UNIVERSITI PUTRA MALAYSIA FATIGUE CRACK …psasir.upm.edu.my/7320/1/FK_2009_9a.pdf · the linear material is ... keputusan tersebut diplotkan dalam graf tegasan ... faktor kekuatan

UNIVERSITI PUTRA MALAYSIA

FATIGUE CRACK PROPAGATION IN ALUMINIUM 6063 TUBES

AZIM ATAOLLAHI OSHKOUR

FK 2009 9


By


Thesis Submitted to the School of Graduate Studies, Universiti Putra Malaysia,

in Fulfilment of the Requirement for the Degree of Master of Science

January 2009

ii

To

My Beloved Parents, Sisters and Brother

iii

Abstract of thesis presented to the Senate of Universiti Putra Malaysia in fulfilment

of the requirement for the degree of Master of Science


By


January 2009

Chairman: Prof. Ir. Barkawi b. Sahari, PhD

Faculty: Engineering

Tubular structures have extensive usage from domestic to aviation. Therefore

estimate of life and safety are essential for design and use. Fatigue is one the most

frequent cause for failure in components. Beside fatigue, external surfaces of

structures are always in contact with environment and due to imperfection during

product fabrication, surface crack may exist. Therefore surface crack is the most

common form of crack in engineering structures. To overcome the fatigue problem

the design approaches should be considered. The fatigue design approaches are

divided in two categories which are safe life approach and damage tolerant design.

Due to the importance of tubular structure and possible effect of fatigue in structural

damage, the present work has focused on fatigue and fatigue crack propagation

behavior in cylindrical structures.

The fatigue design approaches utilized stress based safe life and damage tolerant

design approach. The finite element software, ABAQUS used to analyze fatigue and

determine fracture parameters. At the beginning fatigue test in finite life was carried

out based on Japanese standard. The fatigue tests were done at room temperature and

iv

about 350°C under stress ratio equal to -1 and 0.1. Following experimental part, a

3-D fatigue analysis was carried out by ABAQUS. In fatigue analysis by ABAQUS,

the linear material is considered and the results of finite element analysis are plotted

in maximum stress versus number of cycles to failure graph. Fatigue analysis was

carried out in same condition as experimental part at room temperature and 350°C

under stress ratio equal to -1 and 0.1. Subsequent to fatigue analysis, the fatigue

crack propagation tests were also carried out. The fatigue crack propagation test was

carried out under increasing stress intensity factor or constant amplitude stress. In

fatigue crack propagation test, a cracked tubular specimen was used. The crack is

located in specimen by wire cut. The crack was an external and circumferential with

straight front with depth of 0.37mm. The results of fatigue crack propagation were

plotted in two types of graphs; first is crack length ,a, versus the number of cycles, N

at each crack length and second is the crack growth rate which plotted as function of

rate of crack length upon the number of cycles, da/dN versus the stress intensity

factor as fracture parameter. Moreover ABAQUS was used to derive the fracture

parameters. Two types 3-D tubes with crack were modeled. In first model assumed

as sharp and thin crack, but in the second type the blunt crack considered. Material of

tube in ABAQUS assumed to be a linear elastic and elastic prefect plastic. The

results of crack modeling include fracture parameters as stress intensity factor, and J-

integral which were plotted as a function of crack front.

The experimental results of fatigue showed a good agreement with finite element

fatigue results. Based on fatigue results the fully reversed fatigue is more severe than

fatigue with stress ratio equal to 0.1. Temperature does affect fatigue life which is

shown by a decrease in yield strength and ultimate strength of material which

v

resulted in reduction in the fatigue life of specimens with increasing temperature.

The fatigue crack propagation results indicated crack growth rate in loading with

stress ratio equal to 0.1 is more than stress ratio equal to -1. Crack first grew through

the thickness followed by the surface of specimen. This was verified by the results

from finite element that show maximum fracture parameters in the deepest point of

the crack.

vi

Abstrak tesis yang dikemukakan kepada Senat Universiti Putra Malaysia

sebagai memenuhi keperluan untuk ijazah Master Sains

PERTAMBAHAN KERETAKAN LSUE DALAM TIUB ALUMINIUM

Oleh


Januari 2009

Pengerusi: Prof. Ir. Barkawi b. Sahari, PhD

Fakulti: Kejuruteraan

Kegunaan sturktur berbentuk tiub begitu luas, dari kegunaan rumah hingga bidang

penerbangan. Oleh hal yang demikian, anggaran/kiraan jangka hayat dan

keselamatan adalah penting dalam merekabentuk dan kegunaannya. Factor kelesuan

adalah salah satu penymbang kepada kegagalan sesuatu komponen. Di samping itu

permukaan luar yang sentiasa terdedah kepada persekitaran dan kesan dari fakbrikasi

yang kurang memuaskan, keretakaan permukan boleh wujud. Dalam struktur

kejuruteraan, retakan permukaan merupakan bentuk keretakan yang biasa wujud.

Untuk mengatasi masalah kelesuan ini, pendekatan kelesuan rekabentuk perlu

diambil kira. Pendekatan kelesuan rekabentuk terbahagi kepada dua kategori, iaitu

pendekatan jangkahayat selamat dan ketahanan rekabentuk retak. Oleh yang

demikian, kerja penyelidikan ini tertumpu kepada ujian kelesuan dan kelakuan lesu

dengan penambahan retakan dalam struktur silinder.

Pendekatan kelesuan rekabentuk menggunakan tegasan jangkahayat selamat dan

pendekatan ketahanan rekebentuk di lakukan. ABAQUS merupakan satu perisian

unsur terhingga untuk menganalisis kelesuan struktur dan penghasilan parameter

vii

retakan. Pada permulaan kagian, kaedah terhingga ini berdasarkan Japanese

Standard. Ujian kelesuan telah dijalankan dalam suhu bilik dan pada 350°C di bawah

nisbah tegasan bersamaan dengan -1 dan 1.0.

Untuk ujian yang seterusnya, analisis lesu 3-D dilaksanakan dengan ABAQUS.

Analisis kelesuan dengan perisian unsur terhingga ini, bahan linar diambil kira dan

keputusan tersebut diplotkan dalam graf tegasan maksimum hingga gagal terhadap

bilangan kitaran. Ujian kelesuan dijalankan dengan keadaan yang sama dalam suhu

bilik dan 350°C di bawah nisbah tegasan bersamaan dengan -1 dan 1.0. Setelah ujian

kelesuan dijalankan, ujian kelakuan lesu dengan penambahan retakan dijalankan.

Ujian ini dijalankan di bawah faktor tumpuan tegasan atau amplitud tegasan

berterusan. Dalam ujian kelakuan lesu dengan penambahan retakan, spesimen tiub

bertakuk di buat dengan mesin Wire Cut. Tiub tersebut dilekuk sedalam 0.37mm di

permukaan luar tiub. Ujian kelakuan lesu dengan penambahan retakan ini diplotkan

kepada 2 jenis graf; panjang retakan terhadap jumlah kitaran di setiap panjang

retakan dan berikutnya adalah kadar penambahan keretakan diplotkan sebagai fungsi

kadar panjang retakan keatas bilangan kitaran melawan faktor tegasan sebagai

parameter keretakan. Disamping itu ABAQUS digunakan untuk memperolehi

parameter retakan. Dua jenis tiub 3-D dengan retakan dimodelkan. Model pertama

menganggapkan retakan tajam dan nipis. Model kedua menganggapkan retakan

tumpul. Bahan tiub dalam ABAQUS memperkenalkan kengal linar dan kengal linar

platci unggul. Keputusan model retakan mengandungi parameter retakan sebagai

faktor kekuatan tegasan dan J-intergral yang diplotkan sebagai suatu fungsi

kepelbaaian retakan hadapan.

viii

Keputusan ujian kelesuan dan kelesuan unsur terhingga menunjukkan hwdungan

yang baik. Berdasarkan ujian kelesuan, ujian berbalikan penuh lebih

menyumbangkan tegasan berbanding nisbah tegasan bersamaan dengan 0.1. Kesan

peningkatan suhu jangkahayat menunjiukkan peningkatan suhu mengurangkan

kekuatan dan takat alah bahan tersebut. Keputusan penambahan retakan lesu

menunjukkan penambahan kadar retakan dalam bebanan beserta nisbah tegasan

bersamaan dengan 0.1 adalah lebih berbanding nisbah tegasan bersamaan dengan -1.

Pembesaran retakan bermula dari ketebalan seterusnya ke atas permukaan spesimen.

Ini dibuktikan melalui keputusan unsur terhingga yang menunjunkkan parameter

retakan maksimum berlaku dalam kedalaman yang paling dalam daripada retakan

tersebut.

ix

ACKNOWLEDGEMENTS

First and foremost I would like to express my deep gratefulness to my supervisor

Professor Dr. Ir. Barkawi b. Sahari providing me the wonderful opportunity to

continue my master’s program. I am also very thankful for his tireless, kind

assistance, support, critical advice, encouragement and suggestions during the study

and preparation of this thesis.

I would also like to express my gratitude towards my members of my supervisory

committee Dr. Aidy Ali and Associate Professor Dr. Wong Shaw Voon, whom I am

grateful for their practical experience and knowledge that made an invaluable

contribution to this thesis. I truly appreciate the time they devoted in advising me and

showing me the proper directions to continue this research and for their openness,

honesty and sincerity.

I owe thank to Mr. Wildan Ilyas Mohd Ghazali, technician in Strength of Material

Laboratory, Department of Mechanical and Manufacturing Universiti Putra Malaysia

who has always been willing to provide assistance, give advice, and be a friend.

Last but not least the deepest appreciation goes to my very close friends, Hossin,

Mehdi, Ali and Hamid and for their contentious support and encouragement.

x

I certify that an Examination Committee has met on date of viva to conduct the final

examination of Azim Ataollahi Oshkour on his Master of Science entitled "Fatigue

and Crack Propagation in Aluminium 6063 Tubes" in accordance with Universiti

Pertanian Malaysia (Higher Degree) Act 1980 and Universiti Pertanian Malaysia

(Higher Degree) Regulations 1981. The Committee recommends that the candidate

be awarded the relevant degree. Members of the Examination Committee are as

follows:

Chairman, PhD Associate Prof. Ir. Dr. Nor Mariah Adam

Faculty of Engineering

Universiti Putra Malaysia

(Chairman)

Examiner 1, PhD Prof. Ir. Dr. Mohd Sapuan Salit



(Internal Examiner)

Examiner 2, PhD Dr. Abdul Aziz Jafar



(Internal Examiner)

External Examiner, PhD Associate Prof. Ir. Dr. Shahrum bin Abdullah


Universiti Kebangsaan Malaysia

(External Examiner)

HASANAH MOHD GHAZALI, PhD

Professor/Deputy Dean

School of Graduate Studies


Date:

xi

This thesis was submitted to the Senate of Universiti Putra Malaysia and has been

accepted as fulfilment of the requirement for the degree of Master of Science. The

members of the Supervisory Committee were as follows:

Barkawi b. Sahari, PhD Professor



(Chairman)

Aidy Ali, PhD

Lecturer



(Member)

Wong Shaw Voon, PhD Associate Professor



(Member)

HASANAH MOHD GHAZALI, PhD Professor and Dean

School of Graduate Studies


Date: 14 May 2009

xii

I hereby declare that the thesis is based on my original work except for quotations

and citations which have been duly acknowledged. I also declare that it has not been

previously, and is not concurrently, submitted for any other degree at Universiti

Putra Malaysia or at any other institution

Azim Ataollahi Oshkour

Date: 1 April 2009

xiii

TABLE OF CONTENTS

Page

ABSTRACT iii

ABSTRAK vi

ACKNOWLEDGMENT ix

APPROVAL x

DECLARATION xii

LIST IF TABLES xiii

LIST OF FIGURES xiv

LIST OF ABBREVIATIONS xix

CHAPTER

1

1 INTRODUCTION

2 LITERATURE REVIEW

2.1 Introduction 5

2.2 Material Behaviour 5

2.2.1 Hooke’s Law 6

2.2.2 Failure Criteria 7

2.3 Fatigue 9

2.3.1 Mechanism of Fatigue Damage 10

2.3.2 Stress Based Approach 13

2.3.3 Strain Based Approach 15

2.3.4 Mean Stress Models on Fatigue Life 18

2.3.5 Fracture Mechanics Base Approach 22

2.3.5.1 Fracture Mechanics Approach and

Fatigue Crack Propagation Analysis

22

2.3.5.2 Fracture Mechanics 27

2.4 Numerical Evaluation J-integral 35

2.4.1 J-Integral in 2D 35

2.4.2 J-Integral in 3D 39

2.5 Tubes and Cracks 42

2.6 Discussion 48

3 METHODOLOGY

3.1 Introduction 49

3.2 Finite Element Analysis 50

3.2.1 Finite Element Modelling of Evaluation Fracture

Parameter

50

3.2.2 Construction of 3-D Singular Elements at the

Crack Tip

51

3.2.3 Validation of Finite Element Model 53

3.2.4 Model Geometries 55

3.2.5 Other Model Attributes 56

3.2.6 Material Model 56

3.2.7 Boundary Conditions 56

3.2.8 Fatigue Finite Element Analysis 57

xiv

3.2.9 Model Geometries 58

3.2.10 Mesh Generation 59

3.2.11 Other Model Attributes 59

3.3 Experimental Procedure 60

3.3.1 Test Material - Composition and Properties 60

3.3.2 Test Specimens Preparation 62

3.3.3 Static Test Specimens 62

3.3.4 Fatigue Test Specimens 63

3.3.5 Fatigue and Fatigue Crack Growth Experiments 64

3.3.6 Crack Length Measurement 66

3.3.7 High Temperature Environment and Controlling 67

3.4 Discussion 68

4 Results and discussion

4.1 Introduction 69

4.2 Static Tension Properties 69

4.3 Fatigue Test at Room Temperature with R=-1 72

4.4 Fatigue Test at Room Temperature with R=0.1 75

4.5 Fatigue Test at 350°C with R=-1 78

4.6 Fatigue Test at 350°C with R=0.1 79

4.7 Tube Modelling 83

4.8 Fatigue Crack Propagation Test with R=-1 and Maximum

Load 10kN

95

4.9 Fatigue Crack Propagation Test with R=0.1 and Maximum

Load 10kN

98

4.10 Fatigue Crack Propagation test with R=-1 and Maximum

Load 15kN

101

4.11 Fatigue Crack Propagation test with R=0.1 and Maximum

Load 15kN

103

4.12 Summary 105

5 Conclusion and Recommendations

5.1 Conclusion 108

5.2 Recommendations 112

REFERENCES 114

BIODATA OF THE STUDENT 119

LIST OF TABLES

Table Pages

1 Basic material mechanical property 6

2 Classification of various types of fatigue analysis (Schijve,

2004 )

9

3 Characteristics of the three regimes of fatigue crack

propagation

23

4 Some type of defect in tube 43

5 Typical chemical composition of 6063 60

6 Fatigue and Fatigue crack growth test plan 66

7 Average static mechanical properties of specimens at room

temperature and 350°C

72

8 Exponential equation parameters used in fit curve with crack

propagation under maximum load equal to 10kN with R=-1

97


propagation under maximum load equal to 10kN with R=0.1

99


propagation under maximum load equal to 15kN with R=-1

101


propagation under maximum load equal to 15kNwith R=0.1

103

xvi

LIST OF FIGURES

Figure Pages

1 A typical stress-strain curve (Martin, 2005) 6

2 Comparing failure criteria (Dieter, 1986) 8

3 Schematic of the slip band formation in a single grain. The

external loading is acting vertically, in tension-compression

cycling (Yung, Jwo, Richard, and Mark, 2005)

13

4 The stage I and stage II crack growth processes in a

polycrystalline material (Yung et al., 2005)

13

5 Typical a fNσ − curve plotted on log-log axes (Yung et al.,

2005)

14

6 Typical −ap f

Nε curve plotted on log-log axes (Yung et al.,

2005)

16

7 Comparison of −ae f

Nε , −ap f

Nε and −a f

Nε (Yung et al.,

2005)

17

8 Nomenclature for fatigue stress designators (Jung, 2002) 18

9 Typical stress amplitude-life plots for different mean stress

values (Choi, 2002)

20

10 A plot representing the Goodman, Soderberg, and Gerber

relations for the effect of mean stress on the stress amplitude

for fatigue failure (Choi, 2002)

20

11 A schematic representation of the Haigh diagram showing

constant life curves for different mean stress levels in terms of

the maximum and minimum stresses of the fatigue cycle

(Suresh, 1991)

21

12 Principle propagation behavior of a long crack and Paris’

idealization (Anderson, 1994)

25

13 Strain energy release rate: (a) crack growth da due to applied

load P ; (b) Strain energy released dU with crack growth da

(Anderson, 1994)

29

14 Typical mode of fracture (mode of loading) (Anderson, 1994) 30

15 Crack tip opening displacement (CTOD) Model (Dowling,

1999)

33

xvii

16 Application of J-integral approach to crack growth (Saxena,

1992).

34

17 Contour for evaluation of the J-integral 36

18 Closed contour C C C+ −+ + Γ + encloses a domain A that

includes the crack-tip region as 0Γ →

37

19 Definition of local orthogonal Cartesian coordinates at the

point s on the crack front; the crack is in the 1 3x x− plane

39

20 Surface t o ends cracksA A A A A= + + + encloses a domain

volume V that includes the crack-front region as 0Γ →

41

21 Geometry of an external circumferential crack in a hollow

cylinder (Varfolomeyev, 1998)

44

22 Circumferential external surface flaw in a round pipe(A.

Carpinteri and Brighenti, 1998)

45

23 External surface 2aw in a circumferentially notched pipe:

geometrical parameters (Andrea Carpinteri et al., 2003)

47

24 Internal and external semi-elliptical surface cracks in a

pressurized cylinder (Guozhong et al., 2004)

47

25 General description of the methodology adopted in the study 50

26 20 nodes brick element collapse into wage elastic singular

element

52

27 Flow of making the general finite element model 53

28 Three dimensional geometric of model 54

29 Meshing around crack front in 3D model 55

30 Boundary condition applied on model 57

31 Flow of making the general fatigue finite element model 58

32 Geometry of model in fatigue analysis 59

33 Meshing of model 59

34 Tensile test Specimen 63

35 Tensile test Specimen 63

xviii

36 Fatigue specimen 64

37 Fatigue specimen 64

38 Front view of crack 65

39 Side view of crack 65

40 Microscopy video based on rigid platform 66

41 Graded Specimen with crack 67

42 Crack propagation monitoring 67

43 View of furnace 68

44 Stress-strain behaviour of Aluminium 6063 in tension at room

temperature

70

45 Stress-strain behaviour of Aluminium 6063 in tension at

350°C

71

46 Comparison of static behaviour Aluminium 6063 at room

temperature and 350°C

71

47 Stress ratio 73

48 Stress amplitude 73

49 Fully reversed fatigue S-N Curve at room temperature 73

50 Fully reversed fatigue S-N Curve at room temperature in log-

log scale

74

51 Fatigue S-N curve at room temperature for R=0.1 75

52 Fatigue S-N curve at room temperature for R=0.1 in log-log

scale

76

53 Comparing log-log scale fatigue in room temperature with R =

-1 and 0.1

77

54 A fully reversed fatigue S-N Curve at 350°C 78

55 A fully reversed fatigue S-N Curve at 350°C in log-log scale 79

56 A fatigue S-N curve at 350°C for R=0.1 80

57 A fatigue S-N curve at 350°C for R=0.1 in log-log scale 80

xix

58 Comparing log-log scale fatigue at 350°C with R equal to -1

and 0.1

81

59 Comparing log-log scale fatigue at room temperature and

350°C with R= -1 and 0.1

82

60 3D-Crack surface circumferential crack with straight front

modelling in ABAQUS for tube

84

61 3D-Crack surface circumferential crack with straight front

modelling in ABAQUS for tube

85

62 x

σ at crack front 85

63 y


64 z


65 xy

τ at crack front 87

66 xz

τ direction at crack front 88

67 yz

τ at crack front 89

68 Variation of Min J-integral along the crack front under

107.48MPa

89

69 Variation of maximum J-integral along the crack front under

107.48MPa

90

70 Comparing Minimum J-integral 91

71 Comparing Maximum J-integral 91

72 Variation of stress intensity factor along the crack front under

107.48MPa

92

73 Variation of stress intensity factor along the crack front under

205.95MPa

93

74 Comparing minimum stress intensity factor under different

load

94

75 Comparing maximum stress intensity factor under different

load.

94

xx

76 Fully reversed fatigue crack propagation with amplitude stress

equal to 10kN

96

77 The variation of da/dN versus maximum stress intensity factor

(Kmax) for fully reversed fatigue crack propagation with

amplitude stress equal to 10kN

97

78 Fatigue crack propagation with amplitude stress equal to 10kN

and stress ratio equal to 0.1

98


(Kmax) for fatigue crack propagation with amplitude stress

equal to 10kN with R=0.1

99

80 Comparing fatigue crack growth with R=0.1 and R=-1 under

10kN maximum stress

100

81 Comparing da/dN versus maximum stress intensity factor

(Kmax) under 10kN and load ratio equal to 0.1 & -1

100

82 Fully reversed fatigue crack propagation with amplitude stress

equal to 15kN

102


(Kmax) for fully reversed fatigue crack propagation with

amplitude stress equal to 15kN

102

84 Fatigue crack propagation with amplitude stress equal to

161.96MPa and stress ratio equal to 0.1

103


(Kmax) for fatigue crack propagation with maximum load equal

to 15kN with R=0.1

104

86 Comparing fatigue crack growth with R=0.1 and R=-1 under

15kN maximum load

105

87 Comparing da/dN versus maximum stress intensity factor

(Kmax) under 15kN and load ratio equal to 0.1 & -1

105

xxi

LIST OF ABBREVIATIONS

2D Two dimensional

3D Three dimensional

CHAPTER 1

INTRODUCTION

1.1 Introduction

Many critical mechanical components experience cyclic loadings during their service

life. Fundamental understanding of fatigue problem is important for the reliability

assessment under realistic service condition and is valuable for the design and

maintenance versus fatigue failure. Fatigue life prediction is one of the oldest

problems which concern scientists and engineers. Due to the complicated nature of

fatigue mechanisms and the large number of factors that influence fatigue life, there

is so far no unified approach that can treat all fatigue problems (Weixing, 1993). It

has been estimated that as large as ninety percent of all engineering structural and

component failures may be attributed to fatigue failure (Dieter, 1986).

The well known approaches which concern fatigue and fatigue crack propagation

analysis can be categorized in two groups. These include damage tolerant design and

safe life approaches. Based on damage tolerant design, structures are designed to

allow limited crack propagation so that growing cracks can be located and remedied.

The safe life approaches are such as stress base or strain based approach. Damage

tolerance is based on the likelihood of finding cracks and their estimated propagation

rates (Lawson, Chen, and Meshii, 1999).

2

The accuracy of estimated life and safety of engineering components which in oil,

gas, aviation and nuclear industries requires good understanding of fatigue property

such as fatigue crack propagation rate (da/dN) and fatigue strength curve (S-N) is

essential. In the case of existing crack in structure, damage tolerant design can be

used to obtain accurate crack growth data. It mainly estimates the residual fatigue life

of cracked structures on the basis of known crack propagation rates. To carry out this

assessment, it is necessary to have reliable fracture parameters such as stress

intensity factor and J-integral. There are two ways for calculating fracture

parameters, namely analytical and finite element methods. By using finite element

methods, it provides time and commercial benefit. Hence fracture parameters can be

derived by carrying out crack simulation on two dimensional or three dimensional

numerical modeling. Following which the finite element model was developed for

calculating fracture parameters for cracks. Based on crack growth rate (da/dN) data

obtained from fatigue tests and the results of finite element analysis, fatigue crack

growth behavior can be predicted.

1.2 Problem Statement

Cylindrical components such as pressure vessels, pipes, borers, and driving shafts are

commonly used parts of engineering structure. Due to a wide range of usage of this

type of structures such as transmission or storage of fluid as in pipes and high

pressure vessels, it is necessary that these structures be assessed with different

conditions such as materials, temperature and loading. Fatigue caused a structure to

fail and it is known to be one of the major reasons of failure in engineering

structures. Some of the failures by fatigue happened suddenly and unexpectedly at

stresses lower than their design strength, resulting in extensive property damage and

3

loss of life. A lot of efforts have been done to understand why and how materials fail

by fatigue and what are the main parameters and factors in fatigue. In the present

days these economic and life cost of failure and fracture become greater due to

structures become more complexes. Therefore it required special consideration on

safety and life prediction. Based on the importance of fatigue and tubular structures,

the present work focused on analysis and investigating fatigue and fatigue crack

propagation this structures.

1.3 Research Objectives

The objectives of this study are:

1. To determine the fatigue behavior of Aluminum 6063 tube at room and

350°C by using of experimental and finite element methods.

2. To determine the parameters of fracture in aluminum tubes by using of finite

element methods.

3. To determine the mechanisms of fracture associated with fatigue and develop

a model to predict crack propagation length versus number of cycles.

1.4 Thesis Layout

This thesis is divided into six chapters. Following this introductory chapter, chapter

two gives a critical review of relevant literature, from overview of fatigue, creep and

fracture mechanics. Chapter three outlines the underlying theory as well as

Documents

UNIVERSITI PUTRA MALAYSIA FATIGUE CRACK …psasir.upm.edu.my/7320/1/FK_2009_9a.pdf · the linear material is ... keputusan tersebut diplotkan dalam graf tegasan ... faktor kekuatan