-scales in different

experiments. Subsequently, the observed trends are confirmed by investigating crack propagation

kinetics in a high density polyethylene pipe grade.

4.2 Background

The compliance method is based on that the secant compliance, C, the reciprocal of the slope

force, F , versus the crack opening displacement of the sample, COD, is a unique relationship

as function of the crack length, a.

C =COD

F(4.1)

Using the materials Young’s modulus, E, and the sample thickness, B, the normalized secant

compliance, Ux, is obtained:10

Ux =1√

ECB + 1(4.2)

Subsequently, a calibration curve is required linking Ux to the crack length, a, that is normalized

using the sample width, a/W . The curve can be determined in two ways: analytically or experi-

mentally.

63


By analytically integrating the stress, that is characterised by the expression of the stress in-

tensity over the cross-section of the sample for different values of the crack length a/W , accurate

estimates of the compliance can be made given the measured crack opening displacement.10

When the displacements are obtained via other means, e.g. by using local strain gages, one

has to correct for bending and the rotation of the sample. Saxena et al.10 derived the general

expression of the normalized crack length as function of the normalized secant compliance:

a

W= C0 + C1 · Ux + C2 · U2

x + C3 · U3x + C4 · U4

x + C5 · U5x (4.3)

For a Compact Tension specimen, with displacement measured via Clevis brackets (in line with

the load), the coefficients of Equation 4.3 read:15

C0 = 1.0002, C1 = −4.0632, C2 = 11.242, C3 = −106.04,

C4 = 464.33, C5 = −650.68 (4.4)

The compliance method is based on the uniqueness of the relationship between compliance and

crack length. Thus, for a given geometry, every linear elastic, isotropic, homogeneous material

gives the same curve.

The question rises whether this approach is still correct when applied to non-linear, time-

dependent materials. Experience on polymers indicates that this is not the case,31,32 and the

remaining option is to determine the calibration curve experimentally by measuring the compli-

ance using samples with a pre- or in-situ determined crack length and a polynomial tool is used

to obtain a functional fit. The normalized compliance versus crack length results.25,28,32 How-

ever, since material and specimen dimensions are usually constant per study, the only variable

changing with crack length is the secant compliance, C, and therefore often the evolution of the

non-normalized compliance versus crack length is used.9,22,29,35,36 Since creep crack propagation

measurements are rather time consuming, often fatigue experiments are performed to enhance

the propagation,35,36,45–48 and the dynamic compliance, ∆C, is used within Equation 4.2 to ob-

tain a normalized dynamic compliance, ∆Ux. When a sinusoidal load is applied with a constant

amplitude, the displacements at both the minimum and maximum load of each cycle give the

extremes in crack opening displacement. The dynamic compliance of each cycle is calculated

using the reciprocal of the stiffness between minimum and maximum, from here on referred to

as the dynamic stiffness, or:

∆C =CODmax − CODmin

Fmax − Fmin=

∆COD

∆F(4.5)

where ∆C is the dynamic compliance, CODi the crack opening displacement at either the

minimum or the maximum load of that cycle, Fi, and ∆F is defined by Fmax · (1−R) where R

is the load ratio (see Figure 4.1):

R =FminFmax

(4.6)

64

4.3. Experimental

R = 1

R = 0.55

R = 0.1

Fmax

Fmin

R↑

Figure 4.1: Schematic illustration of the applied dynamic load at different R-values.

According to ASTM E647, for each new test at least one visual reading must be performed

to be able to correct the crack length, measured using the dynamic compliance, to the directly

measured real length. This is ultimately effectuated by defining an effective modulus, E ′, and

substituting this value in Equation 4.2 to adjust all crack size calculations:

∆Ux =1√

E ′∆CB + 1(4.7)

Once the crack length is obtained, the stress field near the crack tip is determined by the stress

intensity factor, K, which for a crack opening load (mode I) is defined by:

KI = Y σ√πa (4.8)

With Y being a geometrical parameter, a the crack length and σ the remotely applied stress. The

stress intensity factor for a Compact Tension specimen, as used in this study, at the maximum

load reads:15

Kmax =Fmax

B√W

2 + a/W

(1− a/W )3/2[0.866 + 4.64(a/W )− 13.32(a/W )2 + 14.72(a/W )3 − 5.60(a/W )4

](4.9)

where B is the sample thickness and W its width, measured from the centre of the fixation holes

(see Figure 4.2).

4.3 Experimental

4.3.1 Materials

Three different polymers are used: polycarbonate, polyamide 4,6, and high density polyethylene.

Polycarbonate (PC) is provided by SABIC Innovative Plastics, Bergen op Zoom (LEXAN™ 101R

65


resin) and received partly in the form of extruded 12 mm thick sheets (width O(m)) and partly as

a granular material. Polyamide 4,6 (PA46) is provided by DSM Geleen (DSM Stanyl® TW300),

and received as injection moulded plaques of 100x65x6 mm3. High density polyethylene (HDPE)

is provided by SABIC Europe, and is a bimodal pipe grade (SABIC® Vestolen A 6060R 10000),

obtained as granules.


Tensile bars are injection moulded from PC granules, according to ASTM D638 Type I test

specimen specifications (cross-sectional area of 3.2x13.13 mm2). PA46 plaques are used as-

received and conditioned. Conditioning occurred in a humidity chamber at 62%RH and at 70◦C,

and lasted until the increase in weight in time became negligible (after approx. 3 months). A

condition similar to end-use conditions of products (23◦C/50%RH) was reached. From HDPE

granules, 200x200x15 mm3 plaques are compression moulded using a hot-press and subsequently

surface machined in two directions to obtain plates with a final thickness of 12 mm.

Compact Tension (CT) specimens are produced by cutting the sheets and plaques using a circular

saw followed by precision machining of the fixation holes and notch. From the injection moulded

plaques, the samples are taken such that the crack grows parallel to the flow direction. The

dimensions of the Compact Tension specimen are determined according to the ASTM standard

E64715 and listed in Table 4.1. For the PC and HDPE the larger samples are used, with 12 mm

thickness, and for PA46 the smaller samples with 6 mm thickness.

Figure 4.2: Illustration of a Compact Tension spec-

imen.

small large

B [mm] 6 12

W [mm] 32 64

H [mm] 38.5 77

Table 4.1: Dimensions for the Compact Tension

specimen.

To fabricate pre-cracks of reproducible size, a custom made tool is used that clamps the speci-

men such that a fresh razor blade can be tapped into the notch root by releasing a pendulum.

The exact initial crack length is subsequently measured using a microscope. The method proves

to realize consistent crack lengths.

66

4.3. Experimental


Fatigue experiments are performed on two servo-hydraulic MTS Elastomer Testing Systems,

equipped with 2.5 kN load cells and temperature chambers. For the fatigue crack propagation

measurements, the specimen is mounted to the tensile stage by a Clevis bracket, with dimensions

according to the ASTM standard E647. The specimen can freely rotate around the pin and the

bracket contains one degree of freedom for axial alignment of the upper and lower part. A

sinusoidal load is applied with a frequency of 5Hz, with for each test a constant load amplitude

(R-value) and maximum load. Time, load and displacement are recorded at every peak. For

transparent PC samples, a light from the top is used to illuminate the crack tip. For opaque

samples, HDPE and PA, a strong light source is placed at the back of the sample. It illuminates

the crack when opened.

4.3.4 Camera data acquisition and processing

Crack propagation is monitored using a digital camera and a customized script based on the

MATLAB Image Acquisition toolbox. Cameras used are Prosilica EC1280 with a resolution of

1280x1024 pixels and equipped with either a 55 mm or a 50 mm lens. The cameras are positioned

perpendicular to the specimen surface at a distance such that the final crack covers the full width

of the image. A calibration image, made using a specimen with clearly visible calibration points,

is applied to correlate the number of pixels in the image to the physical crack length.

After each measurement, the Matlab-script locates the position of the crack tip. This crack

length is used to calculate the stress intensity factor using Equation 4.9. From the crack length

as function of time, the derivative is taken in each point using a linear regression of an inter-

val surrounding this point to obtain the crack propagation rate. Loads and displacements are

recorded at the minima and maxima of the load signals. Before obtaining the dynamic compli-

ance in time, the signals are first interpolated using a new linear time vector, because of the

high frequency (5Hz) (Figure 4.3a). Next, the crack opening displacement (COD) is found

by subtracting the two interpolated signals. The COD is subsequently normalized for the load

amplitude to obtain the dynamic compliance, ∆C. The crack length in time results from the

image acquisition, and the data is interpolated using the same new time vector. These actions

result in data of the dynamic compliance and the corresponding crack length, in each time point

(Figure 4.3b).

To obtain better interpretable graphs, an extra data-reduction step is applied. The dynamic

compliance, as function of the crack length, and the crack propagation rate, as function of the

stress intensity factor, are interpolated over a constantly increasing vector along the x-axis. This

resulted in 75 data points equally divided over the complete x-range (crack length and stress

intensity factor, respectively).

67


0 0.5 1 1.5 2 2.5

x 104

0

0.5

1

1.5

3015

2

3000 3010 3020 3030

0.25

0.35

0.45 0.25

0.35

0.45

time [s]

dis

pla

cem

ent [m

m]

raw signal

xi,max

xi,min

a0 0.5 1 1.5 2 2.5

x 104

0

5

10

15

20

25

time [s]

a [m

m] a

nd ∆

C [µ

m/N

]

ai

∆Ci

b

Figure 4.3: a) Displacement versus time, where the inset shows the difference between the original data and

the interpolated points. b) Interpolated crack length and dynamic compliance versus time. Example used:

polycarbonate (70◦C, Fmax = 500, R = 0.2).

4.4 Results

4.4.1 The influence of load ratio, R, and temperature

The crack propagation rate is strongly influenced by the load amplitude and thus the load ratio

R. In general, for the same maximum load, an increase in amplitude, thus a decrease in R-value

results in an increase in crack propagation rate and consequently a decrease in test time.49 Also

a, limited, effect of temperature is measured, and an increase in environmental temperature

results in higher crack propagation rates. Results of tests on PC are presented in Figure 4.4.

With larger load amplitudes, thus decreasing R-values, the dynamic compliance increases for all

crack lengths and the differences between the curves increase with increasing crack length, see

Figure 4.4a. Figure 4.4b shows that with increasing temperature the calibration curves shift in

vertical direction towards higher compliance values. Clearly, no unique relation of crack length

as function of the dynamic compliance results. Using a single reference curve could easily result

in errors in crack length of 30%. Next ∆C is normalized to yield ∆Ux. An apparent modulus is

chosen such that the initial normalized dynamic compliance equals that of the reference curve

at the initial crack length. The results of this normalization are presented in Figure 4.4c and d;

in both cases the reference curve was chosen at R = 0.1 and 23◦C. To match the normalized

dynamic compliance using this reference curve, the apparent moduli had to vary approximately

30%, which strongly exceeds the allowable range (10%) according to ASTM standard E647. Note

that the 30% variation also exceeds the 10% change expected due to an increase in temperature

as observed in DMTA. Despite the correction by normalization that gives an improvement for the

data at room temperature, see Figure 4.4c, with increasing temperatures deviations increase, see

Figure 4.4d. The use of the apparent modulus corrects for the translations in vertical direction

at higher temperatures, but the shift is by a constant factor, thus also the curvature changes.

68

4.4. Results

The use of the normalized dynamic compliance clearly does not result in a unique relation valid

for the complete range of R-values and temperatures. To illustrate this even more clearly, also

data of other R-values are added in gray in Figure 4.4d. If the reference curve is determined

on a short-term measurement (e.g. R = 0.1 at room temperature), the dynamic compliance

measured for higher R-values and temperatures would yield crack length values that strongly (in

the order of 5 mm) underestimate the physical crack length. Therefore, the use of the dynamic

compliance or the normalized one to calculate crack lengths, give insufficiently accurate results.

4.4.2 The influence of load

The time-scale of an experiment is directly related to the crack propagation rate, which increases

with increasing maximum stress intensity factor, that scales linearly with the (maximum) load

applied. Figure 4.5a plots the dynamic compliance versus crack length for applying maximum

loads of 250N, 350N and 400N, while maintaining R = 0.1.

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

crack length [mm]

∆C [µ

m/N

]

R=0.1R=0.2R=0.3R=0.4R=0.5R=0.6

Fmax = 800NT = 23◦C a

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

crack length [mm]

∆C [µ

m/N

]

23°C,800N70°C,500N90°C,500N

R = 0.1 b

0 5 10 15 20 25 300.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

crack length [mm]

∆Ux [−

]

cT = 23◦C

0 5 10 15 20 25 30

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

crack length [mm]

∆Ux [−

]

23°C70°C90°C d

Figure 4.4: Data on PC: Calibration curves of the dynamic compliance versus crack length at 23◦C (a,c), and

different temperatures at R = 0.1 (b,d). Figures (a,b) are raw data; (c,d) plot the lines normalized with apparent

moduli that make the lines fit to the reference curve (solid markers) at the initial crack length. In (d) the markers

in colour represent the data in (b) and markers in gray all the available data.

69


0 5 10 150

5

10

15

crack length [mm]

∆C [µ

m/N

]

Fmax

=400N

Fmax

=350N

Fmax

=250N

R=0.1

a0 5 10 15

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

crack length [mm]

∆Ux [−

]

Fmax

=400N

Fmax

=350N

Fmax

=250N

R=0.1

b

Figure 4.5: Calibration curves of a) the dynamic compliance and b) the normalized dynamic compliance versus

crack length, for conditioned PA46, measured at 23◦C using R = 0.1. In Fig.4.5b coloured markers indicate the

data corrected using an apparent modulus, E′, and the gray markers data without correction.

At small crack lengths, the dynamic compliance increases with increasing load and the curves

diverge at larger crack lengths. Again, no unique relation is found. Figure 4.5b plots the

normalized dynamic compliances of these curves with in gray the results using a single modulus

for all experiments (1GPa). Normalization of the coloured curves is done such that the compliance

at the start of each experiment equals the one for the 400N test, and the apparent moduli required

were 9% and 25% larger for the 350N and 250N measurements respectively. Where the curves

for the two maximum loads overlap, the one for the lowest load deviates over the entire range of

crack lengths. Using the final value at the largest crack length for normalization would clearly

yield better results, but such a procedure requires to obtain reference crack lengths at multiple

times during the experiment, similar to an approach suggested by Berer et al.31 Many of the

advantages of the use of the compliance method would, however, vanish by doing this.

4.4.3 Variations in initial crack length

Small initial cracks correspond to a lower initial stress intensity factor and, as a result, lower initial

crack propagation rates. Initial crack sizes are varied by releasing the pendulum (which taps the

razor blade into the notch root) from different heights. Results for PA46 samples measured at

140◦C, are shown in Figure 4.6.

As expected, the (via optical tracking obtained) crack propagation rate as function of the stress

intensity factor is, within experimental error, independent of the initial crack length (Figure 4.6a).

However, again no unique relation between the dynamic compliance and crack length is found,

see Figure 4.6b. Differences between curves increase with increasing test times (i.e. decreasing

initial crack length and increasing R-value).

70

4.4. Results

1 1.2 1.4 1.6 1.8 2 2.5 3

10−7

10−6

10−5

10−4

Kmax

[MPa⋅m0.5]

crac

k pr

opag

atio

n ra

te [m

/s]

large aini

small aini

R = 0.1

R = 0.3

a0 5 10 15

0

5

10

15

20

crack length [mm]

∆C [µ

m/N

]

large aini

small aini R = 0.1

R = 0.3

b

Figure 4.6: a) Crack propagation rates versus stress intensity factor, from crack lengths obtained via optical

tracking and b) the calibration curves for the dynamic compliance versus crack length, on PA46 with different

initial crack sizes measured at 140◦C using a maximum load of 300N with R = 0.1 and R = 0.3.

4.4.4 Confirmation: a HDPE pipe grade

To confirm the results obtained, we test a pipe grade of HDPE (PE100) using different initial

crack lengths, maximum loads and R-values. The calibration curves resulting from the fatigue

crack propagation measurements are presented in Figure 4.7.

0 5 10 15 200

0.5

1

1.5

2

crack length [mm]

∆C [µ

m/N

]

Fmax

=1750N

Fmax

=1500N

Fmax

=1400N

Fmax

=1250N

Fmax

=1000N

R = 0.1

a0 5 10 15 20

0

0.5

1

1.5

2

crack length [mm]

∆C [µ

m/N

]

Fmax

=1750N

Fmax

=1500N

Fmax

=1250N

Fmax

=900N

R = 0.3

b

Figure 4.7: Calibration curves of the dynamic compliance versus crack length for HDPE at 23◦C and various

applied maximum loads, for a) R = 0.1 and b) R = 0.3.

This figure shows that:

- A decrease in dynamic compliances is found with increasing R-value and constant maximum

load.

- An increase in dynamic compliance is found with increasing maximum load and constant

R-value.

71


- Deviations between different curves increase for increasing crack lengths with increasing

test time (decreasing forces, increasing R-values and/or decreasing initial crack length).

- For smaller initial crack lengths the differences slightly increase with increasing crack length.

The experiments illustrate that, when performing a study to determine the crack propagation

kinetics under varying experimental conditions (such as maximum load, R-value, initial crack

length, and temperature), large deviations in the dynamic compliance can be expected. There-

fore, it is impossible to use a single calibration curve as a reference to accurately describe the

crack length, neither as function of the dynamic compliance, ∆C, nor as function of the nor-

malized dynamic compliance, ∆Ux.

4.5 Discussion

Results of the experiments on (normalized) calibration curves obtained under different test con-

ditions can be summarized as follows: (i) The dynamic compliance increases with increasing

temperature and/or maximum load and with decreasing R-value, and (ii) Deviations between

different (normalized) calibration curves increase for larger crack lengths, thus with increasing

test times (e.g. via lower loads, higher R-values and/or smaller initial crack lengths). The effect

of temperature is easy to understand, since the modulus decreases with increasing temperature50

and therefore does its reciprocal, the compliance, increase. We will investigate the influence of

non-linearity in stress-strain curve and that of time- and rate-dependency, typical for polymers,

in somewhat more detail.

4.5.1 Changes in compliance

Figure 4.8 shows the force-displacement response of polymers under constant deformation rate:

linear up to a limited strain (approx.0.2%), and non-linear at higher strains. Figure 4.8a illus-

trates that increasing the maximum load (at the same R-value) decreases the modulus, and

therefore increases the compliance. Figure 4.8b illustrates that with increasing R-value while

keeping the maximum load constant the modulus decreases and the compliance increases. In

contrast, all results on the Compact Tension-specimens show exactly the opposite, and the dy-

namic compliances decrease with decreasing amplitude. This could be caused by time- and rate

effects that were completely neglected.

To investigate this further we plot the dynamic compliance as function of time in Figure 4.9a

of standard tensile bars where no crack propagation occurs for different maximum loads and

R-values. Clearly, the dynamic compliances increase with increasing load for all R-values (as ex-

pected according to Fig.4.8a). Further for the lowest two loads (and at short time-scales for the

highest load), the dynamic compliance increases with increasing R-value, also according to the

expectations of Fig.4.8b. However, the dynamic compliances decrease in time, and the rate of

72

4.5. Discussion

0

0

displacement

forc

e

a

F+max

F−max

1∆C

0

0

displacement

forc

e

b

1∆CR+

1∆CR−

Figure 4.8: Influence of non-linear response on the dynamic compliance a) for different maximum loads while

maintaining the same R-value and b) different R-values with the same maximum load.

decrease increases with increasing R-value and with increasing maximum load. In other words,

the rate of decay increases with increasing mean load. Interestingly, for the highest load the

decay in compliance is for large R-values sufficiently fast to result in a decrease in compliance

with increasing R-value, already after 300 seconds of loading.

We can conclude that the dynamic compliance of polymers is strongly loading condition and

time dependent, even when no crack propagation occurs.

a

R↑F ↑max

R↑

1050N

1575N

2100N

b

# cycles

Figure 4.9: Dynamic compliance on tensile specimens: a) Evolution of the dynamic compliance in time for several

maximum loads and R-values and b) the force-displacement response corresponding during the 10th, 100th, 1000th

and 7000th cycle under a maximum load of 2100N and R = 0.4.

After loading for a different number of cycles (R = 0.4; highest maximum load), the permanent

displacement due to cumulative plastic deformation gradually increases, see Figure 4.9b, while

the dynamic stiffness increases with time under load and hysteresis decreases. The material’s

response becomes more elastic with increasing number of cycles, which is in agreement with

73


observations reported in literature.51–53

The increase in modulus and decrease in damping is related to physical ageing, that is known

to cause an increase in resistance against deformation.54 The material gets stiffer. The rate of

physical ageing depends on temperature, but also on the stress.52,55–59 Figure 4.10 illustrates the

occurrence of physical ageing during fatigue loading, by measuring the yield stress after a certain

time under load. Physical ageing is observed already at room temperature after relatively short

time-scales. These conditions are easily reached during fatigue crack propagation measurements,

even possibly before any (significant) crack growth has taken place.

100

102

104

106

58

60

62

64

66

loading time [s]

yiel

d st

ress

[MP

a]

50MPa55MPa

Figure 4.10: Evolution of yield stress of PC 101R after several loading times under R = 0.1 at 1Hz for different

values of the maximum applied stress. Markers represent measurements, lines are added as a guide to the eye.

Reproduced from Janssen et al.52

The occurrence of physical ageing during the experiments sufficiently explains why the dynamic

compliance decreases with increasing R-value and that differences in dynamic stiffness, and there-

fore the dynamic compliance, increase with the time-scale of the experiment (larger R-values,

smaller applied loads and/or initial crack lengths).

4.5.2 Crack propagation rates

Next we focus on the effect of differences in crack length on crack propagation rates as func-

tion of the stress intensity factor. Figure 4.11 shows the crack propagation rates for HDPE, as

obtained from the data presented in Figure 4.7, and determined from crack lengths obtained

from either the direct camera measurements or from crack lengths calculated using dynamic

compliance measurements via a polynomial-based master curve fitted on the compliance, ∆C,

or the normalized compliance, ∆Ux. As shown in Figure 4.12, data obtained with a maximum

load of 1750N and R = 0.1 are used as a reference.

Crack propagation rates obtained from crack length measurements using optical tracking show

significant scatter at low Kmax-values for R = 0.1. In these experiments stepwise crack propa-

74

4.5. Discussion

1.2 2 3 4 5

10−8

10−7

10−6

10−5

10−4

Kmax

[MPa⋅m0.5]

crac

k pr

opag

atio

n ra

te [m

/s]

optical tracking∆U

x

∆C

R = 0.1

a1.2 2 3 4 5

10−8

10−7

10−6

10−5

10−4

Kmax

[MPa⋅m0.5]

crac

k pr

opag

atio

n ra

te [m

/s]

optical tracking∆U

x

∆C

R = 0.3

b

Figure 4.11: Crack propagation rates versus stress intensity factor, of HDPE (PE100), obtained via either

camera or the (normalized) compliance methods. For R = 0.1 (a) and R = 0.3 (b). Lines represent Paris’ law

fits, using the parameters in Table 4.2.

gation is observed, where a plastic zone behind the crack tip causes an arrest of the crack that

lasts until the plastic zone is sufficiently deteriorated. The crack propagates further and a new

plastic zone develops; this process is repeated.46,60,61 To avoid obscuring by too much scatter,

data in this range should be excluded from the presented results. The crack propagation rates

calculated from the compliance spans a smaller range. This is caused by the consequences of

obtaining a functional polynomial description of the reference calibration curve. This description

is only valid within the range validated by the experimental range of the reference calibration

curve, hence extrapolation outside this range could cause incorrect crack lengths (e.g. negative

or extremely large). As shown in Figure 4.12, many of the measured compliance values and crack

lengths are out of the range of applicability.

The differences between the crack propagation rates obtained via the different measurements are

very small, although differences tend to increase somewhat with increasing R-value. Since the

calculated crack length underestimates the actual crack length, also the stress intensity factor

is underestimated for the (normalized) dynamic compliance data. However, note that since the

stress intensity factor scales with the square root of the crack length these differences are less

pronounced. Nonetheless, this is an issue when the crack propagation rates are used e.g. to

determine the (dynamic) fracture toughness.62

Usually the Paris’ law is used to describe crack propagation rates in the range where its logarithm

increases linearly with the logarithm of the stress intensity factor:63

a = A(R) ·Kmmax (4.10)

The pre-factor, A, is defined by the intersection at Kmax = 1, and m is the slope of the line,

and given in Table 4.2. Where for optical tracking the slopes m of the Paris law are identical for

different R-values and loads, see Figure 4.13a, the results obtained using the dynamic compliance

show that slopes increase with increasing testing times (decreasing load, increasing R-value), see

75


R=0.1

R=0.3

0.5

1

1.5

2

∆C

[µ

m/N

]

5 10 1500

20

crack length [mm]

Figure 4.12: Calibration curves of the dynamic compliance versus crack length for HDPE at 23◦C and various

applied maximum loads, for R = 0.1 and R = 0.3. The actual range of applicability of the polynomial fit (line) of

the reference curve (solid markers) is only where experimental data are available. And its boarders are indicated

with dashed lines.

Figure 4.13b. Please note that the deviations are less when using the normalized compliance,

since then the error in crack length is smaller, but the changes in slope remain.

1.5 2 3 410

−8

10−7

10−6

10−5

10−4

Kmax

[MPa⋅m0.5]

crac

k pr

opag

atio

n ra

te [m

/s]

Fmax

=1750N

Fmax

=1500N

Fmax

=1400N

Fmax

=1250N

Fmax

=1000N

optical tracking

R = 0.1

R = 0.3

a1.5 2 3 4

10−8

10−7

10−6

10−5

10−4

Kmax

[MPa⋅m0.5]

crac

k pr

opag

atio

n ra

te [m

/s]

Fmax

=1750N

Fmax

=1500N

Fmax

=1400N

Fmax

=1250N

Fmax

=1000N

dynamic compliance

R = 0.1

R = 0.3

b

Figure 4.13: Crack propagation rates versus stress intensity factor for HDPE as presented in Fig.4.11, obtained

via optical tracking (a) and the dynamic compliance (b) for R = 0.1 (open markers) and R = 0.3 (closed

markers). Each marker type represents a different measurement.

We clearly find that results obtained via the (normalized) compliance method are time-dependent.

Therefore, although the use of the (dynamic) compliance is well-established within the field of

fracture mechanics of linear materials and isotropic properties, application outside this field

should be handled with care and optical tracking of the crack propagation prevails.

76

4.6. Conclusions

4.6 Conclusions

To study the accuracy of the compliance method in fatigue crack propagation studies on poly-

mers, that are non-linear, time dependent materials, we compare optical tracking with use of the

corresponding force and displacements. Calibration curves are obtained for many loading condi-

tions on a number of different polymers. It is shown that the dynamic compliance increases with

increasing temperature and maximum load, and with decreasing R-value. Differences between

different calibration curves increase with increasing test time (lower maximum load, smaller ini-

tial crack, larger R-values), and no unique relation is found for the crack length as function of

dynamic compliance. Normalizing the dynamic compliance, making use of an apparent modulus,

still does not result in a single, accurate functional description. Therefore, it is impossible to

use a single calibration curve as a reference to sufficiently accurate describe the crack length, as

function of the dynamic compliance, ∆C, or normalized dynamic compliance, ∆Ux.

The origin of the deviations from a single calibration curve can be found in stress enhanced

physical ageing during the experiment. Physical ageing proceeds with time and is accelerated

by temperature and stress. Therefore, with increasing R-values, or increasing mean loads, a

decrease in compliance is found. When the crack lengths obtained from the dynamic compliance

are used to find the corresponding crack propagation rates as function of the stress intensity

factor, the differences appear to be minor, but the stress intensity factor is consequently under-

estimated. The parameters of the Paris’ law for each R-value obtained via optical tracking are

independent of the time-scale of the experiment, but, where optical tracking suggest a constant

slope of the Paris’ law, the slopes increase with increasing test time for the by the compliance

method obtained results. From this, it can be concluded that the use of the dynamic compliance

on non-linear, time dependent materials could result in discrepancies between the actual crack

length and crack propagation kinetics. Therefore, we should interpret results on the measure-

ments, both based on the dynamic and based on the normalized dynamic compliance, with care

and optical tracking is preferred.

optical tracking ∆C ∆Ux

R [-] 0.1 0.3 0.1 0.3 0.1 0.3

A [MPa−mm(1−m/2)s−1] 10−8.7 10−9.25 10−8.1 10−8.7 10−8.2 10−9.3

m [-] 6.8 6.8 5.6 7 5.2 6

Table 4.2: Paris’ law coefficients for the data presented in Figure 4.11.

77


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80

CHAPTER 5

Competition between plasticity-controlled and

crack-growth controlled failure in static and

cyclic fatigue of polymer systems

Abstract

The distinction between plasticity and crack growth-controlled failure can be made by comparing

a polymer’s lifetime under static loading with that under cyclic fatigue loading, with equal

load maxima. Changing static to cyclic loading by systematically increasing the load amplitude

has two consequences. Plasticity-controlled failure is postponed by a decreasing rate of strain

accumulation, while crack-growth controlled failure is significantly enhanced by accelerated crack

propagation. Phenomenology and modelling is discussed, to show that distinction between failure

mechanisms via this route is generic, and validated for a multitude of engineering polymers,

including glass-fibre reinforced variants.

Reproduced from: M.J.W. Kanters, T. Kurokawa, and L.E. Govaert. Submitted 81


5.1 Introduction

With polymers increasingly employed in load-bearing applications, the ability to predict lifetime

under specific loading conditions has become progressively important. From efforts in developing

predictive methods, and work on pressurized polyethylene pipes in particular, it is known that

three failure mechanisms restrict the lifetime of polymers (see Figure 5.1) over three different

regions in time: I) ”ductile failure”, caused by accumulation of plastic strain, II) ”brittle failure”,

caused by slow crack propagation, and III) brittle failure caused by molecular degradation.1–4 All

three mechanisms act in parallel, until one causes catastrophic failure. As stabilisation chemistry

improved over the years, region III shifted towards such long failure times that it is no longer

regarded as the limiting factor,5 and, hence, the focus is on regions I and II.

I) ductiletearing

II) brittlefracture




In the ”ductile” failure region (I) the applied stress induces accumulation of plastic deforma-

tion in time. In most cases, this leads to a failure that is accompanied with large local plastic

deformation (e.g. bulging of pipes, see Figure 5.1), followed by a ductile tearing process.6,7 In

region II, precursors of cracks grow in time until one of them becomes unstable or causes func-

tional problems (e.g. leakage by breaching the pipe wall, see Figure 5.1).3,8–12 Due to the small

macroscopic deformations involved, this failure mode is usually referred to as ”brittle”. From

this perspective, is seems reasonable to use the terminology ”ductile” and ”brittle” to decide

which mechanism has led to failure. However, failure within region I actually does not necessarily

manifest itself in large, voluminous plastic deformation before failure,13 and in some cases, e.g.

in severely annealed samples14 or with relatively low molecular weight polymers,15 the plastic

strain localization is extreme and the resulting macroscopic failure strain is low.16,17 Hence, one

could erroneously conclude that the lifetime is dominated by slow crack growth, while its origin

is in the local accumulation of plastic strain. Needless to say that, in order to be able to predict

a product’s lifetime, both failure mechanisms have to be correctly distinguished and understood.

To do so, one has to know which mechanism is actually experimentally accessed.

82

5.2. Background

In the present study, we demonstrate that a critical comparison of the polymer’s response under

cyclic loading with that under static loading (at equal maximum load) allows a direct identifica-

tion of the active failure mechanisms, plasticity versus crack-growth controlled failure. First the

phenomenology of both failure mechanisms is addressed, including the corresponding methods

to predict failure in both regions. Subsequently, we compare the lifetime of multiple engineering

polymers, including glass-fibre reinforced ones, under static and cyclic fatigue loading to iden-

tify the exact failure mechanism. Finally, the definitions of ”ductile” and ”brittle” failure are

scrutinized in somewhat more detail in the discussion section.

5.2 Background

5.2.1 Crack-growth controlled failure

Small initial flaws, induced by either processing or handling, result in stress concentrations inside

loaded materials. They eventually lead to initiation, and subsequent propagation, of a craze or

crack through the material, to finally cause functional problems (like disintegration of a structure,

or leakage of a pressurized pipe).3,8–12

Stresses around a crack tip are quantified using Linear Elasticity Fracture Mechanics (LEFM),18

and scale with the stress intensity factor, K, which for a crack opening loading (mode I) is

defined by:

KI = Y σ√πa (5.1)

where σ is the remote stress, a the crack length, and Y a geometry factor, which usually depends

on the crack length a. The crack propagation rate, a, is related to the stress intensity factor by

a power law; a relation known as the Paris’ law:19

a = A ·KmI (5.2)

Plotting the crack propagation rate versus the stress intensity factor on a double logarithmic

scale, defines the pre-factor, A, by the intersection at KI = 1, while m is the slope of the line.

Both A and m are regarded to be material parameters. From this perspective it is understood

that the time up to failure under a constant load, tf , caused by slow crack growth, can be

calculated by integrating the crack propagation rate, Equation 5.2, between a certain initial flaw

size, ai, and the crack length at which failure occurs, af ,3,9,20 which using Equation 5.1 yields:

tf − ti =1

Aσm

af∫ai

da

(Y√πa)

m (5.3)

Assuming that the time of initiation, ti, is negligible compared to the lifetime, this can be reduced

to:

tf =

(σ

cf

)−mwith cf = A−

1m ·

af∫ai

da

(Y√πa)

m

1m

(5.4)

83


Equation 5.4 illustrates that the time-to-failure is given by a power law, with the normalizing

factor, cf , that defines a lifetime of 1 second. For constant geometries and identical initial flaw

sizes, cf scales with Paris’ law pre-factor A−1m . In a double logarithmic plot of applied stress

versus the time-to-failure typically a linear relation is found, while the slope equals the reciprocal

of the Paris’ law exponent m.

From experimental studies on crack growth kinetics, it is known that in cyclic loading the crack

propagation rate is significantly enhanced.21–28 In cyclic fatigue, one can vary the minimum load,

mean load, maximum load, load amplitude, and of course frequency. In this work, the load signal

is characterised by the frequency, f , the load maximum, and the load amplitude, expressed in

the load ratio, R:

R =FminFmax

(5.5)

As illustrated in Figure 5.2, R = 1 represents static loading conditions, while decreasing the

R-value makes the load amplitude increase. The stress intensity factor at the load maximum,

Kmax, is used to define the load applied, and the corresponding fatigue crack propagation rate,

Equation 5.2:

a = A ·Kmmax (5.6)

The pre-factor A and m are the parameters, but only A depends on the load ratio, the frequency,

but also on the molecular weight of the polymer used and the temperature.24,28–31 Therefore, to

describe the time-to-failure under a constant maximum load, Equation 5.4, only the normalizing

factor cf varies with load ratio and frequency.

R = 1

R = 0.55

R = 0.1

Fmax

Fmin

R↑

Figure 5.2: Schematic illustration of the static and cyclic loading and how the load ratio R effects the load

amplitude.

When cyclic fatigue is performed on Compact Tension specimens (CT-specimens, made of

polyetherimide (PEI 1010), here just as an example), the crack propagation rate increases with

84

5.2. Background

increasing load amplitude (decreasing load ratio, R), see Figure 5.3a. As a result, the time-to-

failure decreases with increasing load amplitude also if the cyclic experiments are performed on

smooth bars (Figure 5.3b).

This enhanced crack propagation is related to failure of fibrillae bridging the craze zone pro-

ceeding the crack tip.31,32 The fibrillae support part of the load, which causes them to, slowly,

deteriorate until they finally fail.33 As a result, the crack propagation rate is largely determined

by the rate of failure of the fibrillae. During static fatigue, the mechanisms leading to fibril failure

are believed to be disentanglement or chain scission.28–30,34–37 During cyclic loading the fibrils

are alternatingly stretched and compressed. It is hypothesised that during crack closure bending

and, for sufficiently large amplitudes, buckling or even crushing of fibrils occurs,38 provoking

enhanced fibril failure and increased crack propagation rates. As a result, the times-to-failure

decrease under cyclic loading with larger load amplitudes (smaller load ratio’s R), and at higher

frequencies.38,39

Figures 5.3a and b clearly show that the slope, determined by the Paris’ law exponent, m, is

independent of R-value. The only variable changing with the load ratio is A. The same value

for m is used to describe the crack propagation measurements on CT-specimens (Equation 5.6)

and the times-to-failure measured on smooth bars (Equation 5.4). The parameters are presented

in Table 5.1.

Material sample parameters m [-]

PEI 1010

CT- R 0.1 0.3 0.5

4.9specimens A 1.31 · 10−7 5.07 · 10−8 1.09 · 10−8

smooth R 0.1 0.2 0.4 0.6

bars cf 339.5 367.5 475 684.5

Table 5.1: Parameters to describe the crack growth rate of CT-specimens (Fig.5.3a) and the crack-growth

controlled failure of smooth bars (Fig.5.3b) for PEI 1010 at 23◦C for each R-value, using Equations 5.6 and 5.4.

R is dimensionless, A in MPa−mm(1−m/2)s−1 and cf in MPa·s1/m.


In solid polymers, the application of a stress results in an increase of the molecular mobility,40,41

which expresses itself in a constant rate of plastic flow.42 The material cannot sustain this flow

indefinitely and eventually failure is observed. Plasticity-controlled failure can accurately be

described via the stress and temperature dependence of the plastic flow rate during secondary

creep combined with a critical amount of plastic strain that triggers failure.14,43–50 The time-

to-failure can therefore be calculated by observing this plastic flow rate in time until the total

85


0.6 0.8 1 2 3 4

10−8

10−7

10−6

10−5

Kmax

[MPa⋅m0.5

]

cra

ck p

rop

ag

atio

n r

ate

[m

/s]

R0.1

R0.3

R0.5

R↑

a b

R↑

Figure 5.3: PEI 1010 at 23◦C: a) Crack propagation rate versus maximum stress intensity factor, measured on

CT-specimens. Markers represent measurements, lines descriptions using Equation 5.6. b) Time-to-failure versus

maximum load applied for several R-values, measured on smooth bars. Markers represent measurements, lines

descriptions using Equation 5.4. Lines and markers in gray indicate failure due to plasticity-controlled failure.

amount of accumulated plastic strain exceeds this critical value:

εpl(t) =

t′∫0


where εpl is the plastic strain at a certain time, εpl the plastic flow rate for the load and

temperature applied, and εcr the plastic strain at failure. This critical value is the amount of

strain that would have been accumulated if the material would deform with the constant plastic

flow rate during secondary creep for its entire lifetime and is smaller than the actual strain at

failure. However, this phenomenological measure enables a quantitative prediction of the times-

to-failure.

As was first demonstrated by Bauwens-Crowet et al.,43 the steady state reached at the yield point

in a constant strain rate experiment is identical to the steady state reached during secondary

creep and, therefore, we can use the stress- and temperature-dependence measured in well-

defined, short-term constant strain rate experiments, to describe the kinetics of the plastic flow

rate in creep loading. In the simplest case, where a single process governs the deformation, this

can be described using Eyring’s activated flow theory:51

εpl (σ, T ) = ε0︸︷︷︸I

exp

(−∆U

RT

)︸︷︷︸

II

sinh

(σV ∗

kT

)︸︷︷︸

III

(5.8)

Part (I) of Equation 5.8 is a rate factor, ε0. The exponential term in part (II) covers the temper-

ature dependence and part (III) captures the stress dependency of the material, where σ is the

yield stress, V ∗ the activation volume, ∆U the activation energy, R the universal gas constant, k

the Boltzmann’s constant and T the absolute temperature. In most cases only the parameter ε0

86

5.2. Background

depends on the thermodynamic state of the material (age, crystallinity). To obtain a descriptive

method for any arbitrary three-dimensional load, equivalent terms can be used for the stress and

strain rate in Equation 5.8 and the hydrostatic pressure is taken into account.48,52

As Figure 5.4a shows for polycarbonate, PC, as an example polymer here, this model allows an

accurate description of the strain rate and temperature dependency of the yield stress, and, in

combination with the critical strain, this enables an accurate prediction of the stress and tem-

perature dependence of the time-to-failure (Figure 5.4b). Note that both plots yield a linear

relation using a semi-logarithmic scale, with the same absolute slope, α, albeit with opposite

sign.

10−5

10−4

10−3

10−2

10−1

40

50

60

70

strain rate [s−1]

yiel

d st

ress

[MP

a]

20°C40°C60°C

α

a10

210

310

410

5

40

50

60

70


appl

ied

stre

ss [M

Pa]

20°C40°C60°C

b

−α

Figure 5.4: PC: Strain rate dependence of the yield stress (a) and time-to-failure versus load applied (b)

for several temperatures, reproduced from Visser et al.48 Markers represent measurements, lines descriptions

according to Equations 5.7 and 5.8, using the parameters as presented in Table 5.2.

Material V ∗ [nm3] ∆U [kJ/mol] ε0 [s−1] εcr [-]

PC 3.52 327 3.2 · 1032 0.0075

Table 5.2: Eyring-parameters and the critical strain for PC obtained from the strain rate and temperature

dependency of the yield stress and the time-to-failure data presented in Figure 5.4, reproduced from Visser et

al.48

Where a constant load yields a constant plastic flow rate, a cyclic load during fatigue loading

results in an oscillating plastic flow rate, as illustrated in Figure 5.5a. For the same maxi-

mum load, the amount of plastic strain accumulated in time therefore decreases with increasing

load amplitude and, since plasticity-controlled failure occurs once a critical amount of plastic

strain is reached, the time-to-failure increases with increasing load amplitude,44,48,53 as clearly

demonstrated in Figure 5.5b. The time-to-failure for both static and cyclic loading conditions is

accurately described using the deformation kinetics, Equation 5.8, in combination with the time

87


dependent load signal and the critical strain in Equation 5.7. Interestingly, the time-to-failure is

independent of frequency.14,44

time

stre

ss /

log(

plas

tic s

trai

n ra

te)

a

dynamic

static

R↓

b

dynamic

static

Figure 5.5: a) Schematic representation of the dynamic load and its resulting plastic flow rate. b) Time-to-

failure versus maximum applied load for PC, reproduced from Janssen et al.44 Markers represent measurements,

lines descriptions according to Equations 5.7 and 5.8.

The situation where a single process governs the deformation in the plasticity-controlled failure

region is the simplest case. For most polymers reality is sometimes more complex, either be-

cause another molecular process is contributing to the deformation, or because the properties of

the polymer change during the experiment (physical aging). The driving force behind physical

ageing is the strive towards thermodynamic equilibrium. An increase in density, modulus and

yield stress results.54,55 The rate at which this process proceeds depends on molecular mobility,

which is enhanced by temperature,1 but also by applied stress.44,48,53,55–57 Under some condi-

tions, ageing alters the material’s properties significantly during the experimental time-scale, a

phenomenon generally referred to as ”progressive ageing”. As a result, the time-to-failure in-

creases.14,15,44,53,55 An example is presented in Figure 5.6a, which shows the evolution of yield

stress of unplasticized poly(vinyl chloride), uPVC, under load. In Figure 5.6b the time-to-failure

is presented for as-manufactured and annealed uPVC; the dashed lines represent the prediction of

the failure times using the basic approach (Equations 5.7 and 5.8), and is a good approximation

for the annealed sample. For the as-manufactured material the fit is suiting for short failure

times, but starts deviating when low stress levels are applied, where the long times-to-failure

allow ageing with its increase in resistance against deformation. The response seems to evolve

towards that of the annealed samples. Progressive ageing can be included in the model58 by

modifying the rate factor, ε0, such that it becomes a function of the effective time, teff :

ε0 = ε0,rej exp (−Sa(teff )) (5.9)

1Of course, at temperatures close to the glass transition temperature, the material can reach equilibrium and

the ageing process will actually be decelerated by a further increase in temperature.54

88

5.2. Background

where ε0,rej is the rate factor for the unaged material, and Sa a state parameter that uniquely

determines the thermodynamic state. The evolution of the state parameter depends on the

effective time, which magnitude is increased by temperature (Arrhenius type time-temperature

superposition) and stress (Eyring type time-stress superposition).55,59 The result of the charac-

terisation for uPVC and combining Equations 5.7, 5.8, and 5.9 results in the solid lines in Figure

5.6b, Visser et al.14 Full characterisation of the ageing kinetics of all materials presented in this

chapter lies beyond the scope of this work, but one should be aware of its effect.

a

σ↑a

102

103

104

105

106

30

40

50

60


appl

ied

stre

ss [M

Pa]

as−manufacturedannealed

b

Figure 5.6: uPVC at 23◦C: a) Evolution of the yield stress versus anneal time for two loads applied. b) Time-to-

failure versus load applied for as-manufactured and annealed samples. Both figures are reproduced from Visser

et al.14 Markers represent measurements, gray lines are descriptions according to Equations 5.7 and 5.8 without

(dashed lines) and including ageing (solid lines).

In case that another molecular process is contributing,60–65 the stress dependency deviates from

a simple linear relation when a sufficiently large range of stresses and temperatures is covered.

The yield stress versus the logarithm of the applied strain rate may display a change in slope,

either by a contribution of a secondary glass transition (partial main-chain or side-chain mo-

bility)61,62,66,67 or, in the case of semi-crystalline polymers, an additional contribution from a

second phase (crystal).68,69 A successful way to model such behaviour was proposed already in

the early 50’s by Ree and Eyring.60 Based on the assumption that the two molecular processes

act in parallel, the resulting stress is just the sum of the stress contributions of both processes,

each described by an Eyring expression:

σ(ε, T ) = σI(ε, T ) + σII(ε, T )

=kT

V ∗Isinh−1

(ε

ε0,I

exp

(∆UI

RT

))+kT

V ∗IIsinh−1

(ε

ε0,II

exp

(∆UII

RT

))(5.10)

Each process has its own activation energy, ∆Ux, activation volume, ∆V ∗x , and rate factor, ε0,x,

where x = I,II. It is now less straightforward to determine the plastic flow rate as function of the

load applied, since the total stress is distributed over two deformation mechanisms. A solution

can be achieved numerically using straight-forward optimization methods which, in combination

89


with Equation 5.7, allow for an accurate description of the deformation kinetics and the stress

and temperature dependence of the time-to-failure, as shown in Figure 5.7 for polyetherimide,

PEI 1000, here just as an example polymer.

10−5

10−4

10−3

10−2

10−1

100

40

60

80

100

120

strain rate [s−1]

yiel

d st

ress

[MP

a]

23°C60°C100°C120°C

a10

110

210

310

410

5

40

60

80

100

120

time−to−failure [s]ap

plie

d st

ress

[MP

a]

23°C60°C100°C

b

Figure 5.7: PEI 1000: Strain rate dependence of the yield stress (a) and time-to-failure versus load applied

(b) for several temperatures. Markers represent measurements, lines descriptions according to Equations 5.7 and

5.10, using the parameters as presented in Table 5.3

Material x V ∗x [nm3] ∆Ux [kJ/mol] ε0,x [s−1] εcr [-]

PEI 1000I 2.85 335 4 · 1029

0.015II 2.9 85 1 · 109

Table 5.3: Ree-Eyring-parameters and the average critical strain for PEI 1000, obtained from the strain rate

and temperature dependency of the yield stress and the time-to-failure data presented in Figure 5.7.

5.2.3 Distinction between failure mechanisms

At a constant value of the maximum load applied, a change from static to cyclic loading has

a different effect on each of the two failure mechanisms, as illustrated in Figure 5.8. Due to

a decrease in rate of plastic strain accumulation, plasticity-controlled failure (region I) shifts

towards longer failure times with increasing load amplitude, or decreasing R-value. In contrast,

the crack-growth controlled failure (region II) shifts towards shorter failure times, due to an

increase in crack propagation rate with increasing load amplitude. This makes the comparison

of the lifetime under cyclic and under static load a useful tool to distinguish which of the two

mechanisms is actually active and determining failure. To do so, at a chosen maximum load,

different R-values should be used, preferably over a very large load amplitude range (for example

R = 0.1 and R = 1). Then simply check whether the time-to-failure is delayed or advanced.

90

5.2. Background

plasticity-controlled

crack-growth

dynamic

static

Figure 5.8: Schematic illustration of the influence of dynamic loading on plasticity-controlled (region I) and

crack-growth controlled (region II) failure.

5.2.4 Characterisation and distinction

Although comparison of the lifetime under static fatigue with that under cyclic fatigue loading

is a rather straight-forward approach, it remains difficult to decide which loads are interesting to

apply. There are some basic steps that one can take when characterising a material:

1. Perform constant rate experiments at several strain rates (and temperatures) to find the

rate (and temperature) dependence of the yield stress. A suitable range would be strain

rates in the order of 10−1 − 10−5 s−1, with typical standardized strain rates being in the

order of 10−2 − 10−3 s−1.

2. Perform static fatigue experiments to determine the critical strain. Start by applying a

stress equal to the yield stress at 10−3 s−1, which typically results in time-to-failures of

approx. 100 seconds (depending on the critical strain), and start decreasing the stress

based on the kinetics from the rate experiments (remember, similar absolute slopes, but

with opposite signs). Be aware of physical ageing and multiple deformation processes.

This offers a description of the plasticity-controlled failure under static fatigue, and offers the

range of loads that are interesting to apply. One of these loads can be taken as the load maxi-

mum and the load ratio should be varied. The observed times-to-failure can be compared.

Although the change in lifetime with varying load ratio for the different failure mechanisms

is valid for the majority of stresses applied, there are exceptions. For example near the transition

zone, where failure switches from plasticity-controlled to crack-growth controlled with a further

decrease of loads applied, it might be that the time-to-failure increases with decreasing R-value,

even though the dominating mechanism is crack growth. In these cases some additional features

of both mechanisms can be used for distinction:

91


• The stress dependency of each mechanism: plasticity-controlled failure yields a linear

relation of the stress dependency in a semi-logarithmic plot, with in cyclic fatigue the

same or smaller (ageing) slope as in static fatigue. Crack-growth controlled failure yields

a linear relation in a double-logarithmic plot, with a much steeper slope.

• The effect of frequency: Plasticity-controlled failure is independent of frequency, while

crack-growth controlled failure is accelerated by frequency. But be aware of viscous heating

of the sample when too high frequencies are applied because this will also cause the

plasticity-controlled failure to shift to shorter time-scales.

5.3 Experimental

5.3.1 Materials

The unfilled polymers used are polycarbonate, polyphenylsulfone and polyetherimide. Addition-

ally a number of fibre reinforced polymers are used: polyetherimide, a polyphenylene-ether/poly

styrene blend, polycarbonate, polyamide 46, polyphthalamide, and polyphenylene sulfide. Two

polyetherimide grades, ULTEM™ 1000 resin (PEI 1000) and ULTEM™ 1010 resin (PEI 1010),

30% glass-fibre reinforced polyetherimide (GFR PEI) (ULTEM™ 2300 resin), 30% glass-fibre re-

inforced polyphenylene-ether/polystyrene blend (GFR PPE/PS) (NORYL™ FE1630PW resin), a

polycarbonate (PC) (LEXAN™ 143R resin) and 30% glass-fibre reinforced polycarbonate (GFR

PC) (LEXAN™ 141R resin with 30% non-adherent glass fibres) are provided by SABIC Innovative

Plastics, Bergen op Zoom. The 30% glass-fibre reinforced PA46 (GFR PA) is provided by DSM

Geleen (Stanyl® TW200F6). The polyphenylsulfone (PPSU) is provided by Solvay Speciality

Polymers (Radel® R-5000). The 40% glass-fibre reinforced polyphthalamide (GFR PPA) and

40% glass-fibre reinforced polyphenylene sulfide (GFR PPS) are commercially obtained (EMS

Grivory® HT1V-4 FWA and Ticona Fortron® 1140L4, respectively). PEI 1000 is obtained as

0.5 mm thick extruded sheets from which dog-bone shaped samples (ISO 527 Type 5A) are

punched. PEI 1010, 30%GFR PEI, and 30%GFR PA are obtained as injection moulded tensile

bars, according to ISO 527 Type 1A test specimen specifications (cross-sectional area of 4x10

mm2). All other materials are obtained as granules from which tensile bars are injection moulded

according to ASTM D638 Type I test specimen specifications (cross-sectional area of 3.2x13.13

mm2).


Uniaxial tensile tests are performed using Z010 Zwick Material Testing Machines, equipped

with 10 kN load-cells. All measurements above room temperature are performed on a machine

equipped with a temperature chamber. To characterise the deformation kinetics, uniaxial tensile

tests are performed, at least in duplicates, at strain rates ranging from 10−5 s−1 up to 10−1 s−1.

92

5.4. Results

The creep measurements are performed using a wide range of applied stresses, that are applied

in 10 seconds. Cyclic experiments are performed on servo-hydraulic MTS Testing Systems,

equipped with 25 kN load cells and temperature chambers. A sinusoidal load is applied up

to failure; for each test the load amplitude (R-value), maximum load, and frequency are kept

constant.

5.4 Results

The time-to-failure in static and cyclic fatigue is measured on a multitude of polymeric materials:

PC, PPSU , uPVC,14 PEI 1000 and 1010, GFR PC, GFR PEI, GFR PA, GFR PPS, and GFR

PPA, and the results are presented in Figures 5.9 and 5.10. For all the presented materials a

clear distinction can be made between a region where failure is caused by accumulation of plastic

strain and a region where failure is caused by crack propagation. The plasticity-controlled region,

with the same, rather low slope as static loading (R = 1), can be observed at high stresses and

short time-scales and the time-to-failure increases for increasing load amplitude or decreasing

R-value. Failure in the crack-growth controlled region shows a higher slope and the lifetime

for equal maximum load decreases significantly with increasing load amplitude or decreasing R-

value. The change in lifetime with varying R-value is strongest for the crack-growth controlled

failure. Where the plasticity-controlled failure might show some curvature in a double logarithmic

plot, a linear trend is found for failure in the crack-growth controlled region. These observations

demonstrate that this distinct response in cyclic and static fatigue appears generic for all polymer

systems investigated.

Physical ageing and multiple deformation processes, as discussed in the background on plasticity-

controlled failure, can also be recognized in the presented figures. For most unfilled materials,

uPVC, PEI 1000, PC, and PPSU, ageing can be observed within the plasticity-controlled failure

region and should be taken into account. Ageing is more pronounced when a cyclic load is

applied, as already discussed in Chapter 3. PEI 1000, GFR PPA, and GFR PPS, display a change

in slope due to multiple deformation processes. The lines added in the figures, except where

the data displays physical ageing, are descriptions using the simple approaches as presented in

Equations 5.4 and 5.7. They show that the stress dependency in each region can accurately be

described. The results from the descriptions in the crack-propagation controlled region validate

that, for each material, the slope m is independent of load amplitude and temperature applied.

93


a

PC at 23◦C

b

PPSU

75◦C

125◦C

175◦C

c

uPVC at 23◦C

d

PEI 1000

23◦C

100◦C

e

PEI 1010 at 23◦C

f

Region I: plasticity-controlled failure

Region II: crack-growth controlled failure

Figure 5.9: Time-to-failure versus maximum load applied (log-log) under static and cyclic loading conditions

using several R-values: a) PC at 23◦C at 1Hz. b) PPSU at several temperatures, at 1Hz. c) uPVC, obtained

from Visser et al.14 During the cyclic experiments, the minimum load is kept constant at 2.5 MPa (1Hz). d)

PEI 1000 at 23 and 100◦C at 0.3Hz. e) PEI 1010 at 23◦C at 1Hz, as presented in Figure 5.3b. f) Samples after

failure of PEI 1010, corresponding to each failure region.

94

5.4. Results

a

GFR PC at 90◦C

b

GFR PPA at 90◦C

c

GFR PA at 23◦C

d

GFR PPS at 90◦C

GFR PEI at 23◦C

e

Region I: plasticity-controlled failure

Region II: crack-growth controlled failuref

Figure 5.10: Time-to-failure versus maximum load applied (log-log) under static and cyclic loading conditions

using several R-values for glass-fibre reinforced (GFR) polymers at 1Hz: a) GFR PC at 90◦C, b) GFR PPA at

90◦C, c) GFR PA at 23◦C, d) GFR PPS at 90◦C, e) GFR PEI at 23◦C, f) Samples after failure of GFR PEI,

corresponding to each failure region.

95


5.5 Discussion

For all polymer systems presented in Figures 5.9 and 5.10, the amplitude and stress dependency

of the time-to-failure clearly display the two distinct failure regions: plasticity-controlled failure

(region I) and crack-growth controlled failure (region II). The macroscopic deformation at failure

in each region is, however, rather different for the different polymer systems, as illustrated in

Figure in Figures 5.9f and 5.10f. Unreinforced polymers, such as PEI 1010, shown in Figure

5.9f, often display large macroscopic deformations (necking) in the plasticity-controlled region.

In the crack-growth controlled region, however, the sample fails at macroscopically small strains

due to (several) small cracks propagating through the specimen. For reinforced polymers, such

as GFR PEI in Figure 5.10b, the strain at break is typically very small,70–72 and the behaviour

appears rather brittle, and the (macroscopic) deformation at failure is the same in both regions.

To clarify this issue, the behaviour of the reinforced system GFR PPE/PS is discussed in more

detail.

a

GFR PPE/PS

ε↑,T ↓

b

GFR PPE/PS at 23◦C

σ↑

Figure 5.11: GFR PPE/PS: a) Stress-strain response in constant strain rate experiments for 23◦C and 90◦C.

b) Sherby-Dorn plots: Evolution of strain rate versus strain at 23◦C for several loads applied.

Figure 5.11a shows the stress-strain response under constant strain rates at two different tem-

peratures. The strain at break is very small for all temperatures and rates applied, and at low

temperatures and high strain rates the material even breaks before reaching the yield point.

However, with increasing temperature and/or decreasing strain rate the material reaches the

yield point, indicating that a steady state of plastic flow is reached, although the polymer breaks

slightly after. Also during static fatigue the strain at break is very small. The evolution of

strain rate with strain during such a static loading experiment can be visualised using a so-called

Sherby-Dorn plot73 (Figure 5.11b). This shows that initially the strain rate decreases with in-

creasing strain (primary creep), after which a constant strain rate is observed (secondary creep),

and subsequently, after a rather small increase in strain, the sample breaks. These observations

prove that, even though the macroscopic deformation remains limited due to stress and strain

concentrations triggered by the presence of the stiff glass fibres, a yield point is reached during

96

5.6. Conclusions

constant strain rate and constant load experiments, indicating plastic flow. Comparison of the

lifetime in static fatigue with the lifetime in cyclic fatigue, for 23◦C and 90◦C, confirms that

for the higher maximum loads applied indeed the time-to-failure increases with increasing load

amplitude, verifying that, regarding all presented above, failure in this region is dominated by

accumulation of plastic strain. So even though failure occurs at macroscopically small defor-

mations, it is still caused by accumulation of plastic strain, albeit on a very local level. This

substantiates that it is more appropriate to distinguish the typical failure regions as plasticity and

crack-growth controlled failure, rather than ”ductile” and ”brittle” failure.

a


b


Figure 5.12: Time-to-failure versus maximum load applied under static and cyclic loading conditions using

several R-values for GFR PPE/PS (1Hz) at 23◦C (a) and 90◦C (b).

5.6 Conclusions

It is demonstrated that comparison of lifetimes under static and cyclic loads, with the same load

maximum, provides a generic tool to distinct between plasticity-controlled and crack-growth

controlled failure. When, for equal maximum load, the cyclic time-to-failure increases relatively

to the static one, plasticity-controlled failure occurs and is postponed, by a decreasing rate of

plastic strain accumulation. In contrast, when failure occurs faster, crack-growth controlled

failure is the mechanism, enhanced by an increase in crack propagation rate.

Plasticity-controlled failure yields a linear relation of the stress dependency in a semi-logarithmic

plot and is independent of frequency, while crack-growth controlled failure yields a linear relation

in a double-logarithmic plot and is accelerated by frequency. The stress dependency of both

failure mechanisms can be accurately described by using simple approaches based on the actual

kinetics of each failure mechanism, and it is shown that this is valid for a wide range of materials.

By applying this procedure on glass-fibre reinforced polymers, it is shown that, even though failure

might occur at very small macroscopic strains, the mechanism causing failure is not necessarily

caused by slow crack growth but can still be dominated by accumulation of plastic strain. It is

97


therefore more appropriate to distinguish between plasticity and crack-growth controlled failure

rather than between ”ductile” and ”brittle” failure regimes.


The authors would like to thank Martijn van Stiphout, Joris van der Sman, Rijn Stovers, Daan

Burgmans, and Nicky Hoofwijk for their efforts and contributions within the experimental work.

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101

CHAPTER 6

Integral approach of crack-growth in static and

cyclic fatigue in a short-fibre reinforced

polymer; a route to accelerated testing

Abstract

The use of accelerated failure in cyclic fatigue experiments to predict long-term static time-to-

failure is investigated. The influence of both frequency and load amplitude on the time-to-failure

in cyclic fatigue is extensively studied. It is shown that the fatigue crack propagation rate con-

sists of a static component (time dependent) and a cyclic component (cycle dependent). With

decreasing frequency or load amplitude, the contribution of the cyclic component in time di-

minishes, revealing the contribution of the static component. As a consequence, the number

of cycles-to-failure is only independent of frequency for large load amplitudes, and the influence

of the load ratio on the time-to-failure depends strongly on frequency. This load ratio depen-

dency, measured in cyclic fatigue, extrapolates to the same lifetime for different frequencies, and

therefore allows prediction of the long-term static performance. To summarize and conclude this

investigation, a phenomenological, crack propagation based equation is provided and validated,

that captures all relevant aspects and allows predicting long-term failure based on short-term

cyclic fatigue experiments only.

Reproduced from: M.J.W. Kanters, T.A.P. Engels, T.B. van Erp, and L.E. Govaert. In preparation 103

6 Integral approach of crack-growth in static and cyclic fatigue

6.1 Introduction

Polymers and polymer based composites are increasingly employed in load-bearing applications.

The conditions can be demanding (high temperatures, humidities), and loading usually contains

a pronounced dynamic component.1,2 From studies on the long-term performance of pressurized

plastic pipes, it is known that three failure regions can be recognized (Figure 6.1):3–6 I) ”ductile

failure”, caused by accumulation of plastic strain, II) ”brittle failure”, caused by slow crack

propagation, and III) brittle failure, caused by molecular degradation. All three mechanisms act

simultaneously until one of them initiates failure. In other words, it is not the question whether

failure will occur, but rather when or on what time-scale. In order to prevent premature failure,

it is of the utmost importance to be able to predict the long-term performance.

With the requirement of having service lifetimes in the order of 10 years or plus, real time loading

is not really an option to test loaded systems and accelerated testing methodologies have been

developed to access the long-term performance in each region within a reasonable timespan. To

access region III, the rate of molecular degradation can be enhanced by changing the service

environment, e.g. elevated temperatures and/or high concentrations of oxidiser,7 however, the

process is of chemical nature and not included in this investigation that focusses on failure due to

mechanical loading. To access region I, the rate of plastic strain accumulation can be enhanced

by stress and temperature, and the mechanism can be accessed using short-term constant rate

experiments,8 that resulted in analytical methods to predict the lifetime under both static fatigue

(as discussed in Chapter 2) and cyclic fatigue9 (as discussed in Chapters 3 and 5). From these in-

vestigations, we learned that under cyclic loading with equal maximum load, times-to-failure shift

towards longer times in region I, caused by a decreasing rate of plastic strain accumulation.9–11 In

contrast, failure in region II is enhanced by cyclic fatigue loading.12–24 For equal maximum load,

increasing the load amplitude significantly enhances the crack propagation rate, decreasing the

time-to-failure. This is believed to be related to accelerated fibril failure at the crack tip due to

alternating crack opening and closure,25,26 as will be considered in more detail in the discussion

section. Either way, applying cyclic fatigue enables studying the crack propagation kinetics of

polymers within a reasonable time-span, and therefore it is not surprisingly that many studies

have been devoted to characterise crack propagation kinetics.27–31 Most studies focus primarily

on crack propagation in cyclic fatigue, extensively summarized by Manson and Hertzberg;28 a

number of them tried correlating cyclic fatigue to static fatigue.12–16,18–22,25,26,32–34 The influence

of the main variables during cyclic loading (load amplitude and frequency) on crack propagation

rate and/or time-to-failure is systematically studied, sometimes using different temperatures.

The resulting frequency and load amplitude dependency is used to investigate the mechanism

causing (cyclic) fatigue crack propagation. Most researchers agree that the fatigue crack prop-

agation rate in polymers can be split in a fatigue and a creep component.12,20,22,25,26,30,35 Some

researchers find that the stress dependency of the crack propagation rate (and therefore the

time-to-failure) varies with frequency, load amplitude and temperature,12,22,28,30,35 others con-

104

6.1. Introduction

I) ductiletearing

II) brittlefracture




clude that it remains constant.19,20,24,33,36,37 Besides this discrepancy, all studies report a decrease

in lifetime with increasing load amplitude and/or frequency, and the majority concludes that time-

to-failure is determined by the number of cycles-to-failure. However, since lifetime under static

fatigue should be independent of frequency, the interpretation of the performance under cyclic

loading to predict that under static loading is not trivial. Lang and co-workers21,22 developed a

phenomenological method where, for constant crack propagation rates, the dependency of the

stress intensity factor on load amplitude is used to predict the crack propagation rate under static

loading, and Baer and co-workers20,36,38 introduced a method that uses the strain rate at the

crack tip, which increases with increasing frequency and load amplitude. However, a generally

accepted method that captures the load amplitude and frequency dependence, and that can ac-

tually predict the static time-to-failure is still lacking. Furthermore, since usually the frequency is

varied for a single load amplitude (and vice versa when varying the load amplitude), the number

of studies extensively varying both is only limited.

In this work, the time-to-failure in cyclic and static fatigue is investigated extensively for a mul-

titude of frequencies and load amplitudes using injection moulded smooth bars of a glass-fibre

reinforced polymer (GFRP) and an unreinforced glassy polymer (both injection moulded and

compression moulded). We realise that fibre orientation has a significant effect on the fatigue

behaviour,39–42 but studying its influence is beyond the scope of the present investigation, and

the smooth bars are regarded as samples with a constant and well-defined fibre orientation. From

the frequency and load amplitude dependency of the time-to-failure a simple, crack-propagation

based descriptive method is developed that accurately predicts the long-term static lifetime, us-

ing solely the time-to-failure measured in cyclic fatigue. This approach is validated on crack

propagation kinetics and its interpretation is compared to methods provided in the literature.

105


6.2 Background


Characterisation of plasticity-controlled failure has been discussed extensively in Chapters 2, 3,

and 5. Subjecting solid polymers to a constant load results in an increase in molecular mobility,

which is expressed in a constant rate of plastic flow.43–45 The material cannot sustain this flow

indefinitely, and eventually failure is observed. By defining a critical strain that triggers failure,8

εcr, the time-to-failure is obtained by integrating plastic flow in time:9,11,46

εpl =

t′∫0


where εpl is the plastic strain at time t′, and εpl the plastic flow rate for the applied load and

temperature, which can be characterised using constant rate experiments.8 The resulting kinetics,

i.e. the stress- and temperature dependence, of the plastic flow rate is described using Eyring’s

activated flow theory:47

εpl (σ, T ) = ε∗0 sinh

(σV ∗

kT

)(6.2)

The first term in Equation 6.2 is a rate constant, ε∗0, and the second term is a hyperbolic sine

capturing the stress dependence, where σ is the stress applied, V ∗ the activation volume, k the

Boltzmann’s constant and T the absolute temperature. Figure 6.2 shows for glass-fibre reinforced

PPE/PS (see Experimental section) that the strain rate dependency of the yield stress and the

lifetime under static fatigue as function of load applied, are accurately described by Equations

6.2 and 6.1, respectively, and using the parameters presented in Table 6.1. Via this route also the

plasticity-controlled time-to-failure under cyclic fatigue loading can be estimated, as presented

in Figure 6.2b.

Material V ∗ [nm3] ε∗0 [s−1] εcr [-]

GFR PPE/PS 1.2 3.45 · 10−20 0.0022

Table 6.1: Eyring-parameters and the critical strain for GFR PPE/PS, corresponding to the fits in Figure 6.2.

106

6.2. Background

10−5

10−4

10−3

10−2

0

50

100

150

strain rate [s−1]

yiel

d st

ress

[MP

a]

23◦C

a

R↓

23◦C

b

Figure 6.2: GFR PPE/PS: yield stress versus strain rate applied (a) and load applied versus static time-to-failure,

including prediction of the lifetime under dynamic loading for several R-values (b), at 23◦C. Markers represent

measurements, lines descriptions using Equations 6.2 and 6.1. For details see text.

6.2.2 Crack-growth controlled failure

Small initial flaws, either by processing or handling, result in stress concentrations inside a

loaded material. These flaws initiate, and subsequent propagate, a craze or crack, finally causing

failure.5,48–52 Linear Elasticity Fracture Mechanics (LEFM) is used to define the stress distribution

near the crack tip for a crack opening load,53 using the coordinate system in Figure 6.3:

σxx (r, θ) =K√2πr· cos

(θ

2

)[1− sin

(θ

2

)sin

(3θ

2

)]

σyy (r, θ) =K√2πr· cos

(θ

2

)[1 + sin

(θ

2

)sin

(3θ

2

)]

σxy (r, θ) =K√2πr· cos

(θ

2

)[sin

(θ

2

)cos

(3θ

2

)]crack

load

load

θ

r

x

y

σxx

σyy

σxy

Figure 6.3: Coordinate systems

under mode I loading.

The stress concentration amplification in each direction is uniquely determined by the stress

intensity factor, K, which, for a crack opening load (mode I), is described using the general

form:

KI = Y σ√πa (6.3)

where σ is the remote stress, a the crack length, and Y a geometry factor, which is usually

dependent on the crack length a. The value of the stress intensity factor, with units MPa√

m,

107


at the stress where critical fracture occurs, KI,c, is a measure for a material’s resistance against

brittle fracture when a crack is present.

time

crac

k le

ngth

dadt

σ = C

aKI,th

KI,c

very slowcrack

propagation

stable crackpropagation

unstablecrack

propagation

b

Figure 6.4: Illustration of the evolution of crack length in time during (sub critical) crack propagation (a) and

the crack propagation rate versus the stress intensity factor, K, illustrating the different regions (b).

At lower values for the stress intensity, the crack also grows slowly in time under the load applied

(sub critical crack propagation), as illustrated in Figure 6.4a for a constant load. In time, the

stress intensity factor increases due to the increasing crack size. The resulting crack propagation

rate, versus the stress intensity factor, typically shows a sigmoidal shape (Figure 6.4b), and three

regions can be discerned. Below a given threshold value of the stress intensity factor, KI,th, no

or very slow crack propagation is observed. On the other extreme of the curve, when the stress

intensity factor approaches the critical value, KI,c, crack propagation is unstable. For values of

the stress intensity factor in between, the crack propagation is stable and the logarithm of the

crack propagation rate, a, increases linearly with the logarithm of the stress intensity factor; this

is known as Paris’ law:54

a = A ·KmI (6.4)

Pre-factor A defines the crack propagation rate for KI = 1, and m characterises the slope of

the line on double logarithmic plot. Both A and m are regarded to be material parameters. The

time up to failure under a constant load, tf , caused by slow crack propagation, can be calculated

by integrating the crack propagation rate, Equation 6.4, between a certain initial flaw size, ai,

and the crack size at which failure occurs, af , which yields:5,28,55–58

tf − ti =1

Aσm

af∫ai

da

(Y√πa)

m (6.5)

108

6.2. Background

Assuming that the time of initiation, ti, is negligible compared to the lifetime, the expression for

the logarithm of the applied stress versus the logarithm of the time-to-failure is:

log (σ) = − 1

mlog (tf )−

1

m

log (A)− log

af∫ai

da

(Y√πa)

m

(6.6)

which illustrates that the slope of the typical linear relation of the logarithm of the time-to-failure

as function of logarithm of applied load scales with the reciprocal of Paris’ law exponent m. This

can be reduced to a simple power law description:26,33,34,48,59

σ = cf · t− 1m

f (6.7)

with

cf = A−1m · C where C =

1√π·

af∫ai

da

(Y√a)m

1m

(6.8)

Rewriting Equation 6.7 yields for the time-to-failure as function of the applied load:

tf =

(σ

cf

)−m(6.9)

The pre-factor cf defines the stress that results in a lifetime of 1 second and, for constant

geometries and flaw sizes, its value scales with Paris’ law pre-factor A−1m . During cyclic fatigue,

besides frequency, one can vary the minimum load, mean load, maximum load, or load amplitude.

Here, the load maximum is kept constant and the load amplitude is varied via the load ratio, R

(see Figure 6.5):

R =FminFmax

(6.10)

Increasing the R-value makes the load amplitude to decrease, while R = 1 represents static

loading conditions. The stress intensity factor at load maximum, Kmax, is used to define the

crack propagation rate, Equation 6.4:

a (R, f) = A (R, f) ·Km(R,f)max (6.11)

and for a constant (maximum) load, σ, frequency, f , and load ratio, R, the time-to-failure can

be calculated via:

tf (R, f) =

(σ

cf (R, f)

)−m(R,f)

(6.12)

109


R = 1

R = 0.55

R = 0.1

Fmax

Fmin

R↑

Figure 6.5: Schematic illustration of the applied dynamic load at different R-values.

6.3 Experimental

6.3.1 Materials

The polymers used are a 30% glass-fibre reinforced polyphenylene-ether (PPE)/polystyrene (PS)

blend, NORYL™ FE1630PW resin (GFR PPE/PS), and a polyetherimide, Ultem™ 1010 resin

(PEI), both provided by SABIC Innovative Plastics (SABIC IP), Bergen op Zoom. GFR PPE/PS

is obtained as granulate and PEI as granulate and tensile bars according to ISO 527 Type 1A

test specimen specifications.


Tensile bars are injection moulded from GFR PPE/PS granules, according to ASTM D638 Type I

test specimen specifications. From PEI granules, 10 mm thick plaques are compression moulded

using a hot-press and, subsequently, surface machined from two sides to obtain plates with a final

thickness of 6 mm. Compact Tension (CT) specimens are produced by cutting the plates using

a circular saw followed by precision machining of the fixation holes and notch. The dimensions

of the Compact Tension specimen are determined according to the ASTM standard E647, with

thickness 6 mm, width 32 mm and height 38.5 mm. Pre-cracks of reproducible size are created

by tapping a fresh razor blade into the notch root of the sample using a pendulum. The exact

initial crack length is measured using a microscope.

6.3.3 Mechanical testing

Uniaxial tensile test are done on a Zwick Z010 Testing Machine, equipped with a 10 kN load-cell.

To characterise the deformation kinetics, tests are performed, at least in duplicates, at strain

rates ranging from 10−5 s−1 up to 10−2 s−1. Static fatigue experiments are done for a wide

110

6.4. Results

range of stresses; the stress is always applied in 10 seconds and subsequently kept constant until

failure. The time-to-failure is corrected for the load application time. Cyclic fatigue experiments

are performed on a servo-hydraulic MTS Testing System, equipped with a 25 kN load cell.

For the fatigue crack propagation measurements on PEI, the CT-specimen is mounted to the

tensile stage by a Clevis bracket, with dimensions according to the ASTM standard E647. The

specimen can freely rotate around the pin and the bracket contains one degree of freedom for

axial alignment of the upper and lower part. A light from the top is used to illuminate the

crack surface and to visualize the crack tip. Propagation is monitored using a digital camera

and a customized script based on the MATLAB Image Acquisition toolbox. From the crack

length as function of time, the derivative is taken in each point using a linear regression of an

interval surrounding this point to obtain the crack propagation rate. During all cyclic fatigue

experiments, a sinusoidal load is applied up to failure, whereas the frequency and load ratio

(R-value) are kept constant for each experiment. To measure the residual strength, presented in

Appendix A, a sinusoidal load is applied with R = 0.4 and a load maximum of 80 MPa at 1Hz

for fixed loading times, after which the samples is unloaded and a constant rate experiment is

performed at a rate of 10−3 s−1.

6.4 Results

6.4.1 Influence of frequency on load ratio dependence

Figure 6.6 presents results of time-to-failure versus maximum stress applied, for four different

frequencies and various load ratios. Two distinct regions are observed: one where the slope

is rather flat and the time-to-failure increases with increasing load amplitude or decreasing R-

value, and one where the slope is steep and the time-to-failure decreases with increasing load

amplitude. The first region, with slopes equal to the one at R = 1, is related to accumulation

of plastic strain. It is accurately described by the predictions of the time-to-failure under cyclic

fatigue loading, presented in Figure 6.2b. From here, the focus is on failure in the second

region, that is caused by crack propagation. The time-to-failure here decreases significantly with

increasing frequency and decreasing R-value. The decrease in lifetime with increasing load ratio

(the R-dependence of the time-to-failure) increases with increasing frequency. For example, for

a constant load maximum, the difference in lifetime between a test with R = 0.1 and one with

R = 0.4 is approximately one decade at 10Hz, and less than one third of a decade at 0.01Hz.

Clearly, the slope m is independent of load ratio and frequency, to a good approximation, and

therefore the influence of R and f on the lifetime can be captured by solely varying pre-factor

cf . A value of m = 8.1 is found that, using Equation 6.9 in combination with a best fit value

for cf for each frequency and R-value, results in the gray dashed lines in Figure 6.6. The values

for cf are presented in Table 6.2.

Next, the influence of frequency and R-value on the time-to-failure, for a given maximum stress

111


101

102

103

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105

106

40

50

60

70

8090

100

120

140


max

imum

str

ess

[MP

a]

R=0.1R=0.4R=1

0.01Hz

a10

110

210

310

410

510

6

40

50

60

70

8090

100

120

140


max

imum

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[MP

a]

R=0.1R=0.25R=0.4R=0.7R=1

0.1Hz

b

101

102

103

104

105

106

40

50

60

70

8090

100

120

140


max

imum

str

ess

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a]

R=0.1R=0.25R=0.4R=0.55R=0.7R=1

1Hz

c10

110

210

310

410

510

6

40

50

60

70

8090

100

120

140


max

imum

str

ess

[MP

a]

R=0.1R=0.25R=0.4R=0.55R=0.7R=1

10Hz

d

Figure 6.6: GFR PPE/PS: Time-to-failure under cyclic fatigue loading versus maximum stress applied, for

different stress ratios and frequencies: a) 0.01Hz b) 0.1Hz c) 1Hz d) 10Hz. Markers represent measurements,

dashed lines descriptions using Equation 6.9 and cf values presented in Table 6.2, solid lines predictions using

Equation 6.15.

applied, is studied in more detail, as shown in Figure 6.7 for a load maximum of 80 MPa. Note

that different load magnitudes only result in a vertical shift, since the slope m is independent of

load ratio and frequency. Figure 6.7a illustrates that the influence of load ratio, defined by the R-

value, on time-to-failure depends on frequency, and is stronger at higher frequencies. It appears

that all data points converge towards the same lifetime for R = 1. As a consequence, it is clear

that the number of cycles-to-failure increases with increasing load ratio, and its dependence on

load ratio is frequency dependent (Figure 6.7b). This implies that the number of cycles-to-failure

is not independent of frequency for the larger R-values, as often suggested in literature.12,25,33–35

This is once more illustrated in Figures 6.7c and d, where the time-to-failure and cycles-to-

failure are plotted versus frequency per load ratio. These figures show that, although the time-

to-failure decreases with increasing frequency, the effect of frequency diminishes when the R-

value is enlarged. As a result, the number of cycles-to-failure grow with increasing frequency

and R-value. Figure 6.7d indicates that for R = 0.1 the number of cycles-to-failure slightly

112

6.4. Results

increases with increasing frequency, although likely to be disguised by the experimental error.

The R and frequency dependency of the time-to-failure are in agreement with those reported in

literature.14,25,33–35

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

1

102

103

104

105

106

107

R−value [−]

time−

to−

failu

re [s

]

0.01Hz0.1Hz1Hz3Hz10Hz

80MPa

a0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

102

103

104

105

106

107

108

R−value [−]cy

cles

−to

−fa

ilure

[−]

0.01Hz0.1Hz1Hz3Hz10Hz

80MPa

b

10−3

10−2

10−1

100

101

101

102

103

104

105

106

107

frequency [Hz]

time−

to−

failu

re [s

]

R=0R=0.1R=0.25R=0.4R=0.55R=0.7R=1

80MPa

c10

−310

−210

−110

010

110

2

103

104

105

106

107

108

frequency [Hz]

cycl

es−

to−

failu

re [−

]

R=0R=0.1R=0.25R=0.4R=0.55R=0.7R=1

80MPa

d

Figure 6.7: GFR PPE/PS: Time-to-failure (a,c) and cycles-to-failure (b-d) under cyclic fatigue loading for a

maximum load applied of 80 MPa, versus load ratio and frequency. Markers represent the time-to-failure data

based on Equation 6.9 and cf values presented in Table 6.2, solid lines model fits using Equation 6.15.

From the results presented in Figure 6.6, one might argue whether the slope m is actually

independent of load ratio and frequency since, in particular for the longer measurements (low

frequencies and large R-values), the stress dependency seems to decrease. However, in these

cases, the stress ranges are limited and, although a clear distinction is made between plasticity-

controlled failure and crack-growth controlled failure, one should keep in mind that it is likely

that the mechanisms causing failure interact in the transition zone (the ”mechanical knee”3).

An evolving crack decreases the effective surface carrying the applied load and therefore increases

the local stress. Consequently, the plastic flow rate increases and the plasticity-controlled time-

to-failure decreases. In the approach presented, interactions are neglected.

113


Material cf [MPa·s1/m] m [-]

GFR PPE/PS

R [-] 0.1 0.25 0.4 0.55 0.7

8.1

0.01Hz 349.8 - 382.4 - -

0.1Hz 270.8 285.9 315.6 - 388.1

1Hz 211.7 234.7 259.1 302.8 361.5

3Hz 192.2 - - - -

10Hz 166.1 189.7 210.4 252.4 322.3

Table 6.2: Pre-factor cf values to describe the crack-growth controlled failure of smooth bars for GFR PPE/PS

using Equation 6.9.

6.5 Discussion

Historically data like those presented in Figure 6.6 are explained by methods based on damage

accumulation. Appendix A shows how this approach proceeds and illustrates that, although

sufficient parameters are present in the model used to allow lifetime predictions, the damage

model itself is physically incorrect. Therefore we offer a new approach based on crack propagation

kinetics.

6.5.1 Phenomenological description

A phenomenological model can be developed to describe crack-growth controlled failure, based

on the data presented in Figure 6.7. Plotting the time-to-failure versus the load ratio R, like

done in Figure 6.8a for a load maximum load of 80 MPa at 1Hz, defines two limits: a lower

limit, tf,cyclic, and an upper limit, tf,static, corresponding to the lifetime under a pure cyclic load,

R = 0, and to the time-to-failure under a static load, R = 1, respectively. The difference

between the two is a measure for the increase in lifetime due to an increase in R-value (decrease

in amplitude). Thus, the time-to-failure under fatigue loading conditions, tf , as function of R

can be obtained via:

log (tf (R)) = log (tf,cyclic) +Rα∆y = Rα log (tf,static) + (1−Rα) log (tf,cyclic) (6.13)

where α captures the (slightly non-linear) dependency on load ratio.

Assuming that for large load amplitudes, R = 0, the number of cycles-to-failure is independent of

frequency, and that the time-to-failure scales with frequency, the power law relations in Equation

6.9, based on crack growth kinetics, provide expressions for the stress dependency of the time-

to-failure at the two limits:

tf,static =

(σ

ctf ,static

)−mand tf,cyclic =

Nf,cyclic

f=

1

f

(σ

cNf ,cyclic

)−m(6.14)

114

6.5. Discussion

Substitution of these expressions for the lifetime in both limiting cases in Equation 6.13, offers a

description of the time-to-failure under fatigue loading as function of R and f for any arbitrary

maximum load applied:

log (tf (R, f)) = Rα · log

((σ

ctf ,static

)−m)+ (1−Rα) · log

(1

f

(σ

cNf ,cyclic

)−m)(6.15)

Equation 6.15 shows that, besides the stress dependency, m, the only parameters of importance

are the upper and lower limit of the lifetime, defined by ctf ,static and cNf ,cyclic respectively, and

the dependency on load ratio, R, via α. These parameters are determined, for GFR PPE/PS,

using a least-squares fit on the data presented in Figure 6.7a, and presented in Table 6.3. Using

these parameters in Equation 6.15, the times-to-failure as function of load ratio R and frequency

are calculated for a maximum load of 80 MPa, the solid lines in Figure 6.7, while the lifetimes

during cyclic fatigue, in combination with a suitable value for m, are presented as the solid lines

in Figure 6.6. Both figures prove that this simple, easy to apply equation offers an accurate

description of both the influence of frequency and load ratio on the lifetime for the entire range

of loads applied.

a

1Hz, 80MPa

∆y

log(tf,cyclic)

log(tf,static)

∆y ·Rα

101

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104

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106

107

108

40

50

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100

120

140


max

imum

str

ess

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a]

R=0.1R=0.25R=0.4R=0.55R=0.7R=1

1Hz

b

Figure 6.8: a) Time-to-failure for 80 MPa and 1Hz, to illustrate the parameters in the phenomenological

description. b) Cyclic fatigue data, measured at 1Hz, combined with (long-term) static fatigue data and the

predicted time-to-failure. Markers represent measurements, lines model fits according to Equations 6.1 (region

I) and 6.15 (region II)

With the corresponding parameters, also the time-to-failure under static loading conditions is

obtained, defined by the pre-factor under purely static loading, ctf ,static. And, as displayed in

Figure 6.8b, the predicted time-to-failure for R = 1 is in good agreement with the measured time-

to-failure. So, even though more validation is required, the use of this simple, phenomenological

approach, with parameters determined on cyclic fatigue experiments only, allows extrapolation

and estimation of the long-term performance under static loading.

115


Material α [-] ctf ,static [MPa·s1/m] cNf ,cyclic [MPa] m [-]

GFR PPE/PS 1.22 481.9 196.4 8.1

Table 6.3: GFR PPE/PS: Parameters to model the time-to-failure as function of R-value and frequency using

Equation 6.15.

Translations to crack propagation rates

In the phenomenological description, summarized in Equation 6.15, it is the crack propagation

kinetics that determines the power laws defining the upper and lower limit of the time-to-failure

under static and cyclic load respectively. Hence, the R-value and frequency dependence of

the time-to-failure contains information about the actual crack propagation rates under these

conditions. Since Equation 6.13 combines logarithms, the time-to-failure is the product of the

lifetimes of both contributions:

tf (R, f) = tRα

f,static · t(1−Rα)f,cyclic (6.16)

Substitution of Equation 6.14 in Equation 6.16 yields:

tf (R, f) =

(σ

ctf ,static

)−mRα· f−(1−Rα) ·

(σ

cNf ,cyclic

)−m(1−Rα)

(6.17)

Rewriting Equation 6.17 such that the time-to-failure, as function of R and f , is represented by

a power law, similar to Equation 6.12:

tf (R, f) =

σ · f(1−Rα)

m

c(1−Rα)Nf ,cyclic

· cRαtf ,static

−m =

(σ

cf (R, f)

)−m(6.18)

provides an expression for the R and f dependence of the pre-factor, cf :

cf (R, f) = f−(1−Rα)

m · c(1−Rα)Nf ,cyclic

· cRαtf ,static (6.19)

According to Equation 6.8, Paris’ law pre-factor, A, that defines the crack propagation rate,

equals A = c−mf · Cm, and therefore:

A (R, f) = f (1−Rα)(c

(1−Rα)Nf ,cyclic

· cRαtf ,static)−m

· Cm = f (1−Rα) · A(1−Rα)cyclic · A

Rα

static (6.20)

where Astatic and Acyclic are the Paris’ law pre-factors corresponding to a pure static (R = 1)

or pure cyclic (R = 0) loading, respectively. Thus, by substituting Equation 6.20 into the Paris’

law in Equation 6.11, the load ratio and frequency dependency of the time-to-failure relate to

that of the crack propagation rate, via:

a (R, f) = f (1−Rα)︸︷︷︸I

·A(1−Rα)cyclic︸︷︷︸II

·ARαstatic︸︷︷︸III

Kmmax (6.21)

116

6.5. Discussion

The total (cyclic) fatigue crack propagation rate can be split into three contributions: I) the

influence of frequency, that diminishes with increasing R-value, II) the cyclic contribution, and

III) the static contribution. For small R-values, the cyclic component prevails and the static

component is (approximately) unity, and for large R-values vice versa.

The effect of frequency and load ratio on the crack propagation rate is shown in Figure 6.9,

displaying an upper (R = 0) and a lower bound (R = 1) for the crack propagation rate versus

maximum stress intensity factor, Kmax. The upper bound scales with frequency, while the lower

bound is independent of frequency. The crack propagation rate decreases with increasing R-

value. The influence of load ratio, R, decreases with decreasing frequency.

upper bound

lower bound

f↑

f↓

scaleswith f

independentof f

a

a

b

c

d

a b c d

R↑

R↓

R↑

R↓

Figure 6.9: Illustration of the influence of frequency, f , and load ratio, R, on the crack propagation rate versus

maximum stress intensity factor (left). Illustration of the mechanism enhancing the crack propagation rate in

cyclic fatigue (right). Stretching during loading (a), bending/crushing during unloading (b), stretching in a later

stage (c), and eventually failure of the fibril, for large amplitudes in the middle, for smaller in the region where

bending is maximum (d). Reproduced from Zhou et al.25,26

The influence of cyclic loading on the rate of crack propagation is related to failure of fibrillae

bridging the craze zone proceeding the crack tip.28,37 The fibrillae in the craze support part

of the load applied, which causes them to slowly deteriorate until they fail.60 As a result, the

crack propagation rate is largely determined by the rate of failure of these fibrillae. During static

fatigue, the mechanisms leading to fibril failure are believed to be disentanglement or chain

scission.20,24,36,61–64 It is hypothesized that, as illustrated in Figure 6.9, during cyclic loading,

even in tension-tension fatigue, these fibrils are continuously stretched and compressed (a-b-c),

which, during crack closure (b), causes bending (R↑) and, for sufficiently large amplitudes, even

buckling or crushing of the fibrils (R↓),25 stimulating fibril failure (d). In other words, alternating

opening and closing of the crack tip enhances the crack propagation rate and the larger the load

amplitude (or the smaller the load ratio R), and the higher the frequency, the stronger this

acceleration is in time.25,26 This indicates that the actual mechanisms causing fibril deterioration

in cyclic and static fatigue are not related. With increasing R-value and decreasing frequency,

the rate of fibril deterioration caused by the cyclic (un)loading mechanism decreases and, as a

117


result, the contribution of static loading to the total crack propagation rate becomes increasingly

dominant. Therefore, it is the diminishing effect of the cyclic failure mechanism that reveals the

contribution of a static component, thus varying the load ratio and R-value offers the crack

propagation rate under static loading conditions, finally enabling access of the static fatigue

lifetime.

Also in literature fatigue crack propagation is reported to consist of two contributions, a cyclic

and a static, which basically are claimed to interact either in an additive or a multiplicative

manner. The additive approach hypothesises that a pure cycle dependent (cyclic) and a pure time

dependent (static) mechanism contribute simultaneously to the total fatigue crack propagation

rate:12,30,65(δa

δN

)total

=

(δa

δN

)cyclic

+1

f·(δa

δt

)static

(6.22)

The cyclic component is usually regarded as the fatigue component due to the opening and closing

of the crack tip, which magnitude depends on the load amplitude, and the static component as

a viscoelastic creep component, likely scaling with mean applied load. This concept successfully

explains the influence of frequency on crack growth rate in various systems.16,40,66,67 However,

interaction in an additive manner has the consequence that each component only contributes

significantly to the total fatigue crack propagation rate when both components are of roughly

equal magnitude. Rearrangement of Equations 6.8 and 6.9 learns that the Paris law pre-factor

A, and thus the crack propagation rate, scales with the reciprocal of the time-to-failure under a

given load. The lifetimes presented in Figure 6.8a show that, at 1Hz, the crack propagation rate

under static load (R = 1) is approximately 1000-10 times lower than those for cyclic loading with

load ratios ranging from 0.1-0.7, indicating that the total rate is determined solely by the cyclic

component. Hence, lifetime should scale with frequency for all these load ratios, but, as already

shown in Figure 6.7, this is clearly not the case. This points to a different interaction between

the two contributions. Interactions in multiplicative manner, as first proposed by Erdogan,68 are

based on a separation of the stress intensity expression in a component that depends on load

magnitude, Kmax, and one that depends on the load amplitude, ∆K:55,69–72

a = A ·Kmmax = B ·∆Kp ·Kn

max (6.23)

where

∆K = (1−R)Kmax (6.24)

B = A · (1−R)−p (6.25)

p+ n = m (6.26)

The frequency influence can be included by introducing a power q to modify the pre-factor:

B = B′f q.55,70 This expression is not suited to extrapolate to large R-values, since than ∆K

approaches zero, which can be circumvented by using the mean stress intensity factor, Kmean,

118

6.5. Discussion

for the amplitude dependent term, instead of ∆K:15,24,38

a = B′ · f q ·Kpmean ·Kn

max (6.27)

where

Kmean = 1/2 (1 +R) ·Kmax (6.28)

B′ = A · (1/2 (1 +R))−p

Combining Equations 6.21, 6.27 and 6.28 yields:

q = 1−Rα (6.29)

B′ = A(1−Rα)cyclic · A

Rα

static ·(

1 +R

2

)−p(6.30)

The power p in Equation 6.30 can have any arbitrary value, as long as the constraint p+n = m

is satisfied. This suggests that splitting the load in a cyclic component, via ∆K or Kmean, and

a load magnitude component is unnecessary. The influence of frequency expressed in the power

q is usually assumed constant.55,70 Because of a lack of data where both frequency and R-value

are varied, it is difficult to fully validate our observations. However, the approach presented in

Equation 6.21 is able to accurately describe the R dependency of the fatigue crack propagation

rates reported in literature of PMMA,55 PVC,36 and PE,14,24 and the influence of both frequency

and R on fatigue crack propagation in PVC.20

6.5.2 Validation

It is difficult to measure representable crack propagation kinetics on fibre-reinforced polymer

systems using compact tension (CT-)specimens. This is due to differences in fibre-orientation

between CT-specimens and smooth bars, and due to curvatures in the crack’s propagation paths

due to the presence of fibres.73 Crack propagation kinetics are easier to access on isotropic,

transparent CT-specimens, and therefore we performed cyclic fatigue experiments for several

load ratios and frequencies on both smooth bars and CT-specimens of polyetherimide (PEI).

The results are presented in Figure 6.10, and offer sufficient information to determine the pa-

rameters of Equation 6.15 to describe the influence of R-value and frequency, see Table 6.5.

Subsequently, it is tried to determine a suitable initial crack length by performing crack propaga-

tion experiments under 1Hz and R = 0.1 (see Appendix B), and correlating the two experiments.

The frequency and R dependence, measured in fatigue on smooth bars, can subsequently be used

to predict that of the crack propagation rate, measured with CT-specimens. As can be concluded

from the results presented in Figure 6.11, the predictions are in excellent agreement with the

experimentally obtained crack propagation rates, proving that the load ratio and R dependency

obtained from fatigue experiments can accurately be related to actual crack propagation kinetics.

119


101

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104

105

106

30

40

50

60

80

100

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ess

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a10

110

210

310

410

510

6

30

40

50

60

80

100

120


max

imum

str

ess

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a]

R=0.1R=0.2R=0.4R=0.6R=1

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b

101

102

103

104

105

106

30

40

50

60

80

100

120


max

imum

str

ess

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a]

R0.1R0.3R0.5R=1

10Hz

c0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

102

104

106

R−value [−]

time−

to−

failu

re [s

]

0.1Hz1Hz10Hz

80MPa

d

Figure 6.10: PEI: Time-to-failure under cyclic fatigue loading versus maximum stress applied, for different stress

ratios at a) 0.1Hz, b) 1Hz, c) 10Hz. Markers represent measurements, dashed lines descriptions using Equation

6.9 and cf values presented in Table 6.4, solid lines model fits using Equation 6.15. b) Time-to-failure in cyclic

fatigue for a maximum load applied of 80 MPa versus load ratio. Markers represent the time-to-failure calculated

using Equation 6.9 and cf values (Table 6.4), solid lines model fits using Equation 6.15.

Material cf [MPa·s1/m] m [-]

PEI

R [-] 0.1 0.2 0.25 0.3 0.4 0.5 0.6

4.90.1Hz 489.6 - 543.8 - 666.6 - -

1Hz 335.0 360.3 - - 477.3 - 664.5

10Hz 203.2 - - 281.5 - 395.2 -

Table 6.4: Pre-factor cf values to describe the crack-growth controlled failure of smooth bars for PEI using

Equation 6.9.

120

6.6. Conclusions

Material α [-] ctf ,static [MPa·s1/m] cNf ,cyclic [MPa] m [-]

PEI 1.685 1825.6 316.95 4.9

Table 6.5: PEI: Parameters to describe the time-to-failure as function of R-value and frequency using the simple

approach as presented in Equation 6.15.

0.6 0.8 1 2 3 410

−8

10−7

10−6

10−5

10−4

Kmax

[MPa⋅m0.5]

crac

k pr

opag

atio

n ra

te [m

/s]

R=0.1R=0.3R=0.5

a

1Hz

0.6 0.8 1 2 3 410

−8

10−7

10−6

10−5

10−4

Kmax

[MPa⋅m0.5]

crac

k pr

opag

atio

n ra

te [m

/s]

R=0.1R=0.3R=0.5

10Hz

b

Figure 6.11: PEI: Crack propagation rates versus stress intensity factor, for 1Hz (a) and 10Hz (b). Markers

represent measurements, lines predictions using Equation 6.21 and the parameters presented in Table 6.5.

6.6 Conclusions

In this investigation, the influence of both frequency and load amplitude on the time-to-failure in

cyclic fatigue is extensively studied. It is shown that the number of cycles-to-failure is independent

of frequency only for large load amplitudes. The load ratio dependency of the time-to-failure

is strongly frequency dependent. The dependence of the load ratio for different frequencies,

extrapolate to the same lifetime under static loading.

A phenomenological, crack propagation based model is provided and applied to successfully

describe the lifetime in cyclic fatigue and the long-term time-to-failure under static loading.

This indicates that the long-term static performance can be predicted based on short-term

experiments by varying load ratio and frequency in cyclic fatigue. The model suggests that the

total crack propagation rate contains a cyclic and a static contribution acting in a multiplicative

manner. Validation using an unfilled, glassy polymer, shows that extrapolation from cyclic

fatigue experiments towards static loading lifetime predictions is generic, and that the R-value

and frequency dependency measured on smooth bars actually relates to the crack propagation

rates measured on compact-tension specimens.

121



The authors would like to thank Hans van der Pas and Rijn Stovers for their efforts and con-

tributions within the experimental work. Special thanks to Jeffrey Christianen from SABIC IP

for performing the long-term static fatigue experiments that allowed validation of the predicted

long-term performance.

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125


Appendix 6A: Damage based approach

The results clearly show that the number of cycles-to-failure is strongly load ratio and frequency

dependent, which makes it difficult to predict fatigue, since often damage accumulation laws are

based on the number of cycles-to-failure,74,75 among which also the best-known Miner’s law,76

and therefore not applicable due to latter dependency. This has resulted in the development of

approaches to determine the fatigue strength of composites by combining time dependent and

cycle dependent processes in the strength degradation.77–79 One of these approaches is based on

Strength Evolution Integral (SEI), and has been successfully used for prediction of composites’

lifetime for almost 30 years.80–82 It results in an expression of the normalized remaining strength,

Fr as function of load ratio, R, and the normalized applied load, Fa:83

Fr = 1−R (1− Fa)(

t

tf,static

)j1− (1−R) (1− Fa)

(t

tf,cyclic

)j2(A.1)

where tf,static and tf,cyclic are the time-to-failure under static fatigue (R = 1) and pure cyclic

fatigue (R = 0) respectively. The powers ji influence the damage progression; if ji < 1 the rate

of degradation is greatest in the beginning, if ji > 1 the rate increases in time, while if ji = 1

there is no explicit time dependence. The failure criterion is given by Fr = Fa.

For large load amplitudes, e.g. R = 0, the number of cycles-to-failure is independent of frequency

and the time-to-failure scales with frequency (see Figure 6.7), and the power law relations pre-

sented in Equation 6.14 offer expressions for the stress dependency of the time-to-failure at the

two limits. Substituting these two expressions for the time-to-failure under static and cyclic load

in Equation A.1, allows calculation of the remaining strength for any arbitrary load maximum,

frequency, and load ratio. The parameters to describe the time-to-failure for the GFR PPE/PS

are determined via a least-squares fit on the crack-growth controlled failure data and presented

in Table A.1. And, as Figure A.1a shows, these parameters offer an accurate description of the

measured time-to-failure for the several frequencies and load ratios. Since also an expression is

obtained for the performance under static loading, predictions can be done and compared with

long-term static failure data, see Figure A.1b. Using the cyclic fatigue data at 1Hz, and extrap-

olating to R = 1 gives the predicted time-to-failure, which is in good agreement with the R = 1

measurements. Therefore, even though more validation is required, the use of this damage based

approach, with the parameters determined only on cyclic fatigue experiments, seems promising

to extrapolate and estimate the long-term performance under static loading.

Once the parameters are determined, also the evolution of the remaining strength in time can

be calculated. As Figure A.2 shows, the prediction of the remaining strength does not correlate

to the remaining strength measured after different loading times. The inset in Figure A.2 shows

that, according to the model, most of the damage accumulates instantaneously (related to small

values for j1 and j2) while the rate of degradation during the remaining life is much lower.

126

Appendix 6A

102

104

106

101

102

103

104

105

106

predicted time−to−failure [s]

mea

sure

d tim

e−to

−fa

ilure

[s]

R=0.1R=0.25R=0.4R=0.55R=0.7

a10

110

210

310

410

510

610

710

8

40

50

60

70

8090

100

120

140


max

imum

str

ess

[MP

a]

R=0.1R=0.25R=0.4R=0.55R=0.7R=1 b

1Hz

Figure A.1: a) Predicted time-to-failure versus measured time-to-failure, using Equation A.1. The colors

indicate the R-value used, each marker corresponds to a frequency as used in the legend in Fig.6.7. b) Fatigue

data measured at 1Hz and the extrapolated long-term performance under static loading conditions, compared

with long-term static failure data.

.

The measurements however, illustrate that the strength remains constant in time and starts to

decrease near the moment of failure. This clearly shows that, even though the time-to-failure is

predicted accurately, the use of this damage based approach is not very physical and basically

no more than a phenomenological description with sufficient fitting parameters to describe this

complex behaviour.

100

101

102

103

104

80

90

100

110

120

130

140

time−under−load [s]

stre

ngth

[MP

a]

failureb

1Hz, R = 0.4, 80MPa

0 1 2 3 4 5110

120

130

140

Figure A.2: Evolution of the residual strength versus logarithm of the time under load, where the inset zooms

in on the evolution at very short times-scales. The maximum load applied was 80 MPa, with R = 0.4 at 1Hz.

Markers represent measurements, the dashed gray line the prediction according to Equation A.1.

.

127


Material ctf ,static [MPa] cNf ,cyclic [MPa] m [-] j1 [-] j2 [-]

GFR PPE/PS 414 189 8.52 0.1247 0.1

Table A.1: GFR PPE/PS: Parameters to describe the time-to-failure as function of R-value and frequency using

the damage based approach as presented in Equation A.1.

Appendix 6B: Estimation of the initial flaw size

In crack propagation within a smooth bar, it is assumed that an (semi-) elliptical crack propagates

from the centre, the surface, or the corner of the sample. In most failed PEI samples, the

crack initiated from the corner, and thus is assumed to be the case for all experiments. Figure

B.1 sketches the situation of such a corner flaw within the cross-section of a smooth bar and

parameters used to calculate the stress intensity factor.

a

b φ

t

W

Figure B.1: Illustration of a corner crack in the cross-section of a smooth bar.

The corresponding stress intensity factor can be found in the Stress Intensity Factors Handbook84

and, for mode I loading, it reads:

KI =σ√πb

E (k)· Fc

(b

a,b

t, φ

)(B.1)

where E (k) is the complete elliptical integral of the second kind:

E (k) =

∫ π/2

0

√1− k2 sin2 θ dθ with k =

(1− b2

a2

) 12

for a ≥ b

E (k) =b

aE (k1) with k1 =

(1− a2

b2

) 12

for a < b

(B.2)

and

Fc =

[M1 +M2

(b

t

)2

+M3

(b

t

)4]· g1 · g2 · fφ (B.3)

128

Appendix 6B

With

M1 = 1.08− 0.03

(b

a

)for b/a ≤ 1

M1 =

√a

b

(1.08− 0.03

(ab

))for b/a > 1

(B.4)

M2 = −0.44 +1.06

0.3 +(ba

) for b/a ≤ 1

M2 = 0.375(ab

)2

for b/a > 1

(B.5)

M3 = −0.5 + 0.25

(b

a

)+ 14.8

(1− b

a

)15

for b/a ≤ 1

M3 = −0.25(ab

)2

for b/a > 1

(B.6)

g1 = 1 +

[0.08 + 0.4

(b

t

)2]

(1− sinφ)3 for b/a ≤ 1

g1 = 1 +

[0.08 + 0.4

(at

)2]

(1− sinφ)3 for b/a > 1

(B.7)

g2 = 1 +

[0.08 + 0.15

(b

t

)2]

(1− cosφ)3 for b/a ≤ 1

g2 = 1 +

[0.08 + 0.15

(at

)2]

(1− cosφ)3 for b/a > 1

(B.8)

fφ =

[(b

a

)2

cos2 φ+ sin2 φ

]1/4

for b/a ≤ 1

fφ =

[(ab

)2

sin2 φ+ cos2 φ

]1/4

for b/a > 1

(B.9)

Equation B.1-B.9 are applicable as long as:

0.2 ≤ b/a ≤ 2, b/t < 1, 0 ≤ φ ≤ π/2, a/W < 0.2 (B.10)

By substituting Equation B.1 into the Paris’ law (Equation 6.4) the time evolution of the crack

length is calculated in two directions, a (φ = 0) and b (φ = π/2), using the crack propagation

kinetics at R = 0.1, obtained from CT-specimens (Figure B.2a). The sample width and thickness

are taken as limiting crack length in each direction. Note that the magnitude of the final crack

length hardly influences the estimated time-to-failure, since the crack propagation rate strongly

accelerates near failure (see Figure 6.4). An initial quarter circular flaw with radius of 55 µm now

proves to give an excellent description of the time-to-failure at R = 0.1 measured on smooth

bars (Figure B.2b). This figure also shows that this initial flaw size in combination with the R-

value and frequency dependence of the crack propagation rate presented in Equation 6.21, using

parameters in Table 6.5, offers accurate description of the time-to-failure for other R-values, as

displayed in gray.

129


0.8 1 2 310

−8

10−7

10−6

10−5

Kmax

[MPa⋅m0.5]

crac

k pr

opag

atio

n ra

te [m

/s]

R=0.1R=0.3R=0.5

1Hz

a10

110

210

310

410

510

6

40

50

60

80

100

120


max

imum

str

ess

[MP

a]

R=0.1R=0.2R=0.4R=0.6R=1

1Hz

b

Figure B.2: PEI: a) Crack propagation rate versus maximum stress intensity factor, measured on CT-specimens.

Markers represent measurements, lines descriptions according to Equation 6.4. b) Time-to-failure in fatigue,

measured on smooth bars. Markers represent measurements, lines descriptions by integrating Equation 6.4 using

a initial flaw size of 55 µm (solid lines). The predictions for R-values other than 0.1, are obtained using the R

dependence in Equation 6.20.

The initial flaw size is used as an arbitrary fitting parameter, that can link the crack propagation

rate to the time-to-failure and vice versa. The fracture surface can also be analysed using an

optical microscope, showing that, although choosing the point of initiation is rather subjective, all

initiation sites contain features in the corner with the dimensions in the same order of magnitude

as the assumed 55 µm, see Figure B.3a. To be more precise, over 30 analysed crack surfaces,

an average size is found of 61.8 µm, with a standard deviation of ± 23.9 µm. Figure B.3b plots

the range of predicted time-to-failures versus the range of measured flaw sizes, for R = 0.1 and

a load maximum of 80 MPa. The lifetime calculated with a flaw size of 55 µm seems to be a

reasonable approximate.

31 µm 60 µm

95 µm 92 µm

R = 0.1,1Hz,80MPa

Figure B.3: PEI: Optical microscopy of the fracture surfaces within a smooth bar (magnification 10x), the

arrows indicate the chosen location of initiation at the corner (left) and the distribution of initial flaw sizes, on 30

specimens, and the (calculated) corresponding lifetime for R = 0.1, 1Hz and a maximum load of 80MPa (right).

130

CHAPTER 7

Conclusions and recommendations

7.1 Main conclusions

This research focussed on qualifying and quantifying the mechanisms that lead to failure in loaded

polymers, to identify the different mechanisms, and develop methods that enable both access

and prediction of the long-term properties, based on short-term measurements only. Predictive

methods result, for both long-term plasticity-controlled and crack-growth controlled failure, that

are validated on long-term failure data. Additionally, a method for distinction between these

two failure mechanisms is provided, and the influence of stress enhanced physical ageing on

plasticity-controlled failure is clarified.

The main conclusions are:

• Long-term plasticity-controlled failure, measured using a procedure that takes approxi-

mately 1.5 years, can accurately be predicted using this protocol that takes approximately

2 weeks on a single tensile testing machine.

• Physical ageing is accelerated by both temperature and stress, and can induce a substan-

tial increase in yield stress during the experiment. This increase results in an apparent

”endurance limit”.

• The activation energy for ageing increases with age and hence also with annealing tem-

perature. In order to prevent overestimation of the activation energy, which results in an

underestimation of ageing at low temperatures, one should focus on the low temperature

data.

131

7 Conclusions and recommendations

• There is a limit to the activation by stress for larger stresses and/or long time-scales,

where mechanical rejuvenation starts to retard, or even reverse the effects of ageing. The

acceleration by stress is determined by the stress history that the material experienced,

and therefore ageing occurs at a lower rate under a cyclic load than that under static

load (with equal maxima). On the other hand, it also reduces the rate of accumulation of

plastic strain, which delays strain softening (mechanical rejuvenation).

• Mechanical rejuvenation has to be taken into account to properly capture the influence of

progressive ageing on plasticity-controlled failure of glassy polymers, which requires a full

3D constitutive approach.

• The non-linear and viscoelastic behaviour of polymers proves to cause a strong dependence

of the calibration curve on loading condition and loading time. As a result, no unique

relation can be found for crack length as function of the compliance. Using the dynamic

compliance on non-linear, time dependent materials can consequently result in serious

discrepancies between the actual crack length and crack propagation kinetics. Results

based on measurements of the dynamic or normalized compliance should be considered

with care.

• A change from static to cyclic fatigue, with the same maximum stress, leads to longer

lifetimes for plasticity-controlled failure and strongly decreases lifetimes in the case of

crack-growth controlled failure. As such, it is a useful, generic, tool to identify the active

failure mechanism.

• The macroscopic appearance of failure is an unreliable indicator of the active failure mode

(e.g. in fibre reinforced systems).

• In crack-growth controlled failure, the number of cycles-to-failure is only independent of

frequency for large load amplitudes. As a result, the dependence of the time-to-failure on

load ratio is strongly frequency dependent.

• The dependence of crack-growth controlled failure on the load ratio at different frequencies

extrapolate to the same lifetime under static loading, implying that the long-term perfor-

mance in static fatigue can be predicted based on short-term experiments by varying the

load ratio and frequency in cyclic fatigue.

• The total crack propagation rate consists of a cyclic and a static component, acting in a

multiplicative manner. Diminishing the cyclic contribution with decreasing frequency and

load amplitude, enables identification of the lifetime in static fatigue.

132


7.2 Recommendations

The work in this thesis provides answers on questions about failure mechanisms within polymers

that are mainly based on experimental observations. But, the observations also result in remaining

questions that could, or should, be further investigated.

• Implementation of the competition between physical ageing and mechanical rejuvenation

in the Eindhoven Glassy Polymer (EGP) model would enable full predictability of the

effect of progressive ageing on plasticity-controlled failure in complex loading conditions.

It is likely that the interaction between plastic strain and thermodynamic state, currently

captured under the assumption that both are fully decoupled, has to be reconsidered. The

results provided in Chapter 3 should offer sufficient experimental validation and can act as

guidelines for model improvement.

• The influence of fibre orientation is not investigated. Since generally processed objects

possess a distribution of fibre orientations, also the influence of fibre orientation on both

failure mechanisms should be studied.

• Validation of the long-term extrapolation of crack-growth controlled failure, by simply

measuring the lifetime in fatigue for various load ratios at a constant frequency, is only

hampered by the lack of availability of long-term data. Therefore, this validation should

be completed and/or performed on additional polymer systems. Once the protocol is

sufficiently validated, the long-term performance can be estimated.

• The predictions of fatigue life are for single loading conditions only (e.g. constant R-

value and frequency). A challenge would be predicting the lifetime under arbitrary loading

conditions. This can easily be done using the approach presented in Chapter 6, that is

based on crack propagation rates. At present it is assumed that the maximum stress

intensity factor determines the crack propagation rate. Its validity for different wave types

(square, triangular, or a combination of all) should be confirmed.

• The influence of temperature, humidity, molecular weight, and filler-content etc. on crack

propagation rates is usually validated by comparing fatigue lifes under predefined loading

conditions (fixed load ratio and frequency). However, as proven in Chapter 6, the load ratio

dependence of the time-to-failure is affected by frequency. If the influence of frequency

varies when altering the temperature etc., it could be that short-term failure data lead

to erroneous extrapolation of the long-term properties. Therefore, these aspects should

be carefully further examined by extended cyclic fatigue experiments for various systems

under different external conditions.

• The tool to predict crack-growth controlled failure, presented in this thesis, is based on

Linear Elastic Fracture Mechanics, which is sufficient for simple geometries. A challenge

133


would be to develop a continuum approach that enables prediction of crack-growth con-

trolled failure on any arbitrary 3D geometry.

134

Samenvatting

Polymeren worden vanwege hun lage dichtheid en hoge specifieke sterkte steeds vaker ge-

bruikt in toepassingen waarin ze voortdurend worden belast, veelal bij hoge temperaturen en

luchtvochtigheden. Het belangrijkste probleem in dit soort toepassingen is dat alle polymeren

uiteindelijk, tijdsafhankelijk, falen. Het is niet de vraag of falen zal optreden, maar eerder wan-

neer. Om prematuur falen te voorkomen, is het van belang het lange-duur gedrag van belaste

polymeren te kunnen voorspellen.

Uit de bepaling van de levensduur via producttesten is het bekend dat drie specifieke faalgebieden,

met verschillende faalmechanismen, kunnen worden onderscheiden: Gebied I met plasticiteit-

gecontroleerd falen, oftewel ductiel of taai falen. Gebied II met falen veroorzaakt door langzame

scheurgroei, beter bekend als bros falen. En gebied III, waarin falen wordt veroorzaakt door

moleculaire degradatie. Dit onderzoek focust specifiek op spannings-geactiveerde fenomenen,

dus op de gebieden I en II.

De huidige methoden om de levensduur van producten af te schatten zijn tijdrovend en beho-

even grote hoeveelheden materiaal, veelal in de vorm van een eindproduct (bijvoorbeeld buizen

onder druk). Daardoor zijn ze niet erg praktisch om nieuwe materialen te classificeren, of te

ontwikkelen. Het doel van het onderzoek beschreven in dit proefschrift is om testmethoden te

ontwikkelen die het mogelijk maken het lange-duur gedrag te voorspellen via korte-duur metingen,

met weinig materiaal. Deze ontwikkelde methoden worden gevalideerd met gebruik van lange-

duur faaldata. Hoofdstukken 2 en 3 focussen op plasticiteits-gecontroleerd falen; hoofdstukken

4 tot 6 op scheurgroei-gecontroleerd falen. Het proefschrift wordt in hoofdstuk 7 afgesloten met

conclusies en aanbevelingen voor verder onderzoek.

Hoofdstuk 2 presenteert een methode om plasticiteits-gecontroleerd falen te voorspellen, ook

voor materialen die meerdere deformatiemechanismes vertonen. De methode is toegepast op

een buistype polyethyleen en gevalideerd op beschikbare lange-duur certificatiedata. Het blijkt

135

Samenvatting

mogelijk met de ontwikkelde methode binnen enkele weken plasticiteits-gecontroleerd falen te

voorspellen.

Hoofdstuk 3 bestudeert tijd-tot-falen onder een uitgebreide range temperaturen en belastingscon-

dities. In de experimenten lijkt een vermoeiingslimiet op te treden, verklaard met een toename in

weerstand tegen deformatie veroorzaakt door fysische veroudering gedurende het experiment. In

eerste instantie lijkt het alsof deze ontwikkeling sneller gaat onder dynamische belasting. Echter,

de evolutie van de vloeispanning in tijd, bepaald onder een grote hoeveelheid temperaturen en

(zowel statische als dynamische) belastingen, leert ons dat er onder dynamische belasting geen

significante versnelling plaatsvindt. De versnelling door spanning is beperkt, hoogstwaarschijnlijk

omdat mechanische verjonging het effect van veroudering vertraagt, of zelfs ongedaan maakt.

Verder is de snelheid waarmee mechanische verjonging optreedt onder dynamische belasting juist

trager.

Scheurgroei in vermoeiing wordt vaak gemeten via de compliantie-methode: de verandering

in stijfheid van een proefstuk, als gevolg van een in de tijd groeiende scheur, wordt gebruikt

om scheuropening te vertalen naar een scheurlengte. Hoofdstuk 4 vergelijkt de via compliantie-

methode verkregen scheurlengtes met de lengtes gemeten via optisch volgen van de scheur-

tip. Het blijkt dat bij polymeren het niet-lineaire en visco-elastische gedrag resulteert in een

sterke belastingconditie- en tijdsafhankelijkheid van de kalibratiecurves. Hierdoor wordt voor

deze materialen geen unieke relatie gevonden tussen scheurlengte en dynamische compliantie.

De verschillen tussen kalibratiecurves duiden tevens op het optreden van fysische veroudering

gedurende het experiment. Vandaar dat de compliantie-methode alleen goede resultaten zou

kunnen opleveren voor metingen met grote belastings-amplitudes, bij hoge frequenties. Beide

randvoorwaarden leiden immers tot kort durende metingen. Het optisch bepalen van de scheurtip

blijft de voorkeur hebben.

In hoofdstuk 5 worden beide faalmechanismen (accumulatie van plastische rek en scheurgroei)

systematisch behandeld en wordt het effect van cyclische belasting op elk mechanisme bestudeerd.

Bij verhoging van de belastings-amplitude, onder gelijkblijvende maximale belasting, wordt: (i)

plasticiteits-gecontroleerd falen vertraagd en uitgesteld vanwege een lagere snelheid van rekaccu-

mulatie, en (ii) scheurgroei-gecontroleerd falen significant versneld, door een verhoogde scheur-

propagatiesnelheid. Het vergelijken van de levensduur van polymeren onder statische en dy-

namische belasting maakt het daardoor mogelijk onderscheid te maken tussen plasticiteits- en

scheurgroei-gecontroleerd falen. Deze methode is in hoofdstuk 5 toegepast op een veelvoud aan

materialen, inclusief hun glasvezel versterkte varianten.

Het laatste hoofdstuk, 6, bestudeert de methode om scheurpropagatie snel te kunnen meten

via cyclische vermoeiing van vezelversterkte trekstaven. De verhouding van minimale en maxi-

136

Samenvatting

male belasting en de frequentie worden gevarieerd, waaruit blijkt dat het aantal cycli tot falen

onafhankelijk is van frequentie, echter alleen voor grote(re) belastings-amplitudes. De am-

plitudeafhankelijkheid van de levensduur verandert dan ook met frequentie. Door de totale

scheurpropagatiesnelheid op te splitsen in twee te onderscheiden bijdragen, een statische en een

cyclische, kan de levensduur voor verschillende belastings-amplitudes nauwkeurig beschreven wor-

den. Hoewel deze methode nog steeds aardig tijdrovend en materiaal behoevend is, hebben we

laten zien dat lange-duur scheurgroei-gecontroleerd falen onder statische belasting kan worden

afgeschat via korte-duur vermoeiingsexperimenten.

137

Dankwoord

Na inmiddels zo’n 10 jaar met veel plezier op de TU/e rondgelopen te hebben, zijn er uiteraard

veel mensen die het verdienen om bedankt te worden. Om de lengte van dit (luchtige) slot van

het proefschrift toch enigszins gelimiteerd te houden, zal ik me beperken tot de laatste 4 jaar,

de promotie. Ik heb het al die tijd enorm naar mijn zin gehad, voornamelijk door de prettige

en informele sfeer op de 4e verdieping. Hoewel er altijd zowel leuke als minder leuke momenten

zullen zijn geweest, vallen die laatsten geheel in het niet bij de talloze plezierige en memorabele

momenten.

De reden dat er uiteindelijk ook een proefschrift afgeleverd is, heb ik te danken aan mijn

drie (co-)promotoren: Han, bedankt! Allereerst voor het creeren van zo’n bijzondere vak-

groep/omgeving. Hoewel ik eventjes heb moeten wennen aan je duidelijke en directe blik op

de wereld, liet je me altijd merken dat je er vertrouwen in had en dankzij jouw input zijn de

hoofdstukken aanzienlijk verbeterd. Tom, wellicht heb je het al verdrongen, maar tijdens mijn

stage heb je de ”polymeermechanica-vlam” in me verder aangewakkerd en mij gemotiveerd om

de deur naar een promotieplek op een kier te houden. Je legt met je doortastende en kritische

blik op de inhoud vaak de vinger op de zere plek, maar altijd met een prettige noot wat zorgt

voor een plezierige samenwerking. Bedankt voor alles. Leon, allereerst dankjewel dat je me de

mogelijkheid geboden hebt om bij jou te komen promoveren. Al sinds mijn afstuderen zorgde

je aanstekelijke (en uitbundige) enthousiasme ervoor dat ik met veel plezier naar Eindhoven

kwam. Dit was de daaropvolgende jaren zeker niet anders en los van het feit dat ik inhoudelijk

enorm veel van je geleerd heb, was het vooral erg gezellig (met de bijbehorende “(v)luchtige”

momenten), dankjewel! Ik weet zeker dat er momenten zullen zijn waarop ik onze (luidruchtige)

samenwerking zal gaan missen, op je gemopper over mijn ”kutmuziek” na uiteraard.

Daarnaast gaat er een speciaal woord van dank uit naar de contacten van de industrie voor

hun steun. Allereerst naar DSM, voor de financiering, en daarnaast in het bijzonder naar Tom

139

Dankwoord

Engels, Jan Stolk en Ruud Hawinkels. Daarnaast naar SABIC Europe, met name naar Klaas Re-

merie, Linda Havermans en Mark Boerakker, en SABIC IP, voornamelijk naar Tim van Erp, Christ

Koevoets, Erik Stam en Jeffrey Christianen. Tot slot tevens naar Vito Leo van Solvay. Ook wil ik

graag alle studenten bedanken die gedurende deze periode, direct of indirect, hebben bijgedragen

aan dit werk: Joris, Daan, Hans, Stijn, Rijn, Janneke, Rene, Nikki, Martijn, Sander, Britte, Jur,

Rob, Marc, Maurits, Roy, Coen, Sandra, Caroline en Bram. En ik wil zeker de volgende mensen

niet vergeten; onze secretaresses: Marleen en Ans, voor het regelen van alle dingen waar ik (en

menig promovendus met mij) eigenlijk geen zin in had. De mannen van de werkplaats, met in

het bijzonder Sjef en Lucien, die altijd voor me klaar stonden voor menig klusje. Marc, voor de

gastvrijheid in zijn lab, en uiteraard Leo, voor de compensatie van mijn ICT onwetendheid. Merci!

Daarnaast wil ik graag het kantoor waar ik de meeste tijd heb doorgebracht benoemen: de

verkapte koffieruimte GEM-Z 4.22. Menig collega wist ons kantoor te vinden wanneer hij zelf

even geen zin had om te werken, hoewel wij eigenlijk daar waren vanwege precies het tegenover

gestelde. Dat is volgens mij een uitstekende indicatie hoe prettig de sfeer vrijwel altijd was.

Vandaar ook dat ik via deze weg alle (oud-)kamergenoten, zowel vaste bewoners als de (vrijwel

permanente) bezoekers graag wil bedanken voor de erg fijne tijd. Hoewel geadopteerde tradi-

ties, zoals het ”15:00 fruitmomentje” en het (uiterst mannelijke) taart bakken met de π-baking

club, vreemde reacties oproepen bij buitenstaanders, heeft dit er denk ik voor gezorgd dat we

(including the Italians) een hechte groep collega’s zijn geworden. Met de nodige flauwe humor

(soms redelijk eenzijdig..) hebben we samen veel leuke momenten gedeeld en erg veel plezier

gehad. Ik hoop dat we dit in de toekomst zeker zullen voortzetten.

Tot slot zou ik ook graag wat mensen willen bedanken die niet direct met het werk te maken

hebben gehad. Veel vrienden, sommigen muziekgerelateerd, hebben voor de nodige afleiding,

ontspanning en gezelligheid gezorgd naast het afronden van dit proefschrift. Daarnaast mijn

(schoon)familie voor alle steun en interesse, zowel richting de promotie als bij andere zaken.

Een speciaal woord van dank naar mijn ouders, voor alle kansen die ik van jullie gekregen heb

en de support bij het verwezenlijken hiervan. En uiteraard mijn lieve Susan, dankjewel voor je

geduld en steun de laatste tijd en daarnaast bedankt dat je er altijd voor me bent. We gaan een

prachtige toekomst tegemoet samen, met hopelijk nog vele mooie reizen!

Marc Kanters

Juni, 2015

140

Curriculum Vitae

Marc Kanters was born on the 21th of January 1987 in Weert, The Netherlands. After finishing

pre-university education at the Philips van Horne SG in Weert in 2005, he studied Mechanical

Engineering at Eindhoven University of Technology. His master thesis was completed in June

2011 in the Polymer Technology group of prof.dr.ir. Han E.H. Meijer, under supervision of dr.ir.

Leon E. Govaert on the prediction of long-term plasticity-controlled failure of polymers based on

short-term testing. His master thesis was awarded with the SPE Benelux Student Award for the

best thesis 2011. As part of his master track he performed an internship at DSM, Geleen.

In July 2011, he took the opportunity to start his PhD project in the same group, under the

guidance of dr.ir. Leon E. Govaert, which has resulted in the present thesis.

During his PhD the author successfully completed the post-graduate course Register Polymer

Science of the National Dutch Research School PTN (Polymeer Technologie Nederland) and was

awarded the title Registered Polymer Scientist as of October 2013. The course consists of the

following modules: A - Polymer Chemistry, B - Polymer Physics, C - Polymer Properties, and

D/E - Polymer Processing & Rheology. He also attended the 14th European School on Rheology

at University of Leuven in the summer of 2013.

From July 2015, Marc is employed at DSM Ahead, Material Science Centre.

141

List of publications

This thesis has resulted in the following publications:

• M.J.W. Kanters, J. Stolk, and L.E. Govaert. ”Direct comparison of the compliance method

with optical tracking of fatigue crack propagation in polymers.” Polymer Testing (accepted)

• M.J.W. Kanters, K. Remerie, and L.E. Govaert. ”A new protocol for accelerated screen-

ing of long-term plasticity-controlled failure of polyethylene pipe grades.” Submitted for

publication

• M.J.W. Kanters, T. Kurokawa, and L.E. Govaert. ”Competition between plasticity-controlled

and crack-growth controlled failure in static and cyclic fatigue of polymer systems.” Sub-

mitted for publication

• M.J.W. Kanters, T.A.P. Engels, T.B. van Erp, and L.E. Govaert. ”Integral approach of

crack-growth in static and cyclic fatigue in a short-fibre reinforced polymer; a route to

accelerated testing.” In preparation

• M.J.W. Kanters, T.A.P. Engels, and L.E. Govaert. ”Prediction of plasticity-controlled

failure in glassy polymers in static and cyclic fatigue: interaction with physical ageing.” In

preparation

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

• A. Sedighiamiri, L.E. Govaert, M.J.W. Kanters, and J.A.W. van Dommelen. ”Microme-

chanics of semicrystalline polymers: Yield kinetics and long-term failure.”, Journal of

Polymer Science Part B: Polymer Physics, 2012, 50, 1664-1679.

• D. Cavallo, M.J.W. Kanters, H.J.M. Caelers, G. Portale, and L.E. Govaert. ”Kinetics

of the polymorphic transition in isotactic poly(1-butene) under uniaxial extension. New

insights from designed mechanical histories.” Macromolecules, 2014, 47, 3033-3040.

143

List of publications

• W.M.H. Verbeeten, M.J.W. Kanters, T.A.P. Engels, and L.E. Govaert. ”Yield stress

distribution in injection-moulded glassy polymers.” Polymer International (in press).

• M.J.W. Kanters, E.J. Stam, T.A.P. Engels, T.B. van Erp, and L.E. Govaert. ”Relating

the short-term burst-pressure to long-term hydrostatic strength: Ductile failure mode.” In

preparation

144

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