Crack GrowthPC

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Department of Construction Sciences

Solid Mechanics

ISRN LUTFD2/TFHF-10/1041-SE(1-77)

On initiation of chemically assisted

crack growth and crack propagation pathsof branching cracks in polycarbonate

Licentiate Dissertation by

Ulf Hejman

Copyright c 2009 by Ulf HejmanPrinted by Media-Tryck AB, Lund, Sweden

ISBN: 978-91-633-6462-4For information, adress:

Div. Material science, Malmo University, SE-205 06 Malmo, SwedenHomepage: http://www2.mah.se/mumat


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Preface

The present work is a product of the part of my PhD studies conducted to the end of

2009 at the Division of Materials science at Malmo University. This work would nothave been possible without the invaluable input and support concerning the experi-mental work as well as the academic upbringing. To that end, I would especially liketo thank my supervisors Professor Per Stahle and Dr. Christina Bjerken for all theirenthusiasm and inspiration. Also, appreciated consultation with Dr. Zoltan Blum de-serves a special attention here. During my time at Malmo University I have also hadthe great pleasure of associating with Ph. D. colleagues at the Faculty of Odontologyand Health and Society who have made the time here very enjoyable and memorable.In addition, I would also like to mention all of the helpful and encouraging coworkersat the School of Technology who have helped directly or indirectly when I was in need

of advice or support. Especially everyone in the B-corridor is very well appreciated,all mentioned and thought of. Finally, I would like to thank my parents for all en-couragement and support at all times, and my two grandmothers who, so generously,has invited me for dinner from time to time. Thank you all!

The financial support from the Knowledge Foundation for the profile Biofilms- Research centre for Biointerfaces and The Swedish Research Council Proj. No.2007-5317 is acknowledged.

Malmo, Januari 2010

Ulf Hejman


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Abstract

Stress corrosion, SC, in some cases gives rise to stress corrosion cracking, SCC, whichdiffers from purely stress intensity driven cracks in many aspects. They initiate andgrow under the influence of an aggressive environment in a stressed substrate. Theygrow at low load and may branch. The phenomenon of SCC is very complex, boththe initiation phase and crack extension itself of SCC is seemingly associated witharbitrariness due to the many unknown factors controlling the process. Such factorscould be concentration of species in the environment, stress, stress concentration,electrical conditions, mass transport, and so on.

In the present thesis, chemically assisted crack initiation and growth is studiedwith special focus on the initiation and branching of cracks. Polycarbonate plates areused as substrates subjected to an acetone environment. Experimental proceduresfor examining initiation and branching in polycarbonate are presented. An opticalmicroscope is employed to study the substrate.

The attack at initiation is quantified from pits found on the surface, and pits thatact as origin for cracks is identified and the distribution is analysed. A growth criterionfor surface cracks is formulated from the observations, and it is used to numericallysimulate crack growth. The cracks are seen to coalesce, and this phenomenon isstudied in detail. Branching sites of cracks growing in the bulk of polycarbonateare inspected at the sample surface. It is found that the total width of the crackbranches are approximately the same as the width of the original crack. Also, anglesof the branches are studied. Further, for comparison the crack growth in the bulk issimulated using a moving boundary problem based algorithm and similar behaviourof crack branching is found.

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Contents

Preface i

Abstract iii

List of papers v

1 Introduction 1

2 Stress corrosion cracking 2

3 Experimental methods 5

4 Numerical simulation methods 7

4.1 Surface crack simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.2 Crack growth simulation of branching stress corrosion cracks . . . . . . 8

5 Results and discussion 9

5.1 SCC initiation and growth . . . . . . . . . . . . . . . . . . . . . . . . . 95.2 Crack branching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Summary of appended papers 16

References 19

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1. Introduction

Over the past century and more, engineering skills have emerged in order to be ableto make constructions as stable and efficient as possible in the sense of load bearingcapabilities, manufacturing and material consumption. In recent years, the aid ofcomputers has made the hunt for the optimal configuration even easier. However, stillmany failures do happen despite the rigorous computational effort. Constructionsmay fail despite that they are not fatally loaded, often when subjected to a certainenvironment. This phenomenon is called stress corrosion or SC and when this leadsto crack growth, it is called stress corrosion cracking or SCC. This happens underthe influence of an external load or from residual stresses and an aggressive specieto help assist crack growth by driving a chemical reaction, often anodic or cathodic,leading to dissolution. Usually, the reaction involved is an oxidation, as is the case formetals. Other materials, such as polymers like polycarbonate used in the experimentspresented in this thesis, may experience SCC from other types of chemical reactions

driven by the surrounding environment. An other cause for SCC is hydrogen crackingin some susceptible metals. Here hydrogen diffuses in the material making it brittleand susceptible to crack growth. All mechanisms are referred to with the common termSCC. All phenomena have the same characteristic feature as fatigue crack growth inthat sense that a crack may propagate despite that the stress intensity factor is muchlower than the fracture toughness if certain conditions are met. In the following, nodistinction between the phenomena will be made; the term stress corrosion cracking,SCC, will be exclusively used irrespective of the underlying decay. In this thesis, thematerial under consideration is polycarbonate, that is a polymer. SCC is liable forcostly failures due to its nature and the field has been an active research area for the

last five decades and involves many disciplines such as fracture mechanics, materialscience, physics and chemistry. A brief introduction to stress corrosion is given inSection 2.

The present thesis presents both experimental work as well as numerical sim-ulations. The material chosen in the experimental work is commercially availablepolycarbonate, PC. This polymer is chosen because it allows for rapid results, com-pared to e.g. metals and an aim is to make it a model material for similar studiesin other materials. When investigating SCC, it is vital to be able to perform manyexperiments in order to establish a statistical significance and SCC experiments inpolycarbonate can be done within a few minutes, which allows for many experiments

for better statistical foundation. The phenomenon is intrinsically non deterministic,a result derived from this investigation. Even if e.g. branching is known to takeplace, the crack path differs from one experiment to another, but it always shows thesame characteristics. The variations may depend on factors such as impurities in thematerial or shape of the crack tip. The experimental activity has been performedin connection to both appended papers and is described in more detail in Section 3.The experiments are evaluated with numerical simulations or stress field distributioncalculations. An introduction to the numerical schemes is given in Section 4.

The most important results are summarized and discussed in Section 5. Finally,the two papers appended in this thesis are outlined in Section 5.2.

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Faster FasterSlower

1

2

3

Figure 1: Diffusion of species in the initiation phase of SCC may take place in theenvironment 1) and at the surface 2) and in the bulk of the substrate3). The surface evolution is illustrated by a dashed line. If the surface isstretched, the stress here indicated with , influence the diffusion rates.

2. Stress corrosion cracking

The ability of a stress corrosion cracks to grow at low loads makes them dangerous.They are also small and hard to detect and distinguish from general corrosion on anattacked surface. Stress corrosion can take place if a loaded construction element isin contact with an environment containing aggressive species, cf. Davis [1]. A closelyrelated phenomenon is chemically assisted crack growth. Here, the material needs notcorrode, which may imply an electrical transition as is the case for e.g. metals. It

may also be some other type of chemical reaction causing dissolution of the substratemaking it possible for cracks to advance. This is usually the case for e.g. polymerssuch as polycarbonate used here. In this thesis, SCC refers to chemically assistedcrack growth.

It takes some time to initiate stress corrosion after the contact with an aggressivespecie is engaged. During this time, the surface is roughened by general corrosion,cf. Jones [2]. If it is also stressed, as in this case, the flat surface is also thermodynamically unstable to perturbations and the surface roughness is dependent of thestress level, cf. Asaro and Tiller [3], Grinfeld [4]. The surface roughens and eventuallythe deepest pits will evolve and may turn into cracks due to stress concentrationsat the bottom of the pit. This is illustrated in Fig. 1. The amount of time needed

in the initiation phase is dependent not only on the concentration of the species inthe environment but just as important is the diffusion of active species and wasteproducts and chemical reaction rates between the substrate and the environment andthe magnitude of the load. Geometry is not negligible if stress concentrations arecreated. If there is some groove or notch that can act as stress concentrator, a stresscorrosion crack may initiate here in less time since no pit needs to be formed, cf. Yu[5].

There have been a number of linear and non linear solutions published describingthe evolution of a surface. Possible evolution laws have been proposed by e.g. Yangand Srolovitz [6], Yu [5], Spencer and Meiron [7]. The surface evolution is seen to

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1

2(a) (b)

Figure 2: (a) Schematic view of a stress corrosion crack. Diffusion of species 1) inthe environment and 2) in the bulk. The bold line marks the active zone.(b) A real stress corrosion crack. In this case it has branched. The arrowshows the direction of crack growth. Also, the direction of load is indicatedin both figures.

be stress dependent, Kim et al. [8] and Yu and Sou [9] presented an unambiguoussolution making it possible to calculate the stress from the shape of an etched surface.The initiation of stress corrosion cracks is investigated in paper A.

The next stage, the crack propagation stage, where pits have grown deep enough tobe cracks follows if the stress concentration is high enough. If a threshold or activationstress for SCC is exceeded, cracks propagate. There has been many models publishedon the crack growth mechanism. Some of which are summarised in Jones [2]. Theactivation energy needed for crack growth is not an intrinsic material parameter, itdepends on the combination of material and environment and corresponds to a level ofstress where chemical reactions commence leading to dissolution of materiel. Cracksgrow when and where the dissolution of material occurs somewhere on a surface. This

happens in regions with sufficiently high stress usually at the bottom of a deep enoughpit. This is indicated in Fig. 2(a) with a bold line and the finite area in the vicinityof, what will be the crack tip, is the active region where dissolution occurs. Here,stress exceeds the activation energy for dissolution. The size of the active region isstress dependent due to the stress concentration. Higher levels of stress will lead toan increase in size of the active region around the crack tip. The crack will widen toadapt to the new load, the growth rate will be unaffected and constant. From thisit can be understood why stress corrosion cracks are blunt, have a finite width andconsequently has no singular stress field close to the crack tip. In practice, the crackscan be seen as a deep and narrow pits and the sides of the crack can be considered

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as part of the body surface. Instead of using the stress, the strain may be moreeasily calculated so a strain threshold value for crack growth may be used instead. Acommonly used model employed in computations of stress corrosion crack growth isthe film breakage model described in Section 4.2 developed and used by e.g. Jivkov

[10], Jivkov and Stahle [11], Stahle et al. [12].Once again consider the schematic sketch of a deep pit depicted in Fig. 2(a). It

is shown how the active species and waste products need time to diffuse to and fromthe crack tip in both the substrate and in the surrounding environment in order tomaintain the chemical reactions that lead to material dissolution and crack advance-ment in the active region. Mass transport is evidently a delimiting factor and as thecrack grows deeper, the distance to the surface increases and the diffusion rate willbe an even more prominent controlling factor. Usually it is expected to find anotherchemical composition in the environment close to the crack tip where reactions aretaking place than compared to the environment outside of the crack. This is the rea-

son why a stress corrosion crack may not continue to grow if a specimen is cleanedand re immersed in the active solution, see e.g. Jones [2].A stress corrosion crack may grow transgranularly or intergranularly depending

on the material properties and the active species in the environment. The grainboundaries are often the preferred path for SCC. It is generally assumed that sharpcracks are prone to grow in a direction so that the stresses acting at the crack tip willopen it in mode I, cf. Broberg [13]. The same assumption is made for stress corrosioncracks.

Stress corrosion cracks are often, but not always, seen to branch. For sharp cracks,this is frequently observed at dynamic crack growth but scarcely observed otherwise.In the case of SCC, it is facilitated by the influence of the environment and the

dissolution process. The branching is thought to occur when the crack tip is bluntedenough to mimic a free edge where two pits or depressions are formed. From theblunted tip, these two initially interacting depressions can grow into two separatecracks from the corroding surface, in this case the bottom of the original crack orpit. A typical stress corrosion crack with branching sites, studied in paper B, isshown in Fig. 2(b). Branching of sharp cracks have been considered and analyticalcalculations of the stress intensity factor, KI, have been laid forward. See e.g. Parletun[14], Viteck [15], Zhang et al. [16], Suresh and Shih [17] for mode I and by Theocaris[18, 19], Theocaris et al. [20] for mixed mode loading. In paper B, it is investigated ifthese results are applicable also to SCC and their special growth mechanism. Also, the

angle dividing two branches of a crack after branching are investigated experimentallyand the result is compared with result found in the literature.

The third and last stage is a rapid mechanical failure due to overload of the re-maining load bearing material, cf. Jones [2]. The crack mechanisms operating in thedifferent stages are reflected as clearly distinguishable crack surfaces in a fractograph.The corrosion active in the first two stages renders a rough surface, often covered withcorrosion products. The final fracture may be brittle or ductile, with correspondingcharacteristic features. Even if it is easy to analyse a fracture surface after failure hashappened, it is desirable to identify a growing crack before a collapse.

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l=100 mm

3 mmSpecimen

Rig

x l

Figure 3: The loading arrangement for the SC initiation experiment.

3. Experimental methods

The experimental activity conducted within the scope of the present thesis involvesthe investigation of SC initiation and SCC with branching. These experiments differin objective and experimental arrangements, but the same material and environment

parameters have been used. To investigate crack initiation, specimens are machinedfrom a 4 mm thick, extruded plate to a rectangular test piece with the length 100mm and width 30 mm. To avoid unintentional crack initiation at the edges of thespecimens, they are covered with a protective, elastic layer of an epoxy glue. Thespecimens are mounted in a three point bend rig, inert to the environment. Theconfiguration, shown in Fig. 3, is submerged into an acetone solution diluted with10% of water. The concentration is chosen to get an appropriate reaction rate so thatit may be controlled and practically handled. The three point bending gives a linearlyvarying stress on the surface of the specimens. After an exposure of approximatelyfive to ten seconds, the specimen is rinsed with distilled water to stop the attack. An

optical microscope is used to analyse the surface for pits and cracks after exposure.Fig. 4(a) shows how a pattern of randomly distributed pits form on the surface. Ona larger length scale some of the pits grow into cracks on the surface. These cracksgrow and coalesce, as shown in Fig. 4(b), and a close up of the coalescence is shownin Fig. 4(c). Coalescence of cracks is studied in paper A, where initiation of SCC isconsidered.

The crack growth with branching is studied using slightly larger specimens. Theseare also cut out from an extruded plate of polycarbonate but with the thickness of 10mm, a width of 100 mm and a height of 50 mm. A short and narrow notch is madeat the middle of the long edge on each sample and two holes, one on each side of thenotch are drilled to facilitate application of the load opening the notch in mode I. The

side opposite to the notch, the plate is glues to an aluminum bar considered to berigid compared to the polycarbonate. The setup is shown in Fig. 5(a). An undiluted,commercial grade of acetone is carefully suspended to the tip of the notch to startthe chemically assisted crack growth. The acetone must be continuously supplied tothe crack for growth and the load is applied by means of a spring system so that theload is approximately constant during one experiment. The crack and the branchingsites are studied from the side of the plate with an optical microscope. Crack widthand crack angles at the branching sites are measured from the images captured with adigital camera connected to the microscope and presented in paper B. A typical crackpattern is shown in Fig. 5(b) and a close up of a branching site is shown in Fig. 2(b).

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(a)

(b) (c)

Figure 4: (a) Typical SCC initiation pattern of small pits. (b) Typical SCC initiationpattern of surface cracks. (c) Typical pattern of surface cracks coalescing.The arrows indicate the direction of crack growth from the center of thecrack, marked with a white dot.

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i) ii)

iii)

iv)

v)

Figure 6: Schematic sketch of the assumed surface crack growth rules found inSection 4.1.

the crack growth rate is assumed to be constant and equal to v0. The total time of asimulation is denoted t0, and is divided into constant time increments, t. Here, tis chosen to equal t0/500. This corresponds to that the crack extension at a tip lequals l0/500 during one time step.

When the distance perpendicular to the crack growth direction between two meet-ing cracks is less than 1/10 of the length of the sum of the two cracks and the tiphave reached the same lateral position, coalescence occur. The distance between thecracks is a parameter chosen to fit experimental data. If the conditions for coalescenceis fulfilled, the two meeting cracks grow towards each other at an assumed angle of45 by growing one length unit in the direction of the original crack and one in the

direction of the meeting crack until they encounter each other. No stress analysis isperformed during this type of simulation. The growth direction is governed solely bygeometrical considerations. The result of the simulations are used to verify the growthrules. The code is written in the commercially available software Matlab [21].

4.2. Crack growth simulation of branching stress corrosion cracks

The bluntness of a stress corrosion crack implies the absence of a stress singularity atthe crack tip and that the concept of linear fracture mechanics may not be applicable.As a remedy, a number of other models describing SCC have been formulated. Themodel used in the simulation of stress corrosion crack growth is similar to the modelfor SCC described in Section 2. Here, the extrenal load is static, and no rupture

of am oxide film is considered. Instead, surface strains are assumed to influence thechemical reactions of the corrosion. The crack grows by dissolution of material on thesurface of a stretched body where a strain threshold value is reached. The amountof crack extension depends on the level of strain exceeding the threshold strain. Thishappens in the active region, see Fig. 2(b). With this method, the crack will havethe ability to grow, choose direction and decide its width and branch autonomouslywithout any need for any criterion for growth direction and branching. Finite elementcalculations is used to find the strain on the surface. To do this, the commerciallyavailable computer program Abaqus [22] is used. The mesh used to simulate thecrack growth shown in Fig. 5 is shown in Fig. 7. This model is used in paper B

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0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

/0

N/N

0

N/N0

Experiments

Linear fit

(a)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

/0

N/N

0

N/N0

Experiments

Linear fit

(b)

Figure 8: Normalised occurrence of pits (a) and length of cracks (b) along a spec-imens surface. D and N are the number of pits and cracks respectively.The load varies linearly and is normalised for convenience.

The crack growth is simulated using an actual and typical pattern of initiation sitesobtained from an experiment. Fig. 9(b) shows the original experimental distributionand the simulation from the initiation sites to a point where coalescence has takenplace at some locations. This simulation is made in accordance with the growth rules

i)-v) in Section 4. Some of the characteristics of the actual crack pattern can be foundin the simulations by comparing Fig. 9(a) and Fig. 9(f). For example, in Fig. 9(f) itis seen that the distance between a crack tip and the location at its flank where theother crack meet increases with the distance where the two cracks started to interact.However, the tendency to line up and create stripes of cracks is not so pronounced,but can be imagined in some areas. Since two meeting crack ends must have reachedthe same lateral position before coalescence can take place, the tips will always misseach other creating a rhomboidal piece between the cracks when they finally approacheach other. This feature is a simple approximation that corresponds to the hook likeshape that is seen experimentally.

Hitherto, the studied area is considered to represent the same spot on a sample,

and the evolution of crack pattern is followed with time. However, it may be possi-ble to regard the result of the simulation to reflect a single moment in time for areasexperiencing different nominal stress. The density of initiation spots increases approx-imately linear with stress, as observed from the experiments. Consider the patternin Fig. 9(d) to represent a part of the sample where stresses are large, and Fig. 9(b)smaller stresses. Since the patterns reflect the situation at one moment, the cracksthat have not coalesced must be equally long for both areas. This means that thedifferent figures represent different length scales, equivalent to different time steps. Inaccordance to the experimental findings, crack coalescence is more probable at larger

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(a) (b)

(c) (d)

(e) (f)

Figure 9: (a) A typical SCC initiation pattern of surface cracks. The initiation pointshave been indicated. (b) The pattern of initiation sites. (c) A part of theinitiation sites used in the simulation. (d)-(f) The numerical simulation ofthe crack growth.

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Figure 10: Overview of a specimen covered with surface cracks. The number ofcracks increase with load to the left in the figure.

Figure 11: The total crack growth rate depends on the number of free crack endsand is seen to decreases when coalesces takes place.

stress levels simply due to more initiation sites. Thus, Fig. 11 also may represent thetotal crack length change, L, with nominal stress. While comparing with the resultsfrom the experiments, see Fig. 8(b), an expected decrease in L/0, is not observed.

Since no new initiation points are created during the simulation, as one of theassumptions, it is not surprising that the mean total crack growth rate is constantuntil crack coalescence takes place and as the more and more cracks coalesce, the slowerthe crack growth. The decrease is assumed to be logarithmic since the longer timethat elapses, the number of free crack ends should decrease exponentially. An assumedlogarithmic approximation to the decay is shown in Fig. 11. Crack coalescence, in thiscase, begins at approximately t = 0.1t0.

The stress field around two meeting cracks is further analysed using a finite elementmodel with two meeting cracks in an infinite plate seen in Fig. 12. Abaqus [22] isused for the finite element analysis. The loading is also shown in the figure. Thecoalescence is divided in three steps, with different crack advances. The stress fielddistribution of the von Mises effective stress is plotted at the vicinity of the crack tips.

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h

a

d

Figure 12: Overview of the finite element configuration used to analyse the crackpath during coalescence. In the figure d


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(a) (b) (c)

(d)

Figure 13: Von Mises effective stress field around two meeting cracks. The numbersrefers to corresponding stress levels.

for the experiments and by 73 9% for the numerical simulations. This correspondswell to results for sharp cracks where , e.g. Suresh [26], Theocaris [18, 19], Kitagawaet al. [27] calculated a drop in stress intensity factor by 70-90% depending on thecrack configuration. It is seen that both branches continue to grow, is contrast to

purely stress intensity driven cracks where only the wider crack can propagate understatic or quasi static conditions.

The angles at branching sites are also studied to see what direction the crackturns when branching takes place. The angles are defined according to Fig. 15. Theangle to the widest crack branch is denoted , the angle dividing the two branchesis denoted . A mean crack path is defined by + /2 and this angle indicates inwhat direction the crack grows in average. A value of 180 indicates that the crack inaverage would continue straight forward, and this is expected for a symmetric branch,see Fig. 15. Figure 16 depicts the results from the experiments and simulations fromthe measurements of the different branching angles and the result is linked to the

ratio of the crack branch widths. From Fig. 16(a) it is seen that the angle has amean value with the standard deviation of 16113 for the experiments and slightlylarger for the simulations 168 11. The corresponding mean values for the angleis 32 12 and 46 14. From both figures it is seen that the scatter is significantand worse in the measurements of the angle. The scatter makes it hard to see anytrend in the dependance on crack widths. It is seen that for all ratios found, from1 to 5, there is a possibility to get almost any angle in the interval from 140-185

for the angle and between 10-70 for the angle. The mean crack path is shownin Fig. 16(c) for the same interval of width ratios and it is seen that even if there isconsiderable scatter in the and angles, the sum + /2 is close to 180 in the

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0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

3

l1

/ l2

(l2

+l1)/L

Experiment(1) Mean = 2.21(2) Mean = 1.24

Simulation(3) Mean = 1.77(4) Mean = 1.06

(3) (1)

(2)(4)

Figure 14: The crack width ratio versus the normalised sum of the crack branchwidths.

L

l

l

Figure 15: The definition of the measured properties at crack branching.

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120 130 140 150 160 170 180 190 200 210 2200

1

2

3

4

5

6

7

(deg.)

l1/l2

Experiment(1) Mean = 161

Simulation

(3) Mean = 168

(2) Prletun = 166

(1) (2)(3)

(a) The ratio of the widest and the narrowest crack branch after branching versusthe angle defined in Fig. 15.

0 10 20 30 40 50 60 70 80 90 100 1100

1

2

3

4

5

6

7

(deg.)

l1/l2

Experiment(2) Mean = 32

Simulation(3) Mean = 46

(1) Prletun = 28

(1) (2) (3)

(b) The ratio of the widest and the narrowest crack branch after branching versusthe angle defined in Fig. 15.

0 1 2 3 4 5 60

50

100

150

200

250

l1

/ l2

+

/2

(deg.

)

Experimen(2) Mean = 177

Simulering

(1) Mean = 192Reference = 180

(1)(2)

(c) The crack path angle defined in Fig. 15 versus the ratio of the branch widthsafter branching.

Figure 16: Normalised results from the measurements of branching properties of astress corrosion crack.

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Paper A

U. Hejman and Christina Bjerken

Environmentally assisted initiation and growth of

multiple surface cracks

Submitted for publication.


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1. Introduction

The phenomenon of stress corrosion, SC, has been studied intensively forthe last five decades. When subjected to aggressive species in the surround-ing environment, stress corrosion cracking, SCC, may occur in many kindof materials. The mechanisms differ for metals, ceramics and polymers, butall materials experience similar signs of attack and cracking. For a list ofmaterial and environment combinations, see e.g. Davis [1]. Stress corrosionof metals and associated crack growth due to oxidation is perhaps the mostcommon SCC process where electrochemical processes cause the degrada-tion. A well accepted model to describe the process is by formation of anoxide film that may or may not be protective to the underlying substrate.

Repeated breakage of the film makes the crack advance in metals. Often,hydrogen induced cracking is also considered as SCC. In polymers, stress in-duced degradation is caused by dissolution of material or weakening by bondbreakage in the polymer structure. Usually, SCC is divided into three phaseswith characteristic crack growth mechanisms and crack growth velocities.The initiation phase, followed by a steady state crack growth stage wherethe crack growth rate is controlled by chemical kinetics and mass transportin the crack and at the crack tip. Crack growth from these mechanisms mayoccur at low loads, compared with the materials fracture toughness. Finally,an accelerating crack growth, leading to final fracture occurs due to mechan-

ical overload. Over the last fifty years research efforts have been conductedto increase the understanding of the underlying processes. The initiationphase is, however, not as well investigated. Also, it has proven hard to builda theory describing the transition between the different stages. For a generalreview of the SCC phenomenon, see e.g. Jones [2, Ch. 1].

One of the first attempts to describe the evolution of a stressed surfacewas made by Asaro and Tiller [3] and perhaps the most famous extension tothis work was made by Grinfeld [4, 5, 6]. They have theoretically shown thata stressed flat surface is thermodynamically unstable, and surface and bulkdiffusion as well as dissolution and deposition of material will inevitably

lead to roughening of a stressed surface. This process is counteracted bythe surface energy that tends to flatten a surface. This leads to a surfaceroughness with a stress dependent characteristic wavelength spectrum. Thephenomenon may appear at interfaces subjected to SCC or etching as Asaro,Tiller and Grinfeld considered. It may also be observed at crystal growth,treated theoretically by e.g. Spencer et al. [7] and at the interface between a

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solid and its melt by e.g. Grinfeld [8].

An extension of the theoretical analysis of the developing surface rough-ness was developed by Yang and Srolovitz [9]. They found that if the surfacewas allowed to continue to evolve, the initial perturbations of the surface growinto cusp like pits and concluded that this may be a connection between theinitiation of cracks and crack growth. In an analysis, Yu [10] found that a flatsurface will develop a roughness and cracks as long as the stress can destabi-lize the surface. At first the surface roughens and blunt pits, that eventuallymay sharpen into cracks, grow. This implies that the process does not havea stress threshold value. However, the time for a development to a crack canbe significant, and they defined a stress threshold value over which a notch

would immediately form and grow into a crack. Sou and Yu [11] made cal-culations on a stressed cycloid shaped surface and found that small grooveswould grow into a shape where the surface evolve invariantly if the stress issmall and, if the stress is large enough, the grooves will instead form sharpcracks. Spencer and Meiron [12] continued this approach with a nonlinearevolution law and found that singular grooves, cusps, form as a direct re-sult from the nonlinear part. The more irregular the initial surface is, themore pronounced the non linearity contribution would be. The solution theyfound is unambiguous which makes it possible to reverse the approach andget a measure of the stress on the surface by examining the surface rough-ness wavelength distribution. An experimental procedure for calculating thestress from the wavelength distribution was developed by Kim et al. [13] andYu and Sou [14]. Grilhe [15], on the other hand, found the solution to bemulti valued when applying a different technique calculating the energeticdriving force for roughness appearance on a free surfaces under stress. Seealso Grilhe [16] and Colin and Grilhe [17]. Finally, Song et al. [18], Yang[19] calculated the surface evolution in a thin elastic layer and found thefrequency of the surface roughness to be dependent not only on the stressand material properties, such as the modulus of elasticity or surface energy,but also on geometrical factors such as the thickness of the layer.

Simulations of surface evolution using different numerical methods, based

on e.g. the finite element (FE) analysis, have been conducted to evaluate thephenomenon. Prevost et al. [20] developed a numerical technique to analyseand simulate the dissolution and mass transport processes. This procedurecan be used for simulating crack initiation under various conditions, e.g.at grain boundaries. However, crack growth is not at all considered. Thesituation with a growing SC crack has been considered by Jivkov and Stahle

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[21], Jivkov [22]. In this case, the growing crack is driven by dissolution of

material due to breakage of a protective oxide film on the body boundarywhere strain exceeds a threshold value. According to this model, the widthof the crack may not be finite, as is found by others mentioned above, andthe crack cannot be sharp unless the growth mechanism is changed. Thisis also seen with SC cracks found in nature. Another interesting result isthat the cracks grow in a self similar manner. Stahle et al. [23] have usednumerical methods for simulation of crack growth from energetic relationsand come to the similar results as Jivkov has.

Shreter et al. [24, 25] performed etching experiments on mechanicallyloaded silicon plates. It was found that on a nanometre scale the surface

was roughened due to oxidation induced stacking faults. On a micrometerscale, small cracks appeared transverse to the direction of loading. Onlycompressive load gave signs of attack. The number and length of the visiblecracks were found to be increasing with increasing load and no thresholdvalue was found in the experimental results.

This work is concerned with the initiation of SCC in polycarbonate, PC,exposed to an acetone solution. Section 2 describes the polycarbonate andacetone properties and the next section 3 describes the experimental setup.In Section 4, results are presented and discussed. Two different signs ofattack are found on the surface; these are pits and cracks. The distributionof pits and crack patterns are analysed. The way the cracks coalesce, tipto side, is investigated and the finding is supported by stress analysis. Asimulation of the growth of multiple cracks is also presented in this section.Last, a few remarks on the present study, followed by the conclusions, arefound.

2. Material and preparation

The material used in this investigation is an industrial grade of MACRO-CLEAR polycarbonate, PC, (Arla plast AB, Sweden), which is a transpar-ent, amorphous and isotropic thermoelastic polymer with Youngs modulus

E= 2.3 GPa, Poisson ratio = 0.38, tensile yield stress y = 63 GPa. Thematerial is used in the as received state, no measures is undertaken to thesurface but rinsing with ethanol that is known to react very slowly with PC.During manufacturing, the material is extruded to a plate, and thus a certaindegree of anisotropy can not be ruled out, but is not likely. In this study, allspecimens are cut in the same direction and no investigation is made for dif-

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ferent directions in the material. Residual stresses present in a material due

to e.g. manufacturing and preparation can be significant and influence theresult. A polariscope is used to determine if any residual stresses are presentfrom the manufacturing and cutting processes. Only specimens without anysign of residual stresses are chosen.

Polycarbonate is chosen due to its comparably high fracture toughnessofK= 3.7MPa m1/2, Sardinha et al. [26], compared to other polymers. It iseasily machined and prepared. Polycarbonate is also know to be resistant tomany chemicals and can in many cases be suitable as substitution to glass inchemical containers. However, a few agents are known to attack and dissolveit. Among the most aggressive is acetone which diffuses easily in PC and

dissolves it in a hydrolysis process. Acetone is used in the polymerisation ofpolycarbonate. The exposure to acetone forces the polymerisation processesto be reversed leading to a defined dissolution of the material. The acetoneis supplied by VWR International, Stockholm, Sweden, and is an industrialgrade with a purity of at least 99.5%.

The load used in this study is insignificant compared with the toughnessof PC and the specimens are only stretched in the elastic region. Withoutthe addition of acetone, no damage can be seen. Polycarbonate is also cho-sen because it has proven usable in similar studies. Gunnars et al. [27, 28]used this material combination to study non branching crack propagationdirections, and Hejman and Bjerken [29] studied crack paths for branchingcracks.

3. Experimental setup and procedure

A rectangular test specimen with a length of 100 mm and a width of 20 mm iscut out from a PC plate with the thickness 3 mm using a hacksaw. To avoidcrack initiation at the edges, they are polished to a reasonable smoothnessand then covered with a protective layer of epoxy glue.

The test specimen is subjected to three point bending. An illustration ofa specimen mounted in the test rig is shown in Fig. 1. With this fixation, the

effective test length of the specimen, l

, is not the same as the geometricallength, l = 100mm. If nothing else is stated, from now on the meaning ofthe specimens length is always referring to the effective length. The wholeconfiguration is submerged into an acetone and water solution. This setupallows study of both sides of the specimen with the exception for the smalldistance covered by the middle support. The rig is considered as stiff as

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compared to the specimens, it is made of stainless steel and is not affected

by the test solution.The concentration of acetone and the exposure time is correlated so that

the specimens would not break immediately. In the hydrolysis, the acetonesolution dissolves the PC, and carbon dioxide and water is produced. Someinvestigations of this phenomenon have been made by Ghorbel et al. [30]and for water by Akele and Thominette [31]. It has been shown that staticstresses in the PC drastically accelerates the hydrolysis, cf. Liu et al. [32]and Ghorbel et al. [33]. Using the acetone in the received purity of 99.5%,the reaction was too fast to be easily controlled. To this end, the acetoneis diluted with distilled water in various proportions to slow the process

down and thereby gain better control of the process. A solution with lessthan 90 weight % acetone renders no visible reaction for times suitable forrepeated experiments. In this study, the test pieces are exposed for a time ofa few seconds. Solutions with 92 weight % acetone is found to be the mostappropriate for this case. This concentration is used merely on empiricalbasis.

Images of the specimens surface are collected and examined with anoptical microscope. Both sides of a specimen is studied for cracks and pits.The surface is divided into equally sized regions representing all magnitudesof the linearly varying stress on the surface, see Fig. 2. The number of pits iscounted in square shaped areas in every region, marked with the number 1,and the number of cracks crossing a straight line, marked with the number2, is counted in every region. Finally, the total length of the cracks presentin every interval is measured. An area density of pits and linear density ofcracks can then be calculated at regular intervals along the surface to geta measure of the attack at different stress levels in both compression andtension. The regions are so narrow that the stress can considered to beconstant within them.

The micrographs are also used in an attempt to investigate the possibilityfor the pits to act as crack initiation sites. To this end, the distribution ofthe cracks is examined and a numerical simulation of the crack growth is

performed. It is also noted that the cracks coalesce to form stripes over thesample surface, and the nature of the meetings are looked upon by a FEanalysis of the geometry at hand.

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l=100 mm

3 mmSpecimen

Rig

x l

Figure 1: A side view of a specimen mounted in the three point bend rig. The width ofthe specimen is 20 mm.

Figure 2: Overview of a sample surface showing the cracks, and how they are quantified.

4. Results and discussion

4.1. Stress distributionThe type of loading used makes it easy to calculate the stresses on the surfaceof the specimens. For long slender beams and ideally sharp supports, threepoint bending gives a linearly varying plane stress state along the surface.The maximum is positioned at the middle of the specimen and zero stressat the outer supports. A FE analysis of the problem at hand confirms thatthis is true also for a beam of the present dimensions except the parts restingon the middle support. Due to symmetry, only half the beam is shown, thecentre part of the beam corresponds to the right hand side of the figures. TheFE analysis has been performed using ABAQUS [34]. The result is shown in

Fig. 3. The variable x is defined in Fig. 1. In all figures and text, , refers tothe stress component at the surface parallel to the load. The maximum valueis denoted 0 and is chosen to be the highest value found at the middle of thebeam. The small dip found in Fig. 3 is an artifact due to the finite width andrigidity of the boundary conditions chosen. In reality, the middle support isnot so firmly attached. This small effect is neglected and the maximum used

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(a) (b)

Figure 4: Typical micrographs of the polycarbonate surface. (a) Before and, (b) after test.The roughness consists of pits seen as dots.

dependent part also.The pattern of pits can be depicted as a pattern of dots. To find any

correlation between the pits, Fourier analysis is employed to search for anycharacteristic distribution. No preferred distance between the pits can befound. The pits appear to be completely randomly distributed and, at leastin areas small enough that the the stress may be considered as constant,no correlation with the ATG phenomenon can be found. This may be inter-preted as a simultaneous initiation of the pits so they do not have any chance

to influence each other and is also an indication that the distribution of pits,in this case, is not an ATG instability phenomena.

4.3. Crack occurrence

On a larger scale, visible to the bare eye, cracks form perpendicular to theload. Figure 6(a) shows one of these surface cracks. Due to polarization of thelight and the stress optical properties of polycarbonate, light or shadow likeartifacts in the vicinity of the cracks is visible. At this low magnification,no dots are clearly seen. The number of cracks formed at the surface iscounted as a linear density, denoted with N, parallel to the orientation of

the load, as described in Section 3. If nothing else is stated, the meaningof a crack is always referring to cracks appearing on the specimens surfaceand can be seen as furrows on the specimen transverse to the direction ofloading. The length of these furrows is measured, but the penetration depthis not considered. On the surface, many cracks form and grow towards eachother. This is seen in Fig. 6(b). Fig. 6(c) shows the corresponding map of

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(a) (b) (c)

Figure 6: (a) One crack on a specimens surface. (b) An area used to analyze the distribu-tion of initiation sites. The corresponding map of the initiation sites are indicated in (b)and explicitly shown in (c).

that the pits described in Section 4.2 act as initiation sites. Only a fractionof all available pits will grow into cracks and the distribution of these pits isinvestigated. To find a reasonable location for a initiation point of a crackthe following assumptions are made: The cracks are taken to extend with thesame rate at both ends, which is a plausible situation not contradicted by anyof the experimental results. The crack growth rate is assumed to be constant

and independent of stress. Since this is a stress corrosion phenomenon, therate is controlled by chemical reactions at the crack tips, and these are notinfluenced by the load but rather by the diffusion rate of spices to and fromthe crack tip region Jones [2]. These assumption will be addressed and usedlater in Section 4.5. Ostensible conclusions from the given postulates arethe choice of taking the centre point of a crack as its origin and define themiddle point of a crack as its initiation site. An example of this can be seen inFig. 6(c) where a map of crack mid points, corresponding to the crack patternin Fig. 6(b) is depicted. It is seen that a similar random pattern of initiationsites is found as for the pit distribution in general described in Section 4.2.

However, a significant difference can be noted in the randomness. A Fourieranalysis of the crack initiation site distribution reveals that the pattern israndom in nature but the initiation sites are also uniformly distributed andis never seen to be very close together. This is not surprising, since a regionaround a growing crack exhibits stress relaxation and this indicates somedependence between pits that grow into cracks.

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0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

/0

N/N0

N/N0

Experiments

Linear fit

Figure 7: The linear density of cracks along the stressed surface.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

/0

L/L0

L/L0

Experiments

Linear fit

Figure 8: The total length of cracks along the stressed surface.

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4.4. Crack coalescence

If the cracks are allowed to grow, they will eventually encounter each other.When this happens, they are seen to coalesce to form stripes of cracks acrossthe specimens surface. The specimen will finally break along these stripes ifthe exposure is not aborted. In this investigation, the test is always termi-nated long before that happens. This Section presents experimental resultsand a finite element stress investigation of the direction of crack growth oftwo meeting cracks. A special feature can be observed at the point of coa-lescence. From Fig. 9, it is seen that the crack tips avoid each other. Theyattract each other first at a point where crack tips from two parallel andmeeting crack are approximately at the same position in the direction of

crack growth, but they need not necessarily be exactly approaching eachother as long as the tips meet close enough to influence each other. Thedistance between two meeting cracks, for this to happen, is dependent on thecrack lengths. The resulting coalescence is formed crack tip to crack side,not tip to tip. This observation is in line with results for co-linear cracksby Melin [35], who showed that meeting crack tips avoid each other. How-ever, Melin did not consider coalescence. From Fig 9(b) it is also apparentthat the cracks never coalesce completely, at least not at the surface. AFE analysis of two meeting cracks in a large plate is performed. Figure 10shows an illustration of the geometry, but where the distance, d, betweenthe two crack planes is exaggerated compared to the height, h. The cracklength a = h/2, and d = a/100. An external load is applied perpendicularto the crack planes. The symmetry of the stress field in the crack tip regionis investigated. Most cracks, irrespective of process, grow in mode I. Thisassumption is adopted also in this case so that the cracks are assumed togrow in the direction where the gradient of the effective stress is largest, seee.g. Broberg [36]. Figure 11 shows contour plots of the effective stress (vonMises) for three different crack configurations. The three steps are chosen torepresent the meeting of two cracks. The crack tips are rounded, since theassumed crack growth mechanism is dissolution, which was discussed earlier.In the first case, where the cracks are approaching each other, = d/2, the

stress distribution may indicate that the cracks during continuous growthwill turn outwards, avoiding each other. When the crack tips are located atthe same horizontal position, = 0, the stress field seems quite symmetricaround the tips. In the next situation, where the crack tips have passedeach other, = d/2, the stress distribution is clearly tilted inwards, andthus the cracks are assumed to continue to growth towards each other. This

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(a) (b)

Figure 12: (a) Nodes used to calculate the displacement ratio. (b) Displacement ratio (x)of nodes marked in (a) at different stages of crack advance, and a linear fit(-).

Figure 13: Contour plot (von Mises) from the FE analysis of two meeting cracks after they

have started to attracted each other. The stress levels indicated with 1 and 2 correspondto the same stress level in Fig. 11

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(a)

(c)

(b)

(d)

Figure 15: ((a) Initiation sites at the beginning of simulation. (b)-(d) Advancing simulatedcrack pattern.

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its flank, where the other crack meet, increases with the distance for which

the two cracks started to interact. However, the tendency to line up andcreate stripes of cracks is not so pronounced, but can be imagined in someareas. Two meeting crack ends must have reached the same lateral position tobe alongside before coalescence can take place, therefore the tips will alwaysmiss each other and creat a rhomboidal piece between the cracks when theyfinally approach each other. This feature is a simple approximation thatcorresponds to the hook like shape that is seen experimentally.

A consequence of the assumption that the crack growth rate is constantis a decreasing total extension rate, dL/dt, where L is the sum of cracklengths. As more and more cracks coalesce, the number of free ends decreases.

According to the assumption of simultaneous initiation, no new cracks initiateto compensate for this decrease. Figure 16 shows that the total crack growthrate first is constant, and then decreases as more and more coalescence takesplace. Crack coalescence, in this case, begins at approximately t = 0.1t0.The decrease is assumed to be logarithmic, and such a curve fitted to thedata points is also included in the figure.

Hitherto, the studied area is considered to represent the same spot on asample, and the evolution of crack pattern is followed with time. However,it may be possible to regard the result of the simulation to reflect a singlemoment in time for areas experiencing different nominal stress. The densityof initiation spots increases approximately linear with stress, as observedfrom the experiments. Consider the pattern in Fig. 15(d) to represent a partof the sample where stresses are large, and Fig. 15(b) smaller stresses. Sincethe patterns reflect the situation at one moment, the cracks that have notcoalesced must be equally long for both areas. This means that the differentfigures represent different length scales, equivalent to different time steps. Inaccordance to the experimental findings, crack coalescence is more probableat larger stress levels simply due to more initiation sites. Thus, Figure 16also may represent the total crack length change, L, with nominal stress.However, comparing with the results from the experiments, see Fig. 8, adecrease in L/0, is not observed.

5. Further remarks

On the PC surface, two signs of attack are discovered. A surface roughness,seen as dots, that appears without a threshold stress and also in compressionbut is seen to have a slight dependence on positive stress. This may mean

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Figure 16: The crack growth rate is constant until coalescence takes place.

that the dissolution and degradation process has a stress independent part.On a larger scale, multiple cracks appear on the surface, perpendicular to thestress, but in this case the stress must exceed a positive threshold value sothe cracks only appear when the material is stretched and is seen to increaselinearly with stress. Since diffusion and dissolution is stress dependent, but

also takes place without mechanical load, see Liu et al. [32], it is plausibleto believe that the processes creating these damages are diffusion controlledin some way. The experiments performed by Shreter et al. [25] in silicon inan etchant shows damage only on one side, as is found in this study also.In that case, attack is not due to dissolution but on dislocation glide andcracks are found on the compressed surfaces instead, as compared with thepresent study, on the stretched surface. Shreter also found two different signsof attack similar to the ones found here. That is a surface roughness in thenanometre domain and cracks on a larger length scale.

This investigation is made using a light microscope with the possibility

to resolve features in the micrometer scale. Shreter used an AFM to detect aroughness in the nanometre scale. It may be possible to further elaborate thisstudy, taking advantage of an AFM, to find even more detailed informationfrom the surface of the PC and possibly here find a wave pattern correlatingto the stress state. In this study, no ATG instability has been found.

When studying and quantifying the surface cracks, only the surface is

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observed. The cracks are growing, not only on the surface, but also down in

to the depth of the specimen until it reaches a critical depth where the finalfailure will occur if the test is not aborted. In these experiments, the crackgrowth is always aborted long before final fracture occurs. At this stage, allsurface cracks are still quite shallow and are not yet significantly affected byany effects due to the depth. This makes it plausible to, as an initial step,analyse only the surface in two dimensions.

The assumptions used in the model for crack growth are subject for cor-rections to make the simulation more close to the results found in the exper-iments. The objective here is only to test a few simple rules and these arealso to be looked closer upon in a future study.

6. Conclusions

In the present study, characteristics of multiple and chemically assistedcrack initiation in stressed polycarbonate are studied experimentally.A smooth polycarbonate stripe is exposed to acetone while loaded inthree point bending. Two signs of attack, in two different length scales,are discovered.

Pits form on the surface leading to a roughness. Multiple cracks formperpendicular to the load.

The pits are randomly distributed on the surface, indicating a simulta-neous formation independent of each other.

No stress threshold value is found for the formation of pits and only aslight stress dependence is seen.

The occurrence of cracks is found to have a positive stress thresholdvalue and after that to be stress dependent following a growth lawapproximately linearly with stress .

Some of the pits are believed to act as crack initiation sites. These pitsare found have a uniform random distribution indicating a dependance.

In the coalescence of cracks, hooks are seen to form when crack tipsmeet the other cracks side and not the tip.

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The stress field ahead of two meeting cracks are analysed at different

stages of advancements and the crack path is investigated. Under theassumption that stress corrosion cracks grow in mode I, it is verifiedthat the cracks grow to form a hook as is seen in the experiments.

A numerical simulation of multiple surface crack growth is performed.The assumed crack growth rules gives a simulation in which the totalcrack growth is seen to be constant until coalescence takes place. Afterthat, it decreases with the number of free crack ends.

Acknowledgements

We would like to thank Prof. Per Stahle for valuable discussions andinput to the fracture mechanical analysis and Dr. Zoltan Blum at MalmoUniversity for assistance with the polymer chemistry. The financial supportfrom the Knowledge Foundation for the Biofilms - Research centre for Bioin-terfaces and the Swedish Research Council (VR 50562401-02,50562402-02) isacknowledged.

References

[1] J. R. Davis (Ed.), ASM handbook, vol. 13, ASM International, Materials

Park, Ohio, ISBN 0-87170-007-7, 2001.

[2] H. R. Jones (Ed.), Stress-corrosion cracking, ASM International, Mate-rials Park, Ohio, ISBN 0-87170-441-2, 1992.

[3] J. R. Asaro, W. A. Tiller, Interface morphology development duringstress corrosion cracking: Part I Via surface diffusion, Met. Trans. 3(1972) 17891796.

[4] M. A. Grinfeld, Thermodynamic methods in the theory of heterogeneoussystems, Harlow : Longman Scientific & Technical, New York, ISBN 0-

582-06700-6, 1991.[5] M. A. Grinfeld, The stress driven instability in elastic crystals: Mathe-

matical models and physical manifestations, J. Nonlinear Sci. 3 (1993)3583.

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[6] M. A. Grinfeld, Stress corrosion cracking of an elastic plate, Acta Mater.

46 (1998) 631638.

[7] B. J. Spencer, P. W. Voorhees, S. H. Davis, G. B. McFadden, The effectof compositionally-generated elastic stresses on morphological instabilityduring directional solidification, Acta Metall. Mater. 40 (1992) 15991616.

[8] M. A. Grinfeld, Instability of the separation boundary between a non-hydrostatically stressed elastic body and a melt, Sov. Phys. Dokl.31 (831-834).

[9] W. H. Yang, D. J. Srolovitz, Crack like surface instabilities in stressedsolids, Phys. Review Let. 71 (1993) 15931596.

[10] H. H. Yu, Crack nucleation from a single notch caused by stress-dependent surface reactions, Int. J. Sol. Struct. 42 (2005) 38523866.

[11] Z. Sou, H. H. Yu, Crack nucleation on an elastic poly crystal surfacein a corrosive environment: Low dimensional dynamical models, ActaMater. 45 (1997) 22352245.

[12] B. J. Spencer, D. I. Meiron, Nonlinear evolution of the stress-driven

morphological instability in a two-dimensional semi-infinite solid, ActaMetall. 42 (1994) 36293641.

[13] K. S. Kim, J. A. Hurtado, H. Tan, Evolution of a surface-roughnessspectrum caused by stress in nanometre-scale chemical etching, Phys.Review Let. 83 (3872-3875).

[14] H. H. Yu, Z. Sou, Stress-dependent surface reactions and implicationsfor a stress measurement technique, J. Appl. Phys. 87 (1211-1218).

[15] J. Grilhe, Surface instabilities and dislocation formation at the free sur-face of stressed solids, Europhys. Lett. 23 (1993) 141146.

[16] J. Grilhe, Study of roughness formation induced by homogeneous stressat the free surface of solids, Acta Metall. 41 (1993) 909913.

[17] J. Colin, J. Grilhe, Nonlinear effects of the stress driven rearrangementinstability of solid free surfaces, J. Elasticity. 77 (2005) 177185.

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[18] W. Song, F. Yang, J. Zhang, Stress-driven evolution of waviness in an

elastic layer, Mater. Sci. Eng. A409 (2005) 195205.

[19] F. Yang, Stress-induced surface instability of an elastic layer, Mech.Mater. 38 (2006) 111118.

[20] J. H. Prevost, T. J. Baker, J. Liang, Z. Sou, A finite element method forstress-assisted surface reaction and delayed fracture, Int. J. Sol. Struct.38 (2001) 51855203.

[21] A. P. Jivkov, P. Stahle, Strain-driven corrosion crack growth - A pilotstudy of intergranular stress corrosion cracking, Engng. Fracture Mech.

69 (2002) 20952111.[22] A. P. Jivkov, Strain-induced passivity breakdown in corrosion crack ini-

tiation, Theor. Appl. Fracture Mech. 42 (2004) 4352.

[23] P. Stahle, C. Bjerken, A. P. Jivkov, On dissolution driven crack growth,Int. J. Sol. Struct. 44 (2007) 18801890.

[24] Y. G. Shreter, D. V. Tarkin, S. A. Khorev, Y. T. Rebane, Instabilityof an elastically compressed silicon surface under etching, Phys. SolidState 42 (1999) 12951297.

[25] Y. G. Shreter, Y. T. Rebane, D. V. Tarkin, B. K. Barakhtin, V. V. Ry-bin, Kinetic mechanism of surface instability evolution during etching,corrosion, and growth of elastically stressed solids, Phys. Solid State 43(2001) 169175.

[26] N. Sardinha, R. P. Weber, J. C. M. Suarez, Toughness behavior ofgamma-irrated polycarbonate, Polymer Testing 26 (2007) 315355.

[27] J. Gunnars, P. Delfin, P. S. hle, Quasi-static curved crack paths in abi-material specimen obtained by chemical assisted fracture in polycar-bonate, Report Div. Sol. Mech. .

[28] J. Gunnars, P. S. hle, T. C. Wang, On crack path stability in a layeredmaterial, Comp. Mech. 19 (1997) 545552.

[29] U. Hejman, C. Bjerken, Branched cracks in a layered material - disso-lution driven crack growth in polycarbonate, Submitted for publication.

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[30] I. Ghorbel, F. Thominette, P. Spiteri, J. Verdu, Hydrolytic aging of

polycarbonate, I. Physical aspects, J. of Appl. Polymer Sci. 55 (1995)163171.

[31] N. Akele, F. Thominette, Physical ageing and water sorption in poly-carbonate, J. Mater. Sci. Lett. 15 (1996) 10011002.

[32] C. K. Liu, C. T. Hu, S. Lee, Effect of compression and thickness onacetone transport in polycarbonate, Polymer Eng. Sci. (2005) 687693.

[33] I. Ghorbel, F. Thominette, P. Spiteri, J. Verdu, Hydrolytic aging ofpolycarbonate, II. Hydrolysis kinetics, effect of static stresses, J. Appl.

Polymer Sci. 55 (1995) 173179.

[34] Abaqus, Abaqus users manual version 6.4, Abaqus Inc., New York,2003.

[35] S. Melin, Why do cracks avoid each other?, Int. J. Fract. 23 (1983)3445.

[36] B. Broberg, Cracks and fracture, Academic press, ISBN 0-12-134130-5,1999.

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Paper B

U. Hejman and Christina Bjerken

Branching cracks in a layered material -

Dissolution driven crack growth in polycarbonate

Submitted for publication.


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Branched cracks in a layered material Dissolution drivencrack growth in polycarbonate

U. HEJMAN1, C. BJERKN1,21

Div. Material Science, Malm University, Malm, SE-20506 Sweden,

2

Div. Materials Engineering, LundInstitute of Technology, P.O. Box 118, SE-22100 Lund, Sweden

Correspondence: U. Hejman, E-mail: [email protected]

ABSTRACTStress corrosion, in the form of chemically assisted crack growth, in polycarbonate isexamined with focus on crack branching characteristics. Cracks with finite width are

observed, could be expected for dissolution driven cracking. The cracks branched repeatedlyand crack widths before and after branching are measured. Both symmetric and asymmetric

branching is found. In the literature, it is proposed that the crack width is proportional to the

square of the mode I stress intensity factor. Energy considerations lead to that the sum of branch widths must equal the width of the unbranched crack. The results from the presentstudy correspond fairly well with this assumption. The branching angle is found to 32 12,

which is in line with results for scarp cracks reported in the literature. The mean growthdirection of the branches is found to deviate slightly from the expected straight. No significant

correlation between angles and crack widths is found. Also numerically simulated crackgrowth using a simple dissolution law is performed. These results show promising agreement

with the experiments. The scatter in results is mainly addressed to the inherent perturbationsensitivity of stress corrosion cracking.

Keywords stress corrosion cracking, polycarbonate, dissolution, branching, experiment, finite element

modelling

INTRODUCTION

Environmentally assisted crack growth is found in a variety of engineering materials.

Especially for metal alloys, stress corrosion cracking (SCC) is a common problem inengineering constructions. Stress corrosion (SC) cracks are normally initiated at surface

irregularities, such as flaws, grain boundaries and corrosion pits. At these sites, it can be hardto detect a growing crack due to its smallness, especially in the presence of general corrosion

on the surface. Stress corrosion cracking can cause failure well below the yield stress of thematerial. These conditions together with the lack of observable indication of an imminent

failure, may lead to unexpected fracture, and eventually cause personal injuries and costlystructural damage. Most research, in the field of SC, has been done to experimentally measure

stress crack propagation rates for both statically loaded specimens and for cyclic loading, cf.Ch. 1 and 14 in Jones

1. Lifetime predictions and evaluation of different material and

environment combinations are the most common interests. Even though a vast number ofstudies have been done during the last decades, there is still a need to undertake research to

understand the nature of the phenomenon, and how it can be prevented, cf. Ibrahim2.

The focus of most SCC studies has been metallic materials, but also ceramics and polymers

may exhibit SCC. The term SCC is here used for all types of environmentally assisted crackgrowth. The degradation of metals in corrosive environments is usually electro-chemical by

nature. For non-metallic materials, SCC is predominately chemically assisted, cf. Ch. 14 inJones1. Stress corrosion sometimes is divided into two categories; dissolution and


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embrittlement driven.3

In the latter case, a decrease in resistance to crack growth is due to

material degradation caused by the uptake of e.g. hydrogen.3,4

The mechanism of SCC considered here is active dissolution and removal of material at the

crack tip. For metallic materials, several mechanisms are proposed, e.g. rupture of a protectiveoxide film. For corrosion fatigue, it is believed that repeated dislocation interaction with the

crack surface results in film rupture and creation of fresh surfaces that is exposed to theenvironment, cf. Suresh

5. Jivkov and Sthle

6suggested a linear elastic model for which the

dissolution is a function of the straining of the surface with a threshold strain that reflects properties of the oxide film. Mass transportation by dissolution during etching of a

mechanically stressed material was considered by Kim et al.7

A stressed plane surface is anunstable configuration and perturbation of the surface will develop a waviness with a limited

frequency content. The phenomenon is often referred to as the ATG instability, cf. Asaro andTiller

8and Grinfeld

9, and it holds both for mass transformation through dissolution and

diffusion. The process is driven by the variation of the straining of the surface and surfaceenergy density that accompanies the irregularity, and neither a varying load nor plastic

deformation is needed for the mass transportation to proceed. In the case of dissolution,

depressions in the exposed surface are sites with larger strains, where the material corrodesfaster. This may lead to pitting and accelerated corrosion. The continuous transition fromsurface roughness via pitting to initial cracks (or deep pits) makes up the initiation phase of

SC crack growth. These cracks, both the initial and the developing, have finite tip radius dueto the dissolution of the material during the corrosion process. Sthle et al.

10showed that a

narrowing blunted crack cannot degenerate into a sharp crack, unless the crack growthmechanism is altered. Figure 1 shows a SC crack having a finite crack tip radius, and also

Creager and Rice11

show a picture of a SC crack with a blunt tip. The distance between crackflanks enables the transport of corrosion products and reactants to leave or reach the crack tip

area, respectively.It is also known that SC may promote branching and continued growth of the branches, also

in the absence of inertia effects. The most common example is that of branched transgranularcracks, but also branching in isotropic materials is observed as e.g. in the present study. A

more general description of SCC can be found in e.g. Jones1.

Fig. 1 Stress corrosion crack in a pressure vessel steel of type SA533C11. Crack length is 7 mm. Reproduced

with permission from Vattenfall AB, Sweden.

In this study, an experiment is presented, where the polymer polycarbonate (PC) is

subjected to a combination of constant load and the solvent acetone. Gunnars et al.12

performed the same type of experiments to study the path of non-branching cracks growing

towards an interface between the PC and a stiffer substrate. (The results from the study are

also presented in Gunnars et al.

13

.) They found that cracks did not follow straight paths, butdeflected gradually avoiding the interface. Several of the tests in their study gave branched


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cracks, but these were disregarded. In this study, the crack branching is of main interest which

motivates the use of the same method. Figure 2 shows a characteristic crack pattern withrepeated branching and cracks having a finite width.

Fig 2 SC crack pattern in a polycarbonate specimen; a) overview, and b) close-up with two branching points.

There are a number of calculations made for branched sharp cracks in the plane stress states.The usual aim is to calculate dynamic crack growth using quasi-static linear elastic fracture

mechanics. Stress intensity factors (SIFs) for branched sharp cracks have been determined forvarious crack branch configurations. Analytical as well as numerical methods have been

proposed by, e.g. Prletun14

,Kitagawa et al.15

, Vitek16

, Theocaris et al.17,18

, and Suresh andShih

19. For symmetric crack branching with equally long crack branches emanating from a

large crack, Prletun14

have calculated the most favourable crack initial dividing angle to 28,and the mode I SIF for a branch was a factor approximately equal 0.7, of the SIF of the

unbranched crack. The other researchers report similar results.The purpose of this study is to investigate branching characteristics of real SC cracks.

Branching angles and crack widths are measured. The dissolution mechanism is not known,but it is assumed that the dissolution rate is a linear function of strain along the crack surface.

Numerical simulations using this strain based evolution law are performed to study if theassumed mechanism is plausible. The obtained angles from both experiments and simulations

are compared with results for branched sharp cracks in order to investigate if Linear ElasticFracture Mechanics (LEFM) could be used also for SC cracks with a finite width. A

relationship between crack width before and after branching based on the preservation of totalenergy release rate is suggested and compared with results for real cracks. The possibility of

using the numerical method to find crack paths for sharp cracks is also discussed.


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EXPERIMENTAL METHOD

Stress corrosion crack growth in a polycarbonate plate is investigated. In Fig. 3, the setup isillustrated. An initial notch is introduced at the upper side of the specimen. The lower side is

fixed to a stiff substrate. The notch is symmetrically loaded by horizontal forces that are

applied near the notch mouth. Acetone is used as the dissolvent.

Fig. 3 Geometry, boundary conditions and loading of the specimen.

Material

Polycarbonate is an amorphous polymer, meaning that the microstructure shows no long-order structure as crystalline materials do. This implies that PC could be a suitable model for

isotropic, homogeneous materials. Polycarbonate is chosen, since it is a stiff materialcompared to other polymers, easily machined and prepared. The material is frequently used

for investigations of stress fields because of its photoelastic properties. The material used inthis investigation is an industrial-grade of MACROCLEAR polycarbonate (Arla plast AB,

Sweden), which is a transparent thermoelastic polymer with Youngs modulus of elasticity E= 2.3 GPa, Poisson ratio = 0.38 and tensile yield stress y= 63 MPa.

Acetone is used as a dissolving agent. It diffuses easily in polycarbonate and dissolves it ina hydrolysis process that is comparably fast, which is attractive when a large number of

experiments are to be made. In the hydrolysis, it is believed that the acetone dissolves the polycarbonate, by producing carbon dioxide and water. Some investigations of this

phenomenon have been made by Ghorbel et al.20,21

and for water by Akele22

. It has beenshown that static stresses in the polycarbonate drastically accelerate the hydrolysis, cf.

Ghorbel21

. The growing crack, in the present experiment, may easily be followed by the nakedeye. The propagation rate is in the order of 1 cm/s. Thus, it is assumed that inertia effects are

insignificant.

Specimen and loading

The polycarbonate plate is cut out from a sheet with the thickness t= 10 mm using a hacksaw.

The edges are polished to a reasonable smoothness. The material is extruded to a plate, so acertain degree of anisotropy could be anticipated. In this study, all specimens are cut in the

same direction and no investigation is made for different directions in the material. However,the experimental results do not indicate any influence from anisotropy.

The dimensions are shown in Fig. 3, where the width w=10t and the height h=5t. Acoordinate system is introduced at one of the corners of the plate, so that the plate covers the

region 0


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width oft/10. Two holes are drilled through the plate on each side of the notch at a distance

a0/2. Figure 3 shows the approximate position of the holes. The holes are used to apply a loadto the specimen.

An even and continuous flow of acetone to the crack tip is obtained by adding acetone by asmall paper strip introduced between the edges of the notch. Two opposed forces, F, are

applied parallel to the x direction at the holes in the plate as Fig. 3 shows. The load and theacetone are applied at the same time to achieve crack growth.

It is found that neither the loading used in the experiments, nor the application of acetone, ifapplied separately, will render cracking. An unloaded piece of PC is dissolved by acetone but

the polymer will swell and whiten. In the absence of acetone, the mechanical load must bemuch higher to give crack growth and it is also accompanied by plastic deformation. In the

experiments, the final fracture of the specimen gave crack surfaces that did not match. Thesynergetic effect of load and acetone gave crack surfaces with intact transparency and an

obvious loss of material along the crack, see e.g. Fig 2b.

Characterisation of crack patterns

The crack branches in the cracked specimen are photographed with an optical microscope.Because of plastic deformation caused by the final fracture, the crack will not fully closewhen the load is removed. A method is developed to account for this effect and remove the

parallax from the images of the specimens surface so that only the crack width due todissolution could be determined. The sides of the specimen are lightly scratched with fine

sandpaper prior to applying the load. With the help of an imaging software, the sides of thecracks are adjusted so hat continuous scratches pass across them. Thus, the dissolved part

could be identified. The procedure is shown in Fig. 4.The crack width and crack angles, defined in the Result section, are measured from the

fitted images. Since the crack branches repeatedly, a large number of branches can beanalysed in each specimen. All branching points in this investigation are treated equally.

Fig 4 Fitting of the cracked surface to eliminate rigid body motion . Straight scratches in two perpendicular

directions at the specimen surfaces are deliberately introduced before testing.

THEORETICAL CONSIDERATIONS

For small scale yielding, the following relation exists between the stress intensity factors and

the energy release rate, G:

, (1)

where E=E for plane stress and E=E/(1-2) for plane strain with E being the materialselastic modulus and Poissons ratio. When the crack branches, it is fair to assume that the


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total energy release rate is, at least initially, unchanged. If this is the case, the sum of the

branches energy release rates, will equal the energy release rate of the crack before branching. This gives the following relation for the stress intensity factors before,KI0, and

after branching, KIi, respectively, fori=1:n with n being the number of branches.

, (2)

It is known that sharp cracks are prone to follow a mode I path, cf. Broberg23

, meaning thata crack will follow a path whereKII=0. With two branches and vanishingKII, Eq. (2) reduces

to

. (3)

For two symmetric branches Prletun14

, Suresh and Shih19

, and Theocaris17,18

reported a

decrease in SIF to approximately 0.7 of the main crack. This is in line with Eq. (3), where the

stress intensity factor scales with the square, and thus renders a decrease of

In the experiments and the numerical analysis by Gunnars et al.12,13

, which are similar to

those performed in the present study, dissolution driven SC cracks were found to grow predominantly along a mode I path. Here, as discussed in the Introduction section, the

dissolution is assumed to be controlled by the straining of the crack surface. No otherassumption is made, except that the surface will dissolve where the strains s along the crack

surface exceed a threshold value, th. With a curvilinear coordinate s following the crack

surface, the crack growth rate in the normal direction to the surface, vn(s) is determined by

, (4)

where C depends on the chemical and mechanical properties of the environment and thematerial and is here considered constant. This surface evolution law is earlier suggested by

Jivkov and Sthle6

and used in their simulation of SC crack growth cracks due to oxide filmrupture during fatigue loading. The degree of film rupture was quantified by the surface

elastic straining of the surface and assumed to cause dissolution proportionally to this. In theirmodel, the very thin film was considered not to influence the mechanical behaviour of the

material. In this study, no protective film is formed and the remote load is kept constant.However, the same evolution law is chosen, since it is very simple, and it is reasonably to

think that the dissolution is larger at locations with large strains.

It is observed that the shape of the crack front was close to half-circular. Figure 5 shows thecrack shape obtained from a simulation in the present study. Half-circular tips are also seenfor real cracks, see e.g. Fig. 2 and in Creager and Rice

11. An idealization of dissolution driven

crack, which is subjected to a constant KI field during growth, is a long notch with parallelflanks and a front with the shape of a half-circle having a radius that equals the half crackwidth L/2, see Fig. 6. During steady-state growth, the shape of the crack front must beunchanged, which requires that the dissolution rate, vtip, is constant in the crack growth

direction at all points of the crack surface. It was earlier shown by Sthle et al10

., that

(5)


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For a deep notch, the largest stress at the crack tip is proportional to , cf. Tada et

al.24

. Since equals L/2, is proportional to L and the surface strain to the stress, a

constant growth rate results that depends only of the constant th. For self-similar crack

growth, this means that for increasing KI the crack will broaden and vice versa, instead of

accelerating or slowing down. This is in line with what was discussed by Carter

25

. Jivkov andSthle6

found that after an initial increase in growth rate, the SC crack reached steady-state

where the propagation rate is constant and insensitive to the load. For real stress corrosioncracks, typically a period of constant crack growth is observed, cf. Ch. 1 in Jones

1.

Fig. 5 The tip of the simulated growth of a crack branch for two successive time steps. Note that the crack

surfaces are slightly swaying and the crack front is close to half-circular.

Fig. 6 Tip of an idealized crack showing steady-state growth.

From Eqs. (3) and (5) it is also concluded that the width of the crack before branching

equals the sum of the branch widths after, l1 and l2, i.e.

(6)

For a case with symmetric branching, each branch has half the width of the main crack. The

sudden change of the magnitude of stress intensity factor at a crack tip affects the state of thecrack growth for sharp cracks. For dissolution driven cracks the growth rate is constant

according to the reasoning above. Figure 2b shows that the experiments can give asymmetric


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branching. Despite the fact that one crack is thinner than the other, both branches continue to

propagate even ifKI is not equal for the branches. This phenomenon is typical for stresscorrosion crack growth, and is in line with the reasoning above. Since the crack patterns

obtained in the experiments were analysed after final fracture, the crack growth rates are notmeasured.

Different criteria for branching have been suggested in the literature. Carter25

states thatconstant crack velocity and a critical stress intensity must be reached before branching can

occur. For high-strength steels, he found that the critical value is about two to four times thethreshold stress intensity below which SCC does not occur. Jivkov and Sthle

6found that the

degree of bluntness of a dissolution driven crack determined the possibility of crack branching. Even if the criteria are fulfilled, the microstructure and other local differences

strongly influence when branching actually will take place in a real material.

SIMULATIONS

A method to numerically simulate dissolution driven crack growth, where the

Crack GrowthPC

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Transcript of Crack GrowthPC