Fatigue crack propagation in

polycarbonate

R.J.H. Hawinkels

MT 11.30

Eindhoven University of Technology

Department of Mechanical Engineering

Polymer Technology

Supervisors:

Dr.ir. L.E. Govaert

Ir. J.G.F. Wismans

Eindhoven, August 30, 2011

Contents

Abstract 2

1 Introduction 3

1.1 Long-term failure . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.1 Linear elastic fracture mechanics . . . . . . . . . . . . . . 81.3.2 Fatigue crack propagation . . . . . . . . . . . . . . . . . . 12

2 Materials and methods 14

2.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.1 Fatigue crack propagation . . . . . . . . . . . . . . . . . . 152.2.2 Compression tests . . . . . . . . . . . . . . . . . . . . . . 182.2.3 Numerical methods . . . . . . . . . . . . . . . . . . . . . . 18

3 Results and discussions 20

3.1 Influence of specimen thickness . . . . . . . . . . . . . . . . . . . 203.2 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . 213.3 Influence of yield stress . . . . . . . . . . . . . . . . . . . . . . . 23

3.3.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.4 Influence of temperature . . . . . . . . . . . . . . . . . . . . . . . 25

3.4.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.5 Influence of the load signal . . . . . . . . . . . . . . . . . . . . . 28

3.5.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.6 Impact-modified polycarbonate . . . . . . . . . . . . . . . . . . . 30

3.6.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 Conclusions and recommendations 32

1

Abstract

A phenomenological study on the fatigue crack propagation behaviour of PC isperformed. For measuring the crack length during experiments, an optical mea-suring technique with a fixed digital camera is employed. The influence of thethickness has a pronounced effect on the fatigue crack propagation behaviour.An increased fracture resistance is observed at lower thicknesses. Therefore, aminimum thickness should be established for accurate measurements. A stronginterrelation between the influence of the thickness and the yield stress on thefatigue crack propagation behaviour is observed. A decreased yield stress resultsin increased fracture resistance. Both effects occur in combination with largeshear lips at the fracture surface.

The amplitude of the load signal influences the fatigue crack propagationrate for positive load values. An increase in amplitude at a fixed maximum loadresults in an increased fatigue crack propagation rate. This result proofs thatfatigue fracture is influenced by more than the applied energy.

The influence of the temperature on the fatigue crack propagation behaviouris proven to be different for two loading conditions. At low stress intensityfactors an increased temperature, increases the fatigue crack propagation rate,while at high stress intensity factors an increased temperature results in highfracture resistance. This complex behaviour is attributed to the influence of thetemperature on the yield stress in combination with the thickness effects.

A brief study on the influence of impact-modifiers on the fatigue crack prop-agation behaviour shows a significant decrease in fatigue crack propagation ratefor low volume percentages of impact-modifier.

2

Chapter 1

Introduction

1.1 Long-term failure

Over the past 15 years glassy polymers are applied in a growing number of load-bearing applications, such as pressurized pipe systems and aquarium windows.In these applications it is essential to prevent catastrophic failure of the polymerproduct. Pressurized pipe systems, such as an unplasticised polyvinyl chloride(uPVC) gas distribution network, are a common application for glassy polymerswhere failure can lead to casualties. To garantee a safe operation of polymerapplications, lifetime predictions are an essential part of the design process.

Predictions of the long-term failure can be based on real-time experiments.However, these tests are extremely expensive and time-consuming. An exam-ple is pressurized pipes, for which it is shown to have failure times exceeding50 years [1]. Modern test methods for pressurized pipes determine the time-to-failure at increased internal pressures and elevated temperatures, which de-creases the experiment duration to approximately 1.5 years [2]. The results aresubsequently extrapolated for normal operating pressures and ambient temper-atures to determine the stress that can be sustained for 50 years. This is knownas the long-term hydrostatic strength (LTHS).

Figure 1.1 (left) schematically shows the results of an internal pressure test.The applied stress is plotted versus the time-to-failure on a double logarith-mic scale. This yields three distinctive regions characterized by their failuremechanism. Region I is related to ductile failure, characterized by large plasticdeformations. This failure mechanism is called ductile tearing. Subsequently, inregion II very little plastic deformation is visible as a minute hairline crack ap-pears in longitudinal direction of the pipe. This quasi-brittle failure behaviour isattributed to slow crack propagation, where a flaw in the pipe slowly grows intoa hairline crack. At low stresses a nearly stress independent failure mechanismis caused by molecular degradation of the polymer, leading to brittle fractureor a multitude of hairline cracks. This mechanism is not further addressed inthis study. Graphical representations of the failure mechanisms are shown inFigure 1.1 (right).

The LTHS method is a thorough way to determine the lifetime, but it isexpensive, as many experiments are required, and 1.5 years of testing is a longperiod in modern day economy. To improve the methods and develop models to

3

Region I

Region II

Region III

Static

log(time−to−failure)

log(

stre

ss)

Figure 1.1: Left: Schematic results of internal pressure tests depicting threeregions of failure for static loading. Reproduced from Lenman [3] . Right:Graphical representation of ductile tearing (I), hairline cracking (II), and mul-tiple cracking (III) [4].

predict lifetimes, long-term failure of glassy polymers has received considerableattention [4–6].

Visser et al. [4] described an engineering approach to predict ductile failureof pressurized pipes based on short-term tensile experiments. The approachis based on the assumption that ductile failure occurs when the accumulatedplastic strain reaches a critical value. A pressure-modified Eyring relation [7, 8]given by:

˙γ(T, p, τ) = γ0 · exp

(−∆U

RT

)

· sinh

(

τν∗

RT

)

· exp

(−µpν∗

RT

)

, (1.1)

was used to calculate the equivalent plastic strain rate ˙γ. In Equation (1.1) γ0 isthe pre-exponential factor related to the thermodynamic state of the polymer,R the universal gas constant, T the temperature, ∆U the activation energy,τ the equivalent stress, ν∗ the activation volume, µ the pressure dependenceparameter and p the hydrostatic pressure. Integrating this equivalent plasticstrain rate over time yields an accumulated plastic strain. Visser et al. [4]validated this approach by using time-to-failure data of dynamically loadedtensile specimens. Unexpectedly, Visser et al. [4] observed region II failure, asshown in Figure 1.2 (left)(open markers), while static tensile creep failure data[4] over a similar applied stress range only showed region I failure.

Slow crack propagation in the second region was investigated by Gray et al. [5].Using linear elastic fracture mechanics (LEFM) to assess short-term failure data,Gray et al. [5] predicted the long-term failure of polyethylene (PE) pipes. Mea-suring the crack propagation rate, statically loaded single edge notched ten-sion specimens, resulted in Figure 1.3 (left). An increase in crack propagationrate is found with an increase in temperature, shown for three temperatures.Gray et al. [5] described this data with a power law given by:

da

dt= β(K)m, (1.2)

where β and m are constants, representing the intercept and the slope of the

4

102

104

106

25

30

35

40

45

50

55

60

Time−to−failure [s]M

axim

um s

tres

s [M

Pa]

uPVC 1 Hz0.1 Hz0.01 Hz

Figure 1.2: Left: Results reproduced from Visser et al. [4] depict the time-to-failure for dynamically loaded uPVC tensile specimens. Specimens that failedas a result of ductile failure are represented by closed markers.

curve, respectively.The stress intensity factor K for a single edge notched specimen is deter-

mined by:

K = Y σ√

πa, (1.3)

with crack length a and applied stress σ. The geometrical parameter Y , de-pends on the ratio of the crack length and the specimen width. SubstitutingEquation (1.3) into Equation (1.2) and intergrating between the initial ai andfinal af values of the crack length, yields the time-to-failure:

tf =2

2 − m(a

1−m/2

f − a1−m/2

i ) · 1

β(Y σ√

π)m(1.4)

Gray et al. [5] used this approach to predict the time-to-failure of pressurizedpipes. The initial flaw size af present in the pipes could not be determined, soGray et al. [5] calculated the initial flaw size based on failure data of pressurizedpipes. This resulted in a range of initial flaw sizes between 10 µm and 100 µm,choosing an initial flaw of 100 µm, pipe failure data were compared to thepredicted values. The results are shown in Figure 1.3 (right) and are in goodagreement with the data.

Gray et al. [5] also tested a tougher grade of PE, with a lower and nonlin-ear crack propagation rate, and conclude that the crack propagation behaviourcould not be described by a power law due to the curvature in the data. Sothe change in material toughness resulted in a more complex crack propagationbehaviour.

Hertzberg et al. [9] and Gerberich et al. [10] tested the fatigue crack propaga-tion in glassy polymers. The fatigue crack propagation mechanism is essentiallycomparable to slow crack propagation, as an initial flaw grows due to a load andeventually leads to failure. Both, Hertzberg et al. [9] and Gerberich et al. [10],tested polycarbonate at various temperatures to see the effect on the fatiguecrack propagation behaviour. An unexplained inconsistency appeared in bothstudies.

5

0.1 0.3 0.5 0.7 0.910

−10

10−9

10−8

10−7

10−6

K [MPa⋅m0.5]

da/d

t [m

/s]

80°C40°C

23°C

PE

102

103

104

3

5

7

91113

Time−to−failure [h]

Str

ess

[MP

a]

23°C40°C

80°C

100 µm prediction

PE

Figure 1.3: Left: Gray et al. [5] determined the crack propagation rate for threetemperatures. Right: Time-to-failure measurements of pressurized pipes withthe corresponding predictions based on crack propagation rates. Reproducedfrom Gray et al. [5]

Hertzberg et al. [9] presented the temperature dependency of fatigue crackpropagation by plotting the fatigue crack propagation rate, at a fixed stress in-tensity factor of 1.5 MPa·m0.5, versus the experimental temperature, as shownin Figure 1.4 (left). It is clear that the maximum fatigue crack propagationrate, at Kmax =1.5 MPa·m0.5, occurs around -60◦C and that the rate decreasesfor lower temperatures as well as for higher temperatures. This anomaly isalso observed by Gerberich et al. [10], who showed an increasing fatigue crackpropagation rate for temperatures down to -50◦C and decreasing rates for tem-peratures decreasing further to -172◦C. Gerberich et al. [10] also reported achange in the slope of the fatigue crack propagation curves. Figure 1.4 (right)depicts a selection of the Paris law’s reported by Gerberich et al. [10]

150 200 250 300

5

10

15

da/d

N x

10−

7 [m/c

ycle

]

Temperature [K]

Kmax

=1.5 MPa⋅m0.5

1 2 3 4

10−6

10−5

10−4

0°C 50°C

100°C

−21°C−50°C

−172°C

∆K [MPa⋅m0.5]

da/d

N [m

/cyc

le]

PC

Figure 1.4: Left: Hertzberg et al. [9] reported a bell-shaped temperature de-pendence of the fatigue crack propagation curve. Right: Gerberich et al. [10]observed changing slopes and similar temperature dependence noticed byHertzberg.

Ward et al. [11] reported the effect of the specimen thickness and the temper-ature on the fatigue crack propagation behaviour. Only two temperatures, 20◦C

6

and -30◦C, were tested and hence the anomaly found by Hertzberg et al. [9] andGerberich et al. [10] was not observed. Ward et al. [11] tested three thicknesses,3 mm, 6 mm, and 9 mm, of polycarbonate compact tension (CT) specimens atroom temperature and found that the 3 mm specimens had the greatest frac-ture resistance, the 9 mm had a fatigue crack propagation rate of an order ofmagnitude higher, while the 6 mm results were inbetween, as shown in Fig-ure 1.5 (left). Note, that the 6 mm results are similar to the 9 mm at low ∆Kvalues and deviate to lower propagation rates at high ∆K values.

1 3 5 7 9 1110

−7

10−6

10−5

10−4

10−3

∆K [MPa⋅m0.5]

da/d

N [m

/cyc

le]

PC

3 mm6 mm9 mm

1 3 5 7 9 1110

−7

10−6

10−5

10−4

10−3

∆K [MPa⋅m0.5]

da/d

N [m

/cyc

le]

PC

3 mm6 mm9 mm

Figure 1.5: Left: Fatigue crack propagation results for three thicknesses of PCat 20◦C. Right: The effect of the specimen thickness at -30◦C. Reproduced fromWard et al. [11]

.

Figure 1.5 (right) shows the results for the same thicknesses at -30◦C. Atlow temperature all results coincided on one straight line until high ∆K val-ues. Ward et al. [11] attributed the differences in the results to the balancebetween plane strain crazing and plane stress yielding. The latter resulted inthe formation of shear lips for different values of thickness and ∆K.

Pruitt et al. [12] measured the fatigue crack propagation rate in polycar-bonate at different load ratios of the dynamic load signal. The load ratio wasdefined as the ratio between the minimal and maximal load of the load cycle.An increase in fracture resistance of 1.5 decades was found for an increase inload ratio from 0.1 to 0.5. So despite an increased mean load, the fatigue crackpropagation rate decreased. This counterintuitive result is attributed to thesame plane strain to plane stress balance reported by Ward et al. [11]

1.2 Objective

Although a considerable amount of research is performed on the fatigue crackpropagation in glassy polymers, no conclusive explanation for the complex phe-nomena is found. In this work a phenomenological approach is used to gaininsight in the mechanisms of fatigue crack propagation in glassy polymers. Atest setup with an optical measuring system is improved to perform all testsrequired in this study. To visualize the effects of the thickness and the yieldstress, a numerical analysis of a cracked body is performed.

7

The objective in this study is to observe the influences of the thickness, theyield stress, the temperature, and the load signal on the fatigue crack prop-agation behaviour and implement the observation in a theory to explain thephenomena. The influence of impact-modifiers is briefly addressed to observethe effect of the resulting decreased yield stress.

1.3 Background

1.3.1 Linear elastic fracture mechanics

Within the field of fracture mechanics, the propagation of cracks in materialsis a widely studied area. LEFM is a popular approach to study crack propaga-tion [4, 5, 11, 12], which started with the work of Griffith, who noticed that thereis a significant discrepancy between the theoretical breaking stress, the stress tobreak atomic bonds, and the actual breaking stress of glass [13]. He discoveredthat small surface scratches, drastically reduced the breaking strength of a ma-terial. Griffith introduced a fracture criterion based on a global energy balance.He posed that in a cracked body the mechanical energy Ue added to a unit vol-ume of material must be transformed into free surface energy Us and internalenergy Ui. Ommitting thermal energy, kinetic energy, and energy dissipation.

Irwin [14] later improved Griffith’s theory by adding energy dissipation to theequation. Besides focusing on the global energy balance, Irwin also introduceda local stress based fracture criterion. He proved that the asymptotic stress fieldnear a crack tip is given by:

σij =

(

k√r

)

fij(θ), (1.5)

with σij being the components of the Cauchy stress tensor, r the distance to thecrack tip, θ the angle with respect to the crack plane as depicted in Figure 1.6,k a proportionality constant, and fij a dimensionless function of θ consisting ofhigher order terms. In linear elastic bodies the dominant term 1√

rapproaches

infinity as r → 0. This introduces a stress singularity near the crack tip, whichresults in infinite stresses at the crack tip. Therefore, a stress criterion cannotbe employed to describe crack propagation. A new constant is introduced toovercome the crack tip singularity which replaces the proportionality constantk.

The stress intensity factor K is defined in the limit r → 0 [15], as describedby:

KI = limr→0

(√2πrσ22|θ=0

)

= k√

2π (1.6)

The stress intensity factor is usually given a subscript to denote the loadingcondition of the crack. Three different loading conditions, depicted in Figure 1.7,are distinguished; the opening mode, the in-plane shear mode and the out-of-plane shear mode. The modes are denoted by roman numbers I, II, and III,respectively. In this study only mode I loading is applied, hence, all furthertheory will focus on mode I loading.

The stress intensity factor is empirically determined for geometries and load-ing conditions common in fracture mechanics testing [17]. With the stress in-

8

��

���

-

6

6?-

�6

?

� -

x1

x2

θ

r

crack

σ11

σ22

σ12

Figure 1.6: Crack with local coordinate system and stress definations. Localx3-axis is normal to the page.

Figure 1.7: Three modes of fracture. Left: Opening mode. Middle: In-planeshear mode. Right: Out-of-plane shear mode [16].

tensity factor as defined in Equation (1.6), the stress components σ11, σ22, andσ12 at the crack tip become:

σ11 =KI√2πr

[

cos(1

2θ)

(

1 − sin(1

2θ) sin(3

2θ)

)]

,

σ22 =KI√2πr

[

cos(1

2θ)

(

1 + sin(1

2θ) sin(3

2θ)

)]

, (1.7)

σ12 =KI√2πr

[

cos(1

2θ) sin(1

2θ) cos(3

2θ)

]

.

A crack will propagate when a material dependent critical stress is reached.Since the stress intensity factor determines the stress at the crack tip, it ispossible to determine a critical stress intensity factor. This critical value of thestress intensity factor is denoted by KIc and is defined as the plane strain stressintensity factor also known as the fracture toughness. KIc is geometry andloading independent and is therefore regarded as a material parameter. Notethat the fracture toughness is only valid under plane strain conditions. If planestress conditions develop, the crack can propagate beyond the critical value.The critical stress intensity factor is determined from standardized fractureexperiments.

In LEFM the stress at the crack tip becomes infinite, while in reality plasticdeformation evolves in front of the crack tip. The size and shape of the plasticzone depend on the stress state at the crack tip.

9

The shape and size of the plastic zone can be determined by employing ayield criterion. Several yield criteria are available and in this study the VonMises yield criterion [18] is used. Von Mises states that yielding occurs whenthe Von Mises stress σv reaches a critical value. The critical value is the yieldstress σy of the material. The Von Mises yield criterion is given by:

σv =√

3

2σ

d : σd,

σ2

v = 1

2

[

(σ1 − σ2)2 + (σ1 − σ3)

2 + (σ2 − σ3)2]

.(1.8)

In this case the Von Mises stress is expressed in terms of the principalstresses. The principal stresses are defined as the eigenvalues of the Cauchystress. For a plane stress condition the components of the Cauchy stress aregiven in Equation (1.7), and the principal stresses are given by:

σ1 = 1

2(σ11 + σ22) +

(

1

4(σ11 − σ22)

2 + σ2

12

)1/2

,

σ2 = 1

2(σ11 + σ22) −

(

1

4(σ11 − σ22)

2 + σ2

12

)1/2

,

σ3 = 0.

(1.9)

The first and second principal stresses are equal for plane stress and plane strainloading conditions. The third principal stress under plane stress conditions iszero, while under plane strain conditions it depends on the poisson’s ratio ν andthe first and second prinicipal stresses.

σ3 = ν(σ1 + σ2) (1.10)

Implementing the Cauchy stresses given by Equation (1.7) in Equations (1.9)and (1.10), the principal stresses at the crack tip become:

σ1 =KI√2πr

cos(1

2θ)

(

1 + sin(1

2θ)

)

,

σ2 =KI√2πr

cos(1

2θ)

(

1 − sin(1

2θ)

)

,

σ3 = 0 for plane stress,

σ3 =2νKI√

2πrcos(1

2θ) for plane strain.

(1.11)

The prinicipal stresses are defined by the stress intensity factor KI , the dis-tance to the crack tip r, and the angle with respect to the crack growth directionθ. Implementing the principal stresses into the Von Mises yield criterion, de-fined in Equation (1.8), results in a distance ry from the crack tip, representingthe boundary of the plastic zone according to the Von Mises criterion. For planestress conditions this boundary equals

ry =K2

I

4πσ2y

[

1 + cos(θ) + 3

2sin2(θ)

]

(1.12)

for plane strain conditions the boundary becomes

ry =K2

I

4πσ2y

[

(1 − 2ν)2(1 + cos(θ)) + 3

2sin2(θ)

]

(1.13)

10

Crack direction

Load

ing

dire

ctio

n

plane strainplane stress

Figure 1.8: Left: Plastic zone boundaries for plane stress and plane strainconditions. Reproduced from [19]. Right: A dog-bone shaped plastic zone inthree dimensions with plane stress conditions near the surface and plane strainconditions at the symmetry plane [20].

Plotting both the plane stress and the plane strain boundaries, with ν =0.31, σy = 70 MPa, and KI = 2 MPa·m0.5 results in two kidney shaped curvesas given in Figure 1.8 (left). It is clear that the plane stress plastic zone issubstantially larger than the plane strain plastic zone. Considering that a planestrain loading condition occurs at the symmetry plane, perpendicular to thethickness direction, of a specimen, since no free contraction in the thicknessdirection is possible, and a plane stress condition occurs near the free surface,where contraction is possible, a three-dimensional representation of the plasticzone will have the shape of a dog-bone, as shown in Figure 1.8 (right).

The dog-bone shaped plastic zone results in a fracture surface with shearlips. Figure 1.9 (left) shows an x-ray computed tomogram of a specimen withshear lips. The two cross-sections indicate the difference at the fracture surfaceat two stress intensity factors.

The fatigue crack propagation rate is closely related to the size and shape ofthe plastic zone and will be maximal when minimal plasticity develops [6], i.e. aplane strain plastic zone. Full plane strain conditions, however, will never occursince a small region of plane stress is present at the specimens free surfaces. Ata certain critical thickness, the relative influence of the plane stress region of theplastic zone, compared to the plane strain region, becomes insignificant. Abovethis thickenss the fatigue crack propagation rate is independent of the specimenthickness. Emperically, the critical thickness is given by [21]:

Wc = 2.5

(

KIc

σy

)2

, (1.14)

with Wc the critical specimen thickness, KIc the critical stress intensity factorfor mode I loading, and σy the yield stress. For polycarbonate, KIc is typically2.24 MPa·m0.5 and σy = 68 MPa [22]. A specimen thickness of over 3 mmshould therefore be sufficient to measure the plane strain propagation rate ofpolycarbonate.

11

Figure 1.9: Left: A semi-transparent x-ray computed tomogram of a fractured 6mm specimen. Two cross-sections indicate the difference of the fracture surfaceat different stress intensity factors. Right: An image of the same specimen. Theshear lips are indicated with arrows and the white lines represent the transitionfrom the plane strain fracture region to the plane stress shear lips.

1.3.2 Fatigue crack propagation

Fatigue failure occurs due to dynamic loading on a cracked body. A constantamplitude load, characterized by stress or force, is most often exerted to studyfatigue crack propagation in materials. Since the stress in a single point nearthe crack tip is fully determined by the stress intensity factor KI , the fatigueload can also be characterized by the stress intensity factor. The applied loadis fully characterized by its maximum value Kmax, and a load ratio R, whichis defined by the ratio between the minimum stress intensity factor Kmin, andthe maximum stress intensity factor Kmax, and are given by:

R =Kmin

Kmax, (1.15)

∆K = Kmax − Kmin. (1.16)

Several crack propagation models are proposed to relate the fatigue crackpropagation rate da

dN , to the maximum stress intensity factor Kmax [23, 24].Some relate the fatigue crack propagation rate to the load amplitude ∆K andsome incorporate the load ratio R, or the mean load Kmean, to improve theresult. In polymer fatigue the LEFM approach with Paris’ law [25] is widelyapplied [10, 12, 26, 27]. Paris’ law relates the crack propagation rate to themaximum stress intensity factor using two parameters, a pre-exponential factorA, and an exponent m, as given by:

da

dN= A(Kmax)m. (1.17)

Figure 1.10 shows a schematic fatigue crack propagation curve with thecorresponding Paris law. Below a certain stress intensity factor no crack propa-gation will occur, this threshold stress intensity factor is often denoted by Kth.Increasing the stress intensity factor leads to exponential growth of the fatigue

12

crack propagation rate. Near the end of the lifetime the fatigue crack propaga-tion rate will increase again until failure occurs. The Paris law only describesthe fatigue crack propagation rate correctly in the exponential growth region,indicated between the two vertical dashed lines, while initially it over-predictsthe propagation rate and in the end it under-predicts the propagation rate.

Very slowcrackpropagation

Stable crackpropagation

Unstablecrackpropagation

log( Kmax

)

log(

da/

dN )

da

dN= A(K)m

Kth

Figure 1.10: Schematic crack propagation curve (solid line) with the corre-sponding Paris law (diagonal dashed line). The vertical dashed lines indicatethe boundaries of the exponential growth region.

13

Chapter 2

Materials and methods

This chapter describes the experimental setups and materials used to explorethe theories set forth in this report. In the Materials section, various gradesof polycarbonate and the specimen preparation methods are described. In theMethods section an explanation of the test setups for various experiments isprovided.

2.1 Materials

In this study a commercially available Lexan extrusion grade of polycarbonateis used as the main material of interest. The material was received in the formof extruded sheets of two thicknesses. For the fatigue crack propagation ex-periments, Compact Tension (CT) specimens of both thicknesses are produced.The general geometry of a CT specimen is shown in Figure 2.1. All dimensionsare determined according to the ASTM standard E647 [28] and are listed inTable 2.1. The extruded grade specimens are cut with a circular saw from alarge extruded plate and the fixation holes and notch are precision machinedinto the specimen.

W [mm] L [mm] H [mm]Small 6 32 38.5Large 12 64 77

Table 2.1: Specimen dimensions.

In order to remove the radius of the machined notch and start the experi-ment with a sharp crack all specimens are precracked. The small specimens areprecracked with a fresh razor blade that is tapped into the notch root. Thisproved to be a very consitent method, with an average crack length of 790 µmand a variation of 20 µm. The large specimens are fatigue precracked accordingto ASTM standard E647 [28]. A fresh specimen is mounted in the experimentalsetup and precracked to a crack length of approximately 3 mm using a dynamicload with a load ratio of 0.1. To accelerate the initial precrack process a highmaximum force Fmax, of 800 N, is employed. To prevent residual stresses affect-ing the experiment, the final millimeter of precracking is done at a maximumforce lower than the maximum force applied during the experiment.

14

Figure 2.1: geometry of a Compact Tension specimen.

Annealing treatments are performed in a hot air circulating oven at a temper-ature of 120◦C for 90 hours. The specimens are placed on a polytetrafluoroethy-lene (PTFE) plate to prevent sticking and after the treatment, the specimensare allowed to cool to room temperature in open air.

The experiments for the study of the effect of rubber particles were per-formed on a Lexan 141R grade containing 4.5% and 9% of impact-modifierby volume. The impact-modifier was a methacrylate-butadiene-styrene (MBS)core-shell copolymer, commercially available as Paraloid EXL-2600. Since theimpact-modified grades are sensitive to thermal degradation, the melting tem-perature was lowered to 240◦C and the air in the mould is substituted fornitrogen. The 4.5% grade was dryed for 210 minutes at 110◦C, the 9% gradeshowed some degradation after this drying process so the drying time was re-duced to 75 minutes. All manually pressed specimens show an uneven surfaceand are therefore surface machined to the correct thickness. Further cuttingand machining to the dimensions listed in Table 2.1 is equal to the extrudedgrade Lexan specimens.

2.2 Methods

2.2.1 Fatigue crack propagation

Fatigue crack propagation experiments are performed at two different tensilemachines. The first tensile machine is the servo-hydraulic MTS 810 ElastomerTesting System with a 2.5 kN load cell. The second servo-hydraulic machine isthe MTS 831 Elastomer Testing System with a 25 kN load cell. These systemsare very similar to each other. On both systems the testing temperature can beregulated by an environmental control chamber.

The specimen is attached to the tensile stage by a Clevis bracket depictedin Figure 2.2 (left). The dimensions of this bracket are prescribed by ASTMstandard E647 [28]. Besides the rotation of the specimen around the pins, thebracket contains an additional degree of freedom for the axial alignment of theupper and lower pin. The pins can be pretensioned in order to eliminate radial

15

play between the bracket and the pins. The additional degree of freedom allowsfor experiments with a changing sign in the applied force.

Figure 2.2: Left: Image of the calibration specimen with white circles aroundthe calibration points. In the upper and lower right corners, parts of the Clevisbrackets are visible. The shape between the brackets is the machined notch.Right: Image of the upper Clevis bracket with an additional degree of freedom.

The fatigue crack propagation tests are carried out with a force controlledsinusoidal signal with a frequency of 5 Hz. PC shows no frequency sensitiv-ity in the range from 1 to 70 Hz [6], hence the frequency is chosen based onpreventing motion blur in the images. The load ratio R, as defined in Equa-tion (1.15), equals 0.1 for all tests, except for tests performed to study the loadratio dependence of the fatigue crack propagation rate. The maximum force inthe load cycle is chosen between 500 N and 800 N for the 12 mm specimens andbetween 250 N and 400 N for the 6 mm specimens. This range in maximumforce increased the range of the maximum stress intensity factors tested, whilethe tests duration remained within reasonable time. All experiments in thisstudy are repeated at least three times.

The fatigue crack propagation is monitored using a digital camera operatedby the MATLAB® Image Acquisition toolbox. Two digital cameras are avail-able, a Prosilica EC1280 with a resolution of 1280x1024 pixels and a ProsilicaCV640 with a resolution of 659x498 pixels. The cameras are fitted with a Pen-tax C52893K 50 mm lens or a Nikon micro-Nikkor 55 mm lens. The camera ispositioned perpendicular to the specimens surface at a distance such that thefinal crack will cover the full width of the image.

The MATLAB® Image Acquisition toolbox provides a well documentedplatform to operate the cameras. In order to reduce the amount of data, aregion of interest is set around the expected final crack and only this regionis stored for post-processing. The image acquisition frequency is set accordingto the length of the experiment. The total number of images is limited by thecomputers random-access memory.

An image of the experiment is shown in Figure 2.3 (upper). The verticalgray line on the left side of the image is the specimens edge. The 90◦ rotatedV-shape on the right is the machined notch, with the extending horizontal lineto the left being the crack. The two small bright areas between the crack andthe edge of the specimen are dust particles.

16

Figure 2.3: Upper: A grayscale image as obtained in an experiment. Lower:The converted binary image, with the crack shown in white.

After an experiment, the image processing is done by using the MATLAB®Image Processing toolbox. The grayscale images are converted into binary im-ages, shown in Figure 2.3 (lower), with the ’im2bw’ function. The edge of thespecimen is still visible on the left side of the image. On the right, the notch andcrack are depicted in white. The dust particles on the specimen are digitallyremoved by filling white areas below a threshold area. The crack tip is repre-sented by the leftmost white pixel of the crack. To locate this pixel, a boundaryrecognition function ’bwtraceboundary’ determines the boundary between theblack and the white area. The leftmost pixel in this area is the crack tip.

Prior to each experiment, a calibration image of a specimen with clearlyvisible marks, as shown in Figure 2.2 (right), is taken to correlate the number ofpixels in the image to the physical crack length. The calibration image providesseveral fixed distances to correlate the number of pixels to the actual distance.The distances between the calibration points (white circles) are measured ac-curatelly and programmed into the software. Before processing, the pixels ofthe calibration points in the image are selected which, enables the software tocorrelate the number of pixels between to points to the physical distances.

0 2 4 6 8x 10

4

10

20

30

40

50

Cycles [−]

Cra

ck le

ngth

[mm

]

da

dN

1 2 3 410

−8

10−6

10−4

Kmax

[MPa⋅m0.5]

da/d

N [m

/cyc

le]

PC

Figure 2.4: Left: N-a curve with five processing points and a correspondingtangent line. Right: The resulting crack propagation curve with the fitted Parislaw.

Processing the images yields the crack length versus number of cycles curveas depicted in Figure 2.4 (left). The fatigue crack propagation curve is subse-quently obtained by deriving the crack length curve with respect to the numberof cycles. Figure 2.4 (left) illustrates this process for five derivation points. Theslope of the tangent represents the fatigue crack propagation rate at the re-spective crack length. The maximum stress intensity factor Kmax is calculatedby substituting the maximum applied force and the crack length into Equa-

17

tion (2.1). The resulting curve is depicted in Figure 2.4 (right), where elevenderivation points are used. The corresponding Paris law is calculated using aleast square fit.

KI =F

W√

L

2 + a/L

(1 − a/L)3/2

[

0.866 + 4.64(a/L)− 13.32(a/L)2 + 14.72(a/L)3 − 5.60(a/L)4]

(2.1)

2.2.2 Compression tests

Compression tests are performed to determine the yield stress at various tem-peratures and the yield stress of PC after annealing. The tests are executed onthe above mentioned MTS 831 Elastomer Testing System and on a Zwick Z010servo-electric tensile stage with a 10 kN load cell. Cylindrical specimens withdimensions of approximately 6 mm heigh and 6 mm in diameter are machinedfrom the fatigue crack propagation specimens. The cylindrical specimens arecompressed between two parallel flat plates and to minimize friction between theplates and the specimen, PTFE spray and PTFE tape are applied. The machinestiffness is measured by pressing the surfaces together without a specimen. Thisstiffness, typically 19,000 N/mm for the Zwick Z010 and 28,000 N/mm for theMTS 831, is used to calculate the true displacement. The tests are performed ata contstant true strain rate of −10−3 s−1. The resulting yield stresses are listedin Table 2.2. The difference in yield stress between a large and a small CTspecimen, is caused by the slower cooling of the thick plate during production.

True yield stress [MPa]Large @ 23◦C 71.0Small @ 23◦C 70.3Small @ 0◦C 81.0Small Annealed 90h @ 120◦C 79.0

Table 2.2: Yield stress of PC for various conditions.

2.2.3 Numerical methods

A finite element model is created for the simulations of the plastic zone in frontof the crack tip. A Center Cracked Tension (CCT) specimen is used, since thegeometry of the CT specimens used in the fatigue crack propagation experimentsis less suitable for numerical simulations. It has only two planes of symmetryand the pinhole to apply the force requires contact surfaces. The CCT geometryhas three planes of symmetry and a distributed applied load without contactsurfaces. The model does not allow for crack propagation so the stress intensityfactor K can not increase due to a larger crack length. In the simulations thestress intensity factor is increased by an increasing load. This method allowsfor simulations in a K range which is comparable to the experiments.

A finite element model of a CCT specimen, depicted in Figure 2.5 (left), iscreated in the finite element software package MSC.Mentat 2005r3. Two three-dimensional meshes, representing 6 mm and 12 mm thickness, with a meshrefinement near the crack tip, are created from 8-node hexagonal elements.

18

In the mesh refinement, the element size equals 0.075 mm, while the meshsurrounding the refinement is auto-generated by the software. The symmetryboundary conditions are fixed displacements applied perpendicular to the threeplanes of symmetry, while a negative pressure on the top surface simulates anapplied distributed stress.

0 0.2 0.4 0.60

20

40

60

80

Comp. true strain [−]

Com

p. tr

ue s

tres

s [M

Pa]

σy = 65.7 [MPa]

σy = 70.0 [MPa]

σy = 75.9 [MPa]

Exp. data

Figure 2.5: Left: Two dimensional finite element mesh of a CCT specimen.Right: Intrinsic material behaviour predicted by the EGP model for three yieldstresses. Experimental data, provided by Klompen et al. [29], is added to verifythe behaviour.

The nonlinear elasto-viscoplastic material model used, is the single modeEindhoven Glassy Polymer (EGP) model as described in [30] and [31]. This con-stitutive model is implemented into the HYPELA2 subroutine. Figure 2.5 (right)shows the intrinsic material behaviour predicted by the EGP model for threedifferent yield stresses. The model is verified with experimental data obtainedfrom Klompen et al. [29].

During the simulation a script, implemented in the HYPELA2 subroutine,writes coordinates of all elements having at least one integration point withan equivalent plastic strain larger than 10−5 to an external text file. Thisfile is processed by MATLAB® which calculates the volume of each individualelement in a summation loop. Since the elements are not exact rectangles whenstress is applied, the volume is determined by calculating the volume of the fourtetrahedra that constitute the element.

19

Chapter 3

Results and discussions

3.1 Influence of specimen thickness

Figure 3.1 displays the fatigue crack propagation rates for 6 mm (closed mark-ers) and 12 mm (open markers) specimens. The solid line represents Paris’ lawfor the 12 mm results. Only the data from two measurements are depicted forclarity. The 6 mm and 12 mm thick specimens should yield the same results,considering the critical thickness of 3 mm. The results, however, show a differ-ence in the fatigue crack propagation behaviour. For a maximum stress intensityfactor up to Kmax = 2.3 MPa·m0.5, the fatigue crack propagation rate is equalfor 6 mm and 12 mm specimens and is accuratelly described by Paris’ law. Byfurther increasing the stress intensity factor, the 6 mm specimens shows a morefracture resistant behaviour. This suggests that the plastic zone in front of thecrack tip of the 6 mm specimens, must consist of a significant plane stress re-gion. The fatigue crack propagation curve of a 12 mm specimen is consistentlyincreasing, which indicates a nearly complete plane strain plastic zone.

1 2 3 4 5 610

−8

10−6

10−4

Kmax

[MPa⋅m0.5]

da/d

N [m

/cyc

le]

23 °C

6 mm12 mmParis law

Figure 3.1: Fatigue crack propagation rates for a 6 mm and a 12 mm specimen.The solid line represents the Paris law of the 12 mm specimen.

The deviation in the results of the 6 mm specimen and the 12 mm specimenis characterized by the plane strain condition of the plastic zone, therefore the

20

point at which the 6 mm results deviate from the 12 mm results, is called theplane strain deviation point.

The deviation in the behaviour of the 6 mm specimens is also visible onthe fracture surface of the specimens. Figure 3.2 shows images of a 6 mmspecimen (upper) and a 12 mm specimen (lower). The solid lines across thesurfaces represent the crack length for a maximum stress intensity factor of2.3 MPa·m0.5. The 6 mm specimen shows considerable shear lips at higherstress intensity factors, indicating a plane stress plastic region. The 12 mmspecimen shows a nearly flat fracture surface until failure, corresponding to aplane strain plastic zone.

Figure 3.2: Upper: Fracture surface of a 6 mm specimen showing significantshear lips. Lower: Fracture surface of a 12 mm specimen which is nearly flatuntil failure. The solid lines represent the crack length for a maximum stressintensity factor of 2.3 MPa·m0.5.

The three-dimensional plastic zone in front of a crack tip is not visible inexperiments. In numerical simulations the plastic zone can be visualized forvarious thicknesses and yield stresses.

3.2 Numerical simulations

Numerical simulations, using finite element models, show the size and shape ofthe plastic zone in front of the crack tip. To visualize the plastic zone and toinvestigate the effect of the specimen thickness and the yield stress, simulationson a 6 mm and a 12 mm thick mesh are performed.

The volume of the plastic zone is limited by a fully plane stress plastic zoneand a fully plane strain plastic zone. These zones form the upper and lowerbound of the plastic zone volume, respectively. The EGP material model is notsuitable to simulate plane stress conditions and hence only the lower bound ofthe plastic zone volume is depicted in Figure 3.3 (left). The mean plastic volume(MPV) depicted in Figure 3.3 (left) is defined as the numerically calculatedvolume of the plastic zone divided by the thickness of the model. The planestrain model has the smallest MPV, since it is a pure plane strain plastic zone.The 12 mm model has a larger MPV than the plane strain model, meaning aplane stress region exists in the plastic zone. The 6 mm model has an even larger

21

MPV, indicating that the 6 mm model has a larger plane stress contribution tothe plastic zone compared with the 12 mm model. The stress intensity factor atwhich the three curves deviate is similar to the plane strain deviation point ofthe 6 mm specimen in Figure 3.1, being approximately Kmax = 2.3 MPa·m0.5.At a lower stress intensity factor no thickness dependence is observed.

2 3 40

5

10

15

20

25

Kmax

[MPa⋅m0.5]

MP

V x

10−

9 [m3 /m

]

6 mm12 mmplane strain

2 3 40

5

10

15

Kmax

[MPa⋅m0.5]

∆MP

V x

10−

9 [m3 /m

]

σy = 65.7 [MPa]

σy = 70.0 [MPa]

σy = 75.9 [MPa]

Figure 3.3: Left: The mean plastic volume in 6 mm and 12 mm thick finiteelement models show a decrease in plastic zone volume per unit thickness. Theplane strain plastic volume represents the lower limit of the plastic volume.Right: ∆MPV for three yield stresses; σy = 65.7 MPa, σy = 70.0 MPa, andσy = 75.9 MPa.

Comparing the shape of the plastic zone as predicted by the numerical sim-ulations to the dog-bone shape predicted by the LEFM equations, reveals char-acteristic differences. Figure 3.4 shows the numerical plastic zone of a 6 mmmodel at two stress intensity factors. The crack is represented by the gray areas,with the crack tip at the end of the black line. The visible side of the dog-boneis the free surface of the model, while the symmetry plane is invisible behindthe dog-bone. The plane stress region near the surface is less pronounced in thenumerical simulations and the size of the plastic zone increases over the totalthickness of the model when the load is increased. This is the result of the factthat the stress field used to calculate the shape of the plastic zone using theLEFM equation, was assumed to be fully elastic, while in the numerical simu-lations plasticity occurs. This has been noticed by Fernandez Zuniga et al. [32]who assumed the plastic zone in the aluminium alloy Al 7075 to be cylindricallyshaped instead of dog-bone shaped.

Simulation on a 6 mm model using three different yield stresses show decreas-ing plane stress regions in the plastic zone. Figure 3.3 (right) shows the ∆MPV,which is defined as the difference between the three-dimensional MPV and thetwo-dimensional plane strain MPV. A smaller ∆MPV indicates a small planestress region in the plastic zone. The decreased plasticity yields an increased fa-tigue crack propagation rate. Since the plastic volume for all three yield stressesremains approximately equal until Kmax = 2.3 MPa·m0.5, increasing the yieldstress seems to have a similar effect on the fatigue crack propagation behaviouras increasing the thickness of the specimen. In the next section this effect isexperimentally tested by increasing the yield stress with heat treatments.

22

Figure 3.4: Plastic zone representation of a 6 mm thick model atKmax = 1.93 MPa·m0.5 (left) and at Kmax = 2.51 MPa·m0.5 (right). Thecolours represent increasing equivalent stress, with blue being 62 MPa and yel-low being 280 MPa. The crack is indicated by the gray area.

3.3 Influence of yield stress

The size of the plastic zone is not only affected by the thickness of the spec-imen but also by the yield stress. The yield stress of a glassy polymer canbe affected by heat treatments. An amorphous polymer below its glass tran-sition temperature Tg is in a non-equilibrium thermodynamic state, resultingin time-dependent physical and mechanical properties. Over time the polymerstrives towards a favourable equilibrium state, which increases the yield stress.This process is called physical aging and it is accelerated by increasing thetemperature. A heat treatment, in which a glassy polymer is aged at elevatedtemperatures below Tg, is generally refered to as annealing. The effects of an-nealing on the yield stress of polycarbonate is given in Figure 3.5 (left), showingthe results of compression tests at −10−3 s−1 strain rate of an annealed and anas-received specimen.

0 0.05 0.1 0.15 0.20

20

40

60

80

100

Comp. true strain [−]

Com

p. tr

ue s

tres

s [M

Pa]

As−received

Annealed

1 2 3 4 5 610

−8

10−6

10−4

Kmax

[MPa⋅m0.5]

da/d

N [m

/cyc

le]

AnnealedAs−received

Figure 3.5: Left: Results of compression tests show an increase in yield stressdue to annealing. Right: Fatigue crack propagation results of an annealed andan as-received specimen.

Fatigue crack propagation experiments on as-received 6 mm specimens, showsignificant shear lips on the fracture surface suggesting a plane stress contribu-

23

tion in the plastic zone, as presented in Section 3.1. By annealing the 6 mm spec-imens, the yield stress increases approximately 9 MPa (Figure 3.5 (left)). Theeffect on the fatigue crack propagation rate is seen in Figure 3.5 (right), whichshows the fatigue crack propagation curves of an as-received 6 mm specimencompared with an annealed 6 mm specimen. More experiments are performed,but are ommited for clarity. The annealed specimen in Figure 3.5 (right), showsa fatigue crack propagation behaviour comparable to the 12 mm specimen shownin Figure 3.1. Comparable to the effect of specimen thickness, annealing has noeffect at low stress intensity factors, while around Kmax = 2.3 MPa·m0.5 theas-received specimen shows an increased fracture resistance.

The shear lips of the annealed 6 mm specimen (Figure 3.6 (upper)) aresignificantly smaller compared to the shear lips of an as-received 6 mm specimen,Figure 3.6 (lower). This indicates less plasticity and therefore a primarily planestrain plastic zone.

Figure 3.6: Upper: Fracture surface of an as-received 6 mm specimen withsignificant shear lip formation. Lower: Fracture surface of an annealed 6 mmspecimen showing very little shear lips.

3.3.1 Discussion

The effects of plasticity in front of a crack tip during fatigue loading are in-vestigated. Based on existing literature and LEFM theory the influence of thespecimen thickness and yield stress are succesfully assessed. The results showno effect at low stress intensity factors. At a certain stress intensity factor,increased fracture resistance is observed for decreased yield stress or similarlyfor decreased specimen thickness. The deviation is caused by the influence ofthe plane stress region in the plastic zone. It is believed that the plane straindeviation point can shift along the plane strain crack propagation curve. Thisis in accordance with the results reported by Ward et al. [11] This conclusionis summarized in Figure 3.7. The solid line represents the plane strain fatiguecrack propagation curve and the dashed lines represent the increased fractureresistance curves for various yield stresses σy or specimen thicknesses W .

24

log( Kmax

)lo

g( d

a/dN

)

Plane strain W, σy

Figure 3.7: Schematic representation of fatigue crack propagation behaviour.Solid line represents plane strain propagation, dashed lines represent increasedfracture resistance.

3.4 Influence of temperature

Understanding the influence of the temperature on fatigue crack propagationbehaviour in glassy polymers, is critical for the prediction of its service life. Thetemperature has a pronounced effect on the material properties of polymers,which in turn affect the fatigue crack propagation behaviour.

To investigate the influence of the temperature on the fatigue crack prop-agation rate, 6 mm specimens are tested at temperatures of 0◦C , 45◦C, and90◦C. The specimens are allowed to equilibrate for 15 minutes prior to testing.

The fatigue crack propagation results of the 6 mm specimens at three tem-peratures are depicted in Figure 3.8 (left). The fatigue crack propagation curvesof the 0◦C and the 45◦C specimens show a small part of similar fatigue crackpropagation behaviour and subsequently deviate. The 90◦C specimens show nosimilar part with the lower temperatures and deviate from the start of the exper-iments. Although the yield stress is significantly lower in the 90◦C experimentswith respect to the 45◦C experiments, the deviation from the plane strain crackpropagation rate seems to occur at a similar fatigue crack propagation rate.

The fracture surfaces of the specimens in the increased temperature tests,show an increased amount of shear lip formation, as expected from the resultsof Ward et al. [11]. The effects of shear lips are shown in the previous sectionand eliminating these effects is possible by increasing the specimen thickness.

12 mm specimens are tested at equal temperatures as the 6 mm specimensand a predominantly temperature affected result is observed, since no shearlip formation is visible on the fracture surfaces, as show in Figure 3.9. Fig-ure 3.8 (right) depicts the results of 12 mm specimens at three different temper-atures. These results show a more complex fatigue crack propagation behaviourthan stated in the previous section. The fatigue crack propagation curves donot originate in a single plane strain propagation rate, which was proposed inthe previous section. It becomes apparent that the slope as well as the pre-exponential factor of the Paris law are temperature dependent. This complexbehaviour makes it difficult to model fatigue crack propagation accurately withParis’ law.

25

1 3 5 7 910

−8

10−6

10−4

Kmax

[MPa⋅m0.5]

da/d

N [m

/cyc

le]

6 mm

0 °C 45 °C90 °C

1 3 5 7 910

−8

10−6

10−4

Kmax

[MPa⋅m0.5]

da/d

N [m

/cyc

le]

12 mm

0 °C45 °C90 °C

Figure 3.8: Left: Fatigue crack propagation curve of 6 mm specimens at tem-peratures of 0◦C, 45◦C, and 90◦C. Right: Fatigue crack propagation curves of12 mm specimen at temperatures of 0◦C, 45◦C, and 90◦C.

Figure 3.9: Upper: Fracture surface tested at 0◦C with the initial crack on theleft side of the image. Lower: Fracture surface tested at 90◦C with the initialcrack at the left side of the image.

26

An alternative approach is suggested by Krausz et al. [33]. Krausz et al. [33]state that each process contributing to the fatigue crack propagation shouldbe represented by an appropriate elementary rate constant. This fracture ki-netics theory is similar to the Eyring approach [7, 8] used to predict ductilefailure in glassy polymers. The fracture kinetics approach is derived from theunderstanding of the physical processes that control crack propagation, such as,disentanglement, chain scissioning, void creation, and fibril buckling. Once aphysical process is identified it must be expressed in a mathematical model. Tothis end, the theory of rate processes [34] is employed.

Each process requires a certain amount of energy to be activated, otherwiseit would happen spontaneously. This energy is externally supplied through workor temperature. The elementary rate constant is generally expressed as

r =kT

hexp

(

−Ur − Us

kT

)

(3.1)

where k and h are Boltzmann’s and Planck’s constants, respectively, T is theabsolute temperature, Ur is the energy required to activate the process, and Us

is the energy supplied [34].Multi-process phenomena require a rate constant for each process that con-

trols the velocity. Note, that rate constants should not be added for curve-fittingpurposes but strictly to describe real, physical phenomena. When phenomenarequire multiple processes, they can occur in a parallel or in a consecutive order.

Analysis of all processes governing crack propagation is time-consuming anddifficult. The resulting constitutive equation, however, is valid for a wide rangeof conditions since it is based on physical processes that contribute to crackpropagation.

In fracture kinetics it is appropriate to plot the fatigue crack propagationcurves on a semi-logarithmic scale with the stress intensity factor divided bythe test temperature [33]. In Figure 3.10 (left) it is shown that the tempera-ture corrected stress intensity factor Kmax/T results in parallel fatigue crackpropagation rates for both temperatures. Appending the results of the 6 mmspecimens at similar temperatures yields Figure 3.10 (right). This figure showsno similar plane strain propagation rates at different temperatures. The changein plane strain fatigue crack propagation rate is propably due to the increasedavailable energy at elevated temperatures and hence the plane strain fatiguecrack propagation rate is temperature dependent.

3.4.1 Discussion

The LEFM approach with the Paris law representation of the fatigue crackpropagation rate is not able to capture the thermal effects to describe the fatiguefracture behaviour of glassy polymers. The more elaborate fracture kineticsapproach, incorporating activation energy and temperature, appears to be apromising approach to describe fatigue crack propagation behaviour.

Elaborating on the hypothesis stated in the previous section by incorpo-rating the ambient temperature as a parameter, the fatigue crack propagationbehaviour changes by shifting the plane strain propagation rate. With increas-ing temperature the plane strain fatigue crack propagation rates increase, whilethe plane strain deviation point decreases due to the decreasing yield stress.Figure 3.11 depicts a schematic representation of the fatigue crack propagation

27

5 10 1510

−8

10−6

10−4

Kmax

/T x10−3 [MPa⋅m0.5/K]

da/d

N [m

/cyc

le]

0 °C, 12 mm90 °C, 12 mm

5 10 1510

−8

10−6

10−4

Kmax

/T x10−3 [MPa⋅m0.5/K]

da/d

N [m

/cyc

le]

0 °C, 12 mm90 °C, 12 mm0 °C, 6 mm90 °C, 6 mm

Figure 3.10: Left: Temperature corrected fatigue crack propagation curves of12 mm specimens. Right: Temperature corrected fatigue crack propagationcurves of 6 mm and 12 mm combined.

rates at different temperatures. The plane strain lines are extended beyond thedeviation point to indicate the fatigue crack propagation behaviour for planestrain conditions. The deviation indicate the behaviour for plane stress condi-tions.

log( Kmax

)

log(

da/

dN )

Plane strain

Low THigh T

Figure 3.11: Schematic representation of the fatigue crack propagation rate withtemperature dependent plane strain curves.

3.5 Influence of the load signal

This section describes the effect of the load signal on the fatigue crack propaga-tion rate. The load ratio R of the applied force, as described in section 2.2.1, ischanged, while the maximum load is fixed. This is tested on 12 mm thick CTspecimens.

The load ratio R, as defined by Equation (1.15), is chosen between 0.4 and-0.4 as the experimental times increase significantly for higher load ratios. Anegative ratio is obtained by prescribing a compressive minimum load. For the

28

negative ratios the radial play is eliminated from the bracket by pretensioningthe pins.

Figure 3.12 (left) shows the results for the indicated load ratios. Only oneresult per load ratio is depicted for clarity. For the positive load ratios (closedmarkers) the corresponding Paris law is added. The decreasing fatigue crackpropagation rate for increasing positive load ratios, is in agreement with resultsreported by Lang [6], who tested ratios up to R = 0.63. The decreasing fatiguecrack propagation rates for equal Kmax levels and increasing mean stress inten-sity factors suggest a strong effect of the load amplitude on the fatigue crackpropagation behaviour.

1 2 3 4 510

−8

10−6

10−4

Kmax

[MPa⋅m0.5]

da/d

N [m

/cyc

le]

23°C, 6 mm

0.10.250.4−0.1−0.25−0.4

5 10 1510

−8

10−6

10−4

Kmax

/T x10−3 [MPa⋅m0.5/K]

da/d

N [m

/cyc

le]

23°C, 6 mm

0.10.250.4−0.1−0.25−0.4

Figure 3.12: Left: Fatigue crack propagation curves for various load ratios showa decreasing trend for increasing positive load ratio. Right: The temperaturecorrected fatigue crack propagation curves show a similar trend.

During a load cycle in the positive loading regime the specimen is alwaysloaded in tension. However, due to plastic deformations behind the crack tip,the material can be in compression locally. At R = 0 the applied load becomeszero in one point of the load cycle. Decreasing the load ratio below R = 0(open markers) yields no effect on the fatigue crack propagation rate, meaninga compressive phase in the load cycle is not contributing to the fatigue crackpropagation behaviour of PC.

3.5.1 Discussion

Altering the load cycle by changing the load ratio R, for fixed maximum loadsKmax, discards two important load parameters being the load amplitude ∆K,and the mean load Kmean. By changing the load ratio, with fixed maximumload, both parameters will change. Changing the load cycle with equal loadamplitude ∆K, or for equal mean load Kmean, would give an elaborate view onthe load cycle dependence.

Another parameter that is not a part of the load cycle but changes as aconsequence of a change in the load cycle is the strain rate at the crack tip.When the load amplitude changes, while the load frequency remains the same,a smaller strain ratio is applied in the same amount of time. Regarding thestrain rate dependence of the yield stress in polycarbonate [29], this parametermight have a significant effect on the fatigue crack propagation behaviour.

29

The difference in the fracture surfaces of specimens loaded by a positive loadcycle can be examined by scanning electron microscopy, which might reveal theeffect of local compression on the crack propagation rate.

3.6 Impact-modified polycarbonate

McGarry et al. [35] where among the first to report the improvement of fractureresistance by incorporating a dispersed rubbery phase into a polymer matrix.Since this study by McGarry et al. [35], several studies provided detailed de-scriptions of the micromechanisms of the enhanced fracture resistance due toimpact-modifiers. Azimi et al. [36, 37] showed a decreased fatigue crack propa-gation rate in impact-modified polymers when the plastic zone size is equal orlarger than the rubber particle size. Azimi et al. [36, 37] used a diglycidyl etherbisphenol-A (DGEBA) polymer filled with a liquid carboxyl-terminated buta-diene acrylonitrile (CTBN) copolymer. The result is schematically depicted inFigure 3.13 (left) with the impact-modified polymer deviating at the transitionstress intensity factor ∆Kth. Above the transition stress intensity factor, rubbercavitation, rubber shear banding, and plastic void growth mechanisms becomeactive. In Figure 3.13 (right) the results of three impact-modified DGEBA spec-imens are compared with the results of an unmodified specimen. Note that the1% and 5% modified polymers show a significant decrease in fatigue crack prop-agation rate, while the 10% modified polymer shows only a slight decrease fromthe 5% modified polymer.

A drawback of impact-modified polymers is the decrease in yield stress.Engels et al. [38] reported a decrease in yield stress of 10 MPa in the 9% modifiedPC and 5 MPa in the 4.5% modified PC.

0.5 1 210

−6

10−4

10−2

∆ K [MPa⋅m0.5]

da/d

N [m

m/c

ycle

]

DGEBA

0% CTBN1% CTBN5% CTBN10% CTBN

Figure 3.13: Left: Schematic representation of an impact-modified epoxy [37].Right: Results reproduced from Azimi et al. [36, 37] show three impact-modifiedDGEBA specimens compared to an unmodified specimen.

It has been shown in previous sections that the yield stress has a pronouncedeffect on the shear lip creation. By impact-modifying polycarbonate with 4.5%and 9% of a methacrylate-butadiene-styrene (MBS) core-shell copolymer theeffects of a rubbery phase on the crack propagation rate and shear lip creationare investigated.

30

Measuring the fatigue crack propagation rate for 6 mm and 12 mm impact-modified specimens yields the results depicted in Figure 3.14. A significantdecrease in fatigue crack propagation rate is found due to modification of poly-carbonate. However, the effect of the volume percentage is not obvious. AsAzimi et al. [36, 37] showed, the influence of the volume percentage is noticablystronger at lower volume percentages, but tends to weaken for higher percent-ages.

1 2 3 4 5 610

−8

10−6

10−4

Kmax

[MPa⋅m0.5]

da/d

N [m

/cyc

le]

6 mm

0% MBS4.5% MBS9% MBS

5 10 15 2010

−8

10−6

10−4

Kmax

/T x10−3 [MPa⋅m0.5/K]da

/dN

[m/c

ycle

]

6 mm

0% MBS4.5% MBS9% MBS

Figure 3.14: Left: Fatigue crack propagation curves of impact-modified PCcompared with an unmodified PC specimen. Right: The temperature correctedfatigue crack propagation results show a similar trend.

On the fracture surface of the impact-modified specimens no shear lip for-mation is visible. This means the plastic zone in front of the crack tip hasno significant plane stress region, although the fracture resistance is increasedsignificantly.

3.6.1 Discussion

Impact-modified polycarbonate shows promising results for fatigue life improve-ments. The fatigue crack propagation rate is an order of magnitude lower forimpact-modified PC at the critical stress intensity factor of PC, while no shearlips have developed. The effect of the volume percentage can not be determinedfrom these results and require further research.

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Chapter 4

Conclusions and

recommendations

The influence of material and test parameters on the fatigue crack propagationbehaviour of polycarbonate have succesfully been tested in this study. Thisresulted in a theory that describes the intricate balance between the influenceof the thickness, the yield stress, the temperature, and the load signal.

The fatigue crack propagation rate evolves according to a power law when aplane strain condition at the crack tip is dominant. This plane strain behaviouris temperature and load ratio dependent, where an increased temperature re-sults in increased fatigue crack propagation rates, while an increased load ratioresults in decreased fatigue crack propagation rates. Due to the influence of thethickness or the yield stress a significant plane stress condition can develop atthe crack tip. This leads to increased fracture resistance and is accompaniedwith the formation of shear lips at the fracture surfaces.

The power law stated by Paris gives an accurate description of the fatiguecrack propagation rate as long as the plane strain conditions are dominant. Todescribe the plane stress fatigue crack propagation behaviour and the influenceof the temperature, a fracture kinetics approach was proposed by Krausz. Thisapproach is briefly mentioned in this study and deserves to be addressed infurther research.

The influence of the load signal is in this study researched with a fixed max-imum load and a changing minimum load. This approach discards the effects ofthe mean load and the load amplitude. Results showed that the fatigue crackpropagation rate decreased for decreasing load amplitude at fixed maximum loadand hence increasing mean load. In succeeding research it is recommendated toincorporate these two parameters. Especially the load amplitude is expected togive interesting insights in the mechanisms of fatigue crack propagation.

In the last section of this work the influence of impact-modification was stud-ied. Impact-modified PC showed an increased fracture resistance comparable todecreased yield stress or thickness, but without any shear lip formation. Furtherresearch is required on the influence of the volume percentage impact-modifierthat is dispersed in the polymer matrix.

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