Post on 11-Jun-2020
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.
Document status and date:Published: 01/01/2015
Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)
Please check the document version of this publication:
• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publication
General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.
• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.
If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverne
Take down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.
Download date: 19. Jun. 2020
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
3.6 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
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
4.3.2 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3.3 Mechanical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
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
5.2.2 Plasticity-controlled failure . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2.3 Distinction between failure mechanisms . . . . . . . . . . . . . . . . . . 90
5.2.4 Characterisation and distinction . . . . . . . . . . . . . . . . . . . . . . 91
5.3 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.3.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.3.2 Mechanical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.7 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
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
6.2.1 Plasticity-controlled failure . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.2.2 Crack-growth controlled failure . . . . . . . . . . . . . . . . . . . . . . 107
6.3 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
iii
Contents
6.3.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.3.2 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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
6.7 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
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
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
III) chemicaldegradation
Figure 2.1: 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 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 Accelerated screening of long-term plasticity-controlled failure
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
time−to−failure [s]
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
2 Accelerated screening of long-term plasticity-controlled failure
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
2 Accelerated screening of long-term plasticity-controlled failure
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
2 Accelerated screening of long-term plasticity-controlled failure
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
2 Accelerated screening of long-term plasticity-controlled failure
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
2 Accelerated screening of long-term plasticity-controlled failure
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
2 Accelerated screening of long-term plasticity-controlled failure
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
time−to−failure [s]
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
time−to−failure [s]
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
time−to−failure [s]
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 Accelerated screening of long-term plasticity-controlled failure
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
2 Accelerated screening of long-term plasticity-controlled failure
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
time−to−failure [s]
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
References
References
[1] 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.
[2] 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.
[3] Schulte, U. “A vision becomes true: 50 years of pipes made from High Density Polyethylene”. In: “Pro-
ceedings of Plastic Pipes XIII, Washington”, 2006 .
[4] Gabriel, L.H. Corrugated polyethylene pipe design manual and installation guide, chap. 1 - History and
Physical Chemistry of HDPE. Plastic Pipe Institute, USA, 2011.
[5] Andersson, U. “Which factors control the lifetime of plastic pipes and how the lifetime can be extrapolated.”
In: “Proceedings of Plastic Pipe XI”, 2001 .
[6] 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.
[7] 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.
[8] Chudnovsky, A., Zhou, Z., Zhang, H., and Sehanobish, K. “Lifetime assessment of engineering thermoplas-
tics”. International Journal of Engineering Science, 2012. 59, 108–139.
[9] 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.
[10] Erdogan, F. “Ductile fracture theories for pressurised pipes and containers”. International Journal of
Pressure Vessels and Piping, 1976. 4, 253–283.
[11] Gotham, K. “Long-term strength of thermoplastics: the ductile-brittle transition in static fatigue”. Plastics
and Polymers, 1972. 40, 59–64.
[12] Gray, A., Mallinson, J.N., and Price, J.B. “Fracture Behavior of Polyethylene Pipes”. Plastics and Rubber
Processing and Applications, 1981. 1, 51–53.
[13] 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.
[14] 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.
[15] Burn, S. Long-term Performance Prediction for PVC Pipes. AWWA Research Foundation, 2005.
[16] Govaert, L.E. and Peijs, T. “Micromechanical modeling of time-dependent transverse failure in composite
systems”. Mechanics Time-Dependent Materials, 2000. 4, 275–291.
[17] Govaert, L.E., Schellens, H.J., Thomassen, H.J.M., Smit, R.J.M., Terzoli, L., and Peijs, T. “A microme-
chanical approach to time-dependent failure in off-axis loaded polymer composites”. Composites - Part A:
Applied Science and Manufacturing, 2001. 32, 1697–1711.
[18] “ISO 1167 Plastics pipes for the transport of fluids - Determination of the resistance to internal pressure”.
[19] “ISO 9080 Plastic piping and ducting systems - Determination of the long-term hydrostatic strength of
thermoplastics materials in pipe form by extrapolation”.
[20] Frank, A., Pinter, G., and Lang, R.W. “Prediction of the remaining lifetime of polyethylene pipes after up
to 30 years in use”. Polymer Testing, 2009. 28, 737–745.
[21] Frank, A., Hutar, P., and Pinter, G. “Numerical Assessment of PE 80 and PE 100 Pipe Lifetime Based on
Paris-Erdogan Equation”. Macromolecular Symposia, 2012. 311, 112–121.
[22] Kramer, E.J. and Hart, E.W. “Theory of slow, steady state crack growth in polymer glasses”. Polymer,
1984. 25, 1667–1678.
29
2 Accelerated screening of long-term plasticity-controlled failure
[23] Mindel, M.J. and Brown, N. “Creep and recovery of polycarbonate”. Journal of Materials Science, 1973.
8, 863–870.
[24] Bauwens-Crowet, C., Ots, J.M., and Bauwens, J.C. “The strain-rate and temperature dependence of yield of
polycarbonate in tension, tensile creep and impact tests”. Journal of Materials Science, 1974. 9, 1197–1201.
[25] Eyring, H. “Viscosity, Plasticity, and Diffusion as Examples of Absolute Reaction Rates”. The Journal of
Chemical Physics, 1936. 4, 283–291.
[26] Ward, I.M. “Review: The yield behaviour of polymers”. Journal of Materials Science, 1971. 6, 1397–1417.
[27] Visser, H.A., Bor, T.C., Wolters, M., Engels, T.A.P., and Govaert, L.E. “Lifetime Assessment of Load-
Bearing Polymer Glasses: An Analytical Framework for Ductile Failure”. Macromolecular Materials and
Engineering, 2010. 295, 637–651.
[28] Engels, T.A.P., Sontjens, S.H., 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.
[29] van Erp, T.B., Reynolds, C.T., Peijs, T., van Dommelen, J.A.W., and Govaert, L.E. “Prediction of yield
and long-term failure of oriented polypropylene: Kinetics and anisotropy”. Journal of Polymer Science Part
B: Polymer Physics, 2009. 47, 2026–2035.
[30] van Erp, T.B., Cavallo, D., Peters, G.W.M., and Govaert, L.E. “Rate-, temperature-, and structure-
dependent yield kinetics of isotactic polypropylene”. Journal of Polymer Science Part B: Polymer Physics,
2012. 50, 1438–1451.
[31] Janssen, R.P.M., Govaert, L.E., and Meijer, H.E.H. “An analytical method to predict fatigue life of ther-
moplastics in uniaxial loading: Sensitivity to wave type, frequency, and stress amplitude”. Macromolecules,
2008. 41, 2531–2540.
[32] Sherby, O.D. and Dorn, J.E. “Anelastic creep of polymethyl methacrylate”. Journal of the Mechanics and
Physics of Solids, 1958. 6, 145–162.
[33] Roetling, J.A. “Yield stress behaviour of polymethylmethacrylate”. Polymer, 1965. 6, 311–317.
[34] Bauwens-Crowet, C., Bauwens, J.C., and Homes, G. “Tensile yield-stress behavior of glassy polymers”.
Journal of Polymer Science Part A-2: Polymer Physics, 1969. 7, 735–742.
[35] Truss, R.W., Clarke, P.L., Duckett, R.A., and Ward, I.M. “The dependence of yield behavior on temperature,
pressure, and strain rate for linear polyethylenes of different molecular weight and morphology”. Journal of
Polymer Science: Polymer Physics Edition, 1984. 22, 191–209.
[36] Boyd, R.H. “Relaxation processes in crystalline polymers: molecular interpretation - a review”. Polymer,
1985. 26, 1123–1133.
[37] Ree, T. and Eyring, H. “Theory of Non-Newtonian Flow. I. Solid Plastic System”. Journal of Applied
Physics, 1955. 26, 793–800.
[38] Roetling, J. “Yield stress behaviour of poly(ethyl methacrylate) in the glass transition region”. Polymer,
1965. 6, 615–619.
[39] Roetling, J.A. “Yield stress behaviour of isotactic polypropylene”. Polymer, 1966. 7, 303–306.
[40] Pepels, M.P.F. Exploring the potential of polymacrolactones as polyethylene-mimics. Ph.D. thesis, Tech-
nische Universiteit Eindhoven, 2015.
[41] Seguela, R., Elkoun, S., and Gaucher-Miri, V. “Plastic deformation of polyethylene and ethylene copolymers:
Part II Heterogeneous crystal slip and strain-induced phase change”. Journal of Materials Science, 1998.
33, 1801–1807.
[42] Seguela, R., Gaucher-Miri, V., and Elkoun, S. “Plastic deformation of polyethylene and ethylene copolymers:
Part I Homogeneous crystal slip and molecular mobility”. Journal of Materials Science, 1998. 33, 1273–
1279.
[43] Boyd, R.H. “Relaxation processes in crystalline polymers: experimental behaviour - a review”. Polymer,
1985. 26, 323–347.
30
References
[44] Parry, E.J. and Tabor, D. “Effect of hydrostatic pressure and temperature on the mechanical loss properties
of polymers: 1. Polyethylene and polypropylene”. Polymer, 1973. 14, 617–622.
[45] Mears, D.R., Pae, K.D., and Sauer, J.A. “Effects of Hydrostatic Pressure on the Mechanical Behavior of
Polyethylene and Polypropylene”. Journal of Applied Physics, 1969. 40, 4229–4237.
[46] Spitzig, W.A. and Richmond, O. “Effect of hydrostatic pressure on the deformation behavior of polyethylene
and polycarbonate in tension and in compression”. Polymer Engineering & Science, 1979. 19, 1129–1139.
[47] Butler, M.F., Donald, A.M., Bras, W., Mant, G.R., Derbyshire, G.E., and Ryan, A.J. “A real-time si-
multaneous small- and wide-angle X-ray scattering study of in-situ deformation of isotropic polyethylene”.
Macromolecules, 1995. 28, 6383–6393.
[48] Butler, M.F., Donald, A.M., and Ryan, A.J. “Time resolved simultaneous small- and wide-angle X-ray
scattering during polyethylene deformation - II. Cold drawing of linear polyethylene”. Polymer, 1998. 39,
39–52.
[49] Butler, M.F., Donald, A.M., and Ryan, A.J. “Time resolved simultaneous small- and wide-angle X-ray
scattering during polyethylene deformation 3. Compression of polyethylene”. Polymer, 1998. 39, 781–792.
[50] Schrauwen, B.A.G., Janssen, R.P.M., Govaert, L.E., and Meijer, H.E.H. “Intrinsic Deformation Behavior of
Semicrystalline Polymers”. Macromolecules, 2004. 37, 6069–6078.
[51] Kholodovych, V. and Welsh, W.J. “Thermal-Oxidative stability and degradation of polymers”. In: “Physical
properties of polymers handbook”, pp. 927–938. Springer, 2007.
[52] “Exova Case Results”. http://polymer.exova.com/wp-content/uploads/2014/09/. Retrieved: 14-4-
2015.
[53] Eby, R.K. “Thermal Generation of Vacancies and Substitutional Sites in Crystalline Polymers”. Journal of
Applied Physics, 1962. 33, 2253–2256.
[54] Schmidt-Rohr, K. and Spiess, H.W. “Chain diffusion between crystalline and amorphous regions in polyethy-
lene detected by 2D exchange 13C NMR”. Macromolecules, 1991. 24, 5288–5293.
[55] Glowinkowski, S., Makrocka-Rydzyk, M., Wanke, S., and Jurga, S. “Molecular dynamics in polyethylene
and ethylene-1-butene copolymer investigated by NMR methods”. European Polymer Journal, 2002. 38,
961–969.
[56] Hendriks, A. “Storingsrapportage gasdistributienetten 2012”. Tech. rep., KIWA, 2013.
31
2 Accelerated screening of long-term plasticity-controlled failure
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
2 Accelerated screening of long-term plasticity-controlled failure
101
102
103
104
105
106
0
5
10
15
20
25
time−to−failure [s]
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 Prediction of plasticity-controlled failure: interaction with physical ageing
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
3 Prediction of plasticity-controlled failure: interaction with physical ageing
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
3 Prediction of plasticity-controlled failure: interaction with physical ageing
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
3 Prediction of plasticity-controlled failure: interaction with physical ageing
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
3 Prediction of plasticity-controlled failure: interaction with physical ageing
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
applied stress/T [MPa/K]
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 Prediction of plasticity-controlled failure: interaction with physical ageing
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
3 Prediction of plasticity-controlled failure: interaction with physical ageing
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
3 Prediction of plasticity-controlled failure: interaction with physical ageing
101
102
103
104
105
106
30
40
50
60
70
80
90
time−to−failure [s]
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
time−to−failure [s]
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
applied stress/T [MPa/K]
a σ [−]
75°C100°C125°C150°C a
σref = 0MPa
101
102
103
104
105
106
30
40
50
60
70
80
90
time−to−failure [s]
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
3 Prediction of plasticity-controlled failure: interaction with physical ageing
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.
3.6 Acknowledgements
The authors would like to thank Rene le Clercq and Stijn Arntz for their efforts and contributions
within the experimental work.
References
[1] Andersson, U. “Which factors control the lifetime of plastic pipes and how the lifetime can be extrapolated.”
In: “Proceedings of Plastic Pipe XI”, 2001 .
[2] 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.
[3] 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.
[4] Chudnovsky, A., Zhou, Z., Zhang, H., and Sehanobish, K. “Lifetime assessment of engineering thermoplas-
tics”. International Journal of Engineering Science, 2012. 59, 108–139.
[5] Loo, L.S., Cohen, R.E., and Gleason, K.K. “Chain mobility in the amorphous region of nylon 6 observed
under active uniaxial deformation”. Science, 2000. 288, 116–119.
[6] Capaldi, F.M., Boyce, M.C., and Rutledge, G.C. “Enhanced mobility accompanies the active deformation
of a glassy amorphous polymer”. Physical Review Letters, 2002. 89, 175505/1–175505/4.
[7] Lee, H.N., Paeng, K., Swallen, S.F., and Ediger, M.D. “Direct measurement of molecular mobility in actively
deformed polymer glasses”. Science, 2009. 323, 231–234.
[8] Struik, L.C.E. Physical Aging in Amorphous Polymers and Other Materials. Elsevier Scientific Publishing
Company, 1978.
[9] McKenna, G.B. “Glass formation and glassy behavior”. Pergamon Press plc, Comprehensive Polymer
Science: the Synthesis, Characterization, Reactions & Applications of Polymers., 1989. 2, 311–362.
[10] Hutchinson, J.M. “Physical aging of polymers”. Progress in Polymer Science, 1995. 20, 703–760.
[11] Sternstein, S.S. “Homogeneous and inhomogeneous properties of glassy polymers”. Polymer Preprints,
1976. 17, 136–141.
[12] Nanzai, Y., Miwa, A., and Cui, S.Z. “Aging in Fully Annealed and Subsequently Strained Poly(methyl
methacrylate)”. Polymer Journal, 2000. 32, 51–56.
55
3 Prediction of plasticity-controlled failure: interaction with physical ageing
[13] Gui, S.Z. and Nanzai, Y. “Aging in Quenched Poly(methyl methacrylate) under Inelastic Tensile Strain”.
Polymer Journal, 2001. 33, 444–449.
[14] Klompen, E.T.J., Engels, T.A.P., Govaert, L.E., and Meijer, H.E.H. “Modeling of the postyield response of
glassy polymers: Influence of thermomechanical history”. Macromolecules, 2005. 38, 6997–7008.
[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
interpretation of the effect on yield stress and differential scanning calorimetry measurements”. Polymer,
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
polylactides in static loading conditions”. Journal of Materials Science: Materials in Medicine, 2010. 21,
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.
[30] Eyring, H. “Viscosity, Plasticity, and Diffusion as Examples of Absolute Reaction Rates”. The Journal of
Chemical Physics, 1936. 4, 283–291.
[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.
56
References
[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.
[35] Bauwens-Crowet, C., Ots, J.M., and Bauwens, J.C. “The strain-rate and temperature dependence of yield of
polycarbonate in tension, tensile creep and impact tests”. Journal of Materials Science, 1974. 9, 1197–1201.
[36] Janssen, R.P.M., Govaert, L.E., and Meijer, H.E.H. “An analytical method to predict fatigue life of ther-
moplastics in uniaxial loading: Sensitivity to wave type, frequency, and stress amplitude”. Macromolecules,
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.
[38] van Erp, T.B., Reynolds, C.T., Peijs, T., van Dommelen, J.A.W., and Govaert, L.E. “Prediction of yield
and long-term failure of oriented polypropylene: Kinetics and anisotropy”. Journal of Polymer Science Part
B: Polymer Physics, 2009. 47, 2026–2035.
[39] Visser, H.A., Bor, T.C., Wolters, M., Engels, T.A.P., and Govaert, L.E. “Lifetime Assessment of Load-
Bearing Polymer Glasses: An Analytical Framework for Ductile Failure”. Macromolecular Materials and
Engineering, 2010. 295, 637–651.
[40] van Erp, T.B., Cavallo, D., Peters, G.W.M., and Govaert, L.E. “Rate-, temperature-, and structure-
dependent yield kinetics of isotactic polypropylene”. Journal of Polymer Science Part B: Polymer Physics,
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
3 Prediction of plasticity-controlled failure: interaction with physical ageing
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
4 Direct comparison of the compliance method with optical tracking
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
4 Direct comparison of the compliance method with optical tracking
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.
4.3.2 Sample preparation
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
4.3.3 Mechanical tests
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
4 Direct comparison of the compliance method with optical tracking
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
4 Direct comparison of the compliance method with optical tracking
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
4 Direct comparison of the compliance method with optical tracking
- 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
4 Direct comparison of the compliance method with optical tracking
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
4 Direct comparison of the compliance method with optical tracking
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
4 Direct comparison of the compliance method with optical tracking
References
[1] Andersson, U. “Which factors control the lifetime of plastic pipes and how the lifetime can be extrapolated.”
In: “Proceedings of Plastic Pipe XI”, 2001 .
[2] 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.
[3] 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.
[4] Gray, A., Mallinson, J.N., and Price, J.B. “Fracture Behavior of Polyethylene Pipes”. Plastics and Rubber
Processing and Applications, 1981. 1, 51–53.
[5] Burn, S. Long-term Performance Prediction for PVC Pipes. AWWA Research Foundation, 2005.
[6] Lu, J.P., Davis, P., and Burn, L.S. “Lifetime Prediction for ABS Pipes Subjected to Combined Pressure
and Deflection Loading”. Polymer Engineering and Science, 2003. 43, 444–462.
[7] Krishnaswamy, R.K. “Analysis of ductile and brittle failures from creep rupture testing of high-density
polyethylene (HDPE) pipes”. Polymer, 2005. 46, 11664–11672.
[8] Brown, N. “Slow crack growth-notches-pressurized polyethylene pipes”. Polymer Engineering and Science,
2007. 47, 1951–1955.
[9] Schmidt, R.A. “Fracture-toughness testing of limestone - KIc of indiana limestone was measured using
three-point-bend specimens, and toughness is seen to increase with crack length much like many aluminum
alloys”. Experimental Mechanics, 1976. 16, 161–167.
[10] Saxena, A. and Hudak Jr., S.J. “Review and extension of compliance information for common crack growth
specimens”. International Journal of Fracture, 1978. 14, 453–468.
[11] Tsai, Y.M. and Kolsky, H. “A study of the fractures produced in glass blocks by impact”. Journal of the
Mechanics and Physics of Solids, 1967. 15, 263–278.
[12] Swan, G. “The observation of cracks propagating in rock plates”. International Journal of Rock Mechanics
and Mining Sciences and Geomechanics Abstracts, 1975. 12, 329–334.
[13] Li, C.Y., TaIda, P.M., and Wei, R.P. “The effect of environments on fatigue-crack propagation in an
ultra-high-strength steel”. International Journal of Fracture Mechanics, 1967. 3, 29–36.
[14] Landes, J.D. and Wei, R.P. “The kinetics of subcritical crack growth under sustained loading”. International
Journal of Fracture, 1973. 9, 277–293.
[15] “ASTM E647 - 13a Standard Test Method for Measurement of Fatigue Crack Growth Rates”.
[16] Neale, B.K., Curry, D.A., Green, G., Haigh, J.R., and Akhurst, K.N. “A procedure for the determination of
the fracture resistance of ductile steels”. International Journal of Pressure Vessels and Piping, 1985. 20,
155–179.
[17] Havel, R., Neale, B.K., and Senior, B.A. “The fracture properties of aged 316 austenitic steel”. International
Journal of Pressure Vessels and Piping, 1988. 31, 387–403.
[18] Salivar, G.C., Heine, J.E., and Haake, F.K. “The effect of stress ratio on the near-threshold fatigue crack
growth behavior of Ti-8A1-1Mo-1V at elevated temperature”. Engineering Fracture Mechanics, 1989. 32,
807–817.
[19] Salivar, G.C. and Haake, F.K. “A comparison of test methods for the determination of fatigue crack growth
rate threshold in titanium at elevated temperature”. Engineering Fracture Mechanics, 1990. 37, 505–517.
[20] Bencher, C.D., Sakaida, A., Rao, K.T.V., and Ritchie, R.O. “Toughening mechanisms in ductile niobium-
reinforced niobium aluminide (Nb/Nb3Al) in situ composites”. Metallurgical and Materials Transactions A,
1995. 26, 2027–2033.
[21] Fett, T., Kamlah, M., Munz, D., and Thun, G. “Crack resistance and fracture toughness of PZT ceramics”.
In: “Proceedings of SPIE - The International Society for Optical Engineering”, vol. 4333. 2001 pp. 221–230.
78
References
[22] Wright, T.M. and Hayes, W.C. “Fracture mechanics parameters for compact bone. Effects of density and
specimen thickness”. Journal of Biomechanics, 1977. 10, 419–430.
[23] Behiri, J.C. and Bonfield, W. “Crack velocity dependence of longitudinal fracture in bone”. Journal of
Materials Science, 1980. 15, 1841–1849.
[24] Behiri, J.C. and Bonfield, W. “Fracture mechanics of bone - The effects of density, specimen thickness and
crack velocity on longitudinal fracture”. Journal of Biomechanics, 1984. 17, 25–34.
[25] Malik, C.L., Gibeling, J.C., Martin, R.B., and Stover, S.M. “Compliance calibration for fracture testing of
equine cortical bone”. Journal of Biomechanics, 2002. 35, 701–705.
[26] Malik, C.L., Stover, S.M., Martin, R.B., and Gibeling, J.C. “Equine cortical bone exhibits rising R-curve
fracture mechanics”. Journal of Biomechanics, 2003. 36, 191–198.
[27] Nalla, R.K., Kruzic, J.J., Kinney, J.H., and Ritchie, R.O. “Mechanistic aspects of fracture and R-curve
behavior in human cortical bone”. Biomaterials, 2005. 26, 217–231.
[28] Creel, J.A., Stover, S.M., Martin, R.B., Fyhrie, D.P., Hazelwood, S.J., and Gibeling, J.C. “Compliance
calibration for fracture testing of anisotropic biological materials”. Journal of the Mechanical Behavior of
Biomedical Materials, 2009. 2, 571–578.
[29] Owen, M.J. and Bishop, P.T. “Crack-growth relationships for glass-reinforced plastics and their application
to design”. Journal of Physics D: Applied Physics, 1974. 7, 1214–1224.
[30] Zhou, Z., Landes, J.D., and Huang, D.D. “J-R curve calculation with the normalization method for
toughened polymers”. Polymer Engineering and Science, 1994. 34, 128–134.
[31] Berer, M. and Pinter, G. “Determination of crack growth kinetics in non-reinforced semi-crystalline ther-
moplastics using the linear elastic fracture mechanics (LEFM) approach”. Polymer Testing, 2013. 32,
870–879.
[32] Varadarajan, R. and Rimnac, C.M. “Compliance calibration for fatigue crack propagation testing of ultra
high molecular weight polyethylene”. Biomaterials, 2006. 27, 4693–4697.
[33] Varadarajan, R., Dapp, E.K., and Rimnac, C.M. “Static fracture resistance of ultra high molecular weight
polyethylene using the single specimen normalization method”. Polymer Testing, 2008. 27, 260–268.
[34] Balika, W., Pinter, G., and Lang, R. “Fatigue Crack Growth and Process Zone Development in a PE-HD
Pipe Grade in Through-Thickness Direction”. Advanced Engineering Materials, 2006. 8, 1146–1150.
[35] 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.
[36] Frank, A., Pinter, G., and Lang, R.W. “Prediction of the remaining lifetime of polyethylene pipes after up
to 30 years in use”. Polymer Testing, 2009. 28, 737–745.
[37] Frank, A., Hutar, P., and Pinter, G. “Numerical Assessment of PE 80 and PE 100 Pipe Lifetime Based on
Paris-Erdogan Equation”. Macromolecular Symposia, 2012. 311, 112–121.
[38] Hashemi, S. and Williams, J.G. “A fracture toughness study on low density and linear low density
polyethylenes”. Polymer, 1986. 27, 384–392.
[39] Landes, J.D. and Zhou, Z. “Application of load separation and normalization methods for polycarbonate
materials”. International Journal of Fracture, 1993. 63, 383–393.
[40] Bernal, C., Cassanelli, A., and Frontini, P. “On the applicability of the load separation criterion to acryloni-
trile/butadiene/styrene terpolymer resins”. Polymer, 1996. 37, 4033–4039.
[41] Che, M., Grellmann, W., Seidler, S., and Landes, J.D. “Application of a normalization method for determin-
ing J-R curves in glassy polymer PVC at different crosshead speeds”. Fatigue and Fracture of Engineering
Materials and Structures, 1997. 20, 119–127.
[42] Bernal, C.R., Montemartini, P.E., and Frontini, P.M. “The use of load separation criterion and normalization
method in ductile fracture characterization of thermoplastic polymers”. Journal of Polymer Science, Part
B: Polymer Physics, 1996. 34, 1869–1880.
79
4 Direct comparison of the compliance method with optical tracking
[43] Bernal, C., Rink, M., and Frontini, P. “Load separation principle in determination of J-R curve for ductile
polymers: Suitability of different material deformation functions used in the normalization method”. In:
“Macromolecular Symposia”, vol. 147. 1999 pp. 235–248.
[44] Frontini, P.M., Fasce, L.A., and Rueda, F. “Non linear fracture mechanics of polymers: Load Separation
and Normalization methods”. Engineering Fracture Mechanics, 2012. 79, 389–414.
[45] Pinter, G., Balika, W., and Lang, R.W. “A correlation of creep and fatigue crack growth in high density
poly(ethylene) at various temperatures”. European Structural Integrity Society, 2002. 29, 267–275.
[46] Parsons, M., Stepanov, E.V., Hiltner, A., and Baer, E. “Effect of strain rate on stepwise fatigue and creep
slow crack growth in high density polyethylene”. Journal of Materials Science, 2000. 35, 1857–1866.
[47] Pinter, G., Haager, M., Balika, W., and Lang, R.W. “Cyclic crack growth tests with CRB specimens for
the evaluation of the long-term performance of PE pipe grades”. Polymer Testing, 2007. 26, 180–188.
[48] Zhou, Z., Hiltner, A., and Baer, E. “Predicting long-term creep failure of bimodal polyethylene pipe from
short-term fatigue tests”. Journal of Materials Science, 2011. 46, 174–182.
[49] Hertzberg, R.W. and Manson, J.A. Fatigue of engineering plastics. Academic Press, 1980.
[50] Boyd, R.H. “Relaxation processes in crystalline polymers: experimental behaviour - a review”. Polymer,
1985. 26, 323–347.
[51] Lesser, A.J. “Changes in mechanical behavior during fatigue of semicrystalline thermoplastics”. Journal of
Applied Polymer Science, 1995. 58, 869–879.
[52] 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.
[53] Berer, M., Major, Z., Pinter, G., Constantinescu, D.M., and Marsavina, L. “Investigation of the dynamic
mechanical behavior of polyetheretherketone (PEEK) in the high stress tensile regime”. Mechanics of
Time-Dependent Materials, 2014. 18, 663–684.
[54] Struik, L.C.E. Physical Aging in Amorphous Polymers and Other Materials. Elsevier Scientific Publishing
Company, 1978.
[55] Bubeck, R.A. and Kramer, E.J. “Effect of water content on stress aging of nylon 6-10”. Journal of Applied
Physics, 1971. 42, 4631–4636.
[56] Myers, F.A., Cama, F.C., and Sternstein, S.S. “Mechanically enhanced aging of glassy polymers”. Annals
of the New York Academy of Sciences, 1976. 279, 94–99.
[57] Klompen, E.T.J., Engels, T.A.P., Govaert, L.E., and Meijer, H.E.H. “Modeling of the postyield response of
glassy polymers: Influence of thermomechanical history”. Macromolecules, 2005. 38, 6997–7008.
[58] Janssen, R.P.M., Govaert, L.E., and Meijer, H.E.H. “An analytical method to predict fatigue life of ther-
moplastics in uniaxial loading: Sensitivity to wave type, frequency, and stress amplitude”. Macromolecules,
2008. 41, 2531–2540.
[59] 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.
[60] Parsons, M., Stepanov, E.V., Hiltner, A., and Baer, E. “Correlation of fatigue and creep slow crack growth
in a medium density polyethylene pipe material”. Journal of Materials Science, 2000. 35, 2659–2674.
[61] Parsons, M., Stepanov, E.V., Hiltner, A., and Baer, E. “The damage zone ahead of the arrested crack in
polyethylene resins”. Journal of Materials Science, 2001. 36, 5747–5755.
[62] Cheng, W.M., Miller, G.A., Manson, J.A., Hertzberg, R.W., and Sperling, L.H. “Mechanical behaviour of
poly (methyl methacrylate) - Part 2 The temperature and frequency effects on the fatigue crack propagation
behaviour”. Journal of Materials Science, 1990. 25, 1924–1930.
[63] Paris, P. and Erdogan, F. “A critical analysis of crack propagation laws”. Journal of Fluids Engineering,
1963. 85, 528–533.
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 Competition between plasticity-controlled and crack-growth controlled failure
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
III) chemicaldegradation
Figure 5.1: 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 (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
5 Competition between plasticity-controlled and crack-growth controlled failure
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.
5.2.2 Plasticity-controlled failure
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
5 Competition between plasticity-controlled and crack-growth controlled failure
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
εpl (σ, T, t′) dt′ with failure once εpl = εcr (5.7)
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
time−to−failure [s]
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
5 Competition between plasticity-controlled and crack-growth controlled failure
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
time−to−failure [s]
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
5 Competition between plasticity-controlled and crack-growth controlled failure
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
5 Competition between plasticity-controlled and crack-growth controlled failure
• 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).
5.3.2 Mechanical tests
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
5 Competition between plasticity-controlled and crack-growth controlled failure
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 Competition between plasticity-controlled and crack-growth controlled failure
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
GFR PPE/PS at 23◦C
b
GFR PPE/PS at 90◦C
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
5 Competition between plasticity-controlled and crack-growth controlled failure
therefore more appropriate to distinguish between plasticity and crack-growth controlled failure
rather than between ”ductile” and ”brittle” failure regimes.
5.7 Acknowledgements
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.
References
[1] Andersson, U. “Which factors control the lifetime of plastic pipes and how the lifetime can be extrapolated.”
In: “Proceedings of Plastic Pipe XI”, 2001 .
[2] 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.
[3] 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.
[4] Chudnovsky, A., Zhou, Z., Zhang, H., and Sehanobish, K. “Lifetime assessment of engineering thermoplas-
tics”. International Journal of Engineering Science, 2012. 59, 108–139.
[5] Schulte, U. “A vision becomes true: 50 years of pipes made from High Density Polyethylene”. In: “Pro-
ceedings of Plastic Pipes XIII, Washington”, 2006 .
[6] Erdogan, F. “Ductile fracture theories for pressurised pipes and containers”. International Journal of
Pressure Vessels and Piping, 1976. 4, 253–283.
[7] Gotham, K. “Long-term strength of thermoplastics: the ductile-brittle transition in static fatigue”. Plastics
and Polymers, 1972. 40, 59–64.
[8] Lu, X. and Brown, N. “The ductile-brittle transition in a polyethylene copolymer”. Journal of Materials
Science, 1990. 25, 29–34.
[9] Gray, A., Mallinson, J.N., and Price, J.B. “Fracture Behavior of Polyethylene Pipes”. Plastics and Rubber
Processing and Applications, 1981. 1, 51–53.
[10] Lu, J.P., Davis, P., and Burn, L.S. “Lifetime Prediction for ABS Pipes Subjected to Combined Pressure
and Deflection Loading”. Polymer Engineering and Science, 2003. 43, 444–462.
[11] Brown, N. “Slow crack growth-notches-pressurized polyethylene pipes”. Polymer Engineering and Science,
2007. 47, 1951–1955.
[12] Brown, N. “Intrinsic lifetime of polyethylene pipelines”. Polymer Engineering and Science, 2007. 47,
477–480.
[13] 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.
[14] 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.
[15] 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.
[16] Govaert, L.E. and Peijs, T. “Micromechanical modeling of time-dependent transverse failure in composite
systems”. Mechanics Time-Dependent Materials, 2000. 4, 275–291.
98
References
[17] Govaert, L.E., Schellens, H.J., Thomassen, H.J.M., Smit, R.J.M., Terzoli, L., and Peijs, T. “A microme-
chanical approach to time-dependent failure in off-axis loaded polymer composites”. Composites - Part A:
Applied Science and Manufacturing, 2001. 32, 1697–1711.
[18] Anderson, T.L. Fracture Mechanics: Fundamentals and Applications, Second Edition. Taylor & Francis,
1994.
[19] Paris, P. and Erdogan, F. “A critical analysis of crack propagation laws”. Journal of Fluids Engineering,
1963. 85, 528–533.
[20] Williams, J.G. “A model of fatigue crack growth in polymers”. Journal of Materials Science, 1977. 12,
2525–2533.
[21] Zhou, Y., Lu, X., and Brown, N. “A fatigue test for controlling the quality of polyethylene copolymers”.
Polymer Engineering & Science, 1991. 31, 711–716.
[22] Parsons, M., Stepanov, E.V., Hiltner, A., and Baer, E. “Effect of strain rate on stepwise fatigue and creep
slow crack growth in high density polyethylene”. Journal of Materials Science, 2000. 35, 1857–1866.
[23] Lesser, A.J. Encyclopedia Of Polymer Science and Technology, chap. Fatigue Behavior of Polymers. John
Wiley & Sons, Inc., 2002.
[24] Pinter, G., Balika, W., and Lang, R.W. “A correlation of creep and fatigue crack growth in high density
poly(ethylene) at various temperatures”. European Structural Integrity Society, 2002. 29, 267–275.
[25] Pinter, G., Haager, M., Balika, W., and Lang, R.W. “Cyclic crack growth tests with CRB specimens for
the evaluation of the long-term performance of PE pipe grades”. Polymer Testing, 2007. 26, 180–188.
[26] 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.
[27] Frank, A., Pinter, G., and Lang, R.W. “Prediction of the remaining lifetime of polyethylene pipes after up
to 30 years in use”. Polymer Testing, 2009. 28, 737–745.
[28] Zhou, Z., Hiltner, A., and Baer, E. “Predicting long-term creep failure of bimodal polyethylene pipe from
short-term fatigue tests”. Journal of Materials Science, 2011. 46, 174–182.
[29] Hu, Y., Summers, J., Hiltner, A., and Baer, E. “Correlation of fatigue and creep crack growth in poly(vinyl
chloride)”. Journal of Materials Science, 2003. 38, 633–642.
[30] Bernal-Lara, T.E., Hu, Y., Summers, J., Hiltner, A., and Baer, E. “Effect of impact modification on slow
crack growth in poly(vinyl chloride)”. Journal of Materials Science, 2004. 39, 2979–2988.
[31] Bernal-Lara, T.E., Hu, Y., Summers, J., Hiltner, A., and Baer, E. “Stepwise fatigue crack propagation in
poly(vinyl chloride)”. Journal of Vinyl and Additive Technology, 2004. 10, 5–10.
[32] Hertzberg, R.W. and Manson, J.A. Fatigue of engineering plastics. Academic Press, 1980.
[33] Kramer, E.J. “Microscopic and molecular fundamentals of crazing”. In: H.H. Kausch (editor), “Crazing in
Polymers”, vol. 52-53 of Advances in Polymer Science, pp. 1–56. Springer Berlin Heidelberg, 1983.
[34] Weaver, J. and Beatty, C.L. “Effect of temperature on compressive fatigue of polystyrene”. Polymer
Engineering and Science, 1977. 18, 1117–1126.
[35] Shah, A., Stepanov, E.V., Capaccio, G., Hiltner, A., and Baer, E. “Stepwise fatigue crack propagation in
polyethylene resins of different molecular structure”. Journal of Polymer Science, Part B: Polymer Physics,
1998. 36, 2355–2369.
[36] Parsons, M., Stepanov, E.V., Hiltner, A., and Baer, E. “The damage zone ahead of the arrested crack in
polyethylene resins”. Journal of Materials Science, 2001. 36, 5747–5755.
[37] Plummer, C.G., Goldberg, A., and Ghanem, A. “Micromechanisms of slow crack growth in polyethylene
under constant tensile loading”. Polymer, 2001. 42, 9551–9564.
[38] Zhou, Y.Q. and Brown, N. “The mechanism of fatigue failure in a polyethylene copolymer”. Journal of
Polymer Science Part B: Polymer Physics, 1992. 30, 477–487.
99
5 Competition between plasticity-controlled and crack-growth controlled failure
[39] Zhou, Y. and Brown, N. “Anomalous fracture behaviour in polyethylenes under fatigue and constant load”.
Journal of Materials Science, 1995. 30, 6065–6069.
[40] Loo, L.S., Cohen, R.E., and Gleason, K.K. “Chain mobility in the amorphous region of nylon 6 observed
under active uniaxial deformation”. Science, 2000. 288, 116–119.
[41] Capaldi, F.M., Boyce, M.C., and Rutledge, G.C. “Enhanced mobility accompanies the active deformation
of a glassy amorphous polymer”. Physical Review Letters, 2002. 89, 175505/1–175505/4.
[42] Lee, H.N., Paeng, K., Swallen, S.F., and Ediger, M.D. “Direct measurement of molecular mobility in actively
deformed polymer glasses”. Science, 2009. 323, 231–234.
[43] Bauwens-Crowet, C., Ots, J.M., and Bauwens, J.C. “The strain-rate and temperature dependence of yield of
polycarbonate in tension, tensile creep and impact tests”. Journal of Materials Science, 1974. 9, 1197–1201.
[44] Janssen, R.P.M., Govaert, L.E., and Meijer, H.E.H. “An analytical method to predict fatigue life of ther-
moplastics in uniaxial loading: Sensitivity to wave type, frequency, and stress amplitude”. Macromolecules,
2008. 41, 2531–2540.
[45] 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.
[46] van Erp, T.B., Reynolds, C.T., Peijs, T., van Dommelen, J.A.W., and Govaert, L.E. “Prediction of yield
and long-term failure of oriented polypropylene: Kinetics and anisotropy”. Journal of Polymer Science Part
B: Polymer Physics, 2009. 47, 2026–2035.
[47] Engels, T.A.P., Sontjens, S.H., 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.
[48] Visser, H.A., Bor, T.C., Wolters, M., Engels, T.A.P., and Govaert, L.E. “Lifetime Assessment of Load-
Bearing Polymer Glasses: An Analytical Framework for Ductile Failure”. Macromolecular Materials and
Engineering, 2010. 295, 637–651.
[49] van Erp, T.B., Cavallo, D., Peters, G.W.M., and Govaert, L.E. “Rate-, temperature-, and structure-
dependent yield kinetics of isotactic polypropylene”. Journal of Polymer Science Part B: Polymer Physics,
2012. 50, 1438–1451.
[50] 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.
[51] Eyring, H. “Viscosity, Plasticity, and Diffusion as Examples of Absolute Reaction Rates”. The Journal of
Chemical Physics, 1936. 4, 283–291.
[52] Ward, I.M. “Review: The yield behaviour of polymers”. Journal of Materials Science, 1971. 6, 1397–1417.
[53] 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.
[54] Hutchinson, J.M. “Physical aging of polymers”. Progress in Polymer Science, 1995. 20, 703–760.
[55] Klompen, E.T.J., Engels, T.A.P., Govaert, L.E., and Meijer, H.E.H. “Modeling of the postyield response of
glassy polymers: Influence of thermomechanical history”. Macromolecules, 2005. 38, 6997–7008.
[56] Bubeck, R.A. and Kramer, E.J. “Effect of water content on stress aging of nylon 6-10”. Journal of Applied
Physics, 1971. 42, 4631–4636.
[57] Myers, F.A., Cama, F.C., and Sternstein, S.S. “Mechanically enhanced aging of glassy polymers”. Annals
of the New York Academy of Sciences, 1976. 279, 94–99.
[58] 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.
[59] Engels, T.A.P., van Breemen, L.C.A., Govaert, L.E., and Meijer, H.E.H. “Predicting the long-term me-
100
References
chanical performance of polycarbonate from thermal history during injection molding”. Macromolecular
Materials and Engineering, 2009. 294, 829–838.
[60] Ree, T. and Eyring, H. “Theory of Non-Newtonian Flow. I. Solid Plastic System”. Journal of Applied
Physics, 1955. 26, 793–800.
[61] Roetling, J.A. “Yield stress behaviour of polymethylmethacrylate”. Polymer, 1965. 6, 311–317.
[62] Bauwens-Crowet, C., Bauwens, J.C., and Homes, G. “Tensile yield-stress behavior of glassy polymers”.
Journal of Polymer Science Part A-2: Polymer Physics, 1969. 7, 735–742.
[63] Bauwens, J.C., Bauwens-Crowet, C., and Homes, G. “Tensile yield-stress behavior of poly(vinyl chloride)
and polycarbonate in the glass transition region”. Journal of Polymer Science Part A-2: Polymer Physics,
1969. 7, 1745–1754.
[64] Truss, R.W., Clarke, P.L., Duckett, R.A., and Ward, I.M. “The dependence of yield behavior on temperature,
pressure, and strain rate for linear polyethylenes of different molecular weight and morphology”. Journal of
Polymer Science: Polymer Physics Edition, 1984. 22, 191–209.
[65] Boyd, R.H. “Relaxation processes in crystalline polymers: molecular interpretation - a review”. Polymer,
1985. 26, 1123–1133.
[66] Bauwens-Crowet, C. “The compression yield behaviour of polymethyl methacrylate over a wide range of
temperatures and strain-rates”. Journal of Materials Science, 1973. 8, 968–979.
[67] Bauwens-Crowet, C. and Bauwens, J.C. “Annealing of polycarbonate below the glass transition: quantitative
interpretation of the effect on yield stress and differential scanning calorimetry measurements”. Polymer,
1982. 23, 1599–1604.
[68] Seguela, R., Elkoun, S., and Gaucher-Miri, V. “Plastic deformation of polyethylene and ethylene copolymers:
Part II Heterogeneous crystal slip and strain-induced phase change”. Journal of Materials Science, 1998.
33, 1801–1807.
[69] Seguela, R., Gaucher-Miri, V., and Elkoun, S. “Plastic deformation of polyethylene and ethylene copolymers:
Part I Homogeneous crystal slip and molecular mobility”. Journal of Materials Science, 1998. 33, 1273–
1279.
[70] Hardy, G.F. and Wagner, H.L. “Tensile behaviour of glass fiber-reinforced acetal polymer”. Journal of
Applied Polymer Science, 1969. 13, 961–975.
[71] Ibarra, L. and Chamorro, C. “Short fiber-elastomer composites. Effects of matrix and fiber level on swelling
and mechanical and dynamic properties”. Journal of Applied Polymer Science, 1991. 43, 1805–1819.
[72] Chung, H. and Das, S. “Processing and properties of glass bead particulate-filled functionally graded Nylon-
11 composites produced by selective laser sintering ”. Materials Science and Engineering: A, 2006. 437,
226–234.
[73] Sherby, O.D. and Dorn, J.E. “Anelastic creep of polymethyl methacrylate”. Journal of the Mechanics and
Physics of Solids, 1958. 6, 145–162.
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
III) chemicaldegradation
Figure 6.1: 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.
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 Integral approach of crack-growth in static and cyclic fatigue
6.2 Background
6.2.1 Plasticity-controlled failure
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
εpl (σ, T, t′) dt′ with failure once εpl = εcr (6.1)
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
6 Integral approach of crack-growth in static and cyclic fatigue
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
6 Integral approach of crack-growth in static and cyclic fatigue
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.
6.3.2 Sample preparation
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
6 Integral approach of crack-growth in static and cyclic fatigue
101
102
103
104
105
106
40
50
60
70
8090
100
120
140
time−to−failure [s]
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
time−to−failure [s]
max
imum
str
ess
[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
time−to−failure [s]
max
imum
str
ess
[MP
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
time−to−failure [s]
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
6 Integral approach of crack-growth in static and cyclic fatigue
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
102
103
104
105
106
107
108
40
50
60
70
8090
100
120
140
time−to−failure [s]
max
imum
str
ess
[MP
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
6 Integral approach of crack-growth in static and cyclic fatigue
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
6 Integral approach of crack-growth in static and cyclic fatigue
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
6 Integral approach of crack-growth in static and cyclic fatigue
101
102
103
104
105
106
30
40
50
60
80
100
120
time−to−failure [s]
max
imum
str
ess
[MP
a]
R=0.1R=0.25R=0.4R=1
0.1Hz
a10
110
210
310
410
510
6
30
40
50
60
80
100
120
time−to−failure [s]
max
imum
str
ess
[MP
a]
R=0.1R=0.2R=0.4R=0.6R=1
1Hz
b
101
102
103
104
105
106
30
40
50
60
80
100
120
time−to−failure [s]
max
imum
str
ess
[MP
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
6 Integral approach of crack-growth in static and cyclic fatigue
6.7 Acknowledgements
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.
References
[1] 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.
[2] 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.
[3] Andersson, U. “Which factors control the lifetime of plastic pipes and how the lifetime can be extrapolated.”
In: “Proceedings of Plastic Pipe XI”, 2001 .
[4] 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.
[5] 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.
[6] Chudnovsky, A., Zhou, Z., Zhang, H., and Sehanobish, K. “Lifetime assessment of engineering thermoplas-
tics”. International Journal of Engineering Science, 2012. 59, 108–139.
[7] Vogt, H., Enderle, H., Schulte, U., and Hessel, J. “Thermal ageing of PE 100 pipes for accelerated lifetime
prediction under service conditions.” In: “Proceedings of Plastic Pipe XIV”, 2008 .
[8] Bauwens-Crowet, C., Ots, J.M., and Bauwens, J.C. “The strain-rate and temperature dependence of yield of
polycarbonate in tension, tensile creep and impact tests”. Journal of Materials Science, 1974. 9, 1197–1201.
[9] Janssen, R.P.M., Govaert, L.E., and Meijer, H.E.H. “An analytical method to predict fatigue life of ther-
moplastics in uniaxial loading: Sensitivity to wave type, frequency, and stress amplitude”. Macromolecules,
2008. 41, 2531–2540.
[10] 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.
[11] Visser, H.A., Bor, T.C., Wolters, M., Engels, T.A.P., and Govaert, L.E. “Lifetime Assessment of Load-
Bearing Polymer Glasses: An Analytical Framework for Ductile Failure”. Macromolecular Materials and
Engineering, 2010. 295, 637–651.
[12] Dumpleton, P. and Bucknall, C.B. “Comparison of static and dynamic fatigue crack growth rates in high-
density polyethylene”. International Journal of Fatigue, 1987. 9, 151–155.
[13] Zhou, Y., Lu, X., and Brown, N. “A fatigue test for controlling the quality of polyethylene copolymers”.
Polymer Engineering & Science, 1991. 31, 711–716.
[14] van der Grinten, F. and Wichers Schreur, P.W.M. “Use of fatigue testing to evaluate long term performance
of polyethylene”. Plastics Rubber and Composites Processing and Applications, 1996. 25, 294–298.
[15] Parsons, M., Stepanov, E.V., Hiltner, A., and Baer, E. “Correlation of stepwise fatigue and creep slow crack
growth in high density polyethylene”. Journal of Materials Science, 1999. 34, 3315–3326.
[16] Parsons, M., Stepanov, E.V., Hiltner, A., and Baer, E. “Effect of strain rate on stepwise fatigue and creep
slow crack growth in high density polyethylene”. Journal of Materials Science, 2000. 35, 1857–1866.
122
References
[17] Lesser, A.J. Encyclopedia Of Polymer Science and Technology, chap. Fatigue Behavior of Polymers. John
Wiley & Sons, Inc., 2002.
[18] Pinter, G., Balika, W., and Lang, R.W. “A correlation of creep and fatigue crack growth in high density
poly(ethylene) at various temperatures”. European Structural Integrity Society, 2002. 29, 267–275.
[19] Hu, Y., Summers, J., Hiltner, A., and Baer, E. “Kinetics of fatigue and creep crack propagation in PVC
pipe”. Journal of Vinyl and Additive Technology, 2002. 8, 251–258.
[20] Hu, Y., Summers, J., Hiltner, A., and Baer, E. “Correlation of fatigue and creep crack growth in poly(vinyl
chloride)”. Journal of Materials Science, 2003. 38, 633–642.
[21] Pinter, G., Haager, M., Balika, W., and Lang, R.W. “Cyclic crack growth tests with CRB specimens for
the evaluation of the long-term performance of PE pipe grades”. Polymer Testing, 2007. 26, 180–188.
[22] 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.
[23] Frank, A., Pinter, G., and Lang, R.W. “Prediction of the remaining lifetime of polyethylene pipes after up
to 30 years in use”. Polymer Testing, 2009. 28, 737–745.
[24] Zhou, Z., Hiltner, A., and Baer, E. “Predicting long-term creep failure of bimodal polyethylene pipe from
short-term fatigue tests”. Journal of Materials Science, 2011. 46, 174–182.
[25] Zhou, Y.Q. and Brown, N. “The mechanism of fatigue failure in a polyethylene copolymer”. Journal of
Polymer Science Part B: Polymer Physics, 1992. 30, 477–487.
[26] Zhou, Y. and Brown, N. “Anomalous fracture behaviour in polyethylenes under fatigue and constant load”.
Journal of Materials Science, 1995. 30, 6065–6069.
[27] Sauer, J.A. and Richardson, G.C. “Fatigue of polymers”. International Journal of Fracture, 1980. 16,
499–532.
[28] Hertzberg, R.W. and Manson, J.A. Fatigue of engineering plastics. Academic Press, 1980.
[29] Sauer, J.A. and Hara, M. “Effect of molecular variables on crazing and fatigue of polymers”. Advances in
Polymer Science, 1990. 91–92, 69–118.
[30] Wyzgoski, M.G., Novak, G.E., and Simon, D.L. “Fatigue fracture of nylon polymers - part 1 effect of
frequency”. Journal of Materials Science, 1990. 25, 4501–4510.
[31] Kim, H.S. and Wang, X.M. “Temperature and frequency effects on fatigue crack growth of uPVC”. Journal
of Materials Science, 1994. 29, 3209–3214.
[32] Bowman, J. and Barker, M.B. “Methodology for Describing Creep-Fatigue Interactions in Thermoplastic
Components.” Polymer Engineering and Science, 1986. 26, 1582–1590.
[33] Nishimura, H., Nakashiba, A., Nakakura, M., and Sasai, K. “Fatigue behavior of medium-density polyethy-
lene pipes for gas distribution”. Polymer Engineering and Science, 1993. 33, 895–900.
[34] Nishimura, H. and Narisawa, I. “Fatigue behavior of medium-density polyethylene pipes”. Polymer Engi-
neering & Science, 1991. 31, 399–403.
[35] Zhou, Y. and Brown, N. “The fatigue behaviour of notched polyethylene as a function of R”. Journal of
Materials Science, 1989. 24, 1458–1466.
[36] Bernal-Lara, T.E., Hu, Y., Summers, J., Hiltner, A., and Baer, E. “Effect of impact modification on slow
crack growth in poly(vinyl chloride)”. Journal of Materials Science, 2004. 39, 2979–2988.
[37] Bernal-Lara, T.E., Hu, Y., Summers, J., Hiltner, A., and Baer, E. “Stepwise fatigue crack propagation in
poly(vinyl chloride)”. Journal of Vinyl and Additive Technology, 2004. 10, 5–10.
[38] Ayyer, R., Hiltner, A., and Baer, E. “A fatigue-to-creep correlation in air for application to environmental
stress cracking of polyethylene”. Journal of Materials Science, 2007. 42, 7004–7015.
[39] Wyzgoski, M.G. and Novak, G.E. “Fatigue fracture of nylon polymers - Part II Effect of glass-fibre rein-
forcement”. Journal of Materials Science, 1991. 26, 6314–6324.
123
6 Integral approach of crack-growth in static and cyclic fatigue
[40] Pegoretti, A. and Ricco, T. “Fatigue fracture of neat and short glass fiber reinforced polypropylene: effect
of frequency and material orientation”. Journal of Composite Materials, 2000. 34, 1009–1027.
[41] Bernasconi, A., Davoli, P., Basile, A., and Filippi, A. “Effect of fibre orientation on the fatigue behaviour
of a short glass fibre reinforced polyamide-6”. International Journal of Fatigue, 2007. 29, 199–208.
[42] Tanaka, K., Kitano, T., and Egami, N. “Effect of fiber orientation on fatigue crack propagation in short-fiber
reinforced plastics”. Engineering Fracture Mechanics, 2014. 123, 44–58.
[43] Loo, L.S., Cohen, R.E., and Gleason, K.K. “Chain mobility in the amorphous region of nylon 6 observed
under active uniaxial deformation”. Science, 2000. 288, 116–119.
[44] Capaldi, F.M., Boyce, M.C., and Rutledge, G.C. “Enhanced mobility accompanies the active deformation
of a glassy amorphous polymer”. Physical Review Letters, 2002. 89, 175505/1–175505/4.
[45] Lee, H.N., Paeng, K., Swallen, S.F., and Ediger, M.D. “Direct measurement of molecular mobility in actively
deformed polymer glasses”. Science, 2009. 323, 231–234.
[46] 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.
[47] Eyring, H. “Viscosity, Plasticity, and Diffusion as Examples of Absolute Reaction Rates”. The Journal of
Chemical Physics, 1936. 4, 283–291.
[48] Lu, X. and Brown, N. “The transition from ductile to slow crack growth failure in a copolymer of polyethy-
lene”. Journal of Materials Science, 1990. 25, 411–416.
[49] Gray, A., Mallinson, J.N., and Price, J.B. “Fracture Behavior of Polyethylene Pipes”. Plastics and Rubber
Processing and Applications, 1981. 1, 51–53.
[50] Lu, J.P., Davis, P., and Burn, L.S. “Lifetime Prediction for ABS Pipes Subjected to Combined Pressure
and Deflection Loading”. Polymer Engineering and Science, 2003. 43, 444–462.
[51] Brown, N. “Slow crack growth-notches-pressurized polyethylene pipes”. Polymer Engineering and Science,
2007. 47, 1951–1955.
[52] Brown, N. “Intrinsic lifetime of polyethylene pipelines”. Polymer Engineering and Science, 2007. 47,
477–480.
[53] Anderson, T.L. Fracture Mechanics: Fundamentals and Applications, Second Edition. Taylor & Francis,
1994.
[54] Paris, P. and Erdogan, F. “A critical analysis of crack propagation laws”. Journal of Fluids Engineering,
1963. 85, 528–533.
[55] Williams, J.G. “A model of fatigue crack growth in polymers”. Journal of Materials Science, 1977. 12,
2525–2533.
[56] Joseph, S.H. and Leevers, P.S. “Failure mechanics of uPVC cyclically pressurized water pipelines”. Journal
of Materials Science, 1985. 20, 237–245.
[57] Sandilands, G.J. and Bowman, J. “An examination of the role of flaw size and material toughness in the
brittle fracture of polyethylene pipes”. Journal of Materials Science, 1986. 21, 2881–2888.
[58] Wyzgoski, M.G. and Novak, G.E. “Predicting fatigue S-N (stress-number of cycles to fail) behavior of
reinforced plastics using fracture mechanics theory”. Journal of Materials Science, 2005. 40, 295–308.
[59] Brown, N., Lu, X., Huang, Y.L., and Qian, R. “Slow crack growth in polyethylene - a review”. Makro-
molekulare Chemie. Macromolecular Symposia, 1991. 41, 55–67.
[60] Kramer, E.J. “Microscopic and molecular fundamentals of crazing”. In: H.H. Kausch (editor), “Crazing in
Polymers”, vol. 52-53 of Advances in Polymer Science, pp. 1–56. Springer Berlin Heidelberg, 1983.
[61] Weaver, J. and Beatty, C.L. “Effect of temperature on compressive fatigue of polystyrene”. Polymer
Engineering and Science, 1977. 18, 1117–1126.
[62] Shah, A., Stepanov, E.V., Capaccio, G., Hiltner, A., and Baer, E. “Stepwise fatigue crack propagation in
124
References
polyethylene resins of different molecular structure”. Journal of Polymer Science, Part B: Polymer Physics,
1998. 36, 2355–2369.
[63] Parsons, M., Stepanov, E.V., Hiltner, A., and Baer, E. “The damage zone ahead of the arrested crack in
polyethylene resins”. Journal of Materials Science, 2001. 36, 5747–5755.
[64] Plummer, C.G., Goldberg, A., and Ghanem, A. “Micromechanisms of slow crack growth in polyethylene
under constant tensile loading”. Polymer, 2001. 42, 9551–9564.
[65] Hertzberg, R.W., Manson, J.A., and Skibo, M. “Frequency sensitivity of fatigue processes in polymeric
solids”. Polymer Engineering & Science, 1975. 15, 252–260.
[66] Pegoretti, A. and Ricco, T. “Fatigue crack propagation in polypropylene reinforced with short glass fibres”.
Composites Science and Technology, 1999. 59, 1055–1062.
[67] Pegoretti, A. and Ricco, T. “Crack growth in discontinuous glass fibre reinforced polypropylene under
dynamic and static loading conditions”. Composites Part A: Applied Science and Manufacturing, 2002. 33,
1539–1547.
[68] Erdogan, F. “Crack propagation theories”. Tech. rep., DTIC Document, 1967.
[69] Walker, K. “The effect of stress ratio during crack propagation and fatigue for 2024-T3 and 7075-T6
aluminum”. Effects of environment and complex load history on fatigue life, ASTM STP, 1970. 462, 1–14.
[70] Mukherjee, B. and Burns, D.J. “Fatigue-crack growth in polymethylmethacrylate - Effect of frequency,
mean and range of stress-intensity factor”. Experimental Mechanics, 1971. 11, 433–439.
[71] Ritchie, R.O. “Mechanisms of fatigue-crack propagation in ductile and brittle solids”. International Journal
of Fracture, 1999. 100, 55–83.
[72] Ritchie, R.O., Gilbert, C.J., and McNaney, J.M. “Mechanics and mechanisms of fatigue damage and crack
growth in advanced materials”. International Journal of Solids and Structures, 2000. 37, 311–329.
[73] Lang, R.W., Manson, J.A., and Hertzberg, R.W. “Mechanisms of fatigue fracture in short glass fibre-
reinforced polymers”. Journal of Materials Science, 1987. 22, 4015–4030.
[74] Hashin, Z. and Rotem, A. “A cumulative damage theory of fatigue failure”. Materials Science and Engi-
neering, 1978. 34, 147–160.
[75] Petitpas, E., Renault, M., and Valentin, D. “Fatigue behaviour of cross-ply CFRP laminates made of T300
or T400 fibres”. International Journal of Fatigue, 1990. 12, 245–251.
[76] Miner, M.A. “Cumulative damage in fatigue”. Journal of applied mechanics, 1945. 12, 159–164.
[77] Miyano, Y., Nakada, M., Kudoh, H., and Muki, R. “Prediction of tensile fatigue life for unidirectional
CFRP”. Journal of Composite Materials, 2000. 34, 538–550.
[78] Miyano, Y., Nakada, M., and Sekine, N. “Accelerated testing for long-term durability of FRP laminates for
marine use”. Journal of Composite Materials, 2005. 39, 5–20.
[79] Epaarachchi, J.A. “Effects of static-fatigue (tension) on the tension-tension fatigue life of glass fibre
reinforced plastic composites”. Composite Structures, 2006. 74, 419–425.
[80] Reifsnider, K.L. and Stinchcomb, W.W. “Critical-element model of the residual strength and life of fatige-
loaded composite coupons.” ASTM, Philadelphia, PA, USA, Dallas, TX, USA, 1986 pp. 298–313.
[81] Halverson, H.G., Curtin, W.A., and Reifsnider, K.L. “Fatigue life of individual composite specimens based
on intrinsic fatigue behavior”. International Journal of Fatigue, 1997. 19, 369–377.
[82] Reifsnider, K., Case, S., and Duthoit, J. “The mechanics of composite strength evolution”. Composites
Science and Technology, 2000. 60, 2539–2546.
[83] Guedes, R.M. “Durability of polymer matrix composites: Viscoelastic effect on static and fatigue loading”.
Composites Science and Technology, 2007. 67, 2574–2583.
[84] Murakami, Y. Stress intensity factors handbook. No. 1-2 in Stress Intensity Factors Handbook. Pergamon,
1987.
125
6 Integral approach of crack-growth in static and cyclic fatigue
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
time−to−failure [s]
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
6 Integral approach of crack-growth in static and cyclic fatigue
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
6 Integral approach of crack-growth in static and cyclic fatigue
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
time−to−failure [s]
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 Conclusions and recommendations
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
7 Conclusions and recommendations
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