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 1.1 Long-term failure . . . . . . . . . . . . . 1.2 Objective . . . . . . . . . . . . . . . . . 1.3 Background . . . . . . . . . . . . . . . . 1.3.1 Linear elastic fracture mechanics 1.3.2 Fatigue crack propagation . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

3 3 7 8 8 12

2 Materials and methods 2.1 Materials . . . . . . . . . . . . . 2.2 Methods . . . . . . . . . . . . . . 2.2.1 Fatigue crack propagation 2.2.2 Compression tests . . . . 2.2.3 Numerical methods . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

14 14 15 15 18 18

3 Results and discussions 3.1 Influence of specimen thickness 3.2 Numerical simulations . . . . . 3.3 Influence of yield stress . . . . 3.3.1 Discussion . . . . . . . . 3.4 Influence of temperature . . . . 3.4.1 Discussion . . . . . . . . 3.5 Influence of the load signal . . 3.5.1 Discussion . . . . . . . . 3.6 Impact-modified polycarbonate 3.6.1 Discussion . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

20 20 21 23 24 25 27 28 29 30 31

. . . . . . . . . .

4 Conclusions and recommendations

1

32

Abstract A phenomenological study on the fatigue crack propagation behaviour of PC is performed. For measuring the crack length during experiments, an optical measuring technique with a fixed digital camera is employed. The influence of the thickness has a pronounced effect on the fatigue crack propagation behaviour. An increased fracture resistance is observed at lower thicknesses. Therefore, a minimum thickness should be established for accurate measurements. A strong interrelation between the influence of the thickness and the yield stress on the fatigue crack propagation behaviour is observed. A decreased yield stress results in increased fracture resistance. Both effects occur in combination with large shear lips at the fracture surface. The amplitude of the load signal influences the fatigue crack propagation rate for positive load values. An increase in amplitude at a fixed maximum load results in an increased fatigue crack propagation rate. This result proofs that fatigue fracture is influenced by more than the applied energy. The influence of the temperature on the fatigue crack propagation behaviour is proven to be different for two loading conditions. At low stress intensity factors an increased temperature, increases the fatigue crack propagation rate, while at high stress intensity factors an increased temperature results in high fracture resistance. This complex behaviour is attributed to the influence of the temperature on the yield stress in combination with the thickness effects. A brief study on the influence of impact-modifiers on the fatigue crack propagation behaviour shows a significant decrease in fatigue crack propagation rate for 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 loadbearing applications, such as pressurized pipe systems and aquarium windows. In these applications it is essential to prevent catastrophic failure of the polymer product. Pressurized pipe systems, such as an unplasticised polyvinyl chloride (uPVC) gas distribution network, are a common application for glassy polymers where failure can lead to casualties. To garantee a safe operation of polymer applications, 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 example is pressurized pipes, for which it is shown to have failure times exceeding 50 years [1]. Modern test methods for pressurized pipes determine the timeto-failure at increased internal pressures and elevated temperatures, which decreases the experiment duration to approximately 1.5 years [2]. The results are subsequently extrapolated for normal operating pressures and ambient temperatures to determine the stress that can be sustained for 50 years. This is known as 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 logarithmic scale. This yields three distinctive regions characterized by their failure mechanism. Region I is related to ductile failure, characterized by large plastic deformations. This failure mechanism is called ductile tearing. Subsequently, in region II very little plastic deformation is visible as a minute hairline crack appears in longitudinal direction of the pipe. This quasi-brittle failure behaviour is attributed to slow crack propagation, where a flaw in the pipe slowly grows into a hairline crack. At low stresses a nearly stress independent failure mechanism is caused by molecular degradation of the polymer, leading to brittle fracture or a multitude of hairline cracks. This mechanism is not further addressed in this study. Graphical representations of the failure mechanisms are shown in Figure 1.1 (right). The LTHS method is a thorough way to determine the lifetime, but it is expensive, as many experiments are required, and 1.5 years of testing is a long period in modern day economy. To improve the methods and develop models to

3

log(stress)

Region I

Static

Region II

Region III log(time−to−failure)

Figure 1.1: Left: Schematic results of internal pressure tests depicting three regions of failure for static loading. Reproduced from Lenman [3] . Right: Graphical representation of ductile tearing (I), hairline cracking (II), and multiple cracking (III) [4]. predict lifetimes, long-term failure of glassy polymers has received considerable attention [4–6]. Visser et al. [4] described an engineering approach to predict ductile failure of pressurized pipes based on short-term tensile experiments. The approach is based on the assumption that ductile failure occurs when the accumulated plastic strain reaches a critical value. A pressure-modified Eyring relation [7, 8] given by:    ∗   −∆U τ¯ν −µpν ∗ γ¯˙ (T, p, τ¯) = γ˙ 0 · exp · sinh · exp , (1.1) RT RT RT was used to calculate the equivalent plastic strain rate γ¯˙ . In Equation (1.1) γ˙ 0 is the 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 dependence parameter and p the hydrostatic pressure. Integrating this equivalent plastic strain rate over time yields an accumulated plastic strain. Visser et al. [4] validated this approach by using time-to-failure data of dynamically loaded tensile specimens. Unexpectedly, Visser et al. [4] observed region II failure, as shown 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. Measuring the crack propagation rate, statically loaded single edge notched tension specimens, resulted in Figure 1.3 (left). An increase in crack propagation rate 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 = β(K)m , (1.2) dt where β and m are constants, representing the intercept and the slope of the

4

Maximum stress [MPa]

60 55

1 Hz 0.1 Hz 0.01 Hz

uPVC

50 45 40 35 30 25 2 10

4

10 Time−to−failure [s]

6

10

Figure 1.2: Left: Results reproduced from Visser et al. [4] depict the time-tofailure for dynamically loaded uPVC tensile specimens. Specimens that failed as a result of ductile failure are represented by closed markers. curve, respectively. The stress intensity factor K for a single edge notched specimen is determined by: √ K = Y σ πa,

(1.3)

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

2 1 1−m/2 1−m/2 √ (a − ai )· 2−m f β(Y σ π)m

(1.4)

Gray et al. [5] used this approach to predict the time-to-failure of pressurized pipes. The initial flaw size af present in the pipes could not be determined, so Gray et al. [5] calculated the initial flaw size based on failure data of pressurized pipes. 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 the predicted values. The results are shown in Figure 1.3 (right) and are in good agreement with the data. Gray et al. [5] also tested a tougher grade of PE, with a lower and nonlinear crack propagation rate, and conclude that the crack propagation behaviour could not be described by a power law due to the curvature in the data. So the change in material toughness resulted in a more complex crack propagation behaviour. Hertzberg et al. [9] and Gerberich et al. [10] tested the fatigue crack propagation in glassy polymers. The fatigue crack propagation mechanism is essentially comparable to slow crack propagation, as an initial flaw grows due to a load and eventually leads to failure. Both, Hertzberg et al. [9] and Gerberich et al. [10], tested polycarbonate at various temperatures to see the effect on the fatigue crack propagation behaviour. An unexplained inconsistency appeared in both studies.

5

−6

10

PE

PE −7

Stress [MPa]

da/dt [m/s]

10

−8

10

80°C 40°C

−9

10

13 11 9 7

23°C

40°C

80°C

5 3

23°C

100 µm prediction

−10

10

0.1

2

0.3 0.5 0.7 0.9 0.5 K [MPa⋅m ]

10

3

4

10 Time−to−failure [h]

10

Figure 1.3: Left: Gray et al. [5] determined the crack propagation rate for three temperatures. Right: Time-to-failure measurements of pressurized pipes with the corresponding predictions based on crack propagation rates. Reproduced from Gray et al. [5] Hertzberg et al. [9] presented the temperature dependency of fatigue crack propagation by plotting the fatigue crack propagation rate, at a fixed stress intensity factor of 1.5 MPa·m0.5 , versus the experimental temperature, as shown in Figure 1.4 (left). It is clear that the maximum fatigue crack propagation rate, at Kmax =1.5 MPa·m0.5 , occurs around -60◦ C and that the rate decreases for lower temperatures as well as for higher temperatures. This anomaly is also observed by Gerberich et al. [10], who showed an increasing fatigue crack propagation rate for temperatures down to -50◦ C and decreasing rates for temperatures decreasing further to -172◦ C. Gerberich et al. [10] also reported a change 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] −4

10

PC da/dN [m/cycle]

da/dN x10−7 [m/cycle]

15

10

5

−50°C 0°C −21°C

50°C

−172°C

−5

10

100°C −6

K

10

0.5

=1.5 MPa⋅m

max

150

200 250 Temperature [K]

300

1

2

3

4

0.5

∆K [MPa⋅m

]

Figure 1.4: Left: Hertzberg et al. [9] reported a bell-shaped temperature dependence of the fatigue crack propagation curve. Right: Gerberich et al. [10] observed changing slopes and similar temperature dependence noticed by Hertzberg. Ward et al. [11] reported the effect of the specimen thickness and the temperature 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] and Gerberich 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 at room temperature and found that the 3 mm specimens had the greatest fracture resistance, the 9 mm had a fatigue crack propagation rate of an order of magnitude higher, while the 6 mm results were inbetween, as shown in Figure 1.5 (left). Note, that the 6 mm results are similar to the 9 mm at low ∆K values and deviate to lower propagation rates at high ∆K values. −3

−3

10

10 PC

PC −4

da/dN [m/cycle]

da/dN [m/cycle]

−4

10

−5

10

−6

10

3 mm 6 mm 9 mm

−7

10

1

3 5 ∆K [MPa⋅m0.5]

10

−5

10

−6

10

3 mm 6 mm 9 mm

−7

10

7 9 11

1

3 5 ∆K [MPa⋅m0.5]

7 9 11

Figure 1.5: Left: Fatigue crack propagation results for three thicknesses of PC at 20◦ C. Right: The effect of the specimen thickness at -30◦C. Reproduced from Ward et al. [11] . Figure 1.5 (right) shows the results for the same thicknesses at -30◦ C. At low temperature all results coincided on one straight line until high ∆K values. Ward et al. [11] attributed the differences in the results to the balance between plane strain crazing and plane stress yielding. The latter resulted in the formation of shear lips for different values of thickness and ∆K. Pruitt et al. [12] measured the fatigue crack propagation rate in polycarbonate at different load ratios of the dynamic load signal. The load ratio was defined 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 in load ratio from 0.1 to 0.5. So despite an increased mean load, the fatigue crack propagation rate decreased. This counterintuitive result is attributed to the same 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 crack propagation in glassy polymers, no conclusive explanation for the complex phenomena is found. In this work a phenomenological approach is used to gain insight in the mechanisms of fatigue crack propagation in glassy polymers. A test setup with an optical measuring system is improved to perform all tests required in this study. To visualize the effects of the thickness and the yield stress, a numerical analysis of a cracked body is performed.

7

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

1.3 1.3.1

Background Linear elastic fracture mechanics

Within the field of fracture mechanics, the propagation of cracks in materials is a widely studied area. LEFM is a popular approach to study crack propagation [4, 5, 11, 12], which started with the work of Griffith, who noticed that there is a significant discrepancy between the theoretical breaking stress, the stress to break atomic bonds, and the actual breaking stress of glass [13]. He discovered that small surface scratches, drastically reduced the breaking strength of a material. Griffith introduced a fracture criterion based on a global energy balance. He posed that in a cracked body the mechanical energy U˙ e added to a unit volume of material must be transformed into free surface energy U˙ s and internal energy U˙ i . Ommitting thermal energy, kinetic energy, and energy dissipation. Irwin [14] later improved Griffith’s theory by adding energy dissipation to the equation. Besides focusing on the global energy balance, Irwin also introduced a local stress based fracture criterion. He proved that the asymptotic stress field near a crack tip is given by:   k fij (θ), (1.5) σij = √ r with σij being the components of the Cauchy stress tensor, r the distance to the crack 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 of higher order terms. In linear elastic bodies the dominant term √1r approaches infinity as r → 0. This introduces a stress singularity near the crack tip, which results in infinite stresses at the crack tip. Therefore, a stress criterion cannot be employed to describe crack propagation. A new constant is introduced to overcome the crack tip singularity which replaces the proportionality constant k. The stress intensity factor K is defined in the limit r → 0 [15], as described by: √  √ KI = lim 2πrσ22 |θ=0 = k 2π (1.6) r→0

The stress intensity factor is usually given a subscript to denote the loading condition 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-ofplane 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 further theory will focus on mode I loading. The stress intensity factor is empirically determined for geometries and loading conditions common in fracture mechanics testing [17]. With the stress in-

8

x2

6

crack

σ22 σ126 6 -σ - ? 11 r ? θ x1

Figure 1.6: Crack with local coordinate system and stress definations. Local x3 -axis is normal to the page.

Figure 1.7: Three modes of fracture. Left: Opening mode. Middle: In-plane shear mode. Right: Out-of-plane shear mode [16]. tensity factor as defined in Equation (1.6), the stress components σ11 , σ22 , and σ12 at the crack tip become: KI σ11 = √ 2πr KI σ22 = √ 2πr KI σ12 = √ 2πr




 cos( 12 θ) 1 − sin( 12 θ) sin( 32 θ) ,




 cos( 12 θ) 1 + sin( 12 θ) sin( 32 θ) ,



 cos( 12 θ) sin( 12 θ) cos( 32 θ) .

(1.7)

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 is possible to determine a critical stress intensity factor. This critical value of the stress intensity factor is denoted by KIc and is defined as the plane strain stress intensity factor also known as the fracture toughness. KIc is geometry and loading independent and is therefore regarded as a material parameter. Note that the fracture toughness is only valid under plane strain conditions. If plane stress conditions develop, the crack can propagate beyond the critical value. The critical stress intensity factor is determined from standardized fracture experiments. In LEFM the stress at the crack tip becomes infinite, while in reality plastic deformation evolves in front of the crack tip. The size and shape of the plastic zone depend on the stress state at the crack tip. 9

The shape and size of the plastic zone can be determined by employing a yield criterion. Several yield criteria are available and in this study the Von Mises yield criterion [18] is used. Von Mises states that yielding occurs when the Von Mises stress σv reaches a critical value. The critical value is the yield stress σy of the material. The Von Mises yield criterion is given by: q σv = 32 σ d : σ d , (1.8)   σv2 = 12 (σ1 − σ2 )2 + (σ1 − σ3 )2 + (σ2 − σ3 )2 . In this case the Von Mises stress is expressed in terms of the principal stresses. The principal stresses are defined as the eigenvalues of the Cauchy stress. For a plane stress condition the components of the Cauchy stress are given in Equation (1.7), and the principal stresses are given by: σ1 = 21 (σ11 + σ22 ) + σ2 = 12 (σ11 + σ22 ) − σ3 = 0.

1 4 (σ11 1 4 (σ11

2 − σ22 )2 + σ12 2 − σ22 )2 + σ12


1/2
1/2

, ,

(1.9)

The first and second principal stresses are equal for plane stress and plane strain loading conditions. The third principal stress under plane stress conditions is zero, while under plane strain conditions it depends on the poisson’s ratio ν and the 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:
KI σ1 = √ cos( 12 θ) 1 + sin( 21 θ) , 2πr
KI cos( 12 θ) 1 − sin( 21 θ) , σ2 = √ 2πr σ3 = 0 for plane stress, 2νKI for plane strain. σ3 = √ cos( 12 θ) 2πr

(1.11)

The prinicipal stresses are defined by the stress intensity factor KI , the distance to the crack tip r, and the angle with respect to the crack growth direction θ. Implementing the principal stresses into the Von Mises yield criterion, defined in Equation (1.8), results in a distance ry from the crack tip, representing the boundary of the plastic zone according to the Von Mises criterion. For plane stress conditions this boundary equals ry =

KI2  1 + cos(θ) + 4πσy2

3 2

 sin2 (θ)

(1.12)

for plane strain conditions the boundary becomes ry =

KI2  (1 − 2ν)2 (1 + cos(θ)) + 4πσy2 10

3 2

 sin2 (θ)

(1.13)

Loading direction

plane strain plane stress

Crack direction

Figure 1.8: Left: Plastic zone boundaries for plane stress and plane strain conditions. Reproduced from [19]. Right: A dog-bone shaped plastic zone in three dimensions with plane stress conditions near the surface and plane strain conditions 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 curves as given in Figure 1.8 (left). It is clear that the plane stress plastic zone is substantially larger than the plane strain plastic zone. Considering that a plane strain loading condition occurs at the symmetry plane, perpendicular to the thickness direction, of a specimen, since no free contraction in the thickness direction is possible, and a plane stress condition occurs near the free surface, where contraction is possible, a three-dimensional representation of the plastic zone 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 shear lips. Figure 1.9 (left) shows an x-ray computed tomogram of a specimen with shear lips. The two cross-sections indicate the difference at the fracture surface at two stress intensity factors. The fatigue crack propagation rate is closely related to the size and shape of the plastic zone and will be maximal when minimal plasticity develops [6], i.e. a plane strain plastic zone. Full plane strain conditions, however, will never occur since a small region of plane stress is present at the specimens free surfaces. At a certain critical thickness, the relative influence of the plane stress region of the plastic zone, compared to the plane strain region, becomes insignificant. Above this thickenss the fatigue crack propagation rate is independent of the specimen thickness. 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 factor for mode I loading, and σy the yield stress. For polycarbonate, KIc is typically 2.24 MPa·m0.5 and σy = 68 MPa [22]. A specimen thickness of over 3 mm should therefore be sufficient to measure the plane strain propagation rate of polycarbonate.

11

Figure 1.9: Left: A semi-transparent x-ray computed tomogram of a fractured 6 mm specimen. Two cross-sections indicate the difference of the fracture surface at different stress intensity factors. Right: An image of the same specimen. The shear lips are indicated with arrows and the white lines represent the transition from 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 constant amplitude load, characterized by stress or force, is most often exerted to study fatigue crack propagation in materials. Since the stress in a single point near the crack tip is fully determined by the stress intensity factor KI , the fatigue load can also be characterized by the stress intensity factor. The applied load is fully characterized by its maximum value Kmax , and a load ratio R, which is defined by the ratio between the minimum stress intensity factor Kmin , and the maximum stress intensity factor Kmax , and are given by: Kmin , Kmax ∆K = Kmax − Kmin . R=

(1.15) (1.16)

Several crack propagation models are proposed to relate the fatigue crack da propagation rate dN , to the maximum stress intensity factor Kmax [23, 24]. Some relate the fatigue crack propagation rate to the load amplitude ∆K and some incorporate the load ratio R, or the mean load Kmean , to improve the result. In polymer fatigue the LEFM approach with Paris’ law [25] is widely applied [10, 12, 26, 27]. Paris’ law relates the crack propagation rate to the maximum stress intensity factor using two parameters, a pre-exponential factor A, and an exponent m, as given by: da = A(Kmax )m . (1.17) dN Figure 1.10 shows a schematic fatigue crack propagation curve with the corresponding Paris law. Below a certain stress intensity factor no crack propagation 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 propagation rate will increase again until failure occurs. The Paris law only describes the fatigue crack propagation rate correctly in the exponential growth region, indicated between the two vertical dashed lines, while initially it over-predicts the propagation rate and in the end it under-predicts the propagation rate.

log( da/dN )

da = A(K)m dN

Very slow crack propagation

Stable crack propagation

Kth

log( Kmax )

Unstable crack propagation

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

13

Chapter 2

Materials and methods This chapter describes the experimental setups and materials used to explore the theories set forth in this report. In the Materials section, various grades of polycarbonate and the specimen preparation methods are described. In the Methods section an explanation of the test setups for various experiments is provided.

2.1

Materials

In this study a commercially available Lexan extrusion grade of polycarbonate is used as the main material of interest. The material was received in the form of extruded sheets of two thicknesses. For the fatigue crack propagation experiments, Compact Tension (CT) specimens of both thicknesses are produced. The general geometry of a CT specimen is shown in Figure 2.1. All dimensions are determined according to the ASTM standard E647 [28] and are listed in Table 2.1. The extruded grade specimens are cut with a circular saw from a large extruded plate and the fixation holes and notch are precision machined into the specimen. Small Large

W [mm] 6 12

L [mm] 32 64

H [mm] 38.5 77

Table 2.1: Specimen dimensions. In order to remove the radius of the machined notch and start the experiment with a sharp crack all specimens are precracked. The small specimens are precracked with a fresh razor blade that is tapped into the notch root. This proved to be a very consitent method, with an average crack length of 790 µm and a variation of 20 µm. The large specimens are fatigue precracked according to ASTM standard E647 [28]. A fresh specimen is mounted in the experimental setup and precracked to a crack length of approximately 3 mm using a dynamic load with a load ratio of 0.1. To accelerate the initial precrack process a high maximum force Fmax , of 800 N, is employed. To prevent residual stresses affecting the experiment, the final millimeter of precracking is done at a maximum force 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 temperature of 120◦ C for 90 hours. The specimens are placed on a polytetrafluoroethylene (PTFE) plate to prevent sticking and after the treatment, the specimens are allowed to cool to room temperature in open air. The experiments for the study of the effect of rubber particles were performed on a Lexan 141R grade containing 4.5% and 9% of impact-modifier by volume. The impact-modifier was a methacrylate-butadiene-styrene (MBS) core-shell copolymer, commercially available as Paraloid EXL-2600. Since the impact-modified grades are sensitive to thermal degradation, the melting temperature was lowered to 240◦C and the air in the mould is substituted for nitrogen. The 4.5% grade was dryed for 210 minutes at 110◦ C, the 9% grade showed some degradation after this drying process so the drying time was reduced to 75 minutes. All manually pressed specimens show an uneven surface and are therefore surface machined to the correct thickness. Further cutting and machining to the dimensions listed in Table 2.1 is equal to the extruded grade Lexan specimens.

2.2 2.2.1

Methods Fatigue crack propagation

Fatigue crack propagation experiments are performed at two different tensile machines. The first tensile machine is the servo-hydraulic MTS 810 Elastomer Testing System with a 2.5 kN load cell. The second servo-hydraulic machine is the MTS 831 Elastomer Testing System with a 25 kN load cell. These systems are very similar to each other. On both systems the testing temperature can be regulated by an environmental control chamber. The specimen is attached to the tensile stage by a Clevis bracket depicted in Figure 2.2 (left). The dimensions of this bracket are prescribed by ASTM standard E647 [28]. Besides the rotation of the specimen around the pins, the bracket contains an additional degree of freedom for the axial alignment of the upper 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 allows for experiments with a changing sign in the applied force.

Figure 2.2: Left: Image of the calibration specimen with white circles around the calibration points. In the upper and lower right corners, parts of the Clevis brackets 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 controlled sinusoidal signal with a frequency of 5 Hz. PC shows no frequency sensitivity in the range from 1 to 70 Hz [6], hence the frequency is chosen based on preventing motion blur in the images. The load ratio R, as defined in Equation (1.15), equals 0.1 for all tests, except for tests performed to study the load ratio dependence of the fatigue crack propagation rate. The maximum force in the load cycle is chosen between 500 N and 800 N for the 12 mm specimens and between 250 N and 400 N for the 6 mm specimens. This range in maximum force increased the range of the maximum stress intensity factors tested, while the tests duration remained within reasonable time. All experiments in this study are repeated at least three times. The fatigue crack propagation is monitored using a digital camera operated by the MATLAB Image Acquisition toolbox. Two digital cameras are available, a Prosilica EC1280 with a resolution of 1280x1024 pixels and a Prosilica CV640 with a resolution of 659x498 pixels. The cameras are fitted with a Pentax C52893K 50 mm lens or a Nikon micro-Nikkor 55 mm lens. The camera is positioned perpendicular to the specimens surface at a distance such that the final crack will cover the full width of the image. The MATLAB Image Acquisition toolbox provides a well documented platform to operate the cameras. In order to reduce the amount of data, a region of interest is set around the expected final crack and only this region is stored for post-processing. The image acquisition frequency is set according to the length of the experiment. The total number of images is limited by the computers random-access memory. An image of the experiment is shown in Figure 2.3 (upper). The vertical gray line on the left side of the image is the specimens edge. The 90◦ rotated V-shape on the right is the machined notch, with the extending horizontal line to the left being the crack. The two small bright areas between the crack and the 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 images, shown in Figure 2.3 (lower), with the ’im2bw’ function. The edge of the specimen is still visible on the left side of the image. On the right, the notch and crack are depicted in white. The dust particles on the specimen are digitally removed by filling white areas below a threshold area. The crack tip is represented by the leftmost white pixel of the crack. To locate this pixel, a boundary recognition function ’bwtraceboundary’ determines the boundary between the black 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 clearly visible marks, as shown in Figure 2.2 (right), is taken to correlate the number of pixels in the image to the physical crack length. The calibration image provides several fixed distances to correlate the number of pixels to the actual distance. The distances between the calibration points (white circles) are measured accuratelly and programmed into the software. Before processing, the pixels of the calibration points in the image are selected which, enables the software to correlate the number of pixels between to points to the physical distances. 50 40 30

da/dN [m/cycle]

Crack length [mm]

PC −4

10

da dN

20 10 0

−6

10

−8

2

4 Cycles [−]

6

8 4

x 10

10

1

2 Kmax [MPa⋅m0.5]

3

4

Figure 2.4: Left: N-a curve with five processing points and a corresponding tangent line. Right: The resulting crack propagation curve with the fitted Paris law. Processing the images yields the crack length versus number of cycles curve as depicted in Figure 2.4 (left). The fatigue crack propagation curve is subsequently obtained by deriving the crack length curve with respect to the number of cycles. Figure 2.4 (left) illustrates this process for five derivation points. The slope of the tangent represents the fatigue crack propagation rate at the respective crack length. The maximum stress intensity factor Kmax is calculated by substituting the maximum applied force and the crack length into Equa17

tion (2.1). The resulting curve is depicted in Figure 2.4 (right), where eleven derivation points are used. The corresponding Paris law is calculated using a least square fit. 2 + a/L F KI = √ W 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

2.2.2

(2.1)

Compression tests

Compression tests are performed to determine the yield stress at various temperatures and the yield stress of PC after annealing. The tests are executed on the above mentioned MTS 831 Elastomer Testing System and on a Zwick Z010 servo-electric tensile stage with a 10 kN load cell. Cylindrical specimens with dimensions of approximately 6 mm heigh and 6 mm in diameter are machined from the fatigue crack propagation specimens. The cylindrical specimens are compressed between two parallel flat plates and to minimize friction between the plates and the specimen, PTFE spray and PTFE tape are applied. The machine stiffness is measured by pressing the surfaces together without a specimen. This stiffness, typically 19,000 N/mm for the Zwick Z010 and 28,000 N/mm for the MTS 831, is used to calculate the true displacement. The tests are performed at a contstant true strain rate of −10−3 s−1 . The resulting yield stresses are listed in Table 2.2. The difference in yield stress between a large and a small CT specimen, is caused by the slower cooling of the thick plate during production. Large @ 23◦ C Small @ 23◦ C Small @ 0◦ C Small Annealed 90h @ 120◦ C

True yield stress [MPa] 71.0 70.3 81.0 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 front of the crack tip. A Center Cracked Tension (CCT) specimen is used, since the geometry of the CT specimens used in the fatigue crack propagation experiments is less suitable for numerical simulations. It has only two planes of symmetry and the pinhole to apply the force requires contact surfaces. The CCT geometry has three planes of symmetry and a distributed applied load without contact surfaces. The model does not allow for crack propagation so the stress intensity factor K can not increase due to a larger crack length. In the simulations the stress intensity factor is increased by an increasing load. This method allows for 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), is created in the finite element software package MSC.Mentat 2005r3. Two threedimensional meshes, representing 6 mm and 12 mm thickness, with a mesh refinement 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 mesh surrounding the refinement is auto-generated by the software. The symmetry boundary conditions are fixed displacements applied perpendicular to the three planes of symmetry, while a negative pressure on the top surface simulates an applied distributed stress.

Comp. true stress [MPa]

80 60 40

σ = 65.7 [MPa] y

20

σy = 70.0 [MPa] σ = 75.9 [MPa] y

Exp. data

0 0

0.2 0.4 Comp. true strain [−]

0.6

Figure 2.5: Left: Two dimensional finite element mesh of a CCT specimen. Right: Intrinsic material behaviour predicted by the EGP model for three yield stresses. Experimental data, provided by Klompen et al. [29], is added to verify the behaviour. The nonlinear elasto-viscoplastic material model used, is the single mode Eindhoven Glassy Polymer (EGP) model as described in [30] and [31]. This constitutive model is implemented into the HYPELA2 subroutine. Figure 2.5 (right) shows the intrinsic material behaviour predicted by the EGP model for three different yield stresses. The model is verified with experimental data obtained from 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 with an equivalent plastic strain larger than 10−5 to an external text file. This file is processed by MATLAB which calculates the volume of each individual element in a summation loop. Since the elements are not exact rectangles when stress is applied, the volume is determined by calculating the volume of the four tetrahedra 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 markers) and 12 mm (open markers) specimens. The solid line represents Paris’ law for the 12 mm results. Only the data from two measurements are depicted for clarity. 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 difference in the fatigue crack propagation behaviour. For a maximum stress intensity factor up to Kmax = 2.3 MPa·m0.5 , the fatigue crack propagation rate is equal for 6 mm and 12 mm specimens and is accuratelly described by Paris’ law. By further increasing the stress intensity factor, the 6 mm specimens shows a more fracture resistant behaviour. This suggests that the plastic zone in front of the crack tip of the 6 mm specimens, must consist of a significant plane stress region. The fatigue crack propagation curve of a 12 mm specimen is consistently increasing, which indicates a nearly complete plane strain plastic zone. 6 mm 12 mm Paris law

−4

da/dN [m/cycle]

10

−6

10

−8

10

1

2

3

23 °C 4 5 6

Kmax [MPa⋅m0.5]

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 specimen is 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 the plane strain deviation point. The deviation in the behaviour of the 6 mm specimens is also visible on the fracture surface of the specimens. Figure 3.2 shows images of a 6 mm specimen (upper) and a 12 mm specimen (lower). The solid lines across the surfaces represent the crack length for a maximum stress intensity factor of 2.3 MPa·m0.5 . The 6 mm specimen shows considerable shear lips at higher stress intensity factors, indicating a plane stress plastic region. The 12 mm specimen shows a nearly flat fracture surface until failure, corresponding to a plane strain plastic zone.

Figure 3.2: Upper: Fracture surface of a 6 mm specimen showing significant shear lips. Lower: Fracture surface of a 12 mm specimen which is nearly flat until failure. The solid lines represent the crack length for a maximum stress intensity factor of 2.3 MPa·m0.5 . The three-dimensional plastic zone in front of a crack tip is not visible in experiments. In numerical simulations the plastic zone can be visualized for various thicknesses and yield stresses.

3.2

Numerical simulations

Numerical simulations, using finite element models, show the size and shape of the plastic zone in front of the crack tip. To visualize the plastic zone and to investigate the effect of the specimen thickness and the yield stress, simulations on 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 zone and a fully plane strain plastic zone. These zones form the upper and lower bound of the plastic zone volume, respectively. The EGP material model is not suitable to simulate plane stress conditions and hence only the lower bound of the 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 calculated volume of the plastic zone divided by the thickness of the model. The plane strain 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 a plane 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 to the plastic zone compared with the 12 mm model. The stress intensity factor at which the three curves deviate is similar to the plane strain deviation point of the 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.

σ = 70.0 [MPa]

3

y

10

σ = 75.9 [MPa] y

−9

∆MPV x10

−9

MPV x10

15 10 5 0

σ = 65.7 [MPa] y

[m /m]

20

15

6 mm 12 mm plane strain

3

[m /m]

25

2 K

max

3 [MPa⋅m0.5]

5

0

4

2

3 K

max

4 0.5

[MPa⋅m

]

Figure 3.3: Left: The mean plastic volume in 6 mm and 12 mm thick finite element models show a decrease in plastic zone volume per unit thickness. The plane 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, and σy = 75.9 MPa. Comparing the shape of the plastic zone as predicted by the numerical simulations to the dog-bone shape predicted by the LEFM equations, reveals characteristic differences. Figure 3.4 shows the numerical plastic zone of a 6 mm model 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-bone is the free surface of the model, while the symmetry plane is invisible behind the dog-bone. The plane stress region near the surface is less pronounced in the numerical simulations and the size of the plastic zone increases over the total thickness of the model when the load is increased. This is the result of the fact that the stress field used to calculate the shape of the plastic zone using the LEFM equation, was assumed to be fully elastic, while in the numerical simulations plasticity occurs. This has been noticed by Fern´andez Z´ un ˜ iga et al. [32] who assumed the plastic zone in the aluminium alloy Al 7075 to be cylindrically shaped instead of dog-bone shaped. Simulation on a 6 mm model using three different yield stresses show decreasing 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 the two-dimensional plane strain MPV. A smaller ∆MPV indicates a small plane stress region in the plastic zone. The decreased plasticity yields an increased fatigue crack propagation rate. Since the plastic volume for all three yield stresses remains approximately equal until Kmax = 2.3 MPa·m0.5 , increasing the yield stress seems to have a similar effect on the fatigue crack propagation behaviour as increasing the thickness of the specimen. In the next section this effect is experimentally tested by increasing the yield stress with heat treatments.

22

Figure 3.4: Plastic zone representation of a 6 mm thick model at Kmax = 1.93 MPa·m0.5 (left) and at Kmax = 2.51 MPa·m0.5 (right). The colours represent increasing equivalent stress, with blue being 62 MPa and yellow 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 specimen but also by the yield stress. The yield stress of a glassy polymer can be affected by heat treatments. An amorphous polymer below its glass transition temperature Tg is in a non-equilibrium thermodynamic state, resulting in time-dependent physical and mechanical properties. Over time the polymer strives towards a favourable equilibrium state, which increases the yield stress. This process is called physical aging and it is accelerated by increasing the temperature. A heat treatment, in which a glassy polymer is aged at elevated temperatures below Tg , is generally refered to as annealing. The effects of annealing on the yield stress of polycarbonate is given in Figure 3.5 (left), showing the results of compression tests at −10−3 s−1 strain rate of an annealed and an as-received specimen.

80 60

Annealed

Annealed As−received

−4

10 da/dN [m/cycle]

Comp. true stress [MPa]

100

As−received

40

−6

10

20 −8

0 0

10 0.05 0.1 0.15 Comp. true strain [−]

0.2

1

2

3

4

5 6

0.5

Kmax [MPa⋅m

]

Figure 3.5: Left: Results of compression tests show an increase in yield stress due to annealing. Right: Fatigue crack propagation results of an annealed and an as-received specimen. Fatigue crack propagation experiments on as-received 6 mm specimens, show significant shear lips on the fracture surface suggesting a plane stress contribu23

tion in the plastic zone, as presented in Section 3.1. By annealing the 6 mm specimens, the yield stress increases approximately 9 MPa (Figure 3.5 (left)). The effect on the fatigue crack propagation rate is seen in Figure 3.5 (right), which shows the fatigue crack propagation curves of an as-received 6 mm specimen compared with an annealed 6 mm specimen. More experiments are performed, but are ommited for clarity. The annealed specimen in Figure 3.5 (right), shows a fatigue crack propagation behaviour comparable to the 12 mm specimen shown in Figure 3.1. Comparable to the effect of specimen thickness, annealing has no effect at low stress intensity factors, while around Kmax = 2.3 MPa·m0.5 the as-received specimen shows an increased fracture resistance. The shear lips of the annealed 6 mm specimen (Figure 3.6 (upper)) are significantly 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 plane strain plastic zone.

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

3.3.1

Discussion

The effects of plasticity in front of a crack tip during fatigue loading are investigated. Based on existing literature and LEFM theory the influence of the specimen thickness and yield stress are succesfully assessed. The results show no effect at low stress intensity factors. At a certain stress intensity factor, increased fracture resistance is observed for decreased yield stress or similarly for decreased specimen thickness. The deviation is caused by the influence of the plane stress region in the plastic zone. It is believed that the plane strain deviation point can shift along the plane strain crack propagation curve. This is in accordance with the results reported by Ward et al. [11] This conclusion is summarized in Figure 3.7. The solid line represents the plane strain fatigue crack propagation curve and the dashed lines represent the increased fracture resistance curves for various yield stresses σy or specimen thicknesses W .

24

W, σy

log( da/dN )

Plane strain

log( Kmax )

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

3.4

Influence of temperature

Understanding the influence of the temperature on fatigue crack propagation behaviour in glassy polymers, is critical for the prediction of its service life. The temperature 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 propagation rate, 6 mm specimens are tested at temperatures of 0◦ C , 45◦ C, and 90◦ 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 temperatures are depicted in Figure 3.8 (left). The fatigue crack propagation curves of the 0◦ C and the 45◦ C specimens show a small part of similar fatigue crack propagation behaviour and subsequently deviate. The 90◦ C specimens show no similar part with the lower temperatures and deviate from the start of the experiments. Although the yield stress is significantly lower in the 90◦ C experiments with respect to the 45◦ C experiments, the deviation from the plane strain crack propagation 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 results of Ward et al. [11]. The effects of shear lips are shown in the previous section and eliminating these effects is possible by increasing the specimen thickness. 12 mm specimens are tested at equal temperatures as the 6 mm specimens and a predominantly temperature affected result is observed, since no shear lip formation is visible on the fracture surfaces, as show in Figure 3.9. Figure 3.8 (right) depicts the results of 12 mm specimens at three different temperatures. These results show a more complex fatigue crack propagation behaviour than stated in the previous section. The fatigue crack propagation curves do not originate in a single plane strain propagation rate, which was proposed in the previous section. It becomes apparent that the slope as well as the preexponential factor of the Paris law are temperature dependent. This complex behaviour makes it difficult to model fatigue crack propagation accurately with Paris’ law.

25

da/dN [m/cycle]

10

0 °C 45 °C 90 °C

−4

10 da/dN [m/cycle]

−4

−6

10

−8

10

1

3

5

6 mm 7 9

−6

10

−8

10

0.5

Kmax [MPa⋅m

0 °C 45 °C 90 °C

]

1 3 5 Kmax [MPa⋅m0.5]

12 mm 7 9

Figure 3.8: Left: Fatigue crack propagation curve of 6 mm specimens at temperatures of 0◦ C, 45◦ C, and 90◦ C. Right: Fatigue crack propagation curves of 12 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 the left side of the image. Lower: Fracture surface tested at 90◦ C with the initial crack 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 should be represented by an appropriate elementary rate constant. This fracture kinetics theory is similar to the Eyring approach [7, 8] used to predict ductile failure in glassy polymers. The fracture kinetics approach is derived from the understanding of the physical processes that control crack propagation, such as, disentanglement, chain scissioning, void creation, and fibril buckling. Once a physical process is identified it must be expressed in a mathematical model. To this end, the theory of rate processes [34] is employed. Each process requires a certain amount of energy to be activated, otherwise it would happen spontaneously. This energy is externally supplied through work or temperature. The elementary rate constant is generally expressed as   kT Ur − Us r= exp − (3.1) h kT where k and h are Boltzmann’s and Planck’s constants, respectively, T is the absolute 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 controls the velocity. Note, that rate constants should not be added for curve-fitting purposes but strictly to describe real, physical phenomena. When phenomena require multiple processes, they can occur in a parallel or in a consecutive order. Analysis of all processes governing crack propagation is time-consuming and difficult. The resulting constitutive equation, however, is valid for a wide range of conditions since it is based on physical processes that contribute to crack propagation. In fracture kinetics it is appropriate to plot the fatigue crack propagation curves on a semi-logarithmic scale with the stress intensity factor divided by the test temperature [33]. In Figure 3.10 (left) it is shown that the temperature corrected stress intensity factor Kmax /T results in parallel fatigue crack propagation rates for both temperatures. Appending the results of the 6 mm specimens at similar temperatures yields Figure 3.10 (right). This figure shows no similar plane strain propagation rates at different temperatures. The change in plane strain fatigue crack propagation rate is propably due to the increased available energy at elevated temperatures and hence the plane strain fatigue crack propagation rate is temperature dependent.

3.4.1

Discussion

The LEFM approach with the Paris law representation of the fatigue crack propagation rate is not able to capture the thermal effects to describe the fatigue fracture behaviour of glassy polymers. The more elaborate fracture kinetics approach, incorporating activation energy and temperature, appears to be a promising approach to describe fatigue crack propagation behaviour. Elaborating on the hypothesis stated in the previous section by incorporating the ambient temperature as a parameter, the fatigue crack propagation behaviour changes by shifting the plane strain propagation rate. With increasing temperature the plane strain fatigue crack propagation rates increase, while the plane strain deviation point decreases due to the decreasing yield stress. Figure 3.11 depicts a schematic representation of the fatigue crack propagation 27

da/dN [m/cycle]

0 °C, 12 mm 90 °C, 12 mm

−4

10 da/dN [m/cycle]

−4

10

−6

10

−8

−6

10

0 °C, 12 mm 90 °C, 12 mm 0 °C, 6 mm 90 °C, 6 mm

−8

10

5 K

10

15

10

/T x10−3 [MPa⋅m0.5/K]

5 K

max

10

15

/T x10−3 [MPa⋅m0.5/K]

max

Figure 3.10: Left: Temperature corrected fatigue crack propagation curves of 12 mm specimens. Right: Temperature corrected fatigue crack propagation curves of 6 mm and 12 mm combined. rates at different temperatures. The plane strain lines are extended beyond the deviation point to indicate the fatigue crack propagation behaviour for plane strain conditions. The deviation indicate the behaviour for plane stress conditions. Low T High T log( da/dN )

Plane strain

log( Kmax )

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

3.5

Influence of the load signal

This section describes the effect of the load signal on the fatigue crack propagation rate. The load ratio R of the applied force, as described in section 2.2.1, is changed, while the maximum load is fixed. This is tested on 12 mm thick CT specimens. 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. A negative ratio is obtained by prescribing a compressive minimum load. For the

28

negative ratios the radial play is eliminated from the bracket by pretensioning the pins. Figure 3.12 (left) shows the results for the indicated load ratios. Only one result per load ratio is depicted for clarity. For the positive load ratios (closed markers) the corresponding Paris law is added. The decreasing fatigue crack propagation rate for increasing positive load ratios, is in agreement with results reported by Lang [6], who tested ratios up to R = 0.63. The decreasing fatigue crack propagation rates for equal Kmax levels and increasing mean stress intensity factors suggest a strong effect of the load amplitude on the fatigue crack propagation behaviour.

0.1 0.25 0.4 −0.1 −0.25 −0.4

da/dN [m/cycle]

10

−4

10 da/dN [m/cycle]

−4

−6

10

1

2

3

4

10

5

0.5

Kmax [MPa⋅m

−6

10

23°C, 6 mm

−8

23°C, 6 mm

−8

10

0.1 0.25 0.4 −0.1 −0.25 −0.4

]

5

10 Kmax/T x10−3 [MPa⋅m0.5/K]

15

Figure 3.12: Left: Fatigue crack propagation curves for various load ratios show a decreasing trend for increasing positive load ratio. Right: The temperature corrected fatigue crack propagation curves show a similar trend. During a load cycle in the positive loading regime the specimen is always loaded in tension. However, due to plastic deformations behind the crack tip, the material can be in compression locally. At R = 0 the applied load becomes zero 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, meaning a compressive phase in the load cycle is not contributing to the fatigue crack propagation behaviour of PC.

3.5.1

Discussion

Altering the load cycle by changing the load ratio R, for fixed maximum loads Kmax , discards two important load parameters being the load amplitude ∆K, and the mean load Kmean . By changing the load ratio, with fixed maximum load, both parameters will change. Changing the load cycle with equal load amplitude ∆K, or for equal mean load Kmean , would give an elaborate view on the load cycle dependence. Another parameter that is not a part of the load cycle but changes as a consequence 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 the strain rate dependence of the yield stress in polycarbonate [29], this parameter might have a significant effect on the fatigue crack propagation behaviour. 29

The difference in the fracture surfaces of specimens loaded by a positive load cycle can be examined by scanning electron microscopy, which might reveal the effect 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 fracture resistance by incorporating a dispersed rubbery phase into a polymer matrix. Since this study by McGarry et al. [35], several studies provided detailed descriptions of the micromechanisms of the enhanced fracture resistance due to impact-modifiers. Azimi et al. [36, 37] showed a decreased fatigue crack propagation rate in impact-modified polymers when the plastic zone size is equal or larger than the rubber particle size. Azimi et al. [36, 37] used a diglycidyl ether bisphenol-A (DGEBA) polymer filled with a liquid carboxyl-terminated butadiene acrylonitrile (CTBN) copolymer. The result is schematically depicted in Figure 3.13 (left) with the impact-modified polymer deviating at the transition stress intensity factor ∆Kth . Above the transition stress intensity factor, rubber cavitation, rubber shear banding, and plastic void growth mechanisms become active. In Figure 3.13 (right) the results of three impact-modified DGEBA specimens are compared with the results of an unmodified specimen. Note that the 1% and 5% modified polymers show a significant decrease in fatigue crack propagation rate, while the 10% modified polymer shows only a slight decrease from the 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% modified PC and 5 MPa in the 4.5% modified PC. DGEBA da/dN [mm/cycle]

−2

10

−4

10

−6

10

0% CTBN 1% CTBN 5% CTBN 10% CTBN 0.5 1 ∆ K [MPa⋅m0.5]

2

Figure 3.13: Left: Schematic representation of an impact-modified epoxy [37]. Right: Results reproduced from Azimi et al. [36, 37] show three impact-modified DGEBA specimens compared to an unmodified specimen. It has been shown in previous sections that the yield stress has a pronounced effect on the shear lip creation. By impact-modifying polycarbonate with 4.5% and 9% of a methacrylate-butadiene-styrene (MBS) core-shell copolymer the effects of a rubbery phase on the crack propagation rate and shear lip creation are investigated.

30

Measuring the fatigue crack propagation rate for 6 mm and 12 mm impactmodified specimens yields the results depicted in Figure 3.14. A significant decrease in fatigue crack propagation rate is found due to modification of polycarbonate. However, the effect of the volume percentage is not obvious. As Azimi et al. [36, 37] showed, the influence of the volume percentage is noticably stronger at lower volume percentages, but tends to weaken for higher percentages.

da/dN [m/cycle]

10

0% MBS 4.5% MBS 9% MBS

−4

10 da/dN [m/cycle]

−4

−6

10

−8

10

1

2 3 Kmax [MPa⋅m0.5]

6 mm 4 5 6

0% MBS 4.5% MBS 9% MBS

−6

10

−8

10

5

10

15

6 mm 20

Kmax/T x10−3 [MPa⋅m0.5/K]

Figure 3.14: Left: Fatigue crack propagation curves of impact-modified PC compared with an unmodified PC specimen. Right: The temperature corrected fatigue crack propagation results show a similar trend. On the fracture surface of the impact-modified specimens no shear lip formation is visible. This means the plastic zone in front of the crack tip has no significant plane stress region, although the fracture resistance is increased significantly.

3.6.1

Discussion

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

31

Chapter 4

Conclusions and recommendations The influence of material and test parameters on the fatigue crack propagation behaviour of polycarbonate have succesfully been tested in this study. This resulted in a theory that describes the intricate balance between the influence of the thickness, the yield stress, the temperature, and the load signal. The fatigue crack propagation rate evolves according to a power law when a plane strain condition at the crack tip is dominant. This plane strain behaviour is temperature and load ratio dependent, where an increased temperature results in increased fatigue crack propagation rates, while an increased load ratio results in decreased fatigue crack propagation rates. Due to the influence of the thickness or the yield stress a significant plane stress condition can develop at the crack tip. This leads to increased fracture resistance and is accompanied with the formation of shear lips at the fracture surfaces. The power law stated by Paris gives an accurate description of the fatigue crack propagation rate as long as the plane strain conditions are dominant. To describe the plane stress fatigue crack propagation behaviour and the influence of the temperature, a fracture kinetics approach was proposed by Krausz. This approach is briefly mentioned in this study and deserves to be addressed in further research. The influence of the load signal is in this study researched with a fixed maximum load and a changing minimum load. This approach discards the effects of the mean load and the load amplitude. Results showed that the fatigue crack propagation rate decreased for decreasing load amplitude at fixed maximum load and hence increasing mean load. In succeeding research it is recommendated to incorporate these two parameters. Especially the load amplitude is expected to give interesting insights in the mechanisms of fatigue crack propagation. In the last section of this work the influence of impact-modification was studied. Impact-modified PC showed an increased fracture resistance comparable to decreased yield stress or thickness, but without any shear lip formation. Further research is required on the influence of the volume percentage impact-modifier that is dispersed in the polymer matrix.

32

Bibliography [1] U. Schulte, “A vision becomes true: 50 years of pipes made from high density polyethylene.,” in Proceedings of Plastic Pipes XIII, 2006. [2] ISO 9080, Plastics piping and ducting systems - Deterimination of the longterm hydrostatic strength of thermoplastic materials in pipe form by extrapolation. Geneva: International Organisation of Standards, 2003. [3] T. Lenman, Water & Pipes. Virsbo: Wirsbo Bruks AB, 1982. [4] H. A. Visser, T. C. Bor, M. Wolters, T. A. P. Engels, and L. E. Govaert, “Lifetime assessment of load-bearing polymer glasses: An analytical framework for ductile failure,” Macromolcular Material and Engineering, vol. 295, pp. 637–651, 2010. [5] A. Gray, J. N. Mallinson, and J. B. Price, “Fracture behaviour of polyethylene pipes,” Plastics and Rubber Processing and Applications, vol. 1, pp. 51–53, 1981. [6] R. W. Lang, Applicability and limitations of linear elastic fracture mechanics to fatigue in polymers and short-fiber composites. PhD thesis, Lehigh University, 1984. [7] I. M. Ward, “Review: The yield behaviour of polymers,” Journal of Materials Science, vol. 6, no. 11, pp. 1397–1417, 1971. [8] R. A. Duckett, B. C. Goswami, L. Stewart, A. Smith, I. M. Ward, and A. M. Zihlif, “The yielding and crazing behaviour of polycarbonate in torsion under superposed hydrostatic pressure,” British Polymer Journal, vol. 10, no. 1, pp. 11–16, 1978. [9] R. W. Hertzberg and J. A. Manson, Fatigue of Engineering Plastics. New York: Academic Press, 1980. [10] G. C. Martin and W. W. Gerberich, “Temperature effects on fatigue crack growth in polycarbonate,” Journal of Materials Science, vol. 11, pp. 231– 238, 1976. [11] G. Pitman and I. M. Ward, “The molecular weight dependence of fatigue crack propagation in polycarbonate,” Journal of Materials Science, vol. 15, pp. 635–645, 1980.

33

[12] L. Pruitt and D. Rondinone, “The effect of specimen thickness and stress ratio on the fatigue behavior of polycarbonate,” Polymer Engineering and Science, vol. 36, no. 9, pp. 1300–1305, 1996. [13] A. A. Griffith, “The phenomena of rupture and flow in solids,” Philosophical Transactions of the Royal Society of London, vol. 221, pp. 163–198, 1921. [14] G. R. Irwin, “Analysis of stresses and strains near the end of a crack traversing a plate,” Journal of Applied Mechanics, vol. 24, pp. 361–364, 1957. [15] D. P. Rooke and D. J. Cartwright, Compendium of Stress Intensity Factors. London: HMSO Ministry of Defence, 1976. [16] Twisp, “http://en.wikipedia.org/wiki/file:fracture modes v2.svg,” Aug. 2011. [17] T. L. Anderson, Fracture Mechanics: Fundamentals and Applications. Boca Raton, Florida: CRC Press, 3 ed., 2005. [18] R. von Mises, “Mechanik der festen K¨orper im plastisch-deformablen Zustand,” Nachrichten von der Gesellschaft der Wissenschaften zu G¨ ottingen, vol. 1, pp. 582–592, 1913. [19] P. J. G. Schreurs, Fracture Mechanics. Eindhoven University of Technology, 2010. [20] D. L. Holt, P. S. Khor, and M. O. Lai, “The relation between the fracture toughness of plates and the thickness of the shear lips,” Engineering Fracture Mechanics, vol. 6, pp. 307–313, 1974. [21] W. F. Brown and J. E. Srawley, “Plane strain crack toughness testing of high strength metallic materials,” ASTM STP 410, 1966. [22] J. G. Williams, Fracture Mechanics of Polymers. Chichester, UK: Ellis Horwood Limited, 1984. [23] F. Erdogan and G. C. Sih, “On the crack extension in plates under plane loading and transverse shear,” Journal of Basic Engineering, vol. 85, pp. 519–527, 1963. [24] R. G. Foreman, V. E. Kearney, and R. M. Engle, “Numerical analysis of crack propagation in cyclic-loaded structures,” Journal of Basic Engineering, vol. 89, pp. 459–464, 1967. [25] P. C. Paris, M. P. Gomez, and W. E. Anderson, “A relational analytic theory of fatigue,” The Trend in Engineering, vol. 13, pp. 9–14, 1961. [26] J. C. Radon and L. E. Culver, “Effect of temperature and frequency in fatigue of polymers,” Polymer, vol. 16, pp. 539–544, 1975. [27] M. D. Skibo, R. W. Hertzberg, J. A. Manson, and S. L. Kim, “On the generality of discontinuous fatigue crack growth in glassy polymers,” Journal of Materials Science, vol. 12, pp. 531–542, 1977. [28] ASTM Standard E647, Standard test method for measurement of fatigue crack growth rates. Pennsylvannia: ASTM International, 2011. 34

[29] E. T. J. Klompen, T. A. P. Engels, L. E. Govaert, and H. E. H. Meijer, “Modeling of the postyield response of glassy polymers: Influence of thermomechanical history,” Macromolecules, vol. 38, pp. 6997–7008, 2005. [30] L. C. A. van Breemen, “Implementation and validation of a 3D model describing glassy polymer behaviour,” Master’s thesis, Eindhoven University of Technology, 2004. [31] T. M. den Hartog, “A multi mode approach to finite, three dimensional, non-linear viscoplastic behaviour of glassy polymers,” Master’s thesis, Eindhoven University of Technology, 2007. [32] D. Fern´ andez Z´ un ˜ iga, J. F. Kalthoff, A. F. Canteli, J. Grasa, and M. Doblar´e, “Three dimensional finite element calculations of crack tip plastic zones and kIC specimen size requirements,” in Proceedings of ECF 15, Advanced fracture mechanics for life and safety assessments, 2004. [33] A. S. Krausz, ed., Fracture Kinetics of Crack Growth. Boston: Kluwer academic publishers, 1988. [34] S. Glasstone, K. J. Leidler, and H. Eyring, The Theory of Rate Processes. New York: McGraw-Hill book company inc., 1941. [35] J. N. Sultan and F. J. McGarry, “Effect of rubber particle size on deformation mechanisms in glassy epoxy,” Polymer Engineering and Science, vol. 13, pp. 29–34, 1973. [36] H. R. Azimi, R. A. Pearson, and R. W. Hertzberg, “Effect of rubber particle-plastic zone interactions on fatigue crack propagation behaviour of rubber-modified epoxy polymers,” Journal of Materials Science, vol. 13, pp. 1460–1464, 1994. [37] H. R. Azimi, R. A. Pearson, and R. W. Hertzberg, “Fatigue of rubbermodified epoxies: Effect of particle size and volume fraction,” Journal of Materials Science, vol. 31, pp. 3777–3789, 1996. [38] T. A. P. Engels, B. A. G. Schrauwen, L. E. Govaert, and H. E. H. Meijer, “Improvement of the long-term performance of impact-modified polycarbonate by selected heat treatments,” Macromolcular Material and Engineering, vol. 294, pp. 114–121, 2009.

35