STRESS INTENSITY FACTORS OF CIRCUMFERENTIAL SEMI ...library.umac.mo/etheses/b2182941x_ft.pdf · STRESS INTENSITY FACTORS OF CIRCUMFERENTIAL SEMI-ELLIPTICAL INTERNAL SURFACE CRACKS

STRESS INTENSITY FACTORS OF

CIRCUMFERENTIAL SEMI-ELLIPTICAL

INTERNAL SURFACE CRACKS OF TUBULAR

MEMBER SUBJECTED TO AXIAL TENSILE

LOADING

by

YANG YANG

Master of Science in Civil Engineering

2010

Faculty of Science and Technology

University of Macau

ii

STRESS INTENSITY FACTORS OF CIRCUMFERENTIAL

SEMI-ELLIPTICAL INTERNAL SURFACE CRACKS OF

TUBULAR MEMBER SUBJECTED TO AXIAL TENSILE

LOADING

by

YANG YANG

A thesis submitted in partial fulfillment of the

requirements for the degree of

Master of Science in Civil Engineering


University of Macau

2010

Approved by __________________________________________________

Assistant Professor Lam Chi Chiu

Supervisor

__________________________________________________

Associate Professor Kou Kun Pang

Co-Supervisor

__________________________________________________

Associate Professor Yuen Ka Veng

Examining Committee

__________________________________________________

Associate Professor Er Guokang

Examining Committee

Date __________________________________________________________

iii

In presenting this thesis in partial fulfillment of the requirements for a Master's

degree at the University of Macau, I agree that the Library and the Faculty of

Science and Technology shall make its copies freely available for inspection.

However, reproduction of this thesis for any purposes or by any means shall not

be allowed without my written permission. Authorization is sought by

contacting the author at

Address: Room NLG101, Choi Kai Yau Building


University of Macau

Taipa

Macau.

Telephone: 00853-62661905

E-mail: [email protected]

Signature ______________________

Date __________________________

iv

University of Macau

Abstract

STRESS INTENSITY FACTORS OF CIRCUMFERENTIAL

SEMI-ELLIPTICAL INTERNAL SURFACE CRACKS OF

TUBULAR MEMBER SUBJECTED TO AXIAL TENSILE

LOADING

by

Yang Yang

Thesis Supervisor: Assistant Professor Lam Chi Chiu

Associate Professor Kou Kun Pang

Geotechnical and Structural Engineering

ABSTRACT

For two dimensional problems such as a through thickness crack, reliable solutions for

stress intensity factor have been reported by many literatures. However, in practice, the

common flaws in many structural members are surface cracks which may propagate to

part through cracks under repeated loading. These two categories of crack are three

dimensional. Exact solution of stress intensity factors for these cracks is not available

due to the complexity of the problem itself. Reliable computational solutions for stress

intensity factors of surface cracks have been reported. However, all serious solutions

have a limited range of validity for the crack depth and crack length. For the part

through crack no solutions have been found in the literature. For investigating the

detailed process of crack growth from surface crack to part through crack, solutions for

stress intensity factor are necessary.

For cylindrical structural components, surface flaws can appear as internal or

external semi-elliptical cracks, in the axial or circumferential direction. The form of

v

flaw being studied in the current work is an internal circumferential semi-elliptical

surface crack in a tubular member. Resulting from improper welding, this kind of

surface crack can occur in tubes, pipes and pressure vessels. To assess the structural

integrity and to predict the fracture strength of such components, determination of the

stress intensity factor is one of the most vital factors.

In this thesis, stress intensity factors for a wide range of long-deep

circumferential semi-elliptical internal surface cracks in tubular members are

presented. The crack configurations in the tubular members are subjected to axial

tension loading and the stress intensity factors (SIFs) were analyzed by considering

the following three main parameters: (1) the crack depth to thickness ratio (a/T), (2)

the outer radius to thickness ratio (R/T) and (3) the crack length to tube circumference

ratio (c/R). For a/T < 0.8, current finite element results compared well with the

results reported in literatures. In order to investigate the detailed process of crack

growth from surface crack to part through crack, it is necessary to have the accurate SIF

for the a/T > 0.8. Therefore, current finite element analysis was extended to investigate

the SIFs by including a wider range of a/T ratio up to 0.99. Finite element analyses of

cracked tubes have also been carried out to determine the stress intensity factors along

the semi-elliptical crack fronts of part through thickness cracks and through thickness

cracks. The range of the crack geometries covered in this study has not been reported

previously in the literatures. The relationships between the stress intensity factors and

the crack configurations such as crack depth ratio and aspect ratio and the size of the

tube have been established. In order to examine the effect of material plasticity on the

effect of crack tip deformation of long-deep circumferential semi-elliptical internal

surface cracks in tubular members, non-linear finite element analyses were carried out

to study the crack deformation as well as the corresponding J-integral value of tubular

members with long-deep circumferential semi-elliptical internal surface cracks. Then

neural network of MATLAB was used to process those finite element analysis results

and suitable equations for predicting the SIFs were proposed.

vi

TABLE OF CONTENTS

ABSTRACT ........................................................................................................................... IV

TABLE OF CONTENTS ..................................................................................................... VI

LIST OF FIGURES ........................................................................................................... VIII

LIST OF TABLES ............................................................................................................... XV

ACKNOWLEDGMENTS ................................................................................................ XVII

CHAPTER 1: INTRODUCTION AND BACKGROUND ................................................... 1

1.1 INTRODUCTION.................................................................................................................. 1

1.2 BACKGROUND ................................................................................................................... 3

1.3 OBJECTS AND SCOPE OF THE WORK .................................................................................. 6

1.4 ORGANIZATION OF THE THESIS ......................................................................................... 8

REFERENCE ......................................................................................................................... 12

CHAPTER 2: LITERATURE REVIEW ON FRACTURE MECHANICS .................... 14

2.1 BACKGROUND ................................................................................................................. 14

2.2 LINEAR ELASTIC FRACTURE MECHANICS ....................................................................... 17

2.2.1 Stress intensity factor .............................................................................................. 17

2.2.2 Plane stress and plane strain ................................................................................... 20

2.3 ELASTIC-PLASTIC FRACTURE MECHANICS ...................................................................... 20

2.3.1 Crack tip opening displacement .............................................................................. 21

2.3.2 J contour integral..................................................................................................... 23

REFERENCE ......................................................................................................................... 30

CHAPTER 3: BACKGROUND OF FINITE ELEMENT METHODS ........................... 33

3.1 INTRODUCTION................................................................................................................ 33

3.2 BASIC PROCEDURE FOR THE FINITE ELEMENT ANALYSIS .................................................. 34

3.3 APPLICATION OF FINITE ELEMENT METHOD TO FRACTURE MECHANICS........................... 35

3.3.1 Crack tip singularity ................................................................................................ 35

3.3.2 The limited displacement extrapolation technique ................................................. 39

3.3.3 Virtual crack extension method .............................................................................. 41

3.3.4 Evaluation of J-integral by the domain integral method ......................................... 43

3.3.5 Newton-Raphson method........................................................................................ 47

REFERENCE ......................................................................................................................... 56

CHAPTER 4: ANALYSIS OF THE CRACKED TUBULAR ........................................... 59

vii

4.1 INTRODUCTION................................................................................................................ 59

4.2 FINITE ELEMENT MODELING AND ANALYSIS .................................................................... 60

4.2.1 FE model with semi-elliptical crack ....................................................................... 61

4.2.2 FE model with part through crack .......................................................................... 61

4.2.3 FE model and analysis with through thickness crack ............................................. 62

4.3 VERIFICATION OF FE MODEL ........................................................................................... 63

4.3.1 Semi-elliptical surface crack ................................................................................... 63

4.3.2 Through thickness crack ......................................................................................... 64

4.4 FEA RESULT OF CRACKED TUBE ...................................................................................... 65

4.4.1 Semi-elliptical surface crack ................................................................................... 66

4.4.2 Part through crack ................................................................................................... 70

4.3.3 Through thickness crack ......................................................................................... 76

4.5 EFFECT OF LENGTH AND WALL THICKNESS OF TUBE ........................................................ 78

4.5.1 Effect of tube length................................................................................................ 79

4.5.2 Effect of wall thickness of tube .............................................................................. 79

4.6 CONCLUSION ................................................................................................................... 80

REFERENCE ....................................................................................................................... 156

CHAPTER 5: ELASTIC-PLASTIC ANALYSIS FOR CRACKED TUBULAR

MEMBERS ........................................................................................................................... 158

5.1 INTRODUCTION.............................................................................................................. 158

5.2 ELASTIC-PLASTIC FE ANALYSIS OF TUBULAR MEMBER WITH SEMI-ELLIPTICAL INTERNAL

SURFACE CRACK .................................................................................................................. 159

5.3 COMPARISON OF FE RESULTS OBTAINED FROM EPFM AND LEFM .............................. 160

5.3.1 Comparison of COD results obtained from EPFM and LEFM ............................ 160

5.3.2 Comparison of J-integral obtained from EPFM and LEFM ................................. 163

5.4 FEA RESULTS OF CRACKED TUBE WITH ELASTIC-PLASTIC ANALYSIS ........................... 165

5.4.1 Distribution of the critical parameters .................................................................. 166

5.4.2 Compare of Ft from Elastic-Plastic analysis and Linear-Elastic analysis ............. 167

5.4.3 Analytical equation for the prediction of FT based on FEA data for surface crack

with Elastic-Plastic analysis ........................................................................................... 168

5.5 CONCLUSION ................................................................................................................. 171

REFERENCE ....................................................................................................................... 228

CHAPTER 6: CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER

WORK .................................................................................................................................. 229

6.1 CONCLUSION ................................................................................................................. 229

6.2 RECOMMENDATIONS FOR FURTHER WORK .................................................................... 231

viii

LIST OF FIGURES

Number Page

Figure 1.1 Cracks in a tube: (1) Surface crack, (2) Partly Through Wall Crack, (3) Full

Through Wall Crack. .......................................................................................11

Figure 2.1 The three modes of loading that can be applied to a crack ..............................27

Figure 2.2 General mode I problem ...................................................................................27

Figure 2.3 Plastic zone shapes according to Von Mises criteria ........................................28

Figure 2.4 Toughness as a function of thickness ...............................................................28

Figure 2.5 Alternative definitions of CTOD ......................................................................29

Figure 2.6 Contour around crack tip ..................................................................................29

Figure 3.1 Contour around crack tip ..................................................................................50

Figure 3.2 Crack tip mesh with special crack tip element .................................................50

Figure 3.3 Schematic illustrating the variables used in the displacement extrapolation

technique ..........................................................................................................51

Figure 3.4 Schematic illustrating the limited displacement extrapolation technique ........51

Figure 3.5 Virtual crack extension in 2D case ...................................................................52

Figure 3.6 Virtual crack extension in 3D case (local extension) .......................................53

Figure 3.7 A closed contour surrounding the crack tip ......................................................53

Figure 3.8 Surface enclosing an increment of the crack front ...........................................54

Figure 3.9 Interpretation of q by the concept of virtual crack extension ...........................54

Figure 3.10 A closed surface enclosing a volume of V* ...................................................55

Figure 3.11 Newton-Raphson Method ...............................................................................55

Figure 4.1: Flaw characterization ......................................................................................95

Figure 4.2: Cracked plate before transformation ...............................................................95

ix

Figure 4.3: Boundary conditions of cracked tube model ...................................................96

Figure 4.4: Quarter point crack tip element used in FEA ..................................................96

Figure 4.5: Typical FE mesh of tube with inner surface crack ..........................................97

Figure 4.6: Triangular mesh near outer face of part through crack ...................................97

Figure 4.7: FE Mesh of tube with semi-elliptical crack (ac = 0.4, aT = 0.8, RT = 10)...98

Figure 4.8: FE Mesh of tube with surface crack (cπR = 0.106, aT = 0.8, RT = 10).......99

Figure 4.9: FE Mesh of tube with part through crack (cπR = 0.106, aT = 1.5, RT = 10)

........................................................................................................................100

Figure 4.10: FE Mesh of tube with through thickness crack (cπR = 0.106, RT = 10) ..101

Figure 4.11: Comparison of Zahoor's results and current FE at deepest point ................102

Figure 4.12: Ft along crack front of SC (c/πR=0.371, R/T=10) ......................................102

Figure 4.13: Distribution of Ft along crack front of tube (R/T=4.0) ................................103

Figure 4.14: Distribution of Ft along crack front of tube (R/T=10.0) ..............................105



Figure 4.17: Variation of Ft at surface and deepest point of crack (R/T=4.0) .................111

Figure 4.18: Variation of Ft at surface and deepest point of crack (R/T=10.0) ...............112



Figure 4.21: Variation of 𝐾𝑚𝑖𝑛𝐾𝑚𝑎𝑥 versus a/T .........................................................115

Figure 4.22: Variation of Ft at deepest point with crack length .......................................117

Figure 4.23: Variation of Ft at surface point with crack length .......................................119

Figure 4.24: Variation of Ft with R/T ..............................................................................121

Figure 4.25: Basic operation of neural network...............................................................122

Figure 4.26: The operation of neural network of current analysis ...................................122

Figure 4.27: Comparison of Fts and FEA data with a/c ...................................................123

Figure 4.28: Comparison of Fts and FEA data at deepest point .......................................124

Figure 4.29: Comparison of Fts and FEA data at surface point .......................................126

x

Figure 4.30: Reference point chosen for part through crack ...........................................128

Figure 4.31: Distribution of Ft along the crack front of a part through crack ..................129

Figure 4.32: Variation of Ft at different reference point on a part through crack ............131

Figure 4.33: Distribution of Ft for part through crack of tube (R/T=4.0) ........................132

Figure 4.34: Distribution of Ft for part through crack of tube (R/T=10.0) ......................134



Figure 4.37: Variation of Ft at 0.99T with a/T for fart through crack .............................140

Figure 4.38: Variation of Ft at surface with a/T for part through crack ...........................142

Figure 4.39: Variation of 𝐾𝑠𝑢𝑟𝐾0.99𝑇 with a/T for part through crack .......................144

Figure 4.40: Variation of Ft at 0.99T with R/T for part through crack ............................146

Figure 4.41: Variation of Ft at surface with R/T for part through crack ..........................148

Figure 4.42: Comparison of Ftp and FEA data at 0.99T point for part through crack .....150

Figure 4.43: Comparison of Ftp and FEA data at inner surface point for part through crack

........................................................................................................................150

Figure 4.44: Distribution of Ft for through thickness crack .............................................151

Figure 4.45: Variation of Ft with c/πR for through thickness crack ................................153

Figure 4.46: Comparison of Ftf and FEA data at outer surface point for full through crack

........................................................................................................................153

Figure 4.47: Comparison of Ftf and FEA data at inner surface point for full through crack

........................................................................................................................154

Figure 4.48: Effect of tube length on Ft (a/T=1.3, c/πR=0.106, R/T=22.5, T=20mm) ....154

Figure 4.49: Effect of wall thickness on Ft (a/T=1.3, c/πR=0.106, R/T=22.5, L/D=10) .155

Figure 5.1 Elastic-Plastic material model definition from tension test ............................181

Figure 5.2 Typical true stress vs. true strain curve ..........................................................181

Figure 5.3 Distribution of Ft along the crack length versus vary crack depth .................182

Figure 5.4 The boundaries of crack free face along x, y axis ..........................................182

xi

Figure 5.5 Comparison COD in x-z direction with R/T=10.0, c/R=0.477, a/T=0.2 .....183



Figure 5.8 Comparison COD in x-z direction with R/T=10.0, c/R=0.477, a/T=0.85 ...184


Figure 5.10 Comparison COD in x-z direction with R/T=10.0, c/R=0.477, a/T=0.93 .185





Figure 5.15 Comparison COD in x-z direction with R/T=10.0, a/T=0.9, c/R=0.106 ...188








Figure 5.23 Comparison COD in x-z direction with a/T=0.9, c/R=0.477, R/T=4.0 .....192

Figure 5.24 Comparison COD in x-z direction with a/T=0.9, c/R=0.477, R/T=10.0 ...192



Figure 5.27 Comparison COD in y-z direction with R/T=10.0, c/R=0.477, a/T=0.2 ...194



Figure 5.30 Comparison COD in y-z direction with R/T=10.0, c/R=0.477, a/T=0.85 .195



xii





Figure 5.37 Comparison COD in y-z direction with R/T=10.0, a/T=0.9, c/R=0.106 ...199








Figure 5.45 Comparison COD in y-z direction with a/T=0.9, c/R=0.477, R/T=4.0 .....203

Figure 5.46 Comparison COD in y-z direction with a/T=0.9, c/R=0.477, R/T=10.0 ...203



Figure 5.49 Comparison J-integral along crack front with R/T=10.0, c/R=0.477, a/T=0.2

........................................................................................................................205


........................................................................................................................205


........................................................................................................................206


........................................................................................................................206


........................................................................................................................207


........................................................................................................................207

xiii


........................................................................................................................208


........................................................................................................................208


........................................................................................................................209


........................................................................................................................209

Figure 5.59 Comparison J-integral along crack front with R/T=10.0, a/T=0.9, c/R=0.106

........................................................................................................................210


........................................................................................................................210


........................................................................................................................211


........................................................................................................................211


........................................................................................................................212


........................................................................................................................212


........................................................................................................................213


........................................................................................................................213

Figure 5.67 Comparison J-integral along crack front with a/T=0.9, c/R=0.477, R/T=4.0

........................................................................................................................214


........................................................................................................................214

xiv


........................................................................................................................215


........................................................................................................................215

Figure 5.71 Comparison Ft along crack front with R/T=10.0, c/R=0.477, a/T=0.2 ......216



Figure 5.74 Comparison Ft along crack front with R/T=10.0, c/R=0.477, a/T=0.85 ....217







Figure 5.81 Comparison Ft along crack front with R/T=10.0, a/T=0.9, c/R=0.106 ......221








Figure 5.89 Comparison Ft along crack front with a/T=0.9, c/R=0.477, R/T=4.0 ........225

Figure 5.90 Comparison Ft along crack front with a/T=0.9, c/R=0.477, R/T=10.0 ......225



Figure 5.93: Comparison of Fts and FEA data at deepest point .......................................227

Figure 5.94: Comparison of Fts and FEA data at surface point .......................................227

xv

LIST OF TABLES

Number Page

Table 1.1 Parameters assigned in the finite element analysis ............................................10

Table 4.1: Ft from current FEA and Mettu’s result (R/T=10) ...........................................82

Table 4.2: Comparison of Ft from present FEA and Zahoor's result (R/T=10) .................82

Table 4.3: Parameters for the analysis of tubes with surface cracks .................................82

Table 4.4: NSIF Ft at surface point for surface crack (R/T=4.0) .......................................83

Table 4.5: NSIF Ft at deepest point for surface crack (R/T=4.0) .......................................83

Table 4.6: NSIF Ft at surface point for surface crack (R/T=10.0) .....................................83

Table 4.7: NSIF Ft at deepest point for surface crack (R/T=10.0) .....................................84

Table 4.8: NSIF Ft at surface point for surface crack (R/T=15.0) .....................................84

Table 4.9: NSIF Ft at deepest point for surface crack (R/T=15.0) .....................................84

Table 4.10: NSIF Ft at surface point for surface crack (R/T=22.5) ...................................85

Table 4.11: NSIF Ft at deepest point for surface crack (R/T=22.5) ...................................85

Table 4.12: Parameters for the analysis of tubes with part through cracks .......................85

Table 4.13: NSIF Ft at surface point for part through crack (R/T=4.0) .............................86

Table 4.14: NSIF Ft at 0.99T point for part through crack (R/T=4.0) ...............................87

Table 4.15: NSIF Ft at surface point for part through crack (R/T=10.0) ...........................88

Table 4.16: NSIF Ft at 0.99T point for part through crack (R/T=10.0) .............................89





Table 4.21: Parameters for the analysis of tubes with part through cracks .......................94

Table 4.22: NSIF Ft at inner surface for through thickness crack .....................................94

xvi

Table 4.23: NSIF Ft at outer surface for through thickness crack .....................................94

Table 5.1 J-integral values at surface point of surface crack for EPFM(R/T=4.0) ..........173

Table 5.2 J integral values at deepest point of surface crack for EPFM (R/T=4.0).........173

Table 5.3 J integral values at surface point of surface crack for EPFM (R/T=10.0) .......173

Table 5.4 J integral values at deepest point of surface crack for EPFM (R/T=10.0).......174





Table 5.9 NSIF Ft at surface point of surface crack for EPFM (R/T=4.0) ......................175

Table 5.10 NSIF Ft at deepest point of surface crack for EPFM (R/T=4.0) ....................176

Table 5.11 NSIF Ft at surface point of surface crack for EPFM (R/T=10.0) ..................176

Table 5.12 NSIF Ft at deepest point of surface crack for EPFM (R/T=10.0) ..................176





Table 5.17 The ratio of NSIF of EPFM to LEFM at surface point (R/T=4.0) ................178

Table 5.18 The ratio of NSIF of EPFM to LEFM at deepest point (R/T=4.0) ................178

Table 5.19 The ratio of NSIF of EPFM to LEFM at surface point (R/T=10.0)...............179

Table 5.20 The ratio of NSIF of EPFM to LEFM at deepest point (R/T=10.0) ..............179





xvii

ACKNOWLEDGMENTS

The author wishes to express her deeply appreciation and gratitude to Prof. Kun Pang Kou and

Dr. Chi Chiu Lam for their excellent guidance, valuable suggestion, comments and

encouragement throughout this master study. Prof. Kun Pang Kou guides her to be not only a

good researcher, but also a responsible person. Dr. Chi Chiu Lam guides her earnest rigorous

scientific research manner. The author will always be proud of being one of their students.

The author would also like to extend her appreciation to Prof. Ka Veng Yuen, Prof. Guo-Kang

Er, and Dr Wai Meng Quach for their teaching and kind help.

The author would also thank the generous studentship support from the University of Macau.

The author desires to thank all the numbers in the Computer Aided Civil Engineering

Laboratory and the Strength of Materials Laboratory. Special thanks go to Hai Tao Zhu,

Shuang Wen Lan, Xing Lu Liu, Ming Chang Wang, Xiu Xiu Guo, Cheong Ionkeong. The other

good friends: Ka Man Tou, He Qing Mu, Zhi Li Zhang, Yi Qin.

Finally, the author would like to express her deep thanks and gratitude to her parents Xiao Li

Yang and Shi Ju Tu for their support and encouragement.

1

CHAPTER 1: INTRODUCTION AND BACKGROUND

1.1 INTRODUCTION

Fracture is a problem that society has faced for as long as there have been man-made structures.

The problem actually goes worse today than previous centuries, because more can go wrong in

our complex technological society. In reality, there are many more factors which can lead to the

structure failures. From investigating the fallen structures, engineers found that most failure

began with microscopic cracks that may be caused by materials defects. As is known that the

materials is never flawless, the flaw is growing up under the external loading or fatigue loading

in service, until become the critical crack size and finally lead to a failure, the same as

dislocation and impurities etc.. In 1983 a section of 4 in diameter PE pipe developed a major

leak. The gas collected beneath a residence where it ignited, resulting in severe damage to the

house. Maintenance records and a visual inspection of the pipe indicated that it had been pinch

clamped 6 years earlier in the region where the leak developed. A failure investigation [1]

concluded that the pinch clamping operation was responsible for failure. Microscopic

examination of the pipe revealed that a small flaw apparently initiated on the inner surface of

the pipe and grew through the wall. Laboratory tests simulated the pinch clamping operation on

sections of PE pipe; small thumbnail-shaped flaws formed on the inner wall of the pipes, as a

result of the severe strains that were applied. Fracture mechanics tests and analyses [1, 2]

indicated that stresses in the pressurized pipe were sufficient to cause the observe

time-dependent crack growth; i.e., growth from a small thumbnail flaw to a through-thickness

crack over a period of 6 years.

All engineering components and structures contain geometrical discontinuities-threaded

connections, windows in aircraft fuselages, keyways in shafts, teeth of gear wheels, etc. The

size and shape of these features are important since they determine the strength of the artifact.

If these discontinuities in assembly may not be perfect or design may not properly so as to

sharp corners, grooves, nicks, voids, etc., appear and which will cause stress concentration and

2

lead to structure failures. Conventionally, the strength of the components or structures

containing defects is assessed by evaluating the stress concentration caused by the

discontinuity features. However, such a conventional approach would give erroneous answers

if the geometrical discontinuity features have very sharp radii.

Moreover, when the structures are during in service, the maintenance of structure may be

poor or not properly; occasionally, damages in service such as impact, fatigue, unexpected

loads and so on are usually happen, and the service life may have to be very long, etc..

Traditionally these concerns always can cause the microscopic cracks, and most microscopic

cracks are arrested inside the material but it takes one run-away crack to destroy the whole

structure.

Then in order to avoid brittle fracture of the structures, analyze the relationship among

stresses, cracks, and fracture toughness, systematic scientific rules were developed to

characterize cracks and their effects and to predict if and when the structures or their

components containing crack(s) may become unsafe during the service lives of the structures,

this science was called Fracture Mechanics.

Fracture mechanics is a set of theories describing the behavior of solids or structures with

geometrical discontinuity at the scale of the structure. The discontinuity features may be in the

form of line discontinuities in two-dimensional media (such as plates, and shells) and surface

discontinuities in three-dimensional media. Fracture mechanics has now evolved into a mature

discipline of science and engineering and has dramatically changed our understanding of the

behavior of engineering materials.

Conventional failure criteria have been developed to explain strength failures of

load-bearing structures which can be classified roughly as ductile at one extreme and brittle at

another. In the first case, breakage of a structure is preceded by large deformation which occurs

over a relatively long time period and may be associated with yielding or plastic flow. The

brittle failure, on the other hand, is preceded by small deformation, and is usually sudden.

Defects play a major role in the mechanism of both these types of failure; those associated with

ductile failure differ significantly from those influencing brittle fracture. For ductile failures,

3

which are dominated by yielding before breakage, the important defects (dislocations, grain

boundary spacing, interstitial and out-of-size substitutional atoms, and precipitates) tend to

distort and warp the crystal lattice planes. Brittle fracture, however, which takes place before

any appreciable plastic flow occurs, initiates at large defects such as inclusions, sharp notches,

and surface scratches of cracks.

Fracture mechanics can be divided into Linear-Elastic Fracture Mechanics (LEFM) and

Elastic-Plastic Fracture Mechanics (EPFM). LEFM give excellent results for brittle-elastic

materials like high-strength steel, glass, ice, concrete, and so on. However, for ductile materials

like low-carbon steel, stainless steel, certain aluminum alloys and polymers, plasticity will

always precede fracture. Nonetheless, when the load is low enough, linear fracture mechanics

continues to provide a good approximation to the physical reality.

This field has become increasingly important to the engineering community. In recent

years, structural failures and the desire for increased safety and reliability of structures have led

to the development of various fracture and fatigue criteria for many types of structures,

including bridges, planes, pipelines, ships, buildings, pressure vessels, and nuclear pressure

vessels.

1.2 BACKGROUND

Recent catastrophic failures in the mining industry have resulted in an interest in the failure

mechanisms in tubular structures. Coal is one of the most important industries in the world,

while mined predominately with the use of Draglines. Currently there are hundreds of

draglines operating in the world. These machines provide an annual income to the economy.

However, with such high economic pressures it is becoming increasingly necessary to maintain

and operate these Draglines longer. More than 40% of draglines in the 69 coalmines around the

world are between 11 and 20 years old, and another 40% have been in operation more than 20

years [3]. Considering the nominal design life is approximately 20 years and with the high

capital cost of replacement there is increased pressure to extend their operational life. A large

4

number of the Draglines fleet contains numerous tubular members prone to fatigue cracking

and thus a greater understanding of the fatigue mechanisms is required.

Predominately, equipment such as Draglines operating in open cut mines experiences

circumferential cracking in critical areas such as the main boom. For this reason, the analysis of

circumferential elliptical surface cracks in tubular members is of significant interest.

Tubular members also have been used extensively in many engineering structures. Such as

the aerospace, offshore structures, pressure vessels, vehicles motor and drilling pipes. Under

repeated loading, cracks may develop at the surface and grow across the section. Some

researches has concentrated on the use of fracture mechanics to determine the residual life in

tubular steel structures. It has been shown that miniature surface cracked pipe specimen offer a

cost-effective way for evaluating fatigue crack propagation properties. The use of fracture

mechanics to examine tubular structures was first attempted in the latter half of the 1970s [4].

The main thrust of research into the fatigue behavior of tubular structures was spurned out of

concern for oil and gas platforms in the North Sea. Offshore structures in the North Sea

frequently experience arduous cyclic loading as a result of the extreme weather conditions. The

collapse of BP's Sea Gem rig in 1965 with the loss of 13 lives is testimony to these extreme

conditions.

To assess the crack growth behavior and structural integrity involving these cracks, their

stress intensity factor solutions must be known. The three-dimensional nature of this kind of

cracks results in a stress intensity that is not only varying along the crack front but is also highly

sensitive to the crack shape. Numerical techniques or approximate analyses were often

employed to estimate the stress intensity for this problem.

Early attempts used a straight edge or a circular arc to idealize the crack front (Wilhem et

al., 1982 [5]; Mackay and Alperin, 1985 [6]; Forman and Shivakumar, 1986 [7]; Raju and

Newman, 1986 [8]). In some works, the angle of intersection of the crack front with the tube

external surface was taken to be 90 degree to facilitate crack shape definition (Forman and

Shivakumar, 1986 [7]; Raju and Newman, 1986 [8]). These idealizations, though close to, do

not as a matter of fact exactly agree with experimental observations. The above discrepancies

5

may have impact on the correctness of the stress intensity solutions. This problem of crack

shape description is largely solved by using an elliptical arc to model the crack front

(Athanassiadis, 1981 [9]; Astiz, 1986 [10]; Shiratori et al., 1986 [11]). It is well known that the

singularity power at the intersection points is no longer- 1/2 and is dependent on the angle of

intersection and Poisson ration (Bazant and Estenssoro, 1979 [12]; Hayashi and Abe, 1980

[13]). The departure from the square root singularity was sometimes pragmatically overcome

by discarding the numerical solution at the end point and replaced it with that for a neighboring

interior point instead.

There are limited solutions of stress intensity factors available in the literature concerning

the problem of circumferential surface flaws in tubular members. The majority of solutions

have been based on the line spring and finite elements models. It is important to note that

solutions based on line spring models for circumferential cracks in a cylinder only concern

flaw in thin-walled structure. However, other researches, for example, Zahoor (1985) [14]

presented the closed from equations for predicting the SIF of deepest point of a finite length

circumferential part-through surface cracked tubes based the finite element analysis results of

Kumar et al. (1984) [15]. Three parameters ratio, (1) crack aspect ratio (c/a), (2) crack deep

ratio (a/T) and (3) tube radius to thickness ratio (R/T) were identified and the covering ranges

of parameters are shown in Table 1.1. Mettu et al. (1992) [16] carried out similar analysis to

obtain the SIFs of both deepest point and surface point with similar parameters which are

shown in Table 1.1 as well. Bergman (1995) [17] have applied the finite element technique to

examine the stress intensities around a partly circumferential crack under various types of

loading with the parameters which are also shown in Table 1.1. The latest research by D. Peng

et al [18] used a simplicity finite element technique to analyze a crack covered the parameters

are shown in Table 1.1.

From these researches, they are available for a limited number of discrete aspect ratios and

crack depth ratios, especially a/T ratios are never exceed 0.8. However, an exact solution for

the crack covering a wide range of crack geometry is essential for fatigue life evaluation and

structural integrity assessment involving surface cracks because practical surface cracks may

6

come in any aspect ratios and crack depths, and such parameters may changes as the crack

grows along. To covering a wider range of geometric parameters, Mettu et al (1992) [16] used

the displacement extrapolation technique extrapolated values of SIF for a/c = 0.2, 0.4 and 2.0

and a/T = 0 and 1 from the existing results. However, the accuracy of the results in the

extrapolated range remains questionable. In order to investigate the detailed process of crack

growth from surface crack, it is necessary to have accurate values of SIF for a/T ≥ 0.8. In the

present study, finite element analysis has been carried out to determine this set of SIF.

In the present work, the stress intensity factors along the crack front are computed for an

elliptical surface crack in a tube member under tension. A wide range of crack aspect ratios that

should be able to cover most practical crack shapes are examined and the geometry parameters

are shown in Table 1.1. Then neural network of MATLAB was used to process those finite

element analysis results and suitable equations for predicting the SIFs were proposed. The

equations are given to facilitate use, with the current proposed equations, the fatigue crack

growth trend can be established which can help for estimating the inspection intervals for

circular tube structures.

1.3 OBJECTS AND SCOPE OF THE WORK

In this thesis, stress intensity factors for a wide range of long-deep circumferential

semi-elliptical internal surface cracks in tubular members are presented. The crack

configurations in the tubular members are subjected to axial tension loading and the SIFs are

predicted by mean of 3-D finite element analysis. In this study, the SIFs were analyzed by

considering the following three main parameters: (1) the crack depth to thickness ratio (a/T), (2)

the outer radius to thickness ratio (R/T) and (3) the crack depth to crack length ratio (a/c). For

a/T < 0.8, current finite element results compared well with the results reported in literatures.

However, finite element results of SIF of deep circumference inner surface elliptical crack with

a/T > 0.8 have not been reported previously. Therefore, current finite element analysis was

extended to investigate the SIFs by including a wider range of a/T ratio up to 0.99.

7

Investigate the details of crack growth starting from a surface crack to a partly

through-wall crack and finally to the critical fully through-wall crack would greatly help to

implement the realistic fracture behavior happened on the tubular members. The partly through

wall crack is defined as the case when a semi-elliptical flaw just breaks through with different

lengths on the two surfaces. From surface crack to the critical fully through-wall crack, the

process can be divided into two stages. The first stage is the growth of surface cracks up to the

point just before the wall is penetrated. The second stage is the growth of partly through wall

cracks immediately after the wall is penetrated and up to a critical fully through wall crack as

shown in Figure 1.4. However, before the calculation of the crack growth can be carried out, it

is necessary to know the stress intensity factors (SIF) of cracks just before and just after

breaking through to the second side. In the present work, the SIFs of surface cracks in tubular

members with very small ligament and of partly through wall crack were investigated used

finite element method.

Then neural network of MATLAB was used to process those finite element analysis

results and suitable equations for predicting the SIFs were proposed. With the current proposed

equations, the fatigue crack growth trend can be established which can help for estimating the

inspection intervals for circular tube structures.

For deep surface crack with small ligament, the large scale plastic deformation are

happened, that is the materials at the crack tip at some combination of stresses and strains are

no longer satisfying Linear Elastic Fracture Mechanics (LEFM). Therefore, application of

Elastic-Plastic Fracture Mechanics to deal with this condition may be more efficient and

realistic. The current research investigate the J integral, normalized stress intensity factors and

crack tip opening displacements (CTOD) with Elastic-Plastic analysis of circumferential

semi-elliptical internal surface crack of tubular members subjected to axial tensile loading and

compared with the results obtained from the Linear-Elastic analysis.

The objects of the present works are as follows:

8

1. To study the stress intensity factors (SIFs) of a circumferential internal surface crack,

especially, for the deep surface cracks (a/T > 0.8), and partly through wall cracks of tubular

members by mean of finite element method.

2. To apply MATLAB neural network method to process those finite element analysis results

and suitable equations were obtained for predicting the normalized stress intensity factors

(NSIFs) of a circumferential internal surface crack of a tubular member.

3. To investigate the J integral, normalized stress intensity factors (NSIF) and crack tip

opening displacement (CTOD) with Elastic-Plastic analysis of the circumferential internal

surface crack and compared with the results obtained from the Linear-Elastic analysis.

The finite element method was used as the tool for determining the crack tip severity. With

these proposed SIF and based on the principle of Linear-Elastic Fracture Mechanics the full

development of growth of a circumferential crack in a tubular member from a long deep

surface crack to final failure has been investigated, including the stage of a partial through wall

crack. The influences of member size, crack configuration, material properties and loading

conditions have also been explored. In addition, the Elastic-Plastic Fracture Mechanics was

also used to investigate the influence of material plasticity on the prediction of J-integral value

of deep circumferential surface crack with large scale plastic deformation.

1.4 ORGANIZATION OF THE THESIS

The thesis consists of six chapters. These can be summarized as follows:

In chapter 1, introduction of the fracture mechanics are briefly discussed, and the

background of this thesis is presented. In addition, the objects and scope of the work are

described.

Chapter 2, literature review on fracture mechanics. First, basic theories and concepts of

Linear Elastic Fracture Mechanics (LEFM) and Elastic-Plastic Fracture Mechanics (EPFM)

9

are described. Afterword, flat plate with surface crack and tube member with surface crack are

mentioned.

In chapter 3, the finite element methods for fracture mechanics are described. First, the

finite element analysis techniques are presented. Moreover, basic equations and finite element

formulation of solid mechanics are described. Finally, application of the finite element method

to Linear Elastic Fracture Mechanics and Elastic-Plastic Fracture Mechanics is introduced.

Chapter 4, details of the finite element models and analysis in Linear Elastic analysis for

tubular members are described. Then verification of the finite element models and results are

discussed. Afterward, the finite element analysis results of the cracked tube are presented and

discussed. Finally, the effect of length and wall thickness of tube is studied.

In chapter 5, details of Elastic-Plastic analysis for cracked tubular members are presented

and discussed. Comparison of the J-integral results obtained from the Elastic-Plastic analysis

and the Linear-Elastic analysis are carried out and discussed.

Chapter 6, summary, conclusions and recommendations for further work.

10

Table 1.1 Parameters assigned in the finite element analysis

Parameter Values assigned in Zahoor's study

a/T 0.2, 0.4, 0.6, 0.8

c/a 1.5, 3.0, 6.0

R/T 5.0, 10.0, 20.0

Parameter Values assigned in Mettu's study

a/T 0, 0.2, 0.5, 0.8

a/c 0.2, 0.4, 0.6, 0.8, 1.0

R/T 1.0, 2.0, 4.0, 10.0

Parameter Values assigned in Bergman's study

a/T 0.2, 0.4, 0.6, 0.8

c/a 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16

R/T 5.00, 10.0

Parameter Values assigned in D. Peng's study

a/T 0.2, 0.4, 0.6, 0.8

c/a 0.125-2.0

Parameter Values assigned in current FE study

a/T 0.2, 0.5, 0.8, 0.85, 0.9, 0.93, 0.95, 0.97, 0.98, 0.99

c/πR 0.106, 0.159, 0.212, 0.265, 0.318, 0.371, 0.424, 0.477

R/T 4.00, 10.0, 15.0, 22.5

11

Figure 1.1 Cracks in a tube: (1) Surface crack, (2) Partly Through Wall Crack, (3) Full Through

Wall Crack.

12

REFERENCE

[1] Jones, R.E. and Bradley, W.L., "Failure Analysis of a Polyethylene Natural Gas Pipeline."

Forensic Engineering, Vol. 1, pp. 47-59，1987.

[2] Jones, R.E. and Bradley, W.L., "Fracture Toughness Testing of Polyethylene Pipe

Materials." ASTM STP 995, Vol. 1, American Society for Testing and Materials, Philadelphia,

pp.447-456，1989.

[3] Gilewicz P. "International dragline population matures." Coal Age; 105(6), 2000.

[4] Dover WD. "Fatigue crack growth in offshore structures." J Soc Environ Engrs; 15(1): 3-9,

1976.

[5] Wilhem, D., FitzGerald, J., and Dittmer, D., "An empirical approach to determining K for

surface cracks." Proceedings of the 5th

international Conference on Fracture Research 11-21,

1982.

[6] Mackay, T.L. and Alperin, B.J.. "Stress intensity factors for fatigue cracking in

high-strength bolts." Engineering Fracture Mechanics 21, 391-397, 1985.

[7] Forman, R.G. and Shivakumar, V.. "Growth behavior of surface cracks in the

circumferential plane of solid and hollow cylinders." Fracture Mechanics: Seventeen volume.

ASTM 905, 59-74, 1986.

[8] Raju, I.S. and Newman, J.C.. "Stress- intensity Factors for circumferential surface cracks in

pipes and rods under tension and bending loads." Fracture Mechanics: Seventeen volume.

ASTM STP 905, 789-805, 1986.

[9] Athanassiadis, Boissenot, J.M., Brevet, P., Francois, D. and Raharinaivo, A.. "Linear

elastic fracture mechanics computations of cracked cylindrical tensioned bodies."

International Journal of Fracture 17, 553-566, 1981.

[10] Astiz, M.A.. "An incompatible singular elastic element for two- and three-dimensional

crack problems." International Journal of Fracture 31, 105-124, 1986.

13

[11] Shiratori, M., Miyoshi, T., Sakai, Y. and Zhang, G.R.. "Analysis of stress intensity factors

for surface cracks subjected to arbitrarily distributed surface stresses." Trans. Japan Soc. Mech.

Engrs. 660-662, 1986.

[12] Bazant, Z.P. and Estenssoro, L.F.. "Surface singularity and rack propagation."

International Journal of Solids and Structures 15, 405-257, 1979.

[13] Hayashi, K. and Abe, H.. "Stress intensity factors for a semi-elliptical crack in the surface

of a semi-infinite solid." International Journal of Fracture 16, 275-285, 1980.

[14] Zahoor, A. (1985), "Closed Form Expressions for Fracture Mechanics Analysis of

Cracked Pipes", Journal of Pressure Vessel Technology, Vol. 107/203.

[15] Kumar, V., Greman, M. D., Wilkening, W. W., Andrews, W. R., deLorenzi, H. G., and

Mowbray, D. F. (1984)., "Advances in Elastic-Plastic Fracture Analysis", EPRI Report

NP-3607.

[16] Mettu, S. R., Raju, I. S., and Forman, R. G. (1992), "Stress Intensity Factors for Part

-Through Surface Cracks in Hollow Cylinders", JSC Report 25685/LESC Report 30124,

NASA Lyndon B. Johnson Space Center/Lockheed Engineering and Sciences Co. Joint

Publication.

[17] Bergman M. "Stress intensity factors for a circumferential cracks in pipes." Fatigue

Fract. Engng Mater. Struct.; 18(10): 1155-1172, 1995.

[18] Peng D., Wallbrink C., Jones R., "An assessment of stress intensity factors for surface

flaws in a tubular member." Engineering Fracture Mechanics 72, 357-371, 2005.

14

CHAPTER 2: LITERATURE REVIEW ON FRACTURE MECHANICS

2.1 BACKGROUND

'Fracture mechanics' is the name coined for the study combines the mechanics of the cracked

bodies and mechanics properties. As indicated by its name, Fracture mechanics deals with

fracture phenomena and events. The establishment of fracture mechanics is closely related to

same well known disasters in recent history. Several hundred liberty ship fractured extensively

during World War II. The failures occurred primarily because by the change from riveted to

welded construction and the major factors were the combination of poor welded properties

with stress concentrations, and poor choice of brittle materials in the construction. Of the

roughly 2700 liberty ships built during World War II, approximately 400 sustained serious

fracture, and some broken in two. The comet accidents in 1954 sparked an extensive

investigation of the causes, leading to significant progress in the understanding of fracture and

fatigue. In July 1962 the Kings Bridge, Melbourne failed as a loaded vehicle of 45 tones

crossing one of the spans caused to collapse suddenly. Four girders collapsed and the fracture

extended completely through the lower flange of the girder, up the web and in some cases

through the upper flange. Remarkably no one was hurt in the accident [1].

The first milestone was set by Griffith in his famous 1920 paper that quantitatively relates

the flaw size to the fracture stresses [2]. He applied a stress analysis of an elliptical hole

(performed by Inglis [3] seven years earlier) to the unstable propagation of a crack. Griffith

invoked the First Law of Thermodynamic to formulate a fracture theory base on a simple

energy balance. According this theory, a flaw becomes unstable, and thus fracture occurs.

When the strain energy change that results from an increment of a crack growth is sufficient to

overcome the surface energy of the material. Griffith's model correctly predicted the

relationship between strength and flaw size in glass specimens. Subsequent efforts to apply the

Griffith model to metals were unsuccessful. Since this model assumes that the work of fracture

comes exclusively from the surface energy of the material. The Griffith approach only applies

15

to ideally brittle solids. A modification to Griffith model that make it applicable to metals did

not come until 1948.

After studying the early work of Inglis, Griffith, and others, Irwin [4] concluded that the

basic tools need to analyzed fracture already available. The first contribution of Irwin's work is

to extent the Griffith approach to metals by including the energy dissipated by local plastic flaw.

Orowan independently proposed a similar modification to the Griffith theory [5]. During the

same period, Mott [6] extended the Griffith theory to a rapidly propagation crack.

In 1956, Irwin [7] developed the energy release rate concept which is related to the Griffith

theory but is in a form that is more useful for solving engineering problems. Shortly afterward,

several of Irwin's colleagues brought to his attention a paper by Westergaard [8] that was

published in 1938. Westergaard had developed a semi-inverse technique for analyzing stresses

and displacements ahead of a sharp crack. Irwin [9] used the Westergaard approach to show

that the stresses and displacements near the crack tip could be described by a single constant

that was related to the energy release rate. This crack tip characterizing parameter later became

known as the stress intensity factor. During this same period of time, Williams [10] applied

somewhat a different technique to drive crack tip solutions that was essentially identical to

Irwin's results.

A number of successful early applications of fracture mechanics bolstered the standing of

this new field in the engineering community. In 1956, Wells [11] used fracture mechanics to

show that the fuselage failures in several Comet jet aircraft resulted from fatigue cracks

reaching a critical size. A second early application of fracture mechanics occurred at General

Electric in 1957. Winne and Wundt [12] applied Irwin's energy release rate approach to the

failure of the large rotors from steam turbines. They were able to predict the bursting behavior

of large disks extracted from rotor forgings, and applied this knowledge to the prevention of

fracture in actual rotors.

It seems that all great ideas encounter stiff opposition initially, and fracture mechanics is

no exception. In 1960, Paris and his co-works [13] failed to find a receptive audience for their

ideas on applying the fracture mechanics principles to fatigue crack growth. Although, Paris et

16

al. provided convincing experimental and theoretical arguments for their approach, it seems

that design engineers were not yet ready to abandon their S-N curves in favor of a more

rigorous approach to fatigue design. They finally opted to publish their work in a University of

Washington periodical entitled The Trend in Engineering.

One possible historical boundary occurs around 1960 when the fundamentals of the Linear

Elastic Fracture Mechanics were fairly well established and researchers turned their attention

to crack tip plasticity.

Wells [14] proposed the displacement of the crack faces as an alternative fracture criterion

when significant plasticity precedes failure. He attempted to apply LEFM to the low and

medium-strength structural steels. These materials were too ductile to LEFM apply, but Wells

noticed that the crack faces moved apart with plasticity deformation. This observation let to

development of the parameter now known as the crack tip opening displacement (CTOD).

In 1968, Rice [15] development another parameters to characterize nonlinear material

behavior ahead of a crack. By idealizing plasticity deformation as nonlinear elastic, Rice was

able to generate the energy release rate to nonlinear materials. He showed that this nonlinear

energy release rate can be expressed as a line integral, which he called the J-integral, evaluated

along an arbitrary contour around the crack.

The same year, Hutchinson [16] and Rice and Rosengren [17] related the J integral to crack tip

stress fields in nonlinear materials. These analyses showed that J can be viewed as a nonlinear

stress intensity parameter as well as an energy release rate.

In 1971, Begley and Landes [18] who were research engineers at Westinghouse, came

across Rice's article and decided, despite skepticism from their coworkers, to characterize

fracture toughness of these steels with the J integral. Their experience were very successful and

let to the publication a standard procedure for J testing of metals ten years later [19].

A fracture design analysis base on the J integral was not available until Shih and

Hutchinson [20] provided the theoretical framework for such an approach in 1976. A few years

later, the Electric Power Research Institute (EPRI) published a fracture design handbook [21]

base on the Shih and Hutchinson methodology.

17

In United Kingdom, Well's CTOD parameter was applied extensively to fracture analysis

of welded structures, beginning in the late 1960s. While fracture research in the U.S. was

driven primarily by the nuclear power industry during 1970s, fracture research in the UK was

motivated largely by the development of oil resources in the North Sea. In 1971, Burdekin and

Dawes [22] applied several ideas proposed by Wells [23] several years earlier and developed

the CTOD design curve, a semi empirical fracture mechanics methodology for welded steel

structures. The nuclear power industry in the UK developed their only fracture design analysis

[24], base on the strip yield model of Dugdale [25] and Barenblatt [26].

Shih [27] demonstrated a relationship between J integral and CTOD, implying that both

parameters are equally valid for characterizing fracture. The J-based material testing and

structural design approaches developed in the U.S. and British CTOD methodology have

begun to merge in recent years, with positive aspects of each approach combined to yield

improved analysis. Both parameters are current applied throughout the world to a range of

materials. Much of the theoretical foundation of dynamic fracture mechanics was developed

during the period between 1960 and 1980.

2.2 LINEAR ELASTIC FRACTURE MECHANICS

Linear Elastic Fracture Mechanics is one of the most important theories of fracture mechanics.

A solid background in the fundamentals of Linear Elastic Fracture Mechanics is essential to

understanding of more advanced concepts in fracture mechanics.

2.2.1 STRESS INTENSITY FACTOR

The fundamental postulate of Irwin's approach to fracture is generally referred to as Linear

Elastic Fracture Mechanics (LEFM) in which all analyses are based on the elastic parameter,

stress intensity factor. Irwin also suggested that the modes in which a crack can be stressed can

be categorized into three distinct types: opening, sliding and tearing as shown in Figure 2.1.

From these three basic modes, the most general crack deformation can be represented by an

18

appropriate superposition. Among these three modes, mode I is technically the most important

and hence the discussions hereafter are limited to mode I.

Based on the works of Westergaard [8] and the assumption of linear elasticity, Irwin [9]

derived the expression for the stresses at a crack tip as shown in Equations 2.1 where KI is the

stress intensity factor of mode I. Variables in Equations 2.1 are shown in Figure 2.2. This

expression reveals that the patterns of all crack tip stress fields are unique and independent of

any crack geometry and body geometry. The stress intensity factor, KI, depends linearly on the

loading and is a function of the crack size and the crack configuration of the cracked body. In

reverse, for a given value of KI and all elastic properties of the material, the stress fields are

determined. Consequently, the stress intensity factor can be considered as a single parameter

which uniquely characterizes the local stress field in the vicinity of a linear elastic crack tip. At

the crack tip (r = 0), the stresses becomes infinite. Therefore, the stress intensity factor is also a

measure of stress singularity and severity at the crack tip.

ςxx

ςyy

ςxy

=KI

2πrcos

θ

2

1 − sin

θ

2sin

3θ

2

1 + sinθ

2sin

3θ

2

sinθ

2cos

3θ

2

(2.1)

The consequence of this derivation is that, infinite stresses will occur at the crack tip.

However this is not possible in reality as no engineering material can sustain under infinite

stress. The stresses at the crack tip are actually limited to the yield stress for the triaxial

conditions present. Consequently, a plastic zone will exist around the crack tip. Based on the

Equation 2.1 and an appropriate yielding condition such as the Von Mises criterion, the shapes

of the plastic zone size in an infinite body under plane strain and plane stress conditions can be

determined. Figure 2.3 shows these two plastic zones in which rp is the distance from the crack

tip to the border of the yielding zone. As a single characterization parameter, the stress intensity

factor is able to predict the stress conditions around the crack tip. However, this ability largely

19

depends on the size of this plastic zone. For the condition of K dominance to be valid and hence

the existence of a K dominant field, the crack tip must be under the condition of small scale

yielding. Under this condition, not only the crack tip stress field, but also the size of the plastic

zone can be predicted by the stress intensity factor.

If a material fails at the crack tip at some combinations of stresses and strains satisfying

linear elasticity and plane strain, the crack extension must occur at a critical value of the stress

intensity factor, KIc. This KIc is a material constant which is a measure of the fracture toughness.

However, only under certain conditions is this critical stress intensity factor a material constant.

Otherwise it can be geometry dependent. These conditions are mainly controlled by the size of

the plastic zone. In an infinite plate having a through thickness crack and small thickness, for

example, plane stress condition prevails. The large plastic zone size (rp) compared to the plate

thickness will let yielding occur freely in the thickness direction. In this case, a higher stress

intensity can be applied before crack extension occurs. On the other hand, with large plate

thickness, plane strain conditions prevail and induce a small plastic zone. As a result, the

yielding cannot take place freely in the thickness direction due to the constraint by the

surrounding elastic material. In this case, a relatively lower stress intensity will lead to crack

propagation because the strains and stresses are large enough. Therefore, specimens with

different thickness will give different critical stress intensity factors, Kc. Figure 2.4 outlines the

variation of the Kc value with the plate thickness. A plate thickness To will give the maximum

Kc and any plate thickness greater than Ts would give the material constant, i.e. the fracture

toughness, KIc. Any plate thickness in between To and Ts would give a Kc under the mixed

condition of plane strain and plane stress. Because of the significant influence of the small

scale yielding, fracture toughness testing has to be carried out under size limitations on the

specimens. Both the SATM [28] and the BSI [29] give guide lines for LEFM fracture

toughness testing in which the ligament, the crack length and the thickness must be greater than

2.5(KIc

ςy)2. The factor (

KIc

ςy)2 is applied because the plastic zone size (rp) is proportional to it.

20

2.2.2 PLANE STRESS AND PLANE STRAIN

In general, the conditions ahead of a crack are neither plane stress nor plane strain, but are

three-dimensional. However, two-dimensional assumption is valid, and provides a good

approximation, for limiting cases. As a result, most of the classical solutions in fracture

mechanics reduce the problems to two dimensions by assuming plane stress or plane strain

condition.

Consider a though-thickness cracked plate subjected to in-plane loading. Because of the

large crack tip stresses normal to the crack plane, the crack tip material tries to contract in the

crack plane, but is prevented from the surrounding material. This constraint causes a triaxial

state of stress near the crack tip. Therefore, plane strain conditions exist in the interior of the

plate. However, because there are no stresses normal to the free surface, plane tress conditions

exist on the surfaces of the plate. Moreover, there is a region near the plate surface where the

state of stress is neither plane stress nor plane strain.

2.3 ELASTIC-PLASTIC FRACTURE MECHANICS

Linear elastic fracture mechanics can successfully characterize the crack tip stress field only if

the K dominant zone exists, i.e. the small scale yielding prevails. In the case when the plastic

zone is large compared with the crack size, plate thickness or ligament, linear elastic fracture

mechanics is no longer able to characterize the crack tip environment. To deal with the case of

large scale yielding at the crack tip, an alternative has been developed which is Elastic-Plastic

Fracture Mechanics (EPFM). In EPFM two parameters have been adopted to take over from

the stress intensity factor in LEFM which are the crack tip opening displacement (CTOD) and

the J contour integral. Their relationship has been shown by Rice [14] as J = λςyCTOD for

non-linear elasticity where 1 ≤ λ ≤ 2. Two Elastic-Plastic parameters are introduced in below:

the crack tip opening displacement (CTOD) and the J contour integral. Both parameters

21

describe crack tip conditions in elastic-plastic materials, and each can be used as a fracture

criterion.

2.3.1 CRACK TIP OPENING DISPLACEMENT

From examining the fractured test specimens, Wells [15] noticed that the crack faces had

moved apart prior to fracture; plastic deformation blunted an initially sharp crack. The degree

of crack blunting increased in proportion to the toughness of the material. This observation

propose the opening at the crack tip as a measure of fracture toughness. Today, this parameter

is known as the crack tip opening displacement (CTOD).

First, an approximate analysis that related CTOD to the stress intensity factor was

performed in the limit of small scale yielding [14]. Consider a crack with a small plastic zone,

crack tip plasticity makes the crack behave as if it were slightly longer. Thus, CTOD was

estimated by solving for the displacement at the physical crack tip, assuming an effective crack

length of a+ry. The displacement ry behind the effective crack tip is given by:

uy =κ + 1

2μKI

ry

2π (2.2)

and the plastic zone correction for plane stress is

ry =1

2π

KI

ςYS

2

(2.3)

Substituting Equation.(2.3) into Equation (2.2) gives

δ = 2uy =4

π

KI2

ςYS E (2.4)

where δ is the CTOD. Alternatively, CTOD can be related to the energy release rate

22

δ =4

π

𝒢

ςYS (2.5)

Thus in the limit of small scale yielding, CTOD is related to 𝒢 and KI. Wells postulated

that CTOD is an appropriated crack tip characterizing parameter when LEFM is no longer

valid. This assumption was shown to be correct several years later when a unique relationship

between CTOD and the J integral was established.

The strip yield model provides an alternate means for analyzing CTOD [32], where the

plastic zone was modeled by yield magnitude closure stresses. The size of the strip yield zone

was defined by the requirement of finite stresses at the crack tip. The CTOD can be defined as

the crack opening displacement at the end of the strip yield zone. According to this definition,

CTOD in a through crack in an infinite plate subject to a remote tensile stress is given by [32]

δ =8ςYS a

πEln sec

π

2

ς

ςYS (2.6)

Series expansion of the ln sec term gives

δ =8ςYS a

πE 1

2 π

2

ς

ςYS

2

+1

12 π

2

ς

ςYS

4

+ ⋯

=KI

2

ςYS E 1 +

1

6 π

2

ς

ςYS

2

+ ⋯ (2.7)

δ =KI

2

ςYS E=

𝒢

ςYS (2.8)

The strip yield model assumes plane stress conditions and a nonhardening material. The

actual relationship between CTOD and KI and 𝒢 depends on stress state and strain hardening.

The more general form of this relationship can be expressed as follows:

23

δ =KI

2

mςYSE′=

𝒢

mςYS (2.9)

where m is a dimensionless constant that is approximately 1.0 for plane stress and 2.0 for plane

strain.

There are a number of alternative definitions of CTOD. The two most common definitions,

which are illustrated in Figure 2.5., are the displacement at the original crack tip and the 90o

intercept. The latter definition was suggested by Rice [15] and is commonly used to infer

CTOD in finite element measurements. Note that these two definitions are equivalent if the

crack blunts in a semicircle.

2.3.2 J CONTOUR INTEGRAL

The J contour integral has enjoyed great success as a fracture characterizing parameter for

nonlinear materials. By idealizing elastic-plastic deformation as nonlinear elastic, Rice [15]

provided the basis for extending fracture mechanics methodology well beyond the validity

limits of LEFM.

The J integral in the EPFM is defined as the line integral

J = W dyΓ

− T∂u

∂x ds (2.10)

for a two dimensional crack as shown in Figure 2.6a. The Г is an arbitrary contour around the

crack tip in the counter clockwise direction, W = ςij dεij , T is the traction perpendicular to Г

in an outward direction (Ti = ςij nj), u is the displacement in the x-direction and ds is an

element on Г. Obviously the term T∂u

∂x is of the same dimension as ςijεij . It has been shown

[15] that this line integral is independent of the path Г.

24

Applying this to fracture mechanics for linear elasticity and non-linear elasticity, Rice has

shown that the J integral as defined in Equation 2.10 is the change in potential energy (V) for a

virtual crack extension (Δa).

J = −∂V

∂a (2.11)

In the case of linear elasticity where small scale yielding prevails, the energy release rate

G = −∂υ

∂a, and hence

J = G =KI

2

E′ (2.12)

where E'=E for plane stress and E′ =E

1−ν2 for plane strain.

Because of this path independence, any paths such as a circular contour around the crack

tip as shown in Figure 2.6b would be able to give a unique J integral. With Г equal to a circle of

radius r substituted into Equation 2.10, the line integral becomes

J = Wcosθ − T∂u

∂x rdθ

π

−π

(2.13)

where θ and r is the polar coordinates. Because of the path independence, the J integral in

Equation 2.13 can not change with r. As the term Wcosθ − T∂u

∂x is proportional to the

strain energy, the strain energy ςijεij must be a function of r−1 such that the effect of r can

be wiped out from the J integral.

Consider the stress-strain relationship proposed by Hutchinson [16] and Rice and

Rosengren [17] for non-linear elasticity:

ε

εy= α

ς

ςy

n

(2.14)

25

where α is a dimensionless constant and n is the strain hardening exponent. Together with the

r−1 singularity in strain energy, the power singularity in stress and in strain is found to be

r−1/(n+1) and r−n/(n+1) respectively. This is called the HPP singularity, named after the

Hutchinson [16], Rice and Rosengren [17]. Based on these results, the J integral was used to

represent the crack tip stress and strain fields for non-linear elasticity and the following

expressions were derived:

ςij = ςy J

αςyεyIr

1n+1

fij θ (2.15a)

εij = αςy J

αςyεyIr

nn+1

gij θ (2.15b)

where I is a numerical constant depending on the stress strain relation. Equation 2.15 shows

that, the J integral is not only an energy parameter but also a parameter domination the stress

and strain fields in a non-linear elastic material. However, it is not as general as stress

intensity factor in LEFM because Equation 2.15 is dependent on the material constant n. For

linear elastic material, n=1, α=1, I=2π and hence KI = E′J (Equation 2.12).

Analogous to the K dominance, there are some requirements for J dominance. In the first

place, the path independence, the power singularity and Equation 2.15 are all derived from the

assumption of non-linear elastic materials. This assumption implies that the stress strain path of

loading and unloading must be identical. For the real elastic plastic material, however, the

unloading will have a different path from that of loading. During crack extension, materials

behind the crack tip will experience unloading. Therefore, for J dominance to be valid for real

elastic plastic material, only stationary cracks can be dealt with. Secondly, the size of the

inelastic region around the crack tip should be small compared to the J dominant zone.

As one of the parameters to measure the severity at the crack tip, the test of the critical

value of JIc at the onset of a stable crack extension, taking amount of the size limitation of

26

thickness (T), crack length (a) and ligament (W − a) has been given by ASTM [30] and BSI

[29]:

T, a, W − a ≤ 25JIc

ςy 2.16

With these size limitations, the plastic zone size is close to that required for the K

dominance [31]. Therefore, the critical stress intensity factor KIc can be converted from the

experimental JIc through Equation 2.12 based on the plane strain condition. The advantage is

that the specimen size can be much smaller than that required for testing KIc especially for high

toughness materials. For example, to test the toughness of a material of KIc=6000N/mm3/2

,

ςy = 350N/mm2 and E=210000N/mm2

, the required thickness for a valid KIc is 735mm

while it is 11.2mm for a valid JIc.

27

Figure 2.1 The three modes of loading that can be applied to a crack

Figure 2.2 General mode I problem

28

Figure 2.3 Plastic zone shapes according to Von Mises criteria

Figure 2.4 Toughness as a function of thickness

29

Figure 2.5 Alternative definitions of CTOD

Figure 2.6 Contour around crack tip

30

REFERENCE

[1] Engineering News Record, Sept., 20, 1962.

[2] Griffith, A.A. "The Phenomena of Rupture and Flow in Solids." Philosophical

Transactions, Series A, Vol. 221, PP. 163-198, 1920.

[3] Inglis, C.E., "Stress in a Plate Due to the Presence of Cracks and Sharp Corners."

Transactions of the Institute of Naval Architects, Vol. 55, pp. 219-241, 1931.

[4] Irwin, G.R., "Fracture Dynamics." Fracturing of Metals, American Society for Metals,

Cleveland, pp. 147-166, 1984.

[5] Orowan, E., "Fracture and Strength of Solids." Reports on Progress in Physics, Vol. XⅡ,

P. 185-232, 1948.

[6] Mott, N.F., "Fracture of Metals : Theoretical Considerations." Engineering, Vol. 165, pp.

16-18, 1948.

[7] Irwin, G.R., "Onset of Fast Crack Propagation in High Strength Steel and Aluminum

Alloys." Sagamore Research Conference Proceedings, Vol. 2, pp. 289-305, 1956.

[8] Westergaard, H.M., "Bearing Pressures and Cracks." Journal of Applied Mechanics, Vol.

6, pp. 49-53, 1939.

[9] Irwin, G.R., "Analysis of Stresses and Strains near the End of a Crack Traversing a Plate."

Journal of Applied Mechanics, Vol. 24, pp. 361-364, 1957.

[10] Williams, M.L., "On the Stress Distribution at the Base of a Stationary Crack." Journal

of Applied Mechanics, Vol. 24, pp. 109-114, 1957.

[11] Wells, A.A., "The Condition of Fast Fracture in Aluminum Alllys with Particular

Reference to Comet Failures." British Welding Research Association Report, April 1955.

[12] Winne, D.H., and Wundt, B.M., "Application of the Griffith-Irwin Theory of Crack

Propagation to the Bursting Behavior of Disks, Including Analytical and Experimental

Studies." Transactions of American Society of Mechanical Engineers, Vol. 80, pp. 1643-1655,

1958.

31

[13] Paris, P.C., Gomez, M.P., and Anderson, W.P., "A Rational Analytic Theory of Fatigue."

The Trend in Engineering, Vol. 13, pp. 9-14, 1961.

[14] Wells, A.A., "Unstable propagation in metals: Cleavage and Fast Fracture." Proceedings

of the Crack Propagation Symposium, Vol. 1, Paper 84, Cranfield, UK, 1961.

[15] Rice, J.R., "A Path Independent Integral and the Approximate Analysis of Strain

Concentration by Notches and Cracks." Journal of Applied Mechanics, Vol. 35, pp. 379-386,

1968.

[16] Hutchinson, J.W., "Singular Behavior at the End of a Tensile Crack Tip in a Hardening

Material." Journal of the Mechanics and Physics of Solids, Vol. 16, pp. 13-31, 1968.

[17] Rice, J.R. and Rosengren, G.F., "Plane Strain Deformation near a Crack Tip in a

PowerLaw Hardening Material." Journal of Mechanics and Physics of Solids, Vol. 16, pp. 1-12,

1968.

[18] Begley, J.A. and Landes, J.D., "The J-Integral as a Fracture Criterion." ASTM STP 514,

American Society for Testing and Materials, Philadelphia, pp. 1-20, 1972.

[19] E 813-81, "Standard Test Method for JIc, a Measure of Fracture Toughness." American

Society for Testing and Materials , Philadelphia, 1981.

[20] Shih, C.F. and Hutchinson, J.W., "Fully Plastic Solutions and Large-Scale Yielding

Estimates for Plane Stress Crack Problems." Journal of Engineering Materials and Technology,

Vol. 98, pp. 289-295, 1976.

[21] Kumar, V., German, M.D., and Shih, C.V., "An Engineering Approach for

Elastic-Plastic Fracture Analysis." EPRI Report NP-1931, Electric Power Research Institute,

Palo Alto, CA, 1981.

[22] Burdekin, F.M. and Dawes, M.G., "Practical Use of Linear Elastic and Yielding Fracture

Mechanics with Particular Reference to Pressure Vessels." Proceedings of the Institute of

Mechanical Engineers Conference , London, pp. 28-37, May 1971.

[23] Wells, A.A., "Application of Fracture Mechanics at and Beyond General Yielding."

British Welding Journal, Vol. 10, pp. 563-570, 1963.

32

[24] Harrison, R.P., Loosemore, K., Milne, I, and Dowling, A.R., "Assessment of the

Integrity of Structures Containing Defects." Central Electricity Generating Board Report

R/H/R6-Rev 2, April 1980.

[25] Dugdale, D.S., "Yielding in Steel Sheets Containing Slits." Journal of the Mechanics

and Physics of Solids, Vol. 8, pp. 100-104, 1960.

[26] Barenblatt, G.I., "The Mathematical Theory of Equilibrium Cracks in Brittle Fracture."

Advances in Applied Mechanics, Vol. VⅡ, Academic Press, pp. 55-129, 1962.

[27] Shih, C.F. "Relationship between the J-Integral and the Crack Opening Displacement for

Stationary and Extending Cracks." Journal of the Mechanics and Physics of Solids, Vol. 29, pp.

305-326, 1981.

[28] E 399-90, STANDARD TEST METHOD FOR PLANE-STRAIN FRACTURE TOUGHNESS OF METALLIC MATERIALS.

ASTM, Philadelphia, 1990.

[29] British Standards Institution, BS7448 METHODS FOR DETERMINATION OF KIC, CRITICAL CTOD

AND CRITICAL J VALUES OF METALLIC MATERIALS. BSI, London, 1991.

[30] E 813-88,STANDARD TEST METHOD FOR JIc, A MEASURE OF FRACTURE TOUGHNESS. Annual Book of

ASTM Standard, pp 698-712, 1998

[31] McMeeking R.M., Parks D.M., "On criteria for J dominance of crack tip fields in large

scale yielding." ASTM STP 668, pp 175-194, 1979.

[32] Burdekin, F.M. and Stone, D.E.W., "The Crack Opening Displacement Approach to

Fracture Mechanics in Yielding Materials." Journal of Strain Analysis, Vol. 1, pp. 145-153,

1966.

33

CHAPTER 3: BACKGROUND OF FINITE ELEMENT METHODS

3.1 INTRODUCTION

The main idea of the finite element method is to represent a given domain as a collection of

discrete parts called elements. Then the elements are connected at nodes where continuities are

enforced. In 1941, Hrenikoff [1] was the first, who used this idea to represent a plane elastic

medium as a collection of bars and beams. In 1943, Courant [2] used an assemblage of

triangular elements and the principle of minimum to total potential energy to study a torsion

problem. However, the formal presentation of the finite element method is attributed to the

works of Turner et al [3] in 1956 and Argyris and Kelsey [4] in 1960. The term "finite element"

was first used by Clough [5] in 1960. Thereafter, the literature on finite element application has

grown rapidly. In 1967, Zienkiewicz and Cheung [6] wrote and published the first finite

element book. Today, there are numerous journals and books, which are primarily devoted to

the theory and application of the finite element method.

The accuracy of the solutions to crack problems obtained by the finite element method

could not be guaranteed [7-10], until several special crack tip elements were developed by

Tracey [11], Blackburn [12], Akin [13] and Yamada et al [14]. However, the disadvantage of

these special crack tip elements is that changes of the stiffness matrix must be made. In 1970s,

Henshell and Shaw [15] and Barsoum [16,17] observed that the collapsed, quarter-point

element exhibits the required stress singularity characteristic emanating from the node at the

crack tip. Since its inception, standard finite element programs can be used to obtain accurate

solutions to crack problems.

In this chapter, main steps involved in the finite element analysis are summarized.

Application of the finite element method to fracture mechanics such as crack tip singularity ,

the limited displacement extrapolation technique, and the virtual crack extension method are

described in follow sections, respectively.

34

3.2 BASIC PROCEDURE FOR THE FINITE ELEMENT ANALYSIS

One of the advantages of the finite element method over other numerical method is its unified

systematic problem solving procedure that can be automated easily for use on digital

computers. Main steps involved in the finite element analysis are summarized as follows:

1. A body is divided into discrete surfaces in a 2D case and discrete volumes in a 3D case.

These surfaces and volumes are the so-called finite elements. A certain number of 'nodes'

are defined on the boundaries or inside the elements.

2. All discrete elements have unique displacements at the common nodes on the boundaries

between elements. By this way, all elements are interconnected. These displacements at

the nodes are the basic unknowns.

3. The variations of the displacements over the elements are given by an interpolation

function based on the nodal displacements. This function is called the displacement

function.

4. By taking the derivatives of the displacements with respect to the coordinates, the strains

over the elements can be defined. Associated with the material properties such as the

elastic module and Poisson's ratio, the stress over elements can also be defined. Both

strains and stresses are expressed in terms of nodal displacements.

5. A set of equations connecting the nodal force and the nodal displacement is obtained as

below where [K] is the stiffness matrix, {F} and {δ} are the total nodal forces and nodal

displacements. This set of equations is called the element stiffness equation.

F = K δ (3.1)

6. A system stiffness equation is then obtained by assembling all element stiffness equations

and taking all boundary conditions such as forces and constraints in to account through the

principle of equilibrium. Solving this system stiffness equation gives the solution of the

35

basic unknowns, the nodal displacements.

3.3 APPLICATION OF FINITE ELEMENT METHOD TO FRACTURE

MECHANICS

The analytical methods for solving crack problems are limited to the relatively simple cases

such as smooth crack shape, regular member geometry and simple loading. Complicated

situations which cannot be deal with by analytical methods may resort to numerical approaches.

As computing power increases rapidly nowadays, various numerical methods such as the finite

element method (FEM), the boundary element method and the finite difference method have

been developed and applied in fracture mechanics. Among all numerical methods, the FEM is

the most widely used as a tool for the solution of practical engineering fracture problems due to

its ability to deal with the awkward geometries and complicated boundary conditions in a

unified manner. A number of techniques have been suggested and incorporated in the FEM in

order to evaluate the most important parameter, the stress intensity factor, as accurately as

possible. Some of these techniques are discussed in this sections which are the simulation of

the crack tip singularity, the limited displacement extrapolation technique and, the virtual

crack extension method. For problem which involves material yielding or large deformation,

non-linear finite element procedure, such as the Newton-Raphson method, should be applied.

Brief description of this non-linear procedure is also included in following section

3.3.1 CRACK TIP SINGULARITY

In order to simulate the crack tip singularity, the original 8 noded isoparametric element in the

ξ-η space as shown in Figure 3.1 should be converted to a special crack tip element as shown

in Figure 3.1b [16]. The major procedures are to collapse three nodes on one side to the same

position and to move two mid side nodes to the quarter point of the side. This element is

mapped from x-y space through the following mapping:

36

x = Ni ξ, η xi

8

i=1

y = Ni ξ, η yi

8

i=1

−1 ≤ ξ, η ≤ 1

(3.2)

where (xi, yi) are the coordinates at node i in x-y space. The Ni in Equation 3.2 are the shape

functions for node i in terms of the corresponding positions (ξi, ηi) in ξ-η space which is given

as:

Ni = 1 + ξξi 1 + ηηi − 1 + η2 1 + ηηi − 1 − η2 1 + ξξi ξi

2ηi2

4

+ 1 − ξ2 1 + ηηi 1 − ξi2

ηi2

2+ 1 − η2 1 + ξξi 1 − ηi

2 ξi

2

2 (3.3)

The displacements u and v in x and y directions are given in terms of the nodal displacement ui

and vi as below:

u = Ni ξ, η ui

8

i=1

v = Ni ξ, η vi

8

i=1

(3.4)

The strain of the element can be obtained by taking the derivative of u and v with respective to

x and y:

εx

εy

εxy

=

∂Ni

∂x0

0∂Ni

∂y∂Ni

∂y

∂Ni

∂x

ui

vi (3.5)

The derivatives of Ni can be expressed in terms of ξ-η,

37

∂Ni

∂x∂Ni

∂y

= J −1

∂Ni

∂ξ∂Ni

∂η

(3.6)

where J is the Jacobian matrix and is defined as

J =

∂x

∂ξ

∂y

∂ξ∂x

∂η

∂y

∂η

(3.7)

Equation 3.6 shows that if the Jacobian matrix vanishes at the crack tip, the strain singularity

can be achieved. This is equivalent to having a zero determinant of J. For the elements outlined

in Figure 3.1, the coordinates of the nodes are,

x1 = x4 = x8 = 0

x5 = x7 =h

4x2 = x3 = x6 = h

y1 = y4 = y8 = y6 = 0

y5 = −y7 = −l

4y2 = −y3 = −l

(3.8)

Substituting the Equation 3.8 into Equation 3.3 and 3.2 gives:

x =

h 1 + ξ 2

4

y =lη 1 + ξ 2

4

(3.9)

The Jacobian matrix can be calculated,

38

J =

∂x

∂ξ

∂y

∂ξ∂x

∂η

∂y

∂η

=

1

2h 1 + ξ

1

2lη 1 + ξ

01

4l 1 + ξ 2

(3.10)

The determinant of J is given by,

J =hl 1 + ξ 3

8 (3.11)

When x=0 or ξ = −1, J = 0. The displacement u along the line 1-2 is given by,

u = −ξ 1 − ξ

2u1 +

ξ 1 − ξ

2u2 + 1 + ξ2 u5 (3.12)

and,

∂u

∂ξ=

2ξ − 1

2u1 +

2ξ + 1

2u2 − 2ξu5

∂ξ

∂x=

1

hx

(3.13)

Therefore, the strain in the x-direction is given by,

εx =∂ξ

∂x

∂u

∂ξ= −

1

2

3

xh−

4

h u1 +

1

2 −

1

xh+

4

h u2 +

2

xh−

4

h u5 (3.14)

The r−1/2 singularity in strain has been achieved. This approach can be extended to the three

dimensional case and achieve the same required singularity. One of the three dimensional

crack tip element is shown in Figure 4.4. In the application in finite element analysis, the crack

tip elements are arranged as a ring surrounding the crack tip as shown in Figure 3.2. Due to the

nature of crack tip elements, there will be many nodes located at the crack tip position. When

boundary constraints are imposed, there are two options. One of them is to constrain just one

node while another one is to constrain all nodes. However, a sensitivity test shows that no

significant difference in the results is given by these two constraint options. Finally, it should

39

be noted that, crack tip elements should be applied for the elastic case. In case of inelastic

analysis, crack tip element should not be used because the stress at crack tip will be limited to

the yield stress and the singularity will not occur.

3.3.2 THE LIMITED DISPLACEMENT EXTRAPOLATION TECHNIQUE

The displacement extrapolation technique was first proposed by Chan et al. [8]. Stress intensity

factors can be estimated by utilizing the nodal displacements along the crack face.

Mathematically, the displacement extrapolation technique is expressed as:

KIDET = limr∗i→0 KI

∗i (3.15)

and KI∗i =

μ

κ + 1

2π

r∗iv r∗i (3.16)

where μ is the shear modulus, κ is (3-ν)/(1+ν) for plane stress and (3-4ν) for plane strain, ν is

Poisson's ratio, r∗i is the distance between ith node and the crack tip, and v r∗i is the

displacement of one crack face with respect to the other in the direction normal to the crack

plane at distance r∗i from the crack tip, as shown in Figure 3.3. By plotting the nodal K∗i

against the distance r∗i, the best straight line is fitted empirically to obtain the stress intensity

factor KDET at r = 0.

Bank-Sills and Einav [18] observed the lack of consistency of the solutions and

suggested the use of linear regression techniques. In essence, all possible sequential

combinations of the nodal K∗i are evaluated. Because the K∗i associated with the

quarter-point node has been found to provide unreliable stress intensity factor when it was

utilized in DET, it must be excluded in the combinations, with each combination set of the

nodal K∗i, the linear correlation coefficient ρ is calculated given that:

40

ρ = xiyi −

1N xiyi

xi2 −

1N xi 2 yi

2 −1N yi 2

12

(3.17)

where xi is equal to r∗i , yi is equal to K∗i, and N is the number of K∗i in the set evaluated.

The best straight line is fitted with the set which produces a value of ρ closest to unity. Its

corresponding intercept at r=0 is KDET . In this approach, 0.5(N-1)(N-2) combination sets

must be evaluated.

The implicit assumption of the displacement extrapolation technique is that the nodal K∗i

should vary linearly along the crack face. Carpenter [19] proved analytically that the

displacements along any ray emanating from the crack tip generally do not vary linearly.

However, it was observed that the displacements do vary linearly along the crack face for the

case of single edge crack configuration under uniform tension. This explains the excellent

results obtained by Chan et al. [8] for this configuration.

The limited displacement extrapolation technique was first proposed by Lim et al. [20]. It

is proposed that only three nodal K∗i values associated with an element immediately

adjacent to the quarter-point element are used to obtain the stress intensity factor, as

illustrated in Figure 3.4. Mathematically, it is expressed by:

KLDET = limr∗i→0

f K∗2 , K∗3 , K∗4 (3.18)

where f is a best fitted line through K∗2 , K∗3 and K∗4. Lim et al. [20] showed that this

approach does reduce the influence of the shortcomings associated with the original

displacement extrapolation technique.

The displacement extrapolation technique was developed originally in two dimensions.

In three dimensions, Banks-Sills [21] suggested that this approach could be employed along

any ray perpendicular to the crack front which lies on the crack face. The displacement (v) in

Equation (3.16) is replaced by the displacement normal to the crack face.

41

3.3.3 VIRTUAL CRACK EXTENSION METHOD

The fundamental result from finite element analysis is the nodal displacements from which

the nodal forces, strains and stresses can be derived. Introducing the special crack tip element

can merely help to have a real stress and strain environment at the crack tip. The stress

intensity factor is not able to be obtained directly from the finite element analysis. Therefore,

post processing of the obtained displacements, strains and stresses is necessary in order to get

the stress intensity factor. Two categories of this post processing are available.

One of them is based on the near-tip solution. An example of this method is the nodal

force method proposed by Newman [22] in which the nodal forces on a radial line orthogonal

to the crack front are deal with. The stress intensity factors are then expressed in terms of the

distance from the crack tip based on Equation (2.1). The stress intensity factor at the crack tip

can be obtained by extrapolation. Another approach based on the nodal displacements in

place of the nodal force can also get the work done. Advantages of this method are that it can

separate the stress intensity factor for mode I, II, and III. For the three dimensional case, no

assumption such as plane strain or plane stress has to be made. The disadvantage is the region

where Equation (2.1) valid is very localized and highly affected by the fineness of the crack

tip mesh. Although this approach works well in a two dimensional case, it is difficult to

extend to three dimensional finite element analysis.

Another one is based on the estimate of the energy change due to a crack advance. There

several advantages of this approach over the previous one. Firstly, the finite element method

is a numerical approach based on minimizing the potential energy. Therefore, considering

energy of a structure will give a better result than considering the stresses. Secondly, the

approach can extended to cover the non-linear material case. Finally, it is easy to extend to

the three dimensional case. The disadvantage is that this approach cannot separate the

different mode of fracture for mixed mode analysis.

An example of this approach is the virtual crack extension method [23, 24, 25]. As

defined in Section 2.3.2, the J and G are the change of potential energy of a structure

42

Equation (2.11). In a cracked body subjected to mode I loading, the potential energy (P) is

given as:

P =1

2 u T K u − u T F (3.19)

where the {u} is the displacement vector, [K] is the stiffness matrix and {F} is the global

force vector. Under fixed load conditions, the energy release rate (G and J) can be obtained

by taking the derivative of potential energy as below:

G = J = − ∂P

∂a

F=

∂ u T

∂a u T K u − F −

1

2 u T

∂ K

∂a u − u T

∂ F

∂a (3.20)

As defined in Equation (3.1), the first term in Equation (3.20) is zero. For fixed load

conditions, the last term is also zero. Equation (3.20) shows that the energy release rate is

related to the derivative of stiffness matrix [K] with respective to the rack length a. Consider

two contours Γ0 and Γ1 and the crack advance δa as outlined in Figure 3.5. During the

virtual crack extension, elements inside Γ0 have a translation of δa without any distortion.

All elements outside Γ1 are kept unmoving. Consequently, only the elements between Γ0

and Γ1 are subjected to deformations. The energy release rate becomes:

G = J = −1

2 u T

∂ Ki

∂a u

n

i

(3.21)

The summation in Equation (3.21) is for the elements between Γ0 and Γ1. In practice, the

energy release rate is very sensitive to the size of the contours. Small one may give an

unstable result. Therefore, several contours should be investigated until the converged values

appear. In the three dimensional case, this virtual crack extension approach still applies by

replacing the crack length increment δa with incrementing the cracked area locally around the

crack front as shown in Figure 3.6. Recently, Shih has derived a new approach, the energy

43

domain integral, which is very similar to virtual crack extension. This approach is being used

by ABAQUS for J-integral evaluation.

3.3.4 EVALUATION OF J-INTEGRAL BY THE DOMAIN INTEGRAL METHOD

The energy domain integral method for the J integral evaluation was proposed by Shih et al.

[28, 29]. This approach can be applied to both quasistatic and dynamic problems with elastic,

plastic, or viscoplastic material responses, as well as thermal loading. Moreover, the domain

integral formulation is relatively simple to implement numerically, and it is very efficient.

Because of these, it was adopted for the J integral evaluation in ABAQUS. The domain

integral formulation is described in the following paragraphs.

The general expression for the J integral, which includes the effects of inertia and the

inelastic material behavior, is given by [29];

J = limΓ0→0

W + T δ1i − ςij

∂uj

∂x1 nidΓ

Γ0

(3.22)

where Γ0 is a vanishingly small counter-clockwise closed path that surrounds the crack tip,

T is the kinetic energy density, W is the stress work, δ1i is the Kronecker delta, ςij are the

stress tensors, uj are the displacement vector components, and ni are the components of the

unit vector normal to Γ0, as shown in Figure 3.7.

Various material behavior can be taken into account through the definition of W. For an

elastic material loaded under quasistatic conditions (T=0), in the absence of thermal loading,

the stress work is given by:

W = ςij dεije (3.23)

44

where εij are strain tensors. Equation (3.22) is not suitable for numerical analysis, because it

is difficult to determine the stresses and strains along a vanishingly small contour.

Consider a closed contour constructed by connection inner and outer contours, as

illustrated in Figure 3.7. The outer contour (Γ1) is finite, while the inner contour (Γ0) is

vanishingly small. For quasistatic condition (T=0), in the absence of body forces, Equation

(3.22) can be written in terms of the following integral around the closed contour

Γ∗ = Γ1 + Γ+ + Γ− − Γ0 [28, 29]:

J = ςij

∂uj

∂x1− Wδ1i qmidΓ − ς2j

∂uj

∂x1qdΓ

Γ++Γ−Γ∗

(3.24)

where Γ+ and Γ− are the upper and lower crack faces, respectively, mi is the outward

normal on Γ∗, and q is an arbitrary but smooth function that is equal to unity on Γ0 and zero

on Γ1 . Note that mi = −ni on Γ0 . In the absence of crack face tractions, the second

integral in Equation (3.24) vanishes.

Assume that the crack faces are traction free. Apply the divergence theorem to Equation

(3.24) yields:

J = ∂

∂xi ςij

∂uj

∂x1− Wδ1i q dA

A∗

= ςij

∂uj

∂x1− Wδ1i

∂q

∂xidA

A∗

+ ∂

∂xi ςij

∂uj

∂x1 −

∂W

∂x1 qdA

A∗

(3.25)

where A∗ is the area enclosed by Γ∗. Moreover, for elastic material, the following relation

exists.

∂

∂xi ςij

∂uj

∂x1 −

∂W

∂x1= 0 (3.26)

45

Hence, for a linear of nonlinear elastic material under quasistatic conditions, in absence of

body forces, thermal strains, and crack face tractions, the J integral can be expressed as:

J = ςij

∂uj

∂x1− Wδ1i

∂q

∂xidA

A∗

(3.27)

In three dimensions, it is necessary to convert Equation (3.22) into a volume integral.

Figure 3.8 shows a planar crack in a three dimensional body; η is concerned, it is convenient

to define a local coordinate system at η, with x1 normal to the crack front, x2 normal to the

crack plane, and x3 tangent to crack front. The J integral at η is given by Equation (3.22),

where the contour Γ0 lies in the x1-x2 plane.

Consider a tube of length ΔL and radius r0 constructed surrounds a segment of the crack

front, as illustrated in Figure 3.8. Assuming quasi-static conditions, a weighted average J over

the crack front segment ΔL can be defined by:

J ∆L = J η qdη

∆L

= limr0→0

Wδ1i − ςij

∂uj

∂x1 qnidS

s0

(3.28)

where J(η) is the point-wise value of J, S0 is the surface area of the tube, and q is a weight

function.

The weight function, q, can be interpreted as a virtual crack advance [23-27]. Figure 3.9

illustrates an incremental crack advance over ΔL, where q is defined by:

∆a η = q η ∆amax (3.29)

and the incremental area of the virtual crack advance is obtained by:

46

∆Ac = ∆amax q η dη

∆L

(3.30)

In fact, the weight function is not necessary to be defined in terms of a virtual crack extension,

but it is easy to understand the physical aspect of this parameter by attaching the concept of

virtual crack extension [23-27].

If a second tube of radius r1 is constructed around the crack front, as illustrated in Figure

3.10, the weight average J in terms of a closed surface can be defined as:

J ∆L = ςij

∂uj

∂x1− Wδ1i qmidS

S∗

− ς2j

∂uj

∂x1qdS

S++S−

(3.31)

where the closed surface S∗ = S1 + S+ + S− − S0 , and S+ and S− are the upper and

lower crack faces, respectively. For a linear or nonlinear elastic material under quasistatic

conditions, in the absence of body forces, thermal strains, and crack face tractions, the weight

average J can be given as the following expression by applying the divergence theorem:

J ∆L = ςij

∂uj

∂x1− Wδ1i

∂q

∂xidV

V∗

(3.32)

Equation (3.32) requires that q=0 at either end of ΔL. The virtual crack advance interpretation

of q fulfills this requirement.

If the point-wise value of the J integral does not vary appreciably over ΔL, as a first

approximation, J(η) can be expressed by:

J η ≈J ∆L

q η, r0 dη∆L

(3.33)

47

If the q gradient along the crack front is steep relative to the variation in J(η), then

Equation (3.33) gives a reasonable approximation of J(η).

Note that Equation (3.32) was derived in terms of a local coordinate system. The domain

integral formulation can be expressed in terms of a fixed coordinate system by replacing q

with a vector quantity, qi, and evaluation the partial derivatives in the integrand with respect

to xi instead of x1, where the vectors qi and xi are parallel to the direction of crack growth.

3.3.5 NEWTON-RAPHSON METHOD

In finite element analysis, for problem which involves material yielding or large deformation,

non-linear finite element procedure, such as the Newton-Raphson method [30], should be

applied. The Newton-Raphson method uses an iterative process to approach one root of a

function.

The idea of the method is as follows: one starts with an initial guess which is reasonably

close to the true root, then the function is approximated by its tangent line, and one computes

the x-intercept of this tangent line. This x-intercept will typically be a better approximation to

the function's root than the original guess, and the method can be iterated.

In Elastic-Plastic Fracture Mechanics, the Jacobian matrix, which is related to the

deformation of the crack, was solved by Newton-Raphson method. In order to consider the

non-linear material properties in the finite element analysis, the non-linear material property,

such as the full range stress versus strain curve of the material, should be included in the

analysis. Based on the input data of the material curve, the initial stiffness matrix [K], which

is identically the same as the initial tangent-stiffness matrix, was calculated. Then, the

deformation of model δ is analyzed by solving K δ = F , where F is proportional to

the actual load but of arbitrary level.

Imagine that applied the first step load F1 and initial stiffness matrix K1 is identically

the same as the initial tangent-stiffness matrix, then the corresponding displacement, δ1, is

determined according to the following equation:

48

K1 δ1 = F1 (3.34)

The load is now increased to the next step F2 and the corresponding displacement δ2 is

sought. From initial tangent-stiffness matrix K1 , δ2 is solved by Equation (3.35):

K1 δ2 = F2 (3.35)

However, δ2 resolved from Equation (3.35) is not the reality displacement in the second

load step. From the given stress versus strain curve of the material property, the

corresponding force of δ2 can be calculated as 𝐹(δ2), and then the residual force is

calculated according to the following equation.

∆𝐹1 = 𝐹2 − 𝐹 δ2 (3.36)

Based on the input data of the full range stress versus strain curve of the material, new

tangent stiffness was obtained K2 . Then， ∆δ1 is seek for which f δ2 + ∆δ1 = F2 . and

∆δ1 = δ3 − δ2. Thus, with F δ2 and K2 , Equation (3.34) becomes

F2 = 𝐹(δ2) + K2 ∆δ1 or K2 ∆δ1 = F2 − 𝐹(δ2) (3.37)

where F2 − 𝐹(δ2) can be interpreted as a load imbalance-that is, as the difference

between the applied load F2 and the force F δ2 is ∆δ1. The solution process is depicted in

Figure 3.11. After computing ∆δ1, the displacement is updated to estimate δ3 = δ2 + ∆δ1,

for the next iteration and the process is repeated.

ABAQUS compares F2 − 𝐹(δ2) to a tolerance value. If F2 − 𝐹(δ2) is less

than this force residual tolerance, this updated results are accepted as the equilibrium solution.

The tolerance value is set to 0.5% of an average force in the structure, averaged over time.

49

ABAQUS automatically calculated this spatially and time-averaged force throughout the

simulation.

If F2 − 𝐹(δ2) is less than the current tolerance value, F2 and 𝐹(δ2) are in

equilibrium, and δ3 is a valid equilibrium configuration under the applied load. However, it

also checks that the displacement correction, ∆δ1, is small relative to the total incremental

displacement, if ∆δ1 is greater than 1% of the incremental displacement, performs another

iteration. Both convergence checks must be satisfied before a solution is said to have

converged for that load increment.

50

Figure 3.1 Contour around crack tip

Figure 3.2 Crack tip mesh with special crack tip element

51

Figure 3.3 Schematic illustrating the variables used in the displacement extrapolation

technique

Figure 3.4 Schematic illustrating the limited displacement extrapolation technique

52

Figure 3.5 Virtual crack extension in 2D case

53

Figure 3.6 Virtual crack extension in 3D case (local extension)

Figure 3.7 A closed contour surrounding the crack tip

54

Figure 3.8 Surface enclosing an increment of the crack front

Figure 3.9 Interpretation of q by the concept of virtual crack extension

55

Figure 3.10 A closed surface enclosing a volume of V*

Figure 3.11 Newton-Raphson Method

56

REFERENCE

[1] Hrenikoff, A., "Solution of Problems in Elasticity by the Framework Method",

Transactions of the ASME, Journal of Applied Mechanics, Vol. 8, pp. 169-175, 1941.

[2] Courant, R., "Variational Methods for the Solution of Problems of Equilibrium and

Vibration", Bulletin of the American Mathematical Society, Vol. 49, pp. 1-43, 1943.

[3] Turner, M., Clough, R.W., Martin, H.H., and Topp, L., "Stiffness and Deflection Analysis

of Complex Structures", Journal of Aeronautical Science, Vol. 23, pp. 805-823, 1956.

[4] Argyris, J.H., and S. Kelsey, "Energy Theorems and Structural Analysis", Butterworth

Scientific Publications, London 1960.

[5] Clough, R.W., "The Finite Element Method in Plane Stress analysis", Journal of

Structures Division, ASCE, Proceedings of 2nd Conference on Electronic Computation,

pp.345-378, 1960.

[6] Zienkiewicz, O.C. and Cheung, Y.K., "The Finite Element Method in Structural and

Continuum Mechanics", McGraw-Hill Publishing Company Limited, London, 1967.

[7] Swedlow, J.L., Williams, M.L. and Yang, W.H., "Elastoplastic Stresses and Strains in

Cracked Plates", Proceedings of 1st Int. Conf. Fracture, Sendai, Japan, Vol.1, pp. 259-282,

1965.

[8] Chan, S.K., Tuba, I.S. and Wilson, W.K., "On the Finite Element Method in Linear Elastic

Fracture Mechanics", Eng. Fracture Mechanics, Vol. 2, pp.1-17, 1970.

[9] Wilson, W.K., "Crack Tip Finite Element for Plane Elasticity", Westinghouse Rep No

71-1E7-FM-PWR-P2, 1971.

[10] Oglesby, J.J. and Lamackey, O., "An Evaluation of Finite Element Methods for the

Computation of Elastic Stress Intensity Factors", NSRDC Rep No 3571, 1972.

[11] Tracey, D.M., "Finite Elements for Determination of Crack Tip Elastic Stress Intensity

Factors", Eng. Fracture Mechanics, Vol.3, pp.255-266, 1971.

57

[12] Blackbure, W.S., Calculation of Stress Intensity Factors at Crack Tips Using Special

Finite Elements", The Mathematics of Finite Elements, edited by Whiteman, J.R., Academic

Press, pp.327-336, 1973.

[13] Akin, J.E., "The Generation of Elements with Singularities", International Journal for

Numerical Methods in Engineering, Vol. 10, pp. 1249-1259, 1976.

[14] Yamada, Y., Ezawa, Y., Nishguchi, I. and Okabe, M., "Reconsiderations on Singularity

or Crack Tip Elements", International Journal for Numerical Methods in Engineering, Vol. 14,

pp. 1525-1544, 1979.

[15] Henshell, R.D. and Shaw, K.G., Crack Tip Finite Elements Are Unnecessary",

International Journal for Numerical Methods in Engineering, Vol. 9, pp.495-507, 1975.

[16] Barsoum, R.S., "On the Use of Isoparametric Finite Elements in Linear Fracture

Mechanics", International Journal for Numerical Methods in Engineering, Vol. 10, pp.25-37,

1976.

[17] Barsoum, R.S., "Triangular Quarter-Point Elements as Elastic and Perfectly-Plastic

Crack Tip Elements", International Journal for Numerical Methods in Engineering, Vol. 11,

pp. 85-98, 1977.

[18] Banks-Sills, L. and Einav, O., "On Singular, Nine-Noded, Distorted, Isoparametric

Elements in Linear Elastic Fracture Mechanics", Int. J. Computers and structures, Vol. 25, pp.

445-449, 1987.

[19] Carpenter, W.C., "Extrapolation Techniques for Determining Stress Intensity Factors",

Eng. Fracture Mechanics, Vol. 18, pp. 193-201, 1992.

[20] Lim, I.L., Johnston, I.W. and Choi, S.K., "Comparison between Various

Displacement-Based Stress Intensity Factor Computation Techniques", International Journal

of Fracture, Vol. 58, pp. 193-201, 1992.

[21] Banks-Sills, L., "Application of the Finite Element Method to Linear Elastic Fracture

Mechanics", Applied Mechanics Rev., Vol. 44, no 10, pp.447-461, October 1991.

58

[22] Raju I.S. and Newman J.C., Jr., "Stress Intensity Factor for A Wide Range of

Semi-elliptical Surface Cracks in Finite-Thickness Plates", Engineering Fracture Mechanics,

Vol. 11, pp.817-829, 1979.

[23] Parks, D.M., "A Stiffness Derivative Finite Element Technique for Determination of

Crack Tip Stress Intensity Factors", International Journal of Fracture, Vol. 10, pp. 487-502,

1974.

[24] Hellen, T.K., "On the Method of Virtual Crack Extensions" International Journal for

Numerical Methods in Engineering, Vol. 9, pp. 187-207, 1975.

[25] Parks, D.M., "The Virtual Crack Extension Method for Nonlinear Material Behavior",

Computer Methods in Applied Mechanics and Engineering, Vol. 12, pp. 353-364, 1977.

[26] deLorenzi, H.G., "On the Energy Release Rate and the J-Integral of 3-D Crack

Configurations" International Journal of Fracture, Vol. 19, pp.183-193, 1982.

[27] deLorenzi, H.G., "Energy Release Rate Calculations by the Finite Element Method",

Eng. Fracture Mechanics, Vol. 21, pp.129-143, 1985.

[28] Shih, C.F., Moran, B., and Nakamur, T., "Energy Release Rate along a

Three-Dimensional Crack Front in a Thermally Stressed Body", International Journal of

Fracture, Vol. 30, pp. 79-102, 1986.

[29] Moran, B. and Shih, C.F., "A General Treatment of Crack Tip Contour Integrals",

International Journal of Fracture, Vol. 35, pp. 295-310, 1987.

[30] Cook, R.D., Malkus, D.S., Plesha, M.E., “Concepts and applications of finite element

analysis”. John Wiley & Sons, Inc., Canada, 1989.

59

CHAPTER 4: ANALYSIS OF THE CRACKED TUBULAR

4.1 INTRODUCTION

Tubular members have been used extensively in many engineering structures. Such as the

aerospace, offshore structure, pressure vessels, vehicles motor and drilling pipes. Cracks can

occur in those structures due to fatigue loading or due to accidental damage such as dropped

objects or vessel impact. However, for offshore structures, the primary loading includes wind

loading, water waves, current, gravitational loading basically from the top. Occasional

loadings such as impact also happen. Among the sources of loading, wind and waves produce

repeated loadings which cause potential fatigue problems in the structure.

It is well known that most structures contain defects to some extent as a result of

fabrication and manufacture although in some cases such imperfections are of less significance.

In some cases however, defects may grow in service under fatigue loading until they become

critical and finally lead to a failure.

Common forms of cracks are through-thickness cracks, surface cracks and embedded

cracks. In the literature, the shapes of through-thickness cracks, surface cracks and embedded

cracks are usually represented by rectangles, semi-ellipses and ellipses, respectively. In real

structure, a crack is often present in an irregular shape. However, it is difficult to obtain

solution to the crack in irregular shape. Therefore, complex crack must be represented by a

relatively simple shape. Several guides such as BS7910 [1] have been published to characterize

the dimensions of flaws. As illustrated in Figure 4.1, planar flaws may be characterized by the

height and length of their containment rectangles.

Surface crack are the most common flaws in tubular structures, either on the outside or

inner surface. Very often it may be sufficient to describe the surface crack as semi-ellipse with

the axes a and c. Semi-elliptical shape cracks develop from initial flaws, commonly occur in

these locations during welding. The abrupt change in the structure and its response to load

amplifies the nominal stresses in certain points around the crack. Therefore, the fatigue

60

sensitivity of the components is dependent on the combination of the cyclic nature of wave

loading (e.g. offshore tubular components) and initial defects. In an offshore environment it

will also depend on the corrosion aspects of the welded locations. For tubular members, surface

flaws can be re-characterized, dependent on the situations, as internal or external

semi-elliptical cracks, and in the axial or circumferential direction [1, 2]. The form of flaw

being studied in the present work is an internal circumferential semi-elliptical surface crack in

a tubular member.

4.2 FINITE ELEMENT MODELING AND ANALYSIS

In this study, three-dimensional linear elastic finite-element models were employed to simulate

the internal circumferential cracked tubes. A mesh generator was developed for generating the

finite element model of a cracked tubular member [3]. Three kinds of circumferential cracks

including (1) semi-elliptical surface crack, (2) semi-elliptical part through crack and (3)

through thickness crack were modeled. By using the mesh generator program, a finite element

model of plate with a semi-elliptical crack as shown in Figure 4.2 was first generated and then

transformed into a tube using Equation (4.1). Due to double symmetry, only one quarter of the

cracked tube was modeled. Then, the cracked tube model was provided with suitable boundary

conditions along the symmetrical boundaries and an arbitrary node of the model was restrained

in the x direction to prevent any rigid body motion in this direction as shown in Figure 4.3.

x′ =w

π+

w

π+ y cos

πx

w

y′ = w

π+ y sin

πx

w

𝑧 ′ = 𝑧

(4.1)

Quadratic brick elements C3D20 [4] with 20 nodes were used for the whole model. In order to

simulate the stress singularity at the crack tip, two kinds of crack tip element as shown in

Figure 4.4 were adopted for surface crack and part through cracks. In the case of surface

61

crack the 20-node isoparametric brick elements around the crack tip were degenerated into

wedge shapes focused at the crack tip, with adjacent mid side nodes moved to quarter as shown

in Figure 4.5 [5]. For the part through crack, due to the awkward geometry formed by the

outer surface of the tubes and the cracks, the quadrilateral isoparametric element with

mid-side nodes moved to the quarter points was used as the crack tip element. Moreover, a

triangular mesh as shown in Figure 4.6 near the intersection of the crack front and the outer

surface was made to cope with this awkward geometry. As required for the evaluation of

stress intensity factor [6, 7], elements around the crack tip were made to be orthogonal to the

crack front. By using the generator, tubes with surface crack and part through crack could be

modeled with all requirements mentioned accomplished.

4.2.1 FE MODEL WITH SEMI-ELLIPTICAL CRACK

For short cracks as shown in Figure 4.7, the transformed crack was very similar to a

semi-elliptical crack. However, for a long crack as shown in Figure 4.8, the difference was very

obvious. A mesh of the FE model of a tube with a crack is shown in Figure 4.8. This surface

crack was originally semi-elliptical in a flat plate. Transforming from the plate into tube would

introduce a curvature to a semi-elliptical crack. Due to this transformation, the surface crack in

the tube model in the present analysis was no longer purely elliptical. The shape of the surface

crack in the current analysis of cracked tube can be seen from Figure 4.8. For a special

purpose like the one in the current analysis, e.g. a tube with a long crack and small ligament, a

model with a original shape of ellipse could not fulfill this purpose as the crack would penetrate

through the wall when it was still short. By using the chosen transformation, a long crack with

an extremely small ligament could be modeled in a tube.

4.2.2 FE MODEL WITH PART THROUGH CRACK

Figure 4.9 shows a mesh of an FE model of a tube with a part through crack. Exactly the same

transformation as that for the surface was performed to obtain this model. Because of the shape

angle formed by the tube surface and the crack front, the triangular crack tip elements used for

62

the surface cracks could not be used in this case. A compromise was made which was to give

up the triangular crack tip element. A very small square crack tip element as shown in Figure

4.9 was adopted to take over the crack tip position. Actually, this replacement did not reduce

the accuracy as it was verified that sufficient small square crack tip elements could give a result

equally accurate to that from triangular crack tip elements. On the other hand, in order to get

the SIF as close to outer surface as possible, a wedge-shaped mesh was created near the outer

surface. The same as those on the remaining part of the crack front, elements in the

wedge-shape mesh were all orthogonal to the crack front even after the transformation.

4.2.3 FE MODEL AND ANALYSIS WITH THROUGH THICKNESS CRACK

Figure 4.10 shows the mesh of the FE model of a tube with a through thickness crack. It was

simply transformed from a flat plate with a through thickness crack. Having created the model

of a cracked tube, an input file for ABAQUS was generated in which the whole geometry,

especially the shape of the crack was described clearly. Depending on the length of the crack, a

suitable number of elements were arranged along the crack front from the inner surface to the

outer surface.

In the finite element analysis, the J-integral along the crack front was first calculated.

Based on the location of each J-integral, the stress intensity factor along the crack front was

then calculated. For both surface crack and the part through crack, the plane stress condition

was applicable at the intersection of the crack and either inner or outer surface of the tube,

and the stress intensity factor (K) was calculated as Equation (4.2). For all other position, the

plane strain condition was applicable and the stress intensity factor (K) was calculated

according to Equation (4.3). However, for part through cracks, due to the awkward geometry

formed by the outer surface and the crack, the J-integral obtained from FE analyses at this

position was not reliable. Therefore, stress intensity factors at this particular point of part

through crack were ignored.

63

For plane stress:

K = JE (4.2)

For plane strain:

K = JE

1 − ν2 (4.3)

Finally, the material properties including Young's modulus (E=210000 N/mm2) and Poisson's

ration (υ=0.3) were introduced. All analyses were based on the assumption of linear elasticity.

Stress intensity factors along the crack front for a cracked tube subjected to remote tension

stress of 100 MPa for various geometric parameters were determined by the J-contour integral

method [8] with ABAQUS.

4.3 VERIFICATION OF FE MODEL

Since the FE model of the cracked tube was transformed from that of the cracked plate, it

needed to consider that if this transformed FE model was suitable for the analysis of the

cracked tube. However, in order to verify the transformed FE model's ability to handle the

curvature of a tube, an FE model of a tube with surface and through thickness cracks was

verified again by comparing the current FEA results with the existing data.

4.3.1 SEMI-ELLIPTICAL SURFACE CRACK

One of the FE models of a cracked tube is shown in Figure 4.7. With a/c=0.4,0.6 or 0.8,

a/T=0.2, 0.5, 0.8 or 1.0, and R/T=10, the normalized stress intensity factor (NSIF =K

ζ πa) was

calculated. Comparisons of the NSIF values of the deepest and surface points were made with

the results reported by Mettu [9]. In order to make this comparison, the NSIF at the surface

point and deepest point was converted from the J-integral exactly on the surface and deepest

point. It is shown that for the case of deepest point, the maximum difference of Ft of current

64

finite element analysis and Mettu’s prediction is 42.83% when a/T = 1.0 and a/c = 0.4. For the

case of surface point, the maximum difference is 10.78% when a/T = 1.0 and a/c = 0.8. It is

mentioned previously that Mettu et al. carried out finite element analysis to obtain the SIFs of

both deepest point and surface point with the following parameters (a/c = 0.6, 0.8 and 1.0, a/T =

0.2, 0.5 and 0.8, and R/T = 1, 2, 4 and 10). Although there was no finite element results of SIFs

for a/T between 0.8 and 1 in their study, extrapolation technique was applied to obtain SIFs for

both surface point and deepest point for a/T between 0.8 and 1. Therefore, for the case when

a/T is larger than 0.8, large difference is observed when comparing those values obtained by

Mettu et al and the values obtained from current finite element study. For the case when a/T is

less than 0.8, the NSIF values obtained from current study agreed well with those of Mettu et al.

All results are presented in Table 4.1.

Another comparison was made by comparing the current finite element results with the

prediction proposed by Zahoor [10] for a/T varied from 0.2 to 0.99. The comparison is shown

in Figure 4.11 and it is shown that the different is significant between the predictions according

to Zahoor and current finite element study for small values of a/c (<0.2) and large values of a/T

(>0.8). It is noticed that the parameters studied by Zahoor did not cover the whole range of a/T

and a/c which is similar to the case of Mettu’s study. Therefore, it is shown that the prediction

of SIFs based on extrapolation did not compare well to the current finite element prediction.

4.3.2 THROUGH THICKNESS CRACK

One of the tube models with a through thickness crack is shown in Figure 4.10. With R/T = 10

and c/πR = 0.106 to 0.424, the normalized stress intensity factor (NSIF =K

ζ πc), was calculated.

Comparisons for the mean values of F at inner and outer surface with the results from Zahoor

[10] were made. The maximum difference is within 5.32% and for most results, the difference

is around 1%. These results are presented in Table 4.2.

65

4.4 FEA RESULT OF CRACKED TUBE

Three dimensional linear elastic FEA have been carried out to investigate the normalized stress

intensity factors, Ft, along the crack front of tubes. The finite element results of the SIFs (KFE)

were normalized with respect to the applied far end stress and crack deep (a) for surface crack

and crack length (c) for part through crack or through thickness crack. The stress intensity

factor, K, was converted from the J-integral from ABAQUS based on the assumption of plane

strain condition along the crack front except at the surface which was based on the assumption

of plane stress condition. It should be noted that the FE J-integral results might be sensitive to

the mesh design, in particular to the number of mesh. To investigate the effect of the finite

element mesh on the prediction of J-integral values, FE calculations with different number of

elements, rang from ~2500 elements to ~4000 elements, were performed. Resulting J-integral

values were almost identical to the results from the present mesh with 3213 elements. Such

sensitivity analysis provides sufficient confidence in the present FE analysis. Confidence in

the present FE analysis was gained from the path independence of the FE J-integral values,

i.e., the J-integral values of the last server contours were a same constant. Thus, the FE

J-integral was chosen from the last contour. The normalized SIFs (Ft) are calculated

according to Equations (4.4) and (4.5).

For surface crack

Ft =K

ζ πa (4.4)

For part through crack and through thickness crack

Ft =K

ζ πc (4.5)

In the current study, by varying analyzed parameters including (i) crack deep ratio (a/T); (ii)

crack length to circumference ratio (c/πR) and (iii) tube radius to thickness ratio (R/T), tubes

66

with a range of different crack aspect have been analyzed. Throughout the analyses, the wall

thickness, T, was taken as 20mm and L/D as 10, where L is the half length of tube and D is the

diameter of the tube.

4.4.1 SEMI-ELLIPTICAL SURFACE CRACK

Details of parameters for the analysis of tubes with surface crack are shown in Table 4.3. With

the full combination of a/T, c/πR and R/T, a total number of the 320 models have been

analyzed.

Distribution of Ft

Distribution of NSIF along the crack front are shown in Figures 4.12 to 4.16 and it is showed

that, for a circular tube with an inner surface crack under tension loading, the maximum

normalized stress intensity factor (Ft) is always at the deepest point of the crack while the

minimum is always on the surface point. Meanwhile, Ft increases as a/T value increases and the

increasing ratio at the deepest point are much greater than that at the surface points, and on the

other hand, the minimum Ft on the surface point decreases as a/T value increases. For long

cracks, e.g. c/πR=0.318, 0.424, 0.477, Ft values at the surface point are almost the same for all

crack depths and approach to zero.

Effect of crack depth to tube thickness (a/T)

For surface cracks in tubes, Ft values at the surface point increase linearly with a/T as shown in

Figure 4.17. With short cracks, e.g. c/πR = 0.106, 0.159, 0.212, the changes of Ft are very

significant. With long cracks, Figures 4.17 to 4.20 show that Ft is not sensitive to a/T.

Nevertheless, at the deepest point, Ft increase gradually with a/T increases from 0.2 to 0.9, and

approaches to a constant value of 0.9. When a/T is larger than 0.9, Ft increases rapidly.

Meanwhile, Ft increases efficiently as crack gets longer as shown in Figures 4.17 to 4.20.

For a semi-elliptical surface crack, the ration of minimum SIF (Kmin or Ftmin) to maximum

SIF (Kmax or Ftmax) indicates the direction of change of crack shape. The crack always grows in

67

such a manner as to make this ratio approach to unity. Figure 4.21 show the variation of Kmin

Kmax

against a/T. For the current ranges of parameters, Kmin

Kmax is always less than unity with most

cases less than 0.5. For short cracks, Kmin

Kmax increases initially and then decreases with

increasing a/T value. For long cracks, Kmin

Kmax always decreases with increasing a/T value.

Effect of crack length to tube circumference (c/πR)

Figures 4.22 to 4.23 show that the lower the R/T values, the higher the Ft values were obtained.

At the surface point, Ft decreases rapidly with increasing c/πR for all a/T values and Ft

approaches to a constant value as shown in Figure 4.23. For a/T = 0.9, Ft becomes constant with

respect to all c/πR values. When a/T is higher than 0.9, Ft keeps increases with increasing crack

length. However, when a/T is below 0.9, Ft decreases with increasing c/πR value. It may be

noted that since Ft is normalized value of SIF by dividing K with 𝜎 𝜋𝑎, and Ft becomes a

constant value for a/T = 0.9. It implies that, the increasing of SIF with respect to c/πR is the

same as the increase of 𝑎 crack depth increases rate. However, when a/T is larger than 0.9, Ft

increases with increasing c/πR. On the other hand, when a/T is less than 0.9, Ft decreases with

increasing c/πR.

Effect of (R/T)

In order to show the effect of R/T value to the prediction of Ft, plot of Ft at deepest point versus

c/πR with four different R/T values are shown in Figure 4.24 with a fixed a/T value. It is

observed that, for short cracks (c/πR < 0.106), Ft values are similar for all cases of R/T values.

Nevertheless, it is observed that the lower the R/T value, the higher the Ft value was obtained.

At the surface point, as shown in Figure 4.24, a larger value of Ft is obtained for lower R/T

value. However, Ft decreases with increasing c/πR value for all cases of R/T values.

68

Analytical equation for the prediction of Ft based on FEA data for surface crack

All Ft values at the surface point and the deepest point are listed in Tables 4.4 to 4.11. Based on

the current finite element results, functions for predicting the normalized SIF for surface cracks

at the deepest point and the surface point was obtained by curve fitting method. The

mathematical software MATLAB [11] and the toolbox NEURAL NETWORK [12] was

adopted to accomplish the curve fitting process.

Neural network is a newly developed mathematical tool which is inspired by the biological

nervous system. It offers a powerful way to explore the patterns existing in data. The basic

operation is to train a network to perform a set of particular functions. A network can be

composed of by more than one layer and each layer can be composed of more than one element.

Each element is considered as a neuron. By adjusting the number of layers, number of neurons

and especially the weights between neurons, an output (fitted function) with desired difference

from the input (data) can be achieved. The basic operation of the neural network is

schematically shown in Figure 4.25. In the present study, two layers of neurons were used for

Fts both at the deepest and surface points. In the first layer, a Log-Sigmoid Transfer Function

(logsig) was used and a linear Transfer Function (purelin) was used for the second layer. Both

functions are shown in Equations (4.6a and 4.6b).

logsig fl x =1

1 + e−x (4.6 a)

purelin fp x = ax + b (4.6 b)

The format of Fts is shown in Equation (4.7), where W1 and W2 are weight matrices and B1 and

B2 are bias matrices. The sizes of the matrices are defined by the number of neurons used. P is

the input vector. The operation of the neural network analysis in the present study is

schematically shown in Figure 4.26.

69

Fts format Fts = fp (fl W1 ∙ P + B1 )

Fts = W2 ∙ fl W1 ∙ P + B1 + B2

𝑓𝑙 𝑥 =1

1+𝑒−𝑥

input vector P =

a

TR

Tc

πR

(4.7)

For Fts at the deepest point, 10 and 1 neurons were used in the first and second layers

respectively and the corresponding weight matrices W1 and W2 and bias matrices B1 and B2

were calculated as shown in Equation (4.8).

W1 =

−64.6869

5.675844.9263

191.5677−2.4230−44.9468 −4.8062−20.3763

7.3836−3.3068

15.86420.06530.04270.05660.1928−0.0440−0.03500.21130.07030.4742

−1.70291.64026.37836.02107.5070−6.57339.12723.4107

−12.9978−3.4826

B1 =

6.5464−6.7030−48.3153−193.2488−0.928648.46888.19182.1357−0.02181.4741

W2 =

117.2567

2.8857135.3286

3.53130.8197

133.160052.1199−0.44400.12800.4750

T

B2 = −302.2674 (4.8)

For Fts at the surface, 10 and 1 neurons were also used in the first and second layers

respectively and the corresponding weight matrices W1 and W2 and bias matrices B1 and B2

were calculated as shown in Equation (4.9).

70

W1 =

−1.1375

−395.64491.2429

−78.91011.5369−4.2726 −46.5086

0.811677.6124−0.6000

−0.03760.21570.0365

−14.3257−0.06040.0131−0.6023−2.818812.78400.0242

−2.069214.25372.02293.9490−4.038022.8809−11.1446−0.6821−3.86011.4458

B1 =

2.8425389.8332−2.8683137.3933−0.46520.4717

28.378218.5037

−131.03273.1610

W2

=

79.82191.0538

77.0885235.1844

9.6081−0.2252−0.1532433.9769668.7117167.0586

T

B2 = −912.5462 (4.9)

A typical result of the normalized SIFs of deepest point (Fts) predicted from Equations (4.6) to

(4.8) versus a/c together with the finite element results is shown in Fig. 4.27. It is shown that

the prediction of normalized SIFs by using Equations (4.6) to (4.9) compared well with the

finite element results. In addition, plots of the prediction of Fts of both deepest point and surface

point versus c/πR are shown in Fig. 4.28 and 4.29. It is shown that the difference between the

predicted values and the FE results at the deepest point and surface point are less than 3% and

6%, respectively. Therefore, the predictions of the normalized SIFs at deepest point and

surface point by using the current proposed equations compare well with the FE results for a

wider range of a/c and a/T values.

4.4.2 PART THROUGH CRACK

In order to investigate in detail the process of a crack growing form a surface crack to a part

through crack, a very detailed scheme was adopted. An imaginary crack depth (a) was

assumed which was greater than the thickness (T) of the tube that makes the crack front stop at

a certain ∅ rather than π/2 when the crack front intersects the outer surface of the tube. For

most part, the distributions near the inner surface are quite similar to those of the surface crack.

71

In the vicinity of the outer surface, however, the factors were found to increase very rapidly

because of the sharp region formed by the crack front and outer surface as shown in Figure 4.30.

Unlike the case of the surface crack, there was not any deepest point available for the part

through crack. Consequently, another point which is as close to the outer surface as possible

should be used to replace the role of the deepest point for the surface crack. Also, the NSIF

obtained at this point had to be reliable. Due to the nature of the mesh used in this region and of

the J-integral, the factor at the precise intersection of the crack front and the outer surface was

not reliable. As a result, three locations close to that intersection point were chosen as the

reference points which are 0.90T, 0.95T and 0.99T measured from the inner surface as shown

in Figure 4.30 and 4.31. NSIF obtained at these three locations were considered as equivalent

to the deepest point in the case of a part through crack and are shown in Figure 4.32. It is

obvious that the NSIF at 0.99T is the highest and that the one at 0.90T is the lowest.

With the same chose value R/T and c/πR as in the case of surface crack, a/T was arranged

so that the details of cracks just after breaking the wall of tube could be obtained. Crack with

a/T= 9 were very close to the though thickness crack case and, therefore, the analysis stopped at

this point. A total of 544 models of tubes with part through cracks was created and analyzed.

Details of parameters for these analyses are shown in Table 4.12.

Distribution of Ft

Some typical results from finite element analyses for a tube with a part through crack are

shown in Figure 4.33 to 4.36. All results show that the SIF increases dramatically near the

outer surface of tube. This dramatic increase is due to the sharp angle formed by the crack

front and the outer surface of the tube. As a/T gets higher, the SIF increase in a diminishing

way and the pattern of distribution approaches that of a through thickness crack. Because of

the nature of a part through crack, the crack front is not orthogonal to the outer surface of

tube, and the results of J-integral at the outer surface were discarded. Ft at 0.99T from the

outer surface was taken from the FE results.

72

Effect of crack depth to tube thickness (a/T)

Figure 4.37 shows the Ft results at 0.99T. Just after the crack breaks through the wall, Ft

increases rapidly with a/T. Having reached a peak, Ft starts decreasing and finally approaches

to the Ft of the through thickness crack. At the inner surface, as shown in Figure 4.38, Ft

increase rapidly with a/T until a/T values of around 2.5. The increase slows down and

approaches the Ft of the through crack at the surface. The Ft at the surface (Ksur) and 0.99T

(K0.99T) have been examined as well. As shown in Figure 4.39, with each case of c/πR value,

Ksur/K0.99T increases with an increasing a/T as well as c'/c values, where c' is the crack length

on the outer surface. When the crack just breaks the tube, the increasing ratio of NSIF at

surface point to 0.99T point are much larger for short cracks, however, when a/T=3 and 4, the

increasing ratio approaches to constants, and the longer the crack length, the higher the

Ksur/K0.99T increase ratio is observed. It may be noted that, when it is approaching to the

through thickness crack condition, the NSIFs at the surface point and at the 0.99T are

different. This is due to the non-uniform distribution of NSIF along the through thickness

crack front in a tube which will be illustrated in Section 4.4.3.

Effect of crack length to tube circumferential (c/πR)

At the position 0.99T on the crack front, the crack length does have a significant effect on Ft.

As shown in Figure 4.37, the longer the crack, the higher the Ft value. For every distribution

of a constant crack length, a peak appears at a certain a/T value. The shorter the crack, the

earlier this peak comes. The existence of this peak Ft suggests that a failure (ductile tearing or

a brittle fracture) may occur even though it is considered safe for a through thickness crack

with the same crack length under the same loading. For a long crack, e.g. c/πR=0.424, 0.477,

this failure is even more likely to happen. This makes the crack length an important measure

of stability of a crack.

The effect of crack length on the NSIF at the surface is not as obvious as that at 0.99T.

As shown in Figure 4.38, although all of them increase in a diminishing way. There is a

transition point of a/T between 1 and 2. When a/T value is smaller than this transition point,

73

the longer the crack, the lower the Ft value, after this transition point, the trend becomes

totally reversed, namely, the longer the crack, the higher the Ft values. The detailed reason for

the existence of this particular a/T is not certain. However, it could be due to the shift of

neutral axis on the cracked plane. Figure 4.39 shows the Ksur/K0.99T ratio, which distribution

tendency is the same with NSIF at the surface. There is a particular value of a/T between 3

and 4, before this particular a/T value, the longer the crack, the lower the Ksur/K0.99T ratio.

When a/T is larger than this particular value, the longer the crack, the higher the Ksur/K0.99T

ratio. This means that, for a long part through crack, the crack will drive all the way to

develop a through crack with very small crack growth on the surface.

Effect of R/T

The effect of R/T ratio on the prediction of J-integral values at the 0.99T position and the

surface point are the similar. For a short period just after the crack breaks through the wall,

the effect of R/T on the surface crack case, depending on the crack length, are showed in

Figure 4.40 and 4.41. Comparatively, this 'period' at the surface point is much longer than

that at the 0.99T position. Thereafter, R/T shows a clear effect that, for a larger diameter or

relatively thin tubes, Ft is always higher for a part through crack. This effect implies that for a

large or thin tube, failure is more likely to happen under the same loading.

Analytical equation for the prediction of Ft based on FEA data for part through crack

All FEA results are listed in Table 4.13 to 4.20. As for the semi-elliptical surface crack case,

a function Ftp defined by Equation (4.10) was obtained by curve fitting. However, three layers

network were used for Ftp both at the 0.99T points and inner surface points. In the first and

second layers, Log-Sigmoid Transfer Functions (logsig) were used and a linear Transfer

Function (purelin) was used for the third layer. Parameters W1, W2, W3, B1, B2, and B3 are

given by Equations (4.11) and (4.12) for 0.99T points and inner surface points of a partly

through crack.

74

𝐹𝑡𝑝 = 𝑊3𝑓𝑙 𝑊2𝑓𝑙 𝑊1𝑃 + 𝐵1 + 𝐵2 + 𝐵3

𝑓𝑙 𝑥 =1

1 + 𝑒−𝑥

input vector P =

c

πRa

TR

T

(4.10)

For Ftp at the 0.99T point, 10 neurons were used in the first and second layers respectively and

1 neuron was used in the third layer. The corresponding weight matrices W1, W2 and W3 and

bias matrices B1, B2 and B3 were calculated as Equation (4.11).

W1 =

12.6234−0.2812−4.85535.66853.3734

−1.3673−5.4980−2.0977−5.1441119.2328

0.2442−69.1539

0.04621.67071.50202.57113.65140.03577.5018−5.9383

0.027111.2405−0.16190.6814−1.6326−0.0086−2.7674−0.0112−0.03720.0149

B1 =

−7.9147−9.4947 −0.9120−9.25135.4970−8.40439.15080.7234−6.083710.0067

𝑊2=

−2.51920.61540.3883−8.38522.4356

−2.8934−0.4756−0.13280.7256−0.4094

24.6728−1.41070.5350−6.101429.1041−0.2182−26.8810−2.0857−1.8382−2.3433

−6.48491.9117−6.960545.832369.2013−3.0790−33.2641−0.16641.8519−1.1864

−13.4010−0.28294.6555

−25.05991.8130

−15.739314.9957−0.9473−0.3175−0.7267

−57.0159−0.3565

−110.3520−30.806332.0115−59.3470−20.8406−0.0216−0.47330.1087

33.02291.6759−7.6699−1.5284

−328.291734.3419−77.0035

0.13801.4795−0.8210

72.4884−0.017542.24473.6725 5.4871

73.5134−29.2798

0.40820.09670.4231

−26.0336−2.800611.4844−65.703234.8970−27.722719.5567−3.3139−4.5736−2.0226

−5.93014.8618−3.3910−4.9603−20.2926

4.360331.19371.7478

10.73591.6277

−21.66820.4666

18.362728.8656−39.5374−21.5938−3.8530−0.56670.4247−0.7678

B2 =

17.6468−7.3884

−25.717725.6819−6.56429.2763 8.04472.7738

−12.18862.3426

W3=

−114.0770−548.7102

10.342747.6153

−172.9498113.7138

3.2661165.1808354.6679−142.1972

T

B3 = −13.7889 (4.11)

75

For Ftp at the inner surface point, 10 neurons were used in the first and second layers

respectively and 1 neuron were used in the third layer and the corresponding weight matrices

W1, W2 and W3 and bias matrices B1, B2 and B3 were calculated as Equation (4.12).

W1 =

−4.0439−5.79519.18560.53704.9676 5.21678.3921

−91.25905.0132

390.1259

5.2632−0.0027−0.94340.0004−1.60100.0020−2.3183−7.47590.77925.9111

−1.43900.03820.04780.43620.02950.28013.84956.77580.0774

−11.7621

B1 =

7.70512.3073

−2.20091.0151−2.9151−6.8130−29.2103−53.0386−2.444447.0788

𝑊2=

0.8964−0.57985.1743

−118.5776377.0237 −0.3679

0.3782−1.6096−1.4044−0.5734

10.3053−3.87172.6234−8.030214.6947−7.05931.59727.40239.3100−3.9400

5.3188−1.37490.80790.43111.5374−0.29150.1248−3.0417−2.6830−1.4268

131.3665160.3013−39.6014−67.103785.3993

121.5692−32.234482.3063−27.7715152.6629

14.0921−9.25109.9565

53.318260.8070−23.9170

4.1515−11.2670−11.3464−9.8587

−5.02791.1669

40.24778.3100

−38.8890−1.321024.722815.418317.05981.3337

−0.4288−1.781733.805544.4337−23.3930 −4.7195

4.6380−4.57594.9646−1.5388

5.5768−3.538513.578620.5495−33.6834

1.4941−0.9347−0.71420.1020−3.5969

−0.1962−2.8565−1.4781−0.99639.5431−3.9081−0.0121−2.4887−2.8891−2.9837

3.8908−3.0904 21.577337.9748−13.4581

0.7680−0.0780−1.6153−0.8675−3.1311

B2 =

−143.9060−151.2718

8.1406152.3848−459.3180−110.5123

28.7778−82.143515.6772

−143.7396

W3=

−1.7127

−250.3896−14.8925

8.62462.6794

12.171230.499919.2372−19.3991242.5759

T

B3 = −21.4531 4.12

The typical result of the normalized SIFs of 0.99T points were predicted from Equation (4.10)

to (4.11) which compared well with the finite element results, the maximum difference

between the predicted values and FE results are 3.72% and most are less 2%. Comparison of

the Ftp values with FEA values at 0.99T is shown in Figure 4.42. In addition, for inner surface

points the predictions of normalized SIFs by using Equations (4.10) and (4.12). The maximum

76

difference between the prediction of Ftp and FE results are 4.78% and most of them are less

than 2%. Comparison of the Ftp values with FEA values at inner surface is shown in Figure 4.43

and it is shown that the predictions of Ftp values compare well with the FE results as well.

Therefore, the predictions of the normalized SIFs at 0.99T points and inner surface points by

using the current proposed equations compare well with the FE results for a wider range of a/c

and a/T values for part through crack.

4.3.3 THROUGH THICKNESS CRACK

FE analysis of tubes with a through thickness crack has been conducted to calculated the

stress intensity factors along the crack front. Previous work has usually assumed that the tube

was a thin wall structure. Stress intensity factors reported according to this assumption were

at the middle of wall thickness or the mean value over the crack front. Since the maximum

value on the crack front was needed for fatigue life calculations, further analysis was

necessary. Values for the relevant parameters were more or less the same as chosen for the

surface crack and part through crack cases. Details of the parameters used for this analysis are

shown in Table 4.21.

Distribution of Ft

The distribution of Ft is shown in Figure 4.42. The distribution of Ft is uniform along the

thickness direction, especially when c/R value is large. For c/R=0.447, the Ft value of the

inner surface point is about 50% larger than that of the outside surface point.

Effect of crack length to tube circumference (c/πR)

It can be seen from Figure 4.42 that the longer the crack length, higher Ft value is observed.

Figure 4.43 shows the variation of Ft on the outer and inner surfaces. For a certain R/T value

(e.g. R/T = 22.5), Ft on the outer surface is always greater than that on the inner surface when

c/πR is less than 0.13. When c/πR is above this particular point, Ft on the outer surface is

77

always smaller than that on the inner surface. As a crack gets longer, the difference between

the minimum and maximum Ft becomes more pronounced.

Effect of R/T

As shown in Figure 4.43, it is clear that the larger R/T ratio of the tube, the higher the Ft

value is observed. As mentioned before, for each R/T value, Ft on the outer surface is higher

than that on the inner surface when c/R is less than 0.13. Figure 4.43 shows that the point

actually decreases as R/T gets larger. Therefore, for a tube with larger R/T value, Ft on the

inner surface is always higher than that on the outer surface for all crack lengths.

Analytical equation for the prediction of Ft based on FEA data for through thickness

crack

Data for the NSIF Ft are presented in Tables 4.22 to 4.23 for this case. As for the

semi-elliptical surface crack case and part through crack case, a function Ftf defined by

Equation (4.13) was obtained by curve fitting. Parameters W1, W2, B1 and B2 are given by

Equations (4.14) and (4.15) for outer surface point and inner surface point of a through

thickness crack.

Ftf = W2 ∙ fl W1 ∙ P + B1 + B2

𝑓𝑙 𝑥 =1

1 + 𝑒−𝑥

input vector P =

𝑐

𝜋𝑅𝑅

𝑇

(4.13)

For Ftf at the outer surface point, 3 and 1 neurons were used in the first and second layers

respectively and the corresponding weight matrices W1, W2 and bias matrices B1, B2 were

calculated as Equation (4.14).

78

𝑊1 = −0.8342 −0.0799−6.0800 −0.0196−18.9112 −0.1636

𝑊2 = 2589.1 −12.7 −0.3

𝐵1 = −9.31335.24893.2189

𝐵2 = 13.8285 (4.14)

For Ftf at the inner surface point, 3 and 1 neurons were used in the first and second layers

respectively and the corresponding weight matrices W1, W2 and bias matrices B1, B2 were

calculated as Equation (4.12).

𝑊1 = −7.1413 −0.0999−1.6083 −0.152614.3861 0.0678

𝑊2 = −4.5628 2.3477 5.3206

𝐵1 = 3.44563.2740−8.7260

𝐵2 = 2.9262 (4.15)

The typical result of the normalized SIFs of outer surface points were predicted from Equations

(4.13) to (4.14) which compared well with the finite element results, the maximum difference

between the predicted values and FE results are 0.58%. The comparison of Ftf values with FEA

values at outer surface are shown in Figure 4.46. In addition, Equations (4.13) and (4.15) were

used to predict the normalized SIFs for inner surface points. The maximum difference between

the prediction Ftf and FE results are 2.35% and most are also less 1%. The comparison of Ftf

values with FEA values at inner surface are shown in Figure 4.47.

4.5 EFFECT OF LENGTH AND WALL THICKNESS OF TUBE

Two parameters, the length of tube (L), and the wall thickness of tube (T) were set to have

either constant ratio (L/D=10) or constant value (T=20mm) in current finite element study.

The effect of these two parameters has not been investigated in the present analysis. In order

79

to have a brief look on their effect, a sensitivity study for these two parameters was carried

out.

4.5.1 EFFECT OF TUBE LENGTH

The sensitivity study was carried out for a tube with a part through crack. A fixed set of

parameters (a/T=1.3, c/πR=0.106, R/T=22.5, T=20mm) was chosen for this study. NSIF

values were then calculated for L/D (=1, 2, 3, 5, 10, 15). Figure 4.44 shows the variation of Ft

with L/D. It can be seen that for L/D>3, Ft approaches to a constant values. Further increase

of length does not make any difference on the Ft. However, when L/D<3, there is a positive

effect on Ft. Therefore, in some locations such as the joint of chord and brace in a structure, a

crack could be existing in brace and close to this joint. In this case, L/D is very small and the

SIF will then be magnified.

All analyses so far were conducted based on the assumption of double symmetry, namely

the tube lengths on both sides of the crack were the same. The effect of having different tube

lengths on each side of the crack is still not known. Further analysis for this case will be

necessary.

4.5.2 EFFECT OF WALL THICKNESS OF TUBE

Another sensitivity study was carried to check the effect of the wall thickness of the tube on

Ft. Keeping all parameters constant (a/T=1.3, c/πR=0.106, R/T=22.5, L/D=10) and changing

the wall thickness T (=5mm, 10mm, 15mm, 20mm, 25mm, 30mm, 35mm), Ft was then

calculated. As shown in Figure 4.45, there is no change in Ft at all. It is important to state

clearly that Ft is a normalized value. The true SIF depends on the absolute size of the crack.

Increasing the wall thickness with a constant crack depth ratio will increase the crack size and

hence the absolute value of SIF.

80

4.6 CONCLUSION

Finite element analyses of cracked tubes have been carried out. Details such as the

distribution of NSIF of each crack configuration i.e. (1) surface crack, (2) part through crack

and (3) through thickness crack have been investigated.

In the case of the surface crack, the distribution of NSIF for a wide range of crack depth

to tube thickness ratio (0.2 ≤ a/T ≤ 0.99) was investigated. The maximum and minimum

NSIF are found always at the deepest point and the surface point respectively. The large

difference of NSIF between these two points drives the crack to grow in the depth direction

much faster than in the circumferential direction. This is more pronounced in the case of a

long crack. Consequently, a deep long crack will break the wall of the tube with a relatively

small crack increment in the circumferential direction.

In the case of a part through crack, due to the sharp angle formed by the crack front and

the inner surface of the tube, the NSIF near the inner surface is much higher than that on the

outer surface. This large difference will derive the crack to grow in a direction of becoming a

through thickness crack. The variation of NSIF near the inner surface with the imaginary

crack depth ratio shows that in the case of a part through crack, a maximum value of NSIF

will be encountered, as a result, failure, either ductile tearing or brittle fracture, may occur. If

disruptive failure does not occur, the part through will safely grow to the through thickness

crack condition.

In the case of a through thickness crack, due to the curvature of the tube, the distribution

of NSIF along the crack front is non-uniform. For the case of long crack, the maximum NSIF

occurs near the inner surface which is higher than those reported in the literature for the same

situation. Due to this non-uniform distribution of NSIF, a through thickness crack in a tube

may develop similar to a part through crack with the inner crack length longer than that of the

outer crack length. This could be an important factor when the crack opening area is

considered.

81

All NSIF for surface cracks determined from this finite element analysis have been fitted

into a curve which is a function of crack depth ratio (a/T), crack length ratio (c/πR) and the

tube size (R/T). It is shown that the difference between the predicted values and the FE results

at the deepest point and surface point are less than 3% and 6%, respectively. Therefore, the

predictions of the normalized SIFs at deepest point and surface point by using the current

proposed equations compare well with the FE results for a wider range of a/c and a/T values.

For part through crack, the results of the normalized SIFs of 0.99T points and surface point

were predicted from a function of crack length ratio (c/πR), crack depth ratio (a/T), and the

tube size (R/T). The results obtained from the current proposed equations compared well with

the finite element results. The maximum difference between the predicted values and FE

results are 3.72% at the 0.99T points and 4.78% at the surface point, respectively. The same

method was used for through thickness crack, the results of the normalized SIFs of outer

surface points and inner surface points were predicted from a function of crack length ratio

(c/πR) and the tube size (R/T). The maximum difference between the predicted values and FE

results are 0.58% at the outer surface. In addition, for inner surface points the maximum

difference between the prediction Ftf and FE results are 2.35% and most of them are less 1%.

Therefore, the current proposed equations provide reasonable predictions of the normalized

SIFs at outer surface points and inner surface points for through thickness crack. For part

through cracks and through cracks, an interpolation method from current results for getting

the NSIF will be accurate enough.

82

Table 4.1: Ft from current FEA and Mettu’s result (R/T=10)

a/T a/c

Surface Point Deepest Point

Ft (FEA) Ft (Mettu) Diff.(%) Ft (FEA) Ft (Mettu) Diff.(%)

0.2 0.4 0.687 0.689 0.27 0.952 0.936 1.68

0.2 0.6 0.728 0.718 1.42 0.837 0.827 1.23

0.2 0.8 0.735 0.732 0.35 0.741 0.732 1.22

0.5 0.4 0.808 0.797 1.32 1.067 1.02 4.43

0.5 0.6 0.822 0.799 2.84 0.899 0.878 2.33

0.5 0.8 0.805 0.791 1.76 0.778 0.761 2.22

0.8 0.4 1.005 0.967 3.81 1.182 1.12 5.21

0.8 0.6 0.988 0.93 5.86 0.961 0.941 2.1

0.8 0.8 0.937 0.891 4.89 0.817 0.801 2.01

1.0 0.4 1.046 1.108 5.89 2.085 1.192 42.83

1.0 0.6 0.967 1.041 7.61 1.592 0.989 37.89

1.0 0.8 0.880 0.975 10.78 1.305 0.831 36.30

Table 4.2: Comparison of Ft from present FEA and Zahoor's result (R/T=10)

c/πR Ft at inner

surface

Ft at outer

surface

Mean

value

Zahoor's

result Error(%)

0.106 1.0107 1.2347 1.1227 1.1824 5.32

0.159 1.2524 1.3438 1.2981 1.3401 3.24

0.212 1.5793 1.4422 1.51075 1.5377 1.78

0.265 1.9709 1.5535 1.7622 1.7817 1.11

0.318 2.4332 1.6966 2.0649 2.084 0.92

0.371 2.9923 1.8864 2.43935 2.4611 0.89

0.424 3.695 2.1377 2.91635 2.934 0.61

Table 4.3: Parameters for the analysis of tubes with surface cracks


a/T 0.2, 0.5, 0.8, 0.85, 0.9, 0.93, 0.95, 0.97, 0.98, 0.99

c/πR 0.106, 0.159, 0.212, 0.265, 0.318, 0.371, 0.424, 0.477

R/T 4.00, 10.0, 15.0, 22.5

83

Table 4.4: NSIF Ft at surface point for surface crack (R/T=4.0)

c/πR 0.2 0.5 0.8 0.85

a/t

0.95 0.97 0.98 0.99 0.9 0.93

0.106 0.1877 0.3867 0.7206 0.7632 0.7953 0.8168 0.8296 0.8417 0.8474 0.8549

0.159 0.1292 0.2976 0.5824 0.6244 0.6654 0.6900 0.7065 0.7233 0.7319 0.7381

0.212 0.0976 0.2367 0.4779 0.5062 0.5598 0.5807 0.6053 0.6248 0.6351 0.6449

0.265 0.0781 0.1978 0.4001 0.4394 0.4799 0.5038 0.5193 0.5447 0.5571 0.5691

0.318 0.0647 0.1694 0.3402 0.3774 0.4151 0.4373 0.4597 0.481 0.4927 0.5044

0.371 0.0549 0.1478 0.2925 0.3276 0.3622 0.3865 0.4042 0.4238 0.4283 0.4429

0.424 0.0474 0.1307 0.252 0.2866 0.3175 0.338 0.3555 0.3743 0.3848 0.3956

0.477 0.0413 0.117 0.2251 0.2512 0.2763 0.3005 0.315 0.3313 0.3406 0.3499

2ф/π=0 (surface point), R/T=4.0

Table 4.5: NSIF Ft at deepest point for surface crack (R/T=4.0)

c/πR 0.2 0.5 0.8 0.85

a/t

0.95 0.97 0.98 0.99 0.9 0.93

0.106 0.4223 0.641 0.7555 0.7827 0.8292 0.88 0.9371 1.0414 1.1429 1.3591

0.159 0.356 0.5886 0.7497 0.7879 0.8495 0.9159 0.9886 1.1204 1.2466 1.5135

0.212 0.3132 0.5446 0.7372 0.7846 0.8594 0.9383 1.024 1.1785 1.3259 1.6361

0.265 0.2827 0.5085 0.7233 0.7787 0.8648 0.9548 1.0519 1.2264 1.3925 1.7413

0.318 0.2595 0.4784 0.7093 0.7711 0.8668 0.9664 1.0733 1.2654 1.448 1.8311

0.371 0.2412 0.4529 0.6952 0.7623 0.8657 0.9733 1.0885 1.2955 1.4922 1.9052

0.424 0.2262 0.4311 0.6812 0.7522 0.8618 0.9756 1.0976 1.3167 1.5254 1.9638

0.477 0.2138 0.412 0.6708 0.741 0.8551 0.9738 1.1009 1.3297 1.548 2.0072

2ф/π=1 (deepest point), R/T=4.0


c/πR 0.2 0.5 0.8 0.85

a/t

0.95 0.97 0.98 0.99 0.9 0.93

0.106 0.0794 0.2214 0.414 0.4475 0.4807 0.5003 0.5133 0.5262 0.5328 0.4542

0.159 0.0508 0.1536 0.2946 0.3237 0.3544 0.3738 0.3872 0.4014 0.4088 0.3971

0.212 0.0387 0.1139 0.2276 0.253 0.2809 0.2991 0.312 0.326 0.3335 0.3082

0.265 0.0302 0.0912 0.1795 0.2 0.2229 0.2382 0.2492 0.2612 0.2678 0.2678

0.318 0.024 0.0755 0.1477 0.1649 0.1844 0.1975 0.2071 0.2175 0.2233 0.225

0.371 0.0215 0.0642 0.1232 0.1371 0.1529 0.1635 0.1713 0.1798 0.1845 0.1864

0.424 0.018 0.0555 0.1053 0.117 0.1303 0.1392 0.1457 0.1528 0.1567 0.1587

0.477 0.017 0.0488 0.091 0.1007 0.1115 0.1188 0.1241 0.1298 0.1329 0.1346


84


c/πR 0.2 0.5 0.8 0.85

a/t

0.95 0.97 0.98 0.99 0.9 0.93

0.106 0.2937 0.5546 0.7293 0.7566 0.8008 0.8513 0.9112 1.0266 1.1414 1.4153

0.159 0.2436 0.4904 0.7017 0.7392 0.7965 0.8586 0.9305 1.0677 1.2034 1.4998

0.212 0.2083 0.443 0.6751 0.7207 0.7886 0.8602 0.9419 1.0965 1.249 1.6049

0.265 0.187 0.407 0.6505 0.7018 0.7777 0.8564 0.9453 1.1128 1.278 1.638

0.318 0.1712 0.3785 0.6275 0.6828 0.7641 0.8475 0.9414 1.1179 1.2919 1.6718

0.371 0.1587 0.3552 0.606 0.6637 0.7483 0.8347 0.9315 1.1135 1.2932 1.6863

0.424 0.1487 0.3357 0.5857 0.6447 0.7312 0.819 0.9173 1.1021 1.2849 1.6858

0.477 0.1402 0.3191 0.5665 0.6261 0.7131 0.8014 0.9001 1.0857 1.2697 1.6743



c/πR 0.2 0.5 0.8 0.85

a/t

0.95 0.97 0.98 0.99 0.9 0.93

0.106 0.0518 0.1578 0.3144 0.3463 0.3794 0.3997 0.4134 0.4275 0.4347 0.4218

0.159 0.0322 0.1056 0.2199 0.2441 0.2704 0.2872 0.299 0.3115 0.3181 0.3152

0.212 0.0256 0.0782 0.1559 0.175 0.1964 0.2106 0.2209 0.2319 0.2379 0.2392

0.265 0.0198 0.0605 0.1256 0.1407 0.1576 0.169 0.1771 0.186 0.1908 0.1927

0.318 0.0161 0.0496 0.0968 0.1083 0.1213 0.1301 0.1364 0.1434 0.1472 0.1493

0.371 0.0149 0.0418 0.0828 0.0924 0.1033 0.1104 0.1156 0.1212 0.1242 0.1259

0.424 0.0117 0.0359 0.0679 0.0752 0.0835 0.089 0.0929 0.0972 0.0995 0.101

0.477 0.0102 0.0315 0.0599 0.0664 0.0736 0.0784 0.0817 0.0853 0.0872 0.0883



c/πR 0.2 0.5 0.8 0.85

a/t

0.95 0.97 0.98 0.99 0.9 0.93

0.106 0.2462 0.5084 0.7132 0.7402 0.7813 0.8285 0.8863 1.001 1.1183 1.3776

0.159 0.2032 0.4425 0.6773 0.7147 0.7682 0.8256 0.8933 1.025 1.1595 1.4568

0.212 0.1732 0.3963 0.6453 0.6901 0.7533 0.8192 0.8956 1.0433 1.1928 1.5224

0.265 0.1553 0.3616 0.616 0.6652 0.7341 0.8044 0.8846 1.0383 1.1946 1.5406

0.318 0.142 0.3348 0.59 0.6422 0.715 0.7887 0.8725 1.0334 1.1961 1.5563

0.371 0.1315 0.3132 0.5653 0.6186 0.6925 0.7666 0.8502 1.0099 1.1725 1.5349

0.424 0.1197 0.2953 0.5435 0.5974 0.6723 0.7471 0.8316 0.9935 1.1577 1.5234

0.477 0.1129 0.28 0.5223 0.5757 0.6496 0.7229 0.8054 0.963 1.124 1.4851


85


c/πR 0.2 0.5 0.8 0.85

a/t

0.95 0.97 0.98 0.99 0.9 0.93

0.106 0.0343 0.1054 0.2247 0.2517 0.2808 0.2993 0.312 0.3252 0.3321 0.3292

0.159 0.0214 0.0684 0.1498 0.1687 0.1897 0.2035 0.2133 0.2237 0.2293 0.2308

0.212 0.0161 0.0499 0.1021 0.1151 0.1299 0.1399 0.1471 0.1549 0.1591 0.1617

0.265 0.012 0.0388 0.0799 0.09 0.1014 0.109 0.1144 0.1204 0.1235 0.1256

0.318 0.0098 0.0317 0.0618 0.0688 0.0769 0.0822 0.0861 0.0903 0.0925 0.0942

0.371 0.009 0.0266 0.0515 0.0575 0.0641 0.0685 0.0716 0.075 0.0767 0.078

0.424 0.0074 0.0225 0.0432 0.0476 0.0525 0.0557 0.058 0.0604 0.0617 0.0627

0.477 0.0063 0.0195 0.037 0.0412 0.0455 0.0482 0.0502 0.0523 0.0534 0.0541



c/πR 0.2 0.5 0.8 0.85

a/t

0.95 0.97 0.98 0.99 0.9 0.93

0.106 0.2015 0.4589 0.6956 0.7236 0.7613 0.8042 0.858 0.9688 1.0849 1.3478

0.159 0.1657 0.394 0.6521 0.6903 0.7402 0.7923 0.8544 0.9791 1.1092 1.4039

0.212 0.1439 0.3502 0.615 0.6596 0.7176 0.7763 0.8449 0.9812 1.1222 1.4411

0.265 0.129 0.3182 0.5818 0.6297 0.6916 0.7528 0.8231 0.9612 1.1047 1.4308

0.318 0.1179 0.2936 0.553 0.6027 0.6668 0.7296 0.8015 0.9427 1.0886 1.4204

0.371 0.1063 0.2739 0.5263 0.5761 0.6403 0.7022 0.7725 0.9099 1.0529 1.3801

0.424 0.0993 0.257 0.5033 0.5529 0.6169 0.6786 0.7486 0.8857 1.0277 1.3525

0.477 0.0936 0.2433 0.4839 0.53 0.5925 0.6523 0.7198 0.8515 0.989 1.3054


Table 4.12: Parameters for the analysis of tubes with part through cracks


a/T 1.01, 1.03, 1.05, 1.07, 1.1, 1.2, 1.3, 1.5, 1.7, 2, 3, 4, 5, 6,

7, 8, 9,

c/πR 0.106, 0.159, 0.212, 0.265, 0.318, 0.371, 0.424, 0.477

R/T 4.00, 10.0, 15.0, 22.5

86

Table 4.13: NSIF Ft at surface point for part through crack (R/T=4.0)

a/T c'/c

c/πR

0.106 0.159 0.212 0.265 0.318 0.371 0.424 0.477

1.01 0.140 0.8411 0.7529 0.6790 0.6154 0.5524 0.5012 0.4516 0.4034

1.03 0.240 0.8233 0.7564 0.6890 0.6520 0.6077 0.5634 0.5172 0.4688

1.05 0.305 0.8298 0.7746 0.7298 0.6931 0.6587 0.6223 0.5814 0.5352

1.07 0.356 0.8355 0.7910 0.7574 0.7316 0.7074 0.6797 0.6455 0.6033

1.10 0.417 0.8824 0.8411 0.8144 0.7991 0.7873 0.7705 0.7453 0.7104

1.20 0.553 0.9057 0.9044 0.9266 0.9567 0.9979 1.0351 1.0561 1.0765

1.30 0.639 0.9207 0.9487 1.0086 1.0875 1.1683 1.2640 1.3533 1.4331

1.50 0.745 0.9431 1.0078 1.1210 1.2689 1.4426 1.6383 1.8554 2.0961

1.70 0.809 0.9537 1.0393 1.1846 1.3768 1.6092 1.8818 2.2005 2.5771

2.00 0.866 0.9629 1.0655 1.2378 1.4693 1.7570 2.1066 2.5325 3.0606

3.00 0.943 0.9792 1.0971 1.2972 1.5710 1.9212 2.3623 2.9233 3.6536

4.00 0.968 0.9830 1.1069 1.3158 1.6022 1.9705 2.4383 3.0391 3.8300

5.00 0.980 0.9847 1.1113 1.3240 1.6158 1.9918 2.4708 3.0882 3.9043

6.00 0.986 0.9856 1.1137 1.3284 1.6230 2.0030 2.4879 3.1139 3.9430

7.00 0.990 0.9862 1.1151 1.3310 1.6273 2.0097 2.4980 3.1291 3.9658

8.00 0.992 0.9866 1.1160 1.3327 1.6300 2.0140 2.5044 3.1388 3.9804

9.00 0.994 0.9868 1.1166 1.3338 1.6319 2.0169 2.5089 3.1454 3.9904

Surface point R/T=4.0

87

Table 4.14: NSIF Ft at 0.99T point for part through crack (R/T=4.0)

a/T c'/c

c/πR

0.106 0.159 0.212 0.265 0.318 0.371 0.424 0.477

1.01 0.140 1.6741 1.8929 2.0901 2.2779 2.4570 2.6235 2.7756 2.9114

1.03 0.240 1.8562 2.1374 4.1136 2.6642 2.9252 3.1832 3.4354 3.6789

1.05 0.305 1.9483 2.2740 2.5844 2.8967 3.2155 3.5406 3.8704 4.2023

1.07 0.356 2.0054 2.3657 2.7112 3.0623 3.4266 3.8068 4.2029 4.6136

1.10 0.417 2.0513 2.4345 2.8150 3.2092 3.6248 4.0671 4.5392 5.0443

1.20 0.553 2.1379 2.5861 3.0319 3.5091 4.0386 4.6371 5.3210 6.1108

1.30 0.639 2.1435 2.6485 3.1293 3.6443 4.2279 4.9116 5.7274 6.7134

1.50 0.745 2.0025 2.4845 2.9999 3.6339 4.1912 5.1872 5.9502 6.1320

1.70 0.809 1.8479 2.3350 2.8038 3.2936 4.0104 4.8649 5.9279 7.2962

2.00 0.866 1.6850 2.1147 2.5706 3.0558 3.6002 4.2486 5.2124 6.5303

3.00 0.943 1.4578 1.7189 2.0124 2.3481 2.7600 3.2928 3.9785 4.8851

4.00 0.968 1.3319 1.5361 1.7617 2.0184 2.3263 2.7119 3.2114 3.8752

5.00 0.980 1.2785 1.4503 1.6373 1.8495 2.1040 2.4230 2.8359 3.3837

6.00 0.986 1.2514 1.4047 1.5696 1.7562 1.9800 2.2604 2.6228 3.1023

7.00 0.990 1.2364 1.3788 1.5304 1.7017 1.9071 2.1642 2.4959 2.9338

8.00 0.992 1.2278 1.3635 1.5070 1.6688 1.8627 2.1052 2.4177 2.8292

9.00 0.994 1.2234 1.3541 1.4924 1.6483 1.8350 2.0682 2.3682 2.7626

0.99T R/T=4.0

88


a/T c'/c

c/πR

0.106 0.159 0.212 0.265 0.318 0.371 0.424 0.477

1.01 0.140 0.6278 0.4900 0.3979 0.3257 0.2676 0.2220 0.1846 0.1544

1.03 0.240 0.6379 0.5207 0.4358 0.3666 0.3077 0.2569 0.2114 0.1770

1.05 0.305 0.6554 0.5489 0.4713 0.4060 0.3467 0.2933 0.2442 0.2032

1.07 0.356 0.6718 0.5756 0.5056 0.4448 0.3825 0.3314 0.2793 0.2328

1.10 0.417 0.7004 0.6169 0.5564 0.5014 0.4453 0.3905 0.3353 0.2808

1.20 0.553 0.7698 0.7312 0.7070 0.6845 0.6509 0.6059 0.5543 0.4971

1.30 0.639 0.8236 0.8252 0.8408 0.8404 0.8482 0.8312 0.8005 0.7499

1.50 0.745 0.8975 0.9644 1.0452 1.1275 1.2009 1.2581 1.3039 1.3447

1.70 0.809 0.8934 1.0197 1.1683 1.3179 1.4612 1.6000 1.7409 1.8915

2.00 0.866 0.9091 1.0846 1.2951 1.5179 1.7478 1.9911 2.2607 2.5745

3.00 0.943 0.8840 1.1156 1.4082 1.7426 2.1191 2.5536 3.0758 3.7311

4.00 0.968 1.0091 1.0726 1.3832 1.7449 2.1619 2.6547 3.2607 4.0377

5.00 0.980 1.0127 1.2481 1.5655 1.6861 2.3851 2.9148 3.5746 4.4309

6.00 0.986 1.0146 1.2525 1.5738 1.5926 2.4071 2.9484 3.6250 4.5061

7.00 0.990 1.0157 1.2551 1.5786 1.9652 2.4197 2.9676 3.6537 4.5488

8.00 0.992 1.0164 1.2568 1.5817 1.9703 2.4278 2.9798 3.6718 4.5755

9.00 0.994 1.0169 1.2579 1.5838 1.9737 2.4332 2.9880 3.6839 4.5934


89


a/T c'/c

c/πR

0.106 0.159 0.212 0.265 0.318 0.371 0.424 0.477

1.01 0.140 1.7673 1.9788 2.1650 2.3215 2.4466 2.5416 2.6104 2.6573

1.03 0.240 2.0231 2.3224 2.6017 2.8567 3.0841 3.2824 3.4531 3.5996

1.05 0.305 2.1715 2.5272 2.8668 3.1889 3.4900 3.7677 4.0224 4.2566

1.07 0.356 2.2746 2.6722 3.0569 3.4307 3.7911 4.1364 4.4660 4.7823

1.10 0.417 2.3611 2.8038 3.2298 3.6496 4.0650 4.4770 4.8857 5.2939

1.20 0.553 2.5386 3.0733 3.6055 4.1551 4.7353 5.3571 6.0319 6.7701

1.30 0.639 2.6208 3.1933 3.7732 4.3928 5.0762 5.8452 6.7222 7.7346

1.50 0.745 2.5473 3.1682 3.7781 4.4394 5.2001 6.1060 7.2068 8.5670

1.70 0.809 2.6771 3.3379 3.9586 4.6393 5.4440 6.4319 7.6709 9.2494

2.00 0.866 2.5341 3.3114 3.9741 4.6671 5.4801 6.4884 7.7758 9.4543

3.00 0.943 2.0853 2.8040 3.5462 4.3152 5.2466 6.3262 7.6600 9.3794

4.00 0.968 1.5322 2.4950 2.9795 3.7544 4.6229 5.6502 6.9004 8.5824

5.00 0.980 1.4225 1.6712 1.9271 3.4154 2.5895 3.0757 3.7219 4.5728

6.00 0.986 1.3603 1.5729 1.7886 3.2885 2.3498 2.7519 3.2787 3.9780

7.00 0.990 1.3230 1.5131 1.7036 1.9233 2.1987 2.5555 3.0235 3.6447

8.00 0.992 1.2997 1.4755 1.6496 1.8496 2.1008 2.4270 2.8552 3.4231

9.00 0.994 1.2849 1.4515 1.6146 1.8012 2.0359 2.3411 2.7419 3.2728

0.99T R/T=10.0

90


a/T c'/c

c/πR

0.106 0.159 0.212 0.265 0.318 0.371 0.424 0.477

1.01 0.140 0.4922 0.3653 0.2807 0.2193 0.1731 0.1382 0.1115 0.0971

1.03 0.240 0.5265 0.4016 0.3155 0.2498 0.1991 0.1585 0.1266 0.1039

1.05 0.305 0.5351 0.4243 0.3454 0.2811 0.2266 0.1812 0.1453 0.1166

1.07 0.356 0.5547 0.4519 0.3769 0.3127 0.2556 0.2061 0.1645 0.1300

1.10 0.417 0.5912 0.4970 0.4265 0.3620 0.2991 0.2465 0.1987 0.1580

1.20 0.553 0.6728 0.6200 0.5776 0.5281 0.4703 0.4073 0.3492 0.2927

1.30 0.639 0.7467 0.7321 0.7207 0.6933 0.6410 0.5972 0.5388 0.4812

1.50 0.745 0.8651 0.9148 0.9643 0.9944 1.0024 0.9968 0.9702 0.9636

1.70 0.809 0.9023 1.0212 1.1377 1.2323 1.3057 1.3664 1.4228 1.4799

2.00 0.866 0.9482 1.1351 1.3266 1.5024 1.6656 1.8283 2.0044 2.2076

3.00 0.943 0.9617 1.2435 1.5619 1.8948 2.2487 2.6457 3.1173 3.7050

4.00 0.968 0.9251 1.2279 1.5812 1.9654 2.3905 2.8841 3.4873 4.2569

5.00 0.980 1.0674 1.3679 1.7319 2.1370 2.5948 3.1374 3.8126 4.6882

6.00 0.986 1.0701 1.3744 1.7443 2.1582 2.6286 3.1892 3.8903 4.8039

7.00 0.990 1.0717 1.3781 1.7514 2.1702 2.6475 3.2179 3.9330 4.8672

8.00 0.992 1.0727 1.3805 1.7559 2.1777 2.6593 3.2356 3.9593 4.9060

9.00 0.994 1.0733 1.3821 1.7589 2.1827 2.6672 3.2475 3.9767 4.9315


91


a/T c'/c

c/πR

0.106 0.159 0.212 0.265 0.318 0.371 0.424 0.477

1.01 0.140 1.8016 2.0093 2.1759 2.2985 2.3809 2.4306 2.4553 2.4617

1.03 0.240 2.1001 2.4116 2.6863 2.9180 3.1069 3.2572 3.3756 3.4693

1.05 0.305 2.2774 2.6544 3.0000 3.3080 3.5766 3.8083 4.0090 4.1849

1.07 0.356 2.4026 2.8279 3.2266 3.5944 3.9290 4.2319 4.5082 4.7644

1.10 0.417 2.5153 2.9818 3.4215 3.8382 4.2315 4.6039 4.9593 5.3043

1.20 0.553 2.7513 3.3289 3.8891 4.4528 5.0327 5.6387 6.2820 6.9718

1.30 0.639 2.8602 3.4894 4.1148 4.7682 5.4735 6.2531 7.1260 8.1173

1.50 0.745 2.8448 3.5084 4.1688 4.8915 5.7245 6.7096 7.8940 9.3365

1.70 0.809 3.0053 3.6794 4.3565 5.1205 6.0276 7.1311 8.4941 10.2009

2.00 0.866 2.9924 3.7182 4.3960 5.1561 6.0763 7.2262 8.6875 10.5737

3.00 0.943 2.5629 3.4146 4.2740 5.1236 6.0783 7.2416 8.7191 10.6507

4.00 0.968 2.3176 2.9740 3.8181 4.6703 5.6633 6.9497 8.5101 10.4910

5.00 0.980 1.5481 1.8422 2.1513 2.5388 3.0272 3.6428 4.4309 5.4579

6.00 0.986 1.4613 1.7077 1.9655 2.2744 2.6664 3.2013 3.8993 4.8102

7.00 0.990 1.4082 1.6240 1.8479 2.1174 2.4622 2.9099 3.4947 4.2902

8.00 0.992 1.3747 1.5703 1.7713 1.9981 2.3259 2.7321 3.2633 3.9643

9.00 0.994 1.3532 1.5352 1.7206 1.9446 2.2334 2.6102 3.1031 3.7530

0.99T R/T=15.0

92


a/T c'/c

c/πR

0.106 0.159 0.212 0.265 0.318 0.371 0.424 0.477

1.01 0.140 0.3826 0.2595 0.1860 0.1372 0.1044 0.0821 0.0665 0.0553

1.03 0.240 0.4074 0.2872 0.2109 0.1572 0.1189 0.0919 0.0728 0.0592

1.05 0.305 0.4308 0.3142 0.2363 0.1784 0.1349 0.1033 0.0805 0.0641

1.07 0.356 0.4534 0.3409 0.2622 0.2010 0.1527 0.1166 0.0899 0.0704

1.10 0.417 0.4858 0.3803 0.3019 0.2370 0.1823 0.1400 0.1074 0.0828

1.20 0.553 0.5839 0.5065 0.4397 0.3696 0.3018 0.2443 0.1948 0.1530

1.30 0.639 0.6678 0.6225 0.5711 0.5119 0.4467 0.3799 0.3219 0.2687

1.50 0.745 0.7996 0.8205 0.8169 0.7948 0.7444 0.7008 0.6553 0.6093

1.70 0.809 0.8927 0.9751 1.0239 1.0511 1.0376 1.0364 1.0333 1.0313

2.00 0.866 0.9800 1.1530 1.2932 1.3957 1.4771 1.5541 1.6375 1.7302

3.00 0.943 1.0571 1.3758 1.6875 1.9817 2.2795 2.6091 2.9998 3.4851

4.00 0.968 1.0389 1.4006 1.7769 2.1572 2.5647 3.0338 3.6048 4.3296

5.00 0.980 1.1534 1.5198 1.9161 2.3312 2.7916 3.3375 4.0180 4.8986

6.00 0.986 1.1574 1.5301 1.9365 2.3664 2.8478 3.4238 4.1471 5.0903

7.00 0.990 1.1597 1.5359 1.9478 2.3855 2.8780 3.4695 4.2154 5.1916

8.00 0.992 1.1612 1.5395 1.9547 2.3971 2.8962 3.5148 4.2560 5.2515

9.00 0.994 1.1621 1.5419 1.9593 2.4048 2.9081 0.0665 4.2823 5.2900


93


a/T c'/c

c/πR

0.106 0.159 0.212 0.265 0.318 0.371 0.424 0.477

1.01 0.140 1.8263 2.0174 2.1472 2.2143 2.2446 2.2462 2.2292 2.2008

1.03 0.240 2.1562 2.4592 2.6993 2.8762 2.9996 3.0818 3.1347 3.1673

1.05 0.305 2.3580 2.7333 3.0498 3.3040 3.5026 3.6565 3.7780 3.8767

1.07 0.356 2.5040 2.9324 3.3070 3.6234 3.8867 4.1066 4.2952 4.4624

1.10 0.417 2.6655 3.1538 3.5947 3.9866 4.3335 4.6456 4.9327 5.2053

1.20 0.553 2.9698 3.5789 4.1549 4.7176 5.2788 5.8491 6.4406 7.0641

1.30 0.639 3.1174 3.7915 4.4458 5.1158 5.8261 6.5972 7.4451 8.3901

1.50 0.745 3.2530 3.9775 4.7106 5.5064 6.4068 7.4470 8.6680 10.1204

1.70 0.809 3.3192 4.0477 4.8024 5.6532 6.6518 7.8459 9.2965 11.0861

2.00 0.866 3.3738 4.1017 4.8461 5.7111 6.7601 8.0547 9.6761 11.7423

3.00 0.943 3.1370 4.1100 4.9210 5.7862 6.8252 8.1292 9.8071 12.0153

4.00 0.968 2.7547 3.7554 4.6433 5.6858 6.8301 8.1950 9.9041 12.1233

5.00 0.980 1.7152 2.0744 2.4894 2.9719 3.5577 4.2846 5.2062 6.4053

6.00 0.986 1.5954 1.8899 2.2109 2.6371 3.1672 3.8287 4.6687 5.7595

7.00 0.990 1.5206 1.7744 2.0517 2.3959 2.8504 3.4598 4.2323 5.2347

8.00 0.992 1.4725 1.6986 1.9452 2.2534 2.6500 2.9913 3.8564 4.7892

9.00 0.994 1.4409 1.6479 1.8727 2.1551 2.5199 2.2292 3.6036 4.4098

0.99T R/T=22.5

94

Table 4.21: Parameters for the analysis of tubes with part through cracks


c/πR 0.106, 0.159, 0.212, 0.265, 0.318, 0.371, 0.424, 0.477

R/T 4.00, 10.0, 15.0, 22.5

Table 4.22: NSIF Ft at inner surface for through thickness crack

c/πR 4

R/T

22.5 10 15

0.106 0.9795 1.0107 1.0675 1.1570

0.159 1.1100 1.2524 1.3775 1.5394

0.212 1.3276 1.5793 1.7565 1.9614

0.265 1.6262 1.9709 2.1838 2.4129

0.318 2.0121 2.4332 2.6731 2.9239

0.371 2.5058 2.9923 3.2601 3.5407

0.424 3.1453 3.6950 3.9994 4.3231

0.477 3.9956 4.6155 4.9701 5.3544

Inner surface

Table 4.23: NSIF Ft at outer surface for through thickness crack

c/πR 4

R/T

22.5 10 15

0.106 1.2153 1.2347 1.2610 1.2897

0.159 1.3237 1.3438 1.3633 1.3841

0.212 1.4332 1.4422 1.4585 1.4820

0.265 1.5524 1.5535 1.5763 1.6149

0.318 1.6934 1.6966 1.7352 1.7962

0.371 1.8696 1.8864 1.9480 2.0358

0.424 2.0970 2.1377 2.2284 2.3481

0.477 2.3945 2.4698 2.5961 2.7564

Outer surface

95

Figure 4.1: Flaw characterization

Figure 4.2: Cracked plate before transformation

96

Figure 4.3: Boundary conditions of cracked tube model

Figure 4.4: Quarter point crack tip element used in FEA

97

Figure 4.5: Typical FE mesh of tube with inner surface crack

Figure 4.6: Triangular mesh near outer face of part through crack

98

Figure 4.7: FE Mesh of tube with semi-elliptical crack (a

c= 0.4,

a

T= 0.8,

R

T= 10)

99

Figure 4.8: FE Mesh of tube with surface crack (c

πR= 0.106,

a

T= 0.8,

R

T= 10)

100

Figure 4.9: FE Mesh of tube with part through crack (c

πR= 0.106,

a

T= 1.5,

R

T= 10)

101

Figure 4.10: FE Mesh of tube with through thickness crack (c

πR= 0.106,

R

T= 10)

102

Figure 4.11: Comparison of Zahoor's results and current FE at deepest point

Figure 4.12: Ft along crack front of SC (c/πR=0.371, R/T=10)

103

Figure 4.13: Distribution of Ft along crack front of tube (R/T=4.0)

104

Figure 4.13: (Continue) Distribution of Ft along crack front of tube (R/T=4.0)

105


106


107


108


109


110


111

Figure 4.17: Variation of Ft at surface and deepest point of crack (R/T=4.0)

112


113


114


115

Figure 4.21: Variation of 𝐾𝑚𝑖𝑛

𝐾𝑚𝑎𝑥 versus a/T

116

Figure 4.21: (Continue) Variation of 𝐾𝑚𝑖𝑛

𝐾𝑚𝑎𝑥 versus a/T

117

Figure 4.22: Variation of Ft at deepest point with crack length

118

Figure 4.22: (Continue) Variation of Ft at deepest point with crack length

119

Figure 4.23: Variation of Ft at surface point with crack length

120

Figure 4.23: (Continue) Variation of Ft at surface point with crack length

121

Figure 4.24: Variation of Ft with R/T

122

Figure 4.25: Basic operation of neural network

Figure 4.26: The operation of neural network of current analysis

123

Figure 4.27: Comparison of Fts and FEA data with a/c

124

Figure 4.28: Comparison of Fts and FEA data at deepest point

125

Figure 4.28: (Continue) Comparison of Fts and FEA data at deepest point

126

Figure 4.29: Comparison of Fts and FEA data at surface point

127

Figure 4.29: (Continue) Comparison of Fts and FEA data at surface point

128

Figure 4.30: Reference point chosen for part through crack

129

Figure 4.31: Distribution of Ft along the crack front of a part through crack

130

Figure 4.31: (Continue) Distribution of Ft along the crack front of a part through crack

131

Figure 4.32: Variation of Ft at different reference point on a part through crack

132

Figure 4.33: Distribution of Ft for part through crack of tube (R/T=4.0)

133

Figure 4.33: (Continue) Distribution of Ft for part through crack of tube (R/T=4.0)

134


135


136


137


138


139


140

Figure 4.37: Variation of Ft at 0.99T with a/T for fart through crack

141

Figure 4.37: (Continue) Variation of Ft at 0.99T with a/T for fart through crack

142

Figure 4.38: Variation of Ft at surface with a/T for part through crack

143

Figure 4.38: (Continue) Variation of Ft at surface with a/T for part through crack

144

Figure 4.39: Variation of 𝐾𝑠𝑢𝑟

𝐾0.99𝑇 with a/T for part through crack

145

Figure 4.39: (Continue) Variation of 𝐾𝑠𝑢𝑟

𝐾0.99𝑇 with a/T for part through crack

146

Figure 4.40: Variation of Ft at 0.99T with R/T for part through crack

147

Figure 4.40: (Continue) Variation of Ft at 0.99T with R/T for part through crack

148

Figure 4.41: Variation of Ft at surface with R/T for part through crack

149

Figure 4.41: (Continue) Variation of Ft at surface with R/T for part through crack

150

Figure 4.42: Comparison of Ftp and FEA data at 0.99T point for part through crack

Figure 4.43: Comparison of Ftp and FEA data at inner surface point for part through crack

151

Figure 4.44: Distribution of Ft for through thickness crack

152

Figure 4.44: (Continue) Distribution of Ft for through thickness crack

153

Figure 4.45: Variation of Ft with c/πR for through thickness crack

Figure 4.46: Comparison of Ftf and FEA data at outer surface point for full through crack

154

Figure 4.47: Comparison of Ftf and FEA data at inner surface point for full through crack

Figure 4.48: Effect of tube length on Ft (a/T=1.3, c/πR=0.106, R/T=22.5, T=20mm)

155

Figure 4.49: Effect of wall thickness on Ft (a/T=1.3, c/πR=0.106, R/T=22.5, L/D=10)

156

REFERENCE

[1] BSI, British Standard 7910-Guide on Methods for Assessing the Acceptability of Flaws in

Metallic Structures, British Standard Institution, UK 1999.

[2] R/H/R6-Revision 3. Assessment of the integrity of structures containing defects. British

Energy Generation Ltd. 1999.

[3] Kou, K.P. and Burdekin, F.M. "Stress intensity factors for a wide range of long-deep

semi-elliptical surface cracks, partly through-wall cracks and fully through-wall cracks in

tubular members", Engineering Fracture Mechanics, 73, p1693-1710, 2006.

[4] Hibbitt, Karlsson, and Sorense, Inc., ABAQUS User's Manual, Version 6.7-1. USA,

2007.

[5] Barsoum, R.S., "On the use of isoparametric finite elements in linear fracture mechanics",

Int J Numer Meth Engng, 10:25-37, 1976.

[6] I.S. Raju and J.C. Newman, Jr, STRESS-INTENSITY FACTOR FOR A WIDE RANGE OF SEMI-ELLIPTICAL

SURFACE CRACKS IN FINITE-THICKNESS PLATES. Engineering Fracture Mechanics. 11, pp817-829,

1979.

[7] L.Banks-Sills, APPLICATION OF THE FINITE ELEMENT METHOD TO LINEAR ELASTIC FRACTURE MECHANICS.

Applied Mechanics Reviews, 44, pp447-461, 1991.

[8] Rice, J.R., "A Path Independent Integral and the Approximate Analysis of Strain

Concentration by Notches and Cracks", Journal of Applied Mechanics, Vol. 35, pp. 379-386,

1986.

[9] Mettu, S. R., Raju, I. S., and Forman, R. G., "Stress Intensity Factors for Part -Through

Surface Cracks in Hollow Cylinders", JSC Report 25685/LESC Report 30124, NASA Lyndon

B. Johnson Space Center/Lockheed Engineering and Sciences Co. Joint Publication, 1992.

[10] Zahoor, A., DUCTILE FRACTURE HANDBOOK, VOLUME I: CIRCUMFERENTIAL

THROUGH-WALL CRACKS. Electric Power Research Institute, Palo Alto, CA, 1984.

[11] MATLAB MATLAB_User_Manual © 1994-2009 The MathWorks, Inc. 2009.

157

[12] Howard D. and Mark B. NEURAL NETWORK TOOLBOX FOR USE WITH

MATLAB. The mathworks inc. 1998.

158

CHAPTER 5: ELASTIC-PLASTIC ANALYSIS FOR CRACKED TUBULAR

MEMBERS

5.1 INTRODUCTION

Aside from the ideally brittle materials, any loading of a cracked engineering structure is

accompanied by inelastic deformation in the neighborhood of the crack tip, due to stress

concentration. Consequently, the ultimate utility of Linear-Elastic Fracture Mechanics (LEFM)

must necessarily depend on the extent of inelastic deformation being small compared with the

size of the crack and any other characteristic length that cannot be considered for high

toughness materials. Theories based on Elastic-Plastic Fracture Mechanics (EPFM) are needed

to obtain realistic measures of the fracture behavior of cracked structural systems with these

materials.

In the previous study, the theorem of LEFM was applied to obtain the SIF of long-deep

circumferential semi-elliptical internal surface crack in tubular members. As it is mentioned

above that there is a limitation on the theorem of LEFM as no plastic behavior is allowed even

though when the through thickness crack is approaching the outside surface of the tube. When

the tubular member is made of ductile material, such as steel, plastic deformation is expected

for the location where the stress is larger than the yield stress of the material. In order to

examine the effect of material plasticity on the effect of crack tip deformation of long-deep

circumferential semi-elliptical internal surface cracks in tubular members, finite element

analyses were carried out to study the crack deformation as well as the corresponding

J-integral value of tubular members with long-deep circumferential semi-elliptical internal

surface cracks.

159

5.2 ELASTIC-PLASTIC FE ANALYSIS OF TUBULAR MEMBER WITH

SEMI-ELLIPTICAL INTERNAL SURFACE CRACK

The same finite element models which were set up in the previous study were used in the

Elastic-plastic FE analysis of tubular member with semi-elliptical internal surface crack. The

geometry of tubular member, boundary condition and parameters to be studied are the same

except the non-linear material properties were used instead of linear material properties. The

non-linear material properties were obtained from the coupon test results of a typical carbon

steel material. Detail material properties are discussed in the following section.

The coupon test results of stress versus strain of a typical carbon steel material is shown

in Figure 5.1 which is obtained from reference [6]. From the test results, it is obtained that the

elastic modulus (E) is 215028.6 MPa and the corresponding static yield stress is 323.4 MPa. In

the finite element method, true stress (ζtrue) versus true plastic strain curve (εtruep

) is needed to

be defined instead of the nominal stress versus strain curve. Therefore, the static nominal stress

(ζnom) and nominal strain (εnom) which was obtained from the coupon test were then

converted into the true stress (ζtrue) and true plastic strain (εtruep

) according to Equations (5.1)

and (5.2):

ςtrue = ςnom 1 + εnom (5.1)

ςturep

= ln 1 + εnom − ςture

E (5.2)

It is known that substantial localized deformation continues to develop in a tension coupon

after reaching the ultimate load and before the specimen eventually breaks. At this stage the

stress and strain are no longer calculated based on initial dimension. Therefore, the material

true plastic strain was extended to 1.0 in order to consider the necking effect conservatively

[7].The typical true stress versus true plastic strain curve is shown in Figure 5.2.

160

In order to include the effect of non-linear material property on the FE analysis,

non-linear analysis procedure should be introduced. The Newton-Raphson method which is

mentioned previously was applied in the non-linear FE analysis. The maximum far end stress

(100 MPa), which is the same as that used in the elastic analysis before, is used in the

elastic-plastic case as well. This maximum stress is applied in a multiple steps not less than

20 steps for each analysis. This steps size is chosen to ensure numerical stability. Nodal

displacement and the corresponding J-integral value of each non-linear step were calculated.

In order to examine the effect of non-linear material property to the prediction of J-integral

value, comparison of the FE results obtained from LEFM case and EPFM case is carried out

in the following section. Then, detail results which were obtained from the EPFM analysis

are discussed and analyzed.

5.3 COMPARISON OF FE RESULTS OBTAINED FROM EPFM AND LEFM

In order to examine the effect of plastic deformation on the prediction of J-integral of

long-deep circumferential semi-elliptical internal surface crack in tubular members,

comparison of the results of, (1) crack opening displacement (COD), (2) J-integral value (J)

along crack front and (3) normalized stress intensity factor (Ft), which were obtained from

EPFM and LEFM are carried out.

5.3.1 COMPARISON OF COD RESULTS OBTAINED FROM EPFM AND LEFM

The crack opening displacement (COD) is one of the important parameters which describes

crack tip behavior in the analysis of fracture mechanic as it is known that COD value can be

related to the calculation of J-integral value. Therefore, in this study, crack opening

displacement (COD) results obtained from FE analysis of both EPFM and LEFM are

compared. Plot of the COD in z-direction along the through thickness direction (x-axis) and

circumferential direction (y-axis) are shown in Figures 5.4, and comparison of COD of

EPFM analysis results and LEFM analysis results are shown in Figures 5.5 to 5.26 and

161

Figures 5.27 to 5.48, respectively. Discussion of the comparison of COD is shown in the

following sections.

COD in z direction along the through thickness direction (x-axis)

The plots of the COD value along the through thickness direction for a/T varies from 0.2 to

0.99 with fix values of R/T=10 and c/R=0.477 are shown in Figures 5.5 to 5.14. In those

figures, COD values obtained from both analysis of EPFM and LEFM are shown. It is found

that, when a/T is less than 0.5, the COD are almost the same for EPFM case and LEFM case.

It implies that plastic deformation is not significant for a/T value which is less than 0.5. When

a/T values become larger, the different between the COD value for the case of EPFM to

LEFM becomes more significant. Larger COD is observed for the case of EPFM and it is

believed that the larger displacement is caused by the yielding of material near the vicinity of

crack tip. For a/T larger than 0.93, the COD value is almost uniform along the through

thickness direction of the crack. For the case of a/T=0.99, the crack mouth displacement

(where x=0) of the EPFM case is almost six times larger than the LEFM case.

The effect of c/R value to the COD for a fix values of R/T=10 and a/T=0.9 are shown in

Figures 5.15 to 5.22. In those figures, c/R varies from 0.106 to 0.477 and both COD

obtained from the EPFM case and LEFM case are shown. It is observed that larger COD

values were obtained for model with larger c/R value with analysis of EPFM. Meanwhile,

the different between the COD value obtained from EPFM and LEFM becomes larger with

increasing c/R value. For the case of c/R=0.477, the crack mouth displacement (where x=0)

of the EPFM case is about 2.2 times the value obtained from the LEFM case.

Similar comparison was made for models with R/T varies from 4.0 to 22.5 and with fix

values of a/T=0.9 and c/R= 0.477 and the corresponding plots are shown in Figures 5.23 to

5.26. It is found that COD increases as R/T value increases. For the case of R/T=22.5, the

crack mouth displacement (where x=0) of the EPFM case is about 2.6 times the value

obtained from the LEFM case.

162

COD in z direction along the circumferential direction (y-axis)

The plots of the COD value along the circumference direction for a/T varies from 0.2 to 0.99

with fix values of R/T=10 and c/R=0.477 are shown in Figures 5.27 to 5.36. In those figures,

COD values obtained from both analysis of EPFM and LEFM are shown. It is found that,

when a/T is less than 0.5, the COD are almost the same for the analysis from EPFM and

LEFM. It implies that plastic deformation is not significant for a/T value is less than 0.5.

When a/T values become larger, the different between the COD value for the case of EPFM

to LEFM becomes more significant. Larger COD is observed for the case of EPFM and it is

believed that the larger displacement is caused by the yielding of material near the vicinity of

crack tip. For the case of a/T=0.99, the crack mouth displacement (where y=0) of the EPFM

case is almost six times larger than the LEFM case.

The effect of c/R value to the COD for a fix values of R/T=10 and a/T=0.9 are shown in

Figures 5.37 to 5.44. In those figures, c/R varies from 0.106 to 0.477 and both COD from

the analysis of EPFM and LEFM are shown. It is observed that larger COD values were

obtained for model with larger c/R value with analysis of EPFM. Meanwhile, the different

between the COD value obtained from EPFM and LEFM becomes larger with increasing

c/R value. However, for the case of c/R=0.477, the crack mouth displacement (where y=0)

of the EPFM case is about 2.7 times the value obtained from the LEFM case.


values of a/T=0.9 and c/R= 0.477 and the corresponding plots are shown in Figures 5.45 to

5.48. It is found that COD increases as R/T value increases. For the case of R/T=22.5, the

crack mouth displacement (where y=0) of the EPFM case is about 2.85 times the value

obtained from the LEFM case.

For the parameters studied, it is found that the effect of a/T ratio to the COD of the

EPFM case is more significant. It is due to the fact that when the through thickness crack

length is approaching the thickness of the tubular member, larger local yielding near the

vicinity of the crack tip is expected. As a result, larger COD value is obtained.

163

5.3.2 COMPARISON OF J-INTEGRAL OBTAINED FROM EPFM AND LEFM

Distributions of J-integral along the crack front obtained from EPFM are shown in Figures

5.49 to Figure 5.70 together with the J-integral results obtained from LEFM along the crack

front. Effects due to different parameters to the prediction of J-integral are discussed in the

following sections.

Effect of crack depth to thickness ratio (a/T)

For Elastic-Plastic analysis of the surface crack, J-integral values distribution tendency is the

same with Linear-Elastic analysis, however, for the plastic analysis the J-integral values are

much greater than the elastics' as the crack depth deep approached to the thickness of the tube.

Comparison of the J-integral values obtained from the LEFM and EPEM analysis for a/T=0.2

and 0.5 are shown in Figures 5.49 and 5.50, respectively with the same definition of

Linear-Elastic analysis, along the crack front, surface point at =0 degree, and deepest point

at =90 degree. For a/T=0.2 and 0.5, the J-integral values obtained from the EPFM analysis

are closed to that obtained from LEFM analysis. It is observed that the J-integral values of the

EPFM case are a bit lower than that of the LEFM case, especially with approaches to 90

degree. As J-integral value is proportional to the crack displacement, it is believed that this

lower value is due to the higher elastic modulus value which is assigned to the FE model of

EPFM case. Nevertheless, the tendency of the J-integral values for both cases is very similar.

It implies that the effect of plasticity is not significant when a/T value is less than 0.5.

Comparison of the J-integral values obtained from the LEFM and EPFM analysis for a/T

value larger than 0.8 are shown in Figures 5.51 to 5.58. It is shown in those figures that

graduate increase of J-integral values was observed for between 0 degree to 45 degree.

However, when is larger than 45 degree, the J-integral values obtained from the EPFM

analysis increase dramatically. For a/T=0.8, the J-integral values obtained from the EPFM

analysis is about 47% higher than that obtained from the LEFM analysis when =90 degree.

It is believed that the increase of J-integral value is due to the plastic deformation near the

164

vicinity of the crack depth. As the crack depth increases, the influence of the plastic

deformation becomes more significant. This could be observed from the results shown in

Figure 5.58 for the case when a/T=0.99. For this case, the J-integral values obtained from the

EPFM analysis is about 95% higher than that obtained from the LEFM analysis when =90

degree.

Effect of crack length to circumference ratio (c/R)

The effect of c/R value to the J-integral for a fix values of R/T=10 and a/T=0.9 are shown in

Figures 5.59 to 5.66. In those figures, c/R varies from 0.106 to 0.477 and both J-integral

from the analysis of EPFM and LEFM are shown. It is observed that larger J-integral values

were obtained for model with larger c/R value with analysis of EPFM. Meanwhile, the

different between the J-integral value obtained from EPFM and LEFM becomes larger with

increasing c/R value. Comparing the J-integral values obtained from the EPFM analysis to

that obtained from LEFM analysis, graduate increase of J-integral values was observed for

between 0 degree to 45 degree. However, when is larger than 45 degree, the J-integral

values obtained from the EPFM analysis increase dramatically. And J-integral values are

increasing with the crack length increasing. For short crack, c/R=0.106, the J-integral values

obtained from the EPFM analysis is about 22% higher than that obtained from the LEFM

analysis when =90 degree. However, for long crack, c/R=0.477, the J-integral values

obtained from the EPFM analysis is about 86% higher than that obtained from the LEFM

analysis when =90 degree.

Effect of R/T


values of a/T = 0.9 and c/R = 0.477 and the corresponding plots are shown in Figures 5.67

to 5.70. It is found that J-integral values increases as R/T value increases. For the case of R/T

= 22.5, the J-integral value of the EPFM case is about 90% higher than the value obtained

from the LEFM case.

165

5.4 FEA RESULTS OF CRACKED TUBE WITH ELASTIC-PLASTIC ANALYSIS

Three dimensional EPFM FEA have been carried out to investigate the J integral value,

normalized stress intensity factor, Ft, and crack tip opening displacement (CTOD), along the

crack front of tube. The FE analyses were performed using the deformation theory of

plasticity and the small strain analysis theory. Form the deformation theory of plasticity, the

fracture response parameters, such as J and crack tip opening displacement (CTOD), δ, can

be split into elastic and plastic components, as

JEP = Je + JP (5.3)

δEP = δe + δP (5.4)

where the subscripts 'e' and 'p' refer to the elastic and plastic contributions, respectively. The

Equation (5.3) can have a series transformation as follows:

JEP

Je=

Je

Je+

JP

Je (5.5)

JEP = (1 +JP

Je )Je (5.6)

JEP = αJe (5.7)

Under the tension loading, Je and δe can be expressed via linear stress intensity factor KI

and CTOD due to tension loads applied.

Je =KI

E′

2

(5.8)

166

δe = δeT (5.9)

From Equation (5.7), JEP is equal to αJe , and Equation (5.8) is used to investigate the

normalized stress intensity factor, (Ft) which, was converted from the J integral value of

Elastic-Plastic analysis.

Ft = αJeE′

ς πa (5.10)

That is

Ft = JEP E′

ς πa (5.11)

where for plane stress E′ = E, and for plane strain E′ =E

1−υ2 .

By varying parameters including crack depth ratio (a/T), ratio of diameter to thickness

(R/T) and ratio of crack length to perimeter (c/R), tubes with a range of different cracks

have been analyzed. Details of parameters for the Elastic-Plastic analysis of tubes with

surface cracks are the same with Liner-Elastic analysis as shown in Table 4.3. With the full

combination of a/T, c/R and R/T a total of 320 models has been dealt with. Throughout the

analyses, the wall thickness, T, was taken as 20mm and L/D as 10, where L is the half length

of tube.

5.4.1 DISTRIBUTION OF THE CRITICAL PARAMETERS

For EPFM analysis, the critical parameters are crack opening displacement (COD), J-integral

value (J) along crack front and normalized stress intensity factor (Ft).These parameters are

used to predict the crack tip behavior with plasticity deformation. The J-integral values at

surface point and deepest point are as shown in Table 5.1 to 5.8, and Ft values at the surface

point and the deepest point are listed in Table 5.9 to 5.16. Form these tables, it can be seen

167

that the critical parameters distributions are similar with LEFM analysis results, however, the

values of EPFM are much larger than LEFM analysis results as shown in Figure 5.3.

5.4.2 COMPARE OF FT FROM ELASTIC-PLASTIC ANALYSIS AND LINEAR-ELASTIC ANALYSIS

In the previous study of the LEFM analysis, normalized stress intensity factor NSIF was

calculated based on the J-integral results obtained from the finite element analysis. Although

the J-integral results which were obtained from both LEFM and EPFM analysis were

compared directly in the previous section, better information could be obtained by comparing

the NSIF since this factor is normalized with respect to the square-root of crack depth.

Therefore, the NSIF (Ft) values were calculated based on the J-integral results obtained from

the EPFM analysis. Comparisons of the NSIF (Ft) obtained from the EPFM analysis and

LEFM analysis were made and the values along the crack tip were showed in Figures 5.71 to

5.92. It may be noted that since Ft is the normalized values of SIF which is related to the

J-integral value, the tendency of Ft are very similar to the results of J-integral. As it is shown

from those figures that the maximum and minimum values of Ft occurred on the deepest point

and surface point of the crack, respectively, detail comparison of the Ft values on the deepest

point and surface point were made for the EPFM case and LEFM case.

The Ft values were obtained from the EPFM analyses are listed in Tables 5.9 to 5.16.

And the ratio of NSIF of EPFM analysis to NSIF of LEFM analysis at surface point and

deepest point are listed in Tables 5.17 to 5.24. In these tables, the ratio which are larger than

1.05 are shadowed. The shadowed values indicate the influence of material non-linear effect

on the prediction of J-integral value. From those tables, it can be seen that, the influence of

material non-linear are more pronounced on deepest points than on surface points. The Ft

values of EPFM analysis are increasing rapidly with crack depth approaches the thickness of

the tube. Similar behavior is observed for increasing c/R value and R/T value. Although the

prediction of the Ft values depend on three parameters, a/T, R/T and c/R, a general

comparison of the Ft values obtained from the EPFM analysis and the LEFM analysis is

discussed in the following section.

168

In general, it is shown from the tables that when a/T is larger than 0.8, the Ft value on

the deepest point obtained from the Elastic-Plastic analysis is larger than that obtained from

the Elastic analysis. For the prediction of Ft value on the surface point, the Ft values obtained

from the Elastic-plastic analysis is larger than that obtained from the Elastic analysis when

a/T is larger than 0.9 and c/R is larger than 0.477. It is also observed that for large R/T value

(R/T > 10), when c/R < 0.212 and a/T > 0.5, the Ft values obtained from the Elastic-Plastic

analysis is smaller than that obtained from the Elastic analysis. However, for large a/T and

c/R values, the predictions of Ft obtained from the Elastic-Plastic analysis are larger than

that obtained from the Elastic analysis.

5.4.3 ANALYTICAL EQUATION FOR THE PREDICTION OF FT BASED ON FEA DATA FOR

SURFACE CRACK WITH ELASTIC-PLASTIC ANALYSIS

All Ft values at the surface point and the deepest point are listed in Table 5.9 to 5.16. Based

on these FE results, a function Fts for predicting the Ft values for surface crack at the deepest

points and the surface points was obtained by fitting the corresponding FEA data into a curve.

The mathematical software MATLAB [8] together with the toolbox NEURAL NETWORK

[9] was adopted to accomplish this curve fitting. In the present study, three layers were used

for Fts both at the deepest points and surface points. In the first and second layers,

Log-Sigmoid Transfer Functions (logsig) were used and a linear Transfer Function (purelin)

was used for the third layer the Equation are shown in (5.12). Parameters W1, W2, W3, B1, B2,

and B3 are given by Equations (5.13) and (5.14) for deepest points and surface points.

𝐹𝑡𝑠 = 𝑊3𝑓𝑙 𝑊2𝑓𝑙 𝑊1𝑃 + 𝐵1 + 𝐵2 + 𝐵3

𝑓𝑙 𝑥 =1

1 + 𝑒−𝑥

input vector P =

a

TR

Tc

πR

(5.12)

169

For Fts at the deepest point, 10 neurons were used in the first and second layers respectively and

1 neuron was used in the third layer. The corresponding weight matrices W1, W2 and W3 and

bias matrices B1, B2 and B3 were calculated as Equation (5.13).

W1 =

−33.2879 −5.3925 −7.843128.3261 0.2756 90.19938.5089 1.0411 4.21692.8839 −0.0514 −3.9703−8.8138 1.8322 11.0352−7.6484 0.0287 3.8663−28.3665 −0.0619 −3.4317−0.0926 −0.1124 −8.861723.6993 0.0487 8.0991−50.0256 −1.8474 13.8336

B1 =

55.7525−9.6848−17.4482−0.6215−6.81965.9024

30.11511.8045

−19.723070.5057

𝑊2=

−8.7460 16.3577 29.4580 −47.4555 −4.0888 −29.2203 25.9044 13.5737 −20.4177 1.3941−44.5523 −3.6991 9.2055 4.2726 39.2543 19.5212 −7.7625 20.4163 −55.0369 −2.148276.7593 −81.2636 89.7773 418.8020−311.8997513.6849−158.1385−392.7255 −8.1271 13.16795.5379 4.6551 −0.3383 −20.9201 −8.2218 −10.7286 −6.0614 3.2057 −0.6063 1.15002.7334 −278.5284 −0.3731 −22.0230 −2.1438 −11.9688 7.5105 5.0075 −1.6338 561.8040

12.5659 −83.2221 −27.8714339.4536 29.8464 181.0010−209.5422 14.1156 −5.0512 −1.35020.0458 −33.8315 −0.2642 16.3361 4.1116 12.3913 −11.3540 −1.7091 66.3362 0.25651.4361 86.3131 −9.2256 −13.5844 8.3690 −15.1940 7.8500 −1.8370 −154.4901−0.19245.4132 11.5244 0.1173 −0.1813 −4.5976 −7.3445 −0.1248 −0.2006 −3.7615 −1.1695−5.1846 −19.6075 11.0570 6.6933 8.2244 8.8823 −3.2714 4.1435 2.6628 15.3628

B2 =

7.48791.1622

−87.746719.4711

−276.0316−87.4780−40.269079.2297−1.8317−5.3801

W3=

26.61304.54950.2213

33.4320−28.562015.60307.4905

−11.9630−29.3912−26.7149

T

B3 = 10.1381 (5.13)

For Fts at the surface point, 10 neurons were used in the first and second layers respectively and

1 neuron were used in the third layer and the corresponding weight matrices W1, W2 and W3

and bias matrices B1, B2 and B3 were calculated as Equation (5.14).

170

W1 =

5.0281−18.9932

8.6188−1.3968−74.7655−1.2082−3.6901−3.637231.62866.8223

−0.0726−0.1060−0.14350.89808.1381 0.10450.16660.1198−2.0298−0.1344

−33.5322−11.1035

7.0530−2.1828

−109.5114−108.5186

24.5943−13.9118−5.0740−19.4802

B1 =

0.626926.1798−8.3530−2.395228.257635.4930−6.82774.6235

−17.37555.0826

𝑊2=

36.9986−5.294019.2731−16.1220

0.0017 28.12541.7734

−45.6061−26.8539 −7.1662

−53.0995−35.279056.84586.8685−2.5899 53.6097−15.614351.48123.3122 1.5363

−16.0617−28.9737−12.8955 43.5639 0.714180.53021.3320

−52.24853.7882−7.0994

5.0443−2.6772−34.0487−20.7580

8.4438−68.810931.710732.9767

123.2406−4.2063

−1.82110.5851−0.5655−10.3416−0.210241.2488−0.10342.3281−2.29520.6423

31.91661.6942−0.35642.7896−0.52135.40040.0957

−10.11163.2402−1.4908

4.6823−5.22424.9310 1.7302

−2.6500 −37.7987−2.2154−16.0339−3.58503.6070

5.20844.8359−8.032062.6855−1.062810.6141−2.8224−2.39130.9124

−3.2781

−0.7977−102.2196−34.7347

4.65865.5067

−81.1708395.3753

6.629485.80451.2078

6.5702−11.6530−34.937512.54230.7923

−29.487110.883568.5475−10.4069

6.4887

B2 =

4.720341.395521.2320−58.0374−12.866916.8994−24.9902−69.8588−117.5377−0.5968

W3=

−0.2247−0.3290−0.18340.9672

−630.72500.51040.30270.0653−0.2749−0.3998

T

B3 = 0.3998 (5.14)

The typical result of the normalized SIFs of deepest points were predicted from Equations 5.12

to 5.13 which compared well with the finite element results, the maximum difference between

the predicted values and FE results are 4.1% and most are less 1%. In addition, for surface

points the prediction of normalized SIFs by using Equations 5.12 and 5.14. The maximum

difference between the prediction Fts and FE results are 6.5% and most are also less 2%.

Therefore, the predictions of the normalized SIFs at deepest points and surface points by using

the current proposed equations compare well with the FE results for a wider range of a/c and

a/T values for surface crack with Elastic-Plastic analysis. The plot of MATLAB equations and

171

FEA results of deepest points and surface points are shown in Figures 5.93 and 5.94,

respectively.

5.5 CONCLUSION

In the present study, in order to examine the effect of material plasticity on the effect of crack

tip deformation of long-deep circumferential semi-elliptical internal surface cracks in tubular

members, finite element analyses were carried out to study the crack deformation as well as

the corresponding J-integral value of tubular members with long-deep circumferential

semi-elliptical internal surface cracks. Comparison of the results of, (1) crack opening

displacement (COD), (2) J-integral value (J) along crack front and (3) normalized stress

intensity factor (Ft), which were obtained from EPFM and LEFM are carried out.

From comparison of COD values of EPFM and LEFM, it can be obtained that, the effect

of a/T ratio to the COD of the EPFM case is more significant. When a/T is less than 0.5, the

COD are almost the same for the analysis from EPFM and LEFM. It implies that plastic

deformation is not significant for a/T values are less than 0.5. As a/T values become larger,

the influence of material plasticity on the COD becomes more significant. Larger COD is

observed for the case of EPFM and it is believed that the larger displacement is caused by the

yielding of material near the vicinity of crack tip.

Comparison of the J-integral values obtained from the LEFM and EPEM analysis

showed, that the effect of plasticity is not significant when a/T value is less than 0.5. When

a/T value is larger than 0.8, comparing the J-integral values obtained from the EPEM analysis

to that obtained from LEFM analysis, graduate increase of J-integral values was observed for

between 0 degree to 45 degree. However, when is larger than 45 degree, the J-integral

values obtained from the EPEM analysis increase dramatically. It is believed that the increase

of J-integral value is due to the plastic deformation near the vicinity of the crack depth. As

the crack depth increases, the influence of the plastic deformation becomes more significant.

As the crack length becomes longer, the effect of plastic deformation to the prediction of

172

J-integral values becomes more significant. It is also found that J-integral values increases as

R/T value increases.

Comparison of the NSIF obtained from the EPFM analysis and the LEFM analysis were

also made. It may be noted that since Ft is the normalized values of SIF form J-integral by

dividing the factor ς πa, the tendency of Ft are the same with J-integral, only the rate of

increasing of Ft is lower than J-integral.

The current finite element results of SIFs with EPFM analysis were used to produce a

comprehensive set of equations which presented an analytical expression for SIFs as a function

of a/T, c/πR, and R/T for long-deep circumferential semi-elliptical internal surface cracks in

tubular members under EPFM analysis. It is shown that the difference between the predicted

values obtained by using current proposed equations and the FE results at the deepest point and

surface point are less than 4.1% and 6.5%, respectively. Therefore, the current proposed

equations provide well predictions of the normalized SIFs at deepest point and surface point for

a wider range of a/c and a/T values. With the current proposed equations, the fatigue crack

growth trend can be established which can help for estimating the inspection intervals for

circular tube structures.

173

Table 5.1 J-integral values at surface point of surface crack for EPFM(R/T=4.0)

c/πR a/t

0.2 0.5 0.8 0.85 0.9 0.93 0.95 0.97 0.98 0.99

0.106 0.1128 0.2687 1.5910 1.7700 1.9020 1.9850 2.0330 2.0780 2.1000 2.0490

0.159 0.0849 0.2603 1.5990 1.7870 1.9790 2.1080 2.2010 2.3000 2.3530 2.2740

0.212 0.0665 0.2420 1.4890 1.4340 1.7180 1.4850 1.9810 2.0990 2.1630 2.1110

0.265 0.0539 0.2266 1.3660 1.3570 1.5870 1.5380 2.0300 2.2690 2.3800 2.3760

0.318 0.0447 0.2115 1.2320 1.2970 1.4100 1.5740 2.0330 2.2460 2.3580 2.3410

0.371 0.0377 0.1977 1.1000 1.1140 1.4290 1.6750 1.8690 2.0910 2.5280 2.6390

0.424 0.0322 0.1847 0.9568 1.0670 1.3190 1.7180 1.9870 2.3080 2.5210 2.6270

0.477 0.0275 0.1727 0.8359 1.0080 1.2790 1.8390 2.1650 2.6180 2.9500 3.1910

2ф/π=0(surface point), R/T=4.0

Table 5.2 J integral values at deepest point of surface crack for EPFM (R/T=4.0)

c/πR a/t

0.2 0.5 0.8 0.85 0.9 0.93 0.95 0.97 0.98 0.99

0.106 0.6317 1.426 2.055 2.231 2.614 3.322 4.49 6.604 7.91 10.04

0.159 0.6738 1.790 3.052 3.465 4.600 7.139 9.812 13.66 15.2 20.21

0.212 0.6957 2.039 3.995 4.791 7.655 12.800 17.14 23.67 26.5 35.39

0.265 0.7085 2.225 4.931 6.468 12.540 20.510 26.59 37.58 43.1 56.95

0.318 0.7166 2.365 5.882 8.807 19.530 30.260 38.83 57.02 67.2 87.1

0.371 0.7222 2.475 6.841 12.050 28.600 43.130 56.16 84.83 101 128.3

0.424 0.7262 2.564 7.904 16.500 38.920 61.010 81.51 125.5 151 185

0.477 0.7294 2.639 9.155 22.170 51.240 84.670 117.3 185.8 222 267.1

2ф/π=1(deepest point), R/T=4.0

Table 5.3 J integral values at surface point of surface crack for EPFM (R/T=10.0)

c/πR a/t

0.2 0.5 0.8 0.85 0.9 0.93 0.95 0.97 0.98 0.99

0.106 0.0558 0.2479 0.5464 0.5789 0.5962 0.6010 0.6000 0.5943 0.5914 0.4443

0.159 0.0345 0.2100 0.4064 0.4378 0.4506 0.4388 0.4159 0.3867 0.3722 0.4537

0.212 0.0268 0.1696 0.4620 0.5296 0.6100 0.6560 0.6982 0.7377 0.7522 0.5059

0.265 0.0205 0.1474 0.3419 0.3950 0.4607 0.4938 0.5030 0.5179 0.5379 0.6381

0.318 0.0155 0.1289 0.3741 0.4525 0.5782 0.6821 0.7758 0.9327 0.6754 1.0830

0.371 0.0145 0.1133 0.2813 0.3439 0.4420 0.5208 0.5956 0.7505 0.8615 1.0510

0.424 0.0116 0.1003 0.3015 0.3806 0.4235 0.6617 0.8462 1.2430 1.5400 1.6810

0.477 0.0116 0.0893 0.2334 0.2879 0.3933 0.5101 0.6699 1.0900 1.4090 1.7090


174


c/πR a/t

0.2 0.5 0.8 0.85 0.9 0.93 0.95 0.97 0.98 0.99

0.106 0.7631 2.753 4.779 5.334 7.483 11.47 15.47 20.95 24.43 33.21

0.159 0.7875 3.248 6.868 8.370 15.600 24.24 31.58 42.22 52.47 67.53

0.212 0.7384 3.391 8.899 12.670 27.330 40.86 56.25 77.07 95.46 128.50

0.265 0.7440 3.589 11.140 19.720 43.370 63.51 91.09 130.30 158.00 194.40

0.318 0.7475 3.735 13.860 29.670 62.430 95.07 139.40 207.60 243.20 297.00

0.371 0.7500 3.847 17.020 41.270 85.850 138.10 207.00 313.60 356.00 441.40

0.424 0.7519 3.935 20.770 52.250 114.600 193.70 298.90 455.90 508.40 645.00

0.477 0.7519 4.006 24.930 63.500 147.900 264.20 422.10 644.00 715.90 936.50



c/πR a/t

0.2 0.5 0.8 0.85 0.9 0.93 0.95 0.97 0.98 0.99

0.106 0.0360 0.2171 0.5422 0.5845 0.6243 0.6491 0.6574 0.6624 0.6657 0.6694

0.159 0.0209 0.1710 0.3936 0.4344 0.4869 0.4869 0.4893 0.4891 0.4924 0.5693

0.212 0.0176 0.1382 0.3996 0.4708 0.5746 0.6502 0.7164 0.8047 0.8679 0.8841

0.265 0.0131 0.1097 0.2990 0.3605 0.4473 0.5138 0.5729 0.6812 0.7858 0.9150

0.318 0.0104 0.0920 0.2959 0.3711 0.4965 0.6098 0.7364 0.9837 1.1890 1.2910

0.371 0.0079 0.0783 0.2313 0.2850 0.3814 0.4802 0.6036 0.8964 1.1620 1.4000

0.424 0.0074 0.0675 0.2211 0.2836 0.3933 0.5196 0.6933 1.1510 1.5630 1.8710

0.477 0.0063 0.0589 0.1779 0.2213 0.2973 0.3979 0.5591 1.0990 1.6420 1.9990



c/πR a/t

0.2 0.5 0.8 0.85 0.9 0.93 0.95 0.97 0.98 0.99

0.106 0.8057 3.498 6.944 7.909 11.94 19.70 25.26 35.66 41.25 54.94

0.159 0.8233 3.999 9.819 13.340 39.99 39.99 51.12 74.89 91.79 122.10

0.212 0.7637 4.303 12.710 22.370 45.88 70.91 94.53 141.90 169.50 203.50

0.265 0.7676 4.287 15.930 34.830 68.77 111.30 155.30 235.50 284.60 348.10

0.318 0.7701 4.427 19.630 48.380 97.90 167.30 242.40 371.90 420.70 492.30

0.371 0.6956 4.534 24.170 61.770 131.90 235.00 351.70 542.80 623.00 753.20

0.424 0.6957 4.617 29.920 76.010 171.70 323.80 502.70 766.20 844.10 1051

0.477 0.6958 4.682 35.670 90.580 215.90 429.30 690.10 1075 1217 1537


175


c/πR a/t

0.2 0.5 0.8 0.85 0.9 0.93 0.95 0.97 0.98 0.99

0.106 0.0236 0.1699 0.4970 0.5618 0.6237 0.6652 0.7026 0.7343 0.7462 0.7280

0.159 0.0139 0.1232 0.3423 0.3989 0.4632 0.5105 0.5377 0.5795 0.6142 0.6845

0.212 0.0104 0.0939 0.3189 0.3923 0.5082 0.6058 0.6965 0.8471 0.9643 1.0210

0.265 0.0073 0.0734 0.2348 0.2885 0.3794 0.4674 0.5617 0.7478 0.9114 1.0510

0.318 0.0057 0.0597 0.2142 0.2757 0.3775 0.4875 0.6238 0.9042 1.1650 1.3730

0.371 0.0058 0.0495 0.1641 0.2066 0.2771 0.3597 0.4757 0.7700 1.0900 1.3510

0.424 0.0044 0.0409 0.1490 0.1897 0.2665 0.3633 0.4996 0.8669 1.2740 1.7560

0.477 0.0036 0.0346 0.1161 0.1470 0.1981 0.2670 0.3697 0.7267 1.2560 1.8620



c/πR a/t

0.2 0.5 0.8 0.85 0.9 0.93 0.95 0.97 0.98 0.99

0.106 0.7759 4.230 10.04 11.84 20.82 32.43 42.91 59.51 74.10 95.38

0.159 0.7867 4.720 14.36 21.41 44.40 65.92 93.39 134.40 166.80 210.40

0.212 0.7915 5.005 19.14 37.64 74.61 117.60 174.00 261.00 299.70 362.00

0.265 0.7942 5.053 24.64 54.66 113.60 188.10 283.80 430.50 490.20 588.40

0.318 0.7961 5.185 30.93 71.33 160.00 282.20 440.50 654.40 711.10 903.90

0.371 0.7209 5.282 37.35 88.54 211.20 397.80 628.50 946.80 1042 1297

0.424 0.7199 5.164 43.80 105.30 268.70 549.70 875.20 1300 1417 1895

0.477 0.7195 5.217 52.23 121.60 330.60 732.40 1183 1814 2044 2736


Table 5.9 NSIF Ft at surface point of surface crack for EPFM (R/T=4.0)

c/πR a/t

0.2 0.5 0.8 0.85 0.9 0.93 0.95 0.97 0.98 0.99

0.106 0.4606 0.4496 0.8648 0.8849 0.8915 0.8959 0.8971 0.8976 0.8977 0.8822

0.159 0.3996 0.4425 0.8670 0.8892 0.9094 0.9233 0.9334 0.9443 0.9502 0.9294

0.212 0.3536 0.4266 0.8366 0.7965 0.8473 0.7749 0.8856 0.9021 0.9111 0.8955

0.265 0.3183 0.4128 0.8013 0.7749 0.8143 0.7886 0.8964 0.9379 0.9557 0.9500

0.318 0.2899 0.3988 0.7610 0.7575 0.7676 0.7978 0.8971 0.9332 0.9513 0.9430

0.371 0.2663 0.3856 0.7191 0.7021 0.7727 0.8230 0.8602 0.9004 0.9849 1.0012

0.424 0.2459 0.3727 0.6707 0.6871 0.7424 0.8335 0.8869 0.9460 0.9836 0.9990

0.477 0.2272 0.3604 0.6269 0.6678 0.7311 0.8624 0.9258 1.0075 1.0640 1.1010


176

Table 5.10 NSIF Ft at deepest point of surface crack for EPFM (R/T=4.0)

c/πR a/t

0.2 0.5 0.8 0.85 0.9 0.93 0.95 0.97 0.98 0.99

0.106 1.0899 1.0356 0.9829 0.9935 1.0451 1.1590 1.3332 1.6001 1.7423 1.9529

0.159 1.1256 1.1603 1.1978 1.2382 1.3864 1.6991 1.9709 2.3013 2.4120 2.7708

0.212 1.1438 1.2384 1.3704 1.4559 1.7885 2.2751 2.6048 3.0294 3.1877 3.6666

0.265 1.1542 1.2937 1.5225 1.6917 2.2891 2.8799 3.2444 3.8171 4.0683 4.6512

0.318 1.1608 1.3337 1.6629 1.9740 2.8567 3.4981 3.9207 4.7018 5.0763 5.7521

0.371 1.1653 1.3644 1.7933 2.3090 3.4570 4.1762 4.7151 5.7349 6.2318 6.9813

0.424 1.1686 1.3887 1.9276 2.7019 4.0328 4.9670 5.6804 6.9755 7.6198 8.3831

0.477 1.1711 1.4089 2.0745 3.1319 4.6272 5.8514 6.8144 8.4874 9.2300 10.0730



c/πR a/t

0.2 0.5 0.8 0.85 0.9 0.93 0.95 0.97 0.98 0.99

0.106 0.3238 0.4318 0.5068 0.5061 0.4991 0.4930 0.4874 0.4800 0.4764 0.4108

0.159 0.2548 0.3974 0.4371 0.4401 0.4339 0.4212 0.4058 0.3872 0.3779 0.4151

0.212 0.2244 0.3572 0.4660 0.4841 0.5049 0.5150 0.5257 0.5348 0.5373 0.4384

0.265 0.1961 0.3330 0.4009 0.4180 0.4388 0.4469 0.4462 0.4481 0.4543 0.4923

0.318 0.1706 0.3114 0.4194 0.4474 0.4915 0.5252 0.5542 0.6013 0.5091 0.6414

0.371 0.1650 0.2919 0.3636 0.3901 0.4298 0.4589 0.4856 0.5394 0.5750 0.6319

0.424 0.1477 0.2747 0.3765 0.4104 0.4207 0.5173 0.5788 0.6942 0.7687 0.7991

0.477 0.1477 0.2592 0.3312 0.3569 0.4054 0.4542 0.5150 0.6501 0.7353 0.8057



c/πR a/t

0.2 0.5 0.8 0.85 0.9 0.93 0.95 0.97 0.98 0.99

0.106 1.1979 1.4390 1.4989 1.5362 1.7683 2.1537 2.4747 2.8500 3.0619 3.5518

0.159 1.2169 1.5630 1.7968 1.9244 2.5532 3.1309 3.5358 4.0459 4.4872 5.0649

0.212 1.1783 1.5970 2.0453 2.3676 3.3794 4.0649 4.7189 5.4663 6.0525 6.9867

0.265 1.1828 1.6430 2.2884 2.9538 4.2571 5.0678 6.0050 7.1076 7.7867 8.5935

0.318 1.1856 1.6761 2.5525 3.6232 5.1076 6.2004 7.4286 8.9715 9.6606 10.6218

0.371 1.1876 1.7010 2.8286 4.2731 5.9894 7.4730 9.0523 11.0265 11.6882 12.9490

0.424 1.1891 1.7204 3.1247 4.8081 6.9200 8.8504 10.8777 13.2949 13.9678 15.6531

0.477 1.1891 1.7358 3.4234 5.3005 7.8614 10.3362 12.9266 15.8014 16.5749 18.8614


177


c/πR a/t

0.2 0.5 0.8 0.85 0.9 0.93 0.95 0.97 0.98 0.99

0.106 0.2602 0.4041 0.5049 0.5085 0.5108 0.5123 0.5101 0.5068 0.5054 0.5043

0.159 0.1980 0.3586 0.4302 0.4384 0.4511 0.4437 0.4401 0.4355 0.4347 0.4650

0.212 0.1820 0.3224 0.4334 0.4564 0.4900 0.5128 0.5325 0.5586 0.5771 0.5795

0.265 0.1572 0.2872 0.3749 0.3994 0.4323 0.4558 0.4762 0.5139 0.5491 0.5896

0.318 0.1397 0.2631 0.3730 0.4052 0.4555 0.4966 0.5399 0.6176 0.6755 0.7003

0.371 0.1220 0.2427 0.3297 0.3551 0.3992 0.4407 0.4888 0.5895 0.6678 0.7293

0.424 0.1182 0.2253 0.3224 0.3542 0.4054 0.4584 0.5239 0.6680 0.7745 0.8431

0.477 0.1086 0.2104 0.2892 0.3129 0.3525 0.4011 0.4705 0.6528 0.7938 0.8714



c/πR a/t

0.2 0.5 0.8 0.85 0.9 0.93 0.95 0.97 0.98 0.99

0.106 1.2309 1.6220 1.8067 1.8706 2.2337 2.8225 3.1622 3.7183 3.9787 4.5684

0.159 1.2442 1.7343 2.1485 2.4294 4.0878 4.0213 4.4985 5.3884 5.9350 6.8105

0.212 1.1984 1.7990 2.4444 3.1460 4.3785 5.3549 6.1173 7.4172 8.0651 8.7923

0.265 1.2014 1.7957 2.7365 3.9256 5.3606 6.7088 7.8408 9.5554 10.4506 11.4993

0.318 1.2034 1.8248 3.0378 4.6266 6.3960 8.2252 9.7958 12.0078 12.7061 13.6752

0.371 1.1437 1.8467 3.3708 5.2278 7.4240 9.7483 11.7995 14.5068 15.4621 16.9151

0.424 1.1438 1.8635 3.7504 5.7991 8.4704 11.4429 14.1069 17.2355 17.9979 19.9812

0.477 1.1438 1.8766 4.0949 6.3306 9.4982 13.1758 16.5284 20.4153 21.6107 24.1633



c/πR a/t

0.2 0.5 0.8 0.85 0.9 0.93 0.95 0.97 0.98 0.99

0.106 0.2108 0.3575 0.4834 0.4986 0.5105 0.5186 0.5274 0.5336 0.5351 0.5259

0.159 0.1614 0.3044 0.4011 0.4201 0.4399 0.4544 0.4614 0.4740 0.4855 0.5099

0.212 0.1402 0.2657 0.3872 0.4166 0.4608 0.4949 0.5251 0.5731 0.6083 0.6228

0.265 0.1170 0.2350 0.3322 0.3573 0.3982 0.4348 0.4716 0.5384 0.5914 0.6319

0.318 0.1038 0.2120 0.3173 0.3493 0.3972 0.4440 0.4969 0.5921 0.6686 0.7222

0.371 0.1042 0.1929 0.2777 0.3023 0.3403 0.3814 0.4340 0.5464 0.6468 0.7164

0.424 0.0907 0.1754 0.2647 0.2897 0.3337 0.3833 0.4447 0.5797 0.6992 0.8167

0.477 0.0821 0.1614 0.2336 0.2550 0.2877 0.3286 0.3826 0.5308 0.6943 0.8410


178


c/πR a/t

0.2 0.5 0.8 0.85 0.9 0.93 0.95 0.97 0.98 0.99

0.106 1.2079 1.7837 2.1725 2.2888 2.9496 3.6213 4.1215 4.8034 5.3325 6.0193

0.159 1.2163 1.8842 2.5982 3.0778 4.3073 5.1630 6.0803 7.2186 8.0006 8.9401

0.212 1.2200 1.9402 2.9996 4.0809 5.5836 6.8960 8.2995 10.0594 10.7243 11.7267

0.265 1.2220 1.9495 3.4034 4.9177 6.8898 8.7215 10.5994 12.9193 13.7155 14.9505

0.318 1.2235 1.9748 3.8131 5.6178 8.1767 10.6825 13.2053 15.9284 16.5192 18.5302

0.371 1.1643 1.9932 4.1902 6.2589 9.3943 12.6832 15.7735 19.1593 19.9967 22.1968

0.424 1.1635 1.9708 4.5376 6.8256 10.5962 14.9094 18.6136 22.4504 23.3190 26.8302

0.477 1.1632 1.9809 4.9551 7.3349 11.7535 17.2096 21.6405 26.5198 28.0069 32.2387


Table 5.17 The ratio of NSIF of EPFM to LEFM at surface point (R/T=4.0)

c/πR a/t

0.2 0.5 0.8 0.85 0.9 0.93 0.95 0.97 0.98 0.99

0.106 0.9503 0.7119 0.9296 0.9258 0.9210 0.9161 0.9128 0.9096 0.9082 0.8892

0.159 0.9779 0.7434 0.9415 0.9284 0.9168 0.9124 0.9106 0.9092 0.9088 0.8859

0.212 0.9922 0.7805 0.9589 0.8884 0.8793 0.7881 0.8732 0.8708 0.8696 0.8461

0.265 0.9982 0.8084 0.9812 0.8905 0.8817 0.8268 0.9216 0.9289 0.9302 0.9098

0.318 1.0018 0.8324 1.0004 0.9253 0.8771 0.8797 0.9510 0.9554 0.9556 0.9301

0.371 1.0040 0.8540 1.0179 0.9146 0.9369 0.9506 0.9602 0.9686 1.0538 1.0412

0.424 1.0047 0.8732 1.0307 0.9571 0.9605 1.0298 1.0529 1.0778 1.0957 1.0880

0.477 1.0045 0.8892 1.0169 1.0006 1.0247 1.1298 1.1694 1.2227 1.2625 1.2781


Table 5.18 The ratio of NSIF of EPFM to LEFM at deepest point (R/T=4.0)

c/πR a/t

0.2 0.5 0.8 0.85 0.9 0.93 0.95 0.97 0.98 0.99

0.106 0.9995 0.9894 1.0077 1.0135 1.0355 1.1000 1.2009 1.3106 1.3069 1.2382

0.159 0.9999 0.9857 1.0105 1.0245 1.0948 1.2650 1.3740 1.4305 1.3544 1.2880

0.212 1.0001 0.9847 1.0182 1.0477 1.2090 1.4319 1.5183 1.5503 1.4575 1.3655

0.265 1.0001 0.9853 1.0312 1.0970 1.3754 1.5932 1.6466 1.6790 1.5841 1.4557

0.318 1.0002 0.9857 1.0484 1.1801 1.5633 1.7454 1.7802 1.8298 1.7352 1.5628

0.371 1.0002 0.9861 1.0680 1.2927 1.7537 1.9155 1.9544 2.0182 1.9138 1.6877

0.424 1.0004 0.9863 1.0959 1.4340 1.9223 2.1260 2.1842 2.2593 2.1413 1.8392

0.477 1.0001 0.9872 1.1293 1.5908 2.0958 2.3657 2.4630 2.5664 2.4097 2.0385


179


c/πR a/t

0.2 0.5 0.8 0.85 0.9 0.93 0.95 0.97 0.98 0.99

0.106 0.9990 0.7554 0.5997 0.5711 0.5395 0.5205 0.5069 0.4921 0.4848 0.4929

0.159 1.0032 0.8182 0.5935 0.5606 0.5195 0.4860 0.4568 0.4249 0.4093 0.4652

0.212 1.0041 0.8588 0.7093 0.6832 0.6604 0.6432 0.6361 0.6258 0.6177 0.5481

0.265 1.0062 0.8943 0.6920 0.6676 0.6469 0.6267 0.6046 0.5853 0.5818 0.6337

0.318 1.0052 0.9222 0.8031 0.7911 0.7997 0.8110 0.8248 0.8611 0.7137 0.8970

0.371 1.0046 0.9413 0.7729 0.7680 0.7807 0.7925 0.8089 0.8651 0.9032 0.9875

0.424 1.0047 0.9584 0.8758 0.8856 0.8427 0.9814 1.0603 1.2254 1.3300 1.3721

0.477 1.0030 0.9696 0.8406 0.8437 0.8906 0.9519 1.0443 1.2736 1.4142 1.5379



c/πR a/t

0.2 0.5 0.8 0.85 0.9 0.93 0.95 0.97 0.98 0.99

0.106 0.9990 1.0049 1.0068 1.0253 1.1474 1.3363 1.4499 1.4976 1.4545 1.3677

0.159 0.9991 1.0079 1.0243 1.0734 1.3600 1.5726 1.6563 1.6690 1.6508 1.5027

0.212 0.9798 0.9873 1.0495 1.1731 1.5745 1.7650 1.8912 1.9016 1.8579 1.6776

0.265 0.9799 0.9888 1.0900 1.3442 1.7989 1.9768 2.1448 2.1791 2.0894 1.8083

0.318 0.9794 0.9902 1.1505 1.5470 2.0053 2.2311 2.4322 2.4995 2.3409 1.9991

0.371 0.9798 0.9914 1.2223 1.7378 2.2231 2.5277 2.7731 2.8554 2.6195 2.2369

0.424 0.9793 0.9924 1.3068 1.8830 2.4588 2.8540 3.1653 3.2537 2.9471 2.5301

0.477 0.9793 0.9932 1.3956 2.0153 2.7004 3.2115 3.6142 3.7011 3.3367 2.8941



c/πR a/t

0.2 0.5 0.8 0.85 0.9 0.93 0.95 0.97 0.98 0.99

0.106 1.0048 0.8098 0.6423 0.6055 0.5712 0.5528 0.5379 0.5221 0.5148 0.5320

0.159 1.0043 0.8769 0.6389 0.6046 0.5779 0.5441 0.5239 0.5027 0.4940 0.5360

0.212 1.0052 0.9219 0.7863 0.7604 0.7485 0.7425 0.7431 0.7502 0.7594 0.7623

0.265 1.0043 0.9496 0.7551 0.7402 0.7361 0.7357 0.7413 0.7697 0.8059 0.8610

0.318 1.0021 0.9684 0.8898 0.8907 0.9198 0.9504 0.9962 1.0952 1.1729 1.2050

0.371 0.8754 0.9816 0.8515 0.8470 0.8764 0.9202 0.9852 1.1452 1.2723 1.3777

0.424 1.0103 0.9924 0.9496 0.9711 1.0299 1.1106 1.2290 1.5135 1.7230 1.8571

0.477 1.0042 0.9959 0.9103 0.9159 0.9578 1.0402 1.1832 1.5889 1.8998 2.0701


180


c/πR a/t

0.2 0.5 0.8 0.85 0.9 0.93 0.95 0.97 0.98 0.99

0.106 0.9999 1.0089 1.0133 1.0420 1.2129 1.4692 1.5552 1.6361 1.5751 1.4756

0.159 0.9999 1.0120 1.0360 1.1444 1.8434 1.7152 1.7923 1.8906 1.8503 1.6985

0.212 0.9785 1.0151 1.0714 1.3291 1.7437 1.9934 2.1053 2.2142 2.1167 1.8172

0.265 0.9785 0.9932 1.1239 1.5389 1.9594 2.2749 2.4436 2.5636 2.4495 2.1006

0.318 0.9785 0.9951 1.1890 1.7150 2.1912 2.5967 2.8255 2.9549 2.7153 2.2574

0.371 0.9298 0.9966 1.2749 1.8625 2.4312 2.9315 3.2336 3.3819 3.1207 2.6212

0.424 0.9555 0.9978 1.3801 2.0012 2.6727 3.3028 3.6971 3.8205 3.4413 2.9182

0.477 0.9552 0.9991 1.4784 2.1373 2.9243 3.7055 4.2169 4.4017 4.0126 3.4129



c/πR a/t

0.2 0.5 0.8 0.85 0.9 0.93 0.95 0.97 0.98 0.99

0.106 1.0038 0.8757 0.7026 0.6668 0.6298 0.6102 0.6016 0.5901 0.5825 0.5804

0.159 1.0057 0.9382 0.7141 0.6845 0.6560 0.6419 0.6286 0.6222 0.6249 0.6554

0.212 1.0053 0.9723 0.8758 0.8616 0.8690 0.8809 0.8983 0.9408 0.9773 0.9895

0.265 1.0072 0.9889 0.8589 0.8452 0.8603 0.8883 0.9278 1.0172 1.0948 1.1560

0.318 0.9989 0.9969 0.9682 0.9867 1.0329 1.0982 1.1859 1.3614 1.5086 1.6082

0.371 1.0105 1.0010 0.9415 0.9462 0.9830 1.0480 1.1530 1.4004 1.6293 1.7836

0.424 1.0006 1.0064 1.0004 1.0245 1.1009 1.2116 1.3645 1.7259 2.0482 2.3663

0.477 1.0027 1.0075 0.9721 0.9823 1.0326 1.1316 1.2786 1.7206 2.2154 2.6625



c/πR a/t

0.2 0.5 0.8 0.85 0.9 0.93 0.95 0.97 0.98 0.99

0.106 0.9789 1.0036 1.0200 1.0648 1.3421 1.5857 1.7096 1.7831 1.7767 1.6226

0.159 0.9787 1.0082 1.0625 1.2256 1.6459 1.8736 2.0680 2.1649 2.1289 1.8891

0.212 0.9789 1.0115 1.1264 1.4728 1.9059 2.2119 2.4721 2.6071 2.4427 2.0905

0.265 0.9784 1.0005 1.2083 1.6628 2.1826 2.5802 2.8986 3.0571 2.8384 2.4010

0.318 0.9784 1.0027 1.3002 1.8117 2.4525 2.9767 3.3854 3.5083 3.1670 2.7365

0.371 0.9560 1.0043 1.3899 1.9550 2.7167 3.3997 3.8844 4.0477 3.6696 3.1234

0.424 0.9567 0.9900 1.4723 2.0780 2.9751 3.8684 4.4247 4.5579 4.1011 3.6037

0.477 0.9566 0.9910 1.5765 2.1963 3.2394 4.3795 5.0441 5.2800 4.8255 4.2298


181

Figure 5.1 Elastic-Plastic material model definition from tension test

Figure 5.2 Typical true stress vs. true strain curve

182

Figure 5.3 Distribution of Ft along the crack length versus vary crack depth

Figure 5.4 The boundaries of crack free face along x, y axis

183

Figure 5.5 Comparison COD in x-z direction with R/T=10.0, c/R=0.477, a/T=0.2


184



185



186



187



188

Figure 5.15 Comparison COD in x-z direction with R/T=10.0, a/T=0.9, c/R=0.106


189



190



191



192

Figure 5.23 Comparison COD in x-z direction with a/T=0.9, c/R=0.477, R/T=4.0


193



194

Figure 5.27 Comparison COD in y-z direction with R/T=10.0, c/R=0.477, a/T=0.2


195



196



197



198



199

Figure 5.37 Comparison COD in y-z direction with R/T=10.0, a/T=0.9, c/R=0.106


200



201



202



203

Figure 5.45 Comparison COD in y-z direction with a/T=0.9, c/R=0.477, R/T=4.0


204



205



206



207



208



209



210



211



212



213



214



215



216

Figure 5.71 Comparison Ft along crack front with R/T=10.0, c/R=0.477, a/T=0.2


217



218



219



220



221

Figure 5.81 Comparison Ft along crack front with R/T=10.0, a/T=0.9, c/R=0.106


222



223



224



225

Figure 5.89 Comparison Ft along crack front with a/T=0.9, c/R=0.477, R/T=4.0


226



227

Figure 5.93: Comparison of Fts and FEA data at deepest point

Figure 5.94: Comparison of Fts and FEA data at surface point

228

REFERENCE

[1] Kou, K.P. and Burdekin, F.M. "Stress intensity factors for a wide range of long-deep

semi-elliptical surface cracks, partly through-wall cracks and fully through-wall cracks in

tubular members", Engineering Fracture Mechanics, 73, p1693-1710, 2006.

[2] Hibbitt, Karlsson, and Sorense, Inc., ABAQUS User's Manual, Version 6.7-1. USA,

2007.

[3] Barsoum, R.S., "On the use of isoparametric finite elements in linear fracture mechanics",

Int J Numer Meth Engng, 10:25-37, 1976.

[4] I.S. Raju and J.C. Newman, Jr, STRESS-INTENSITY FACTOR FOR A WIDE RANGE OF SEMI-ELLIPTICAL

SURFACE CRACKS IN FINITE-THICKNESS PLATES. Engineering Fracture Mechanics. 11, pp817-829,

1979.

[5] L.Banks-Sills, APPLICATION OF THE FINITE ELEMENT METHOD TO LINEAR ELASTIC FRACTURE MECHANICS.

Applied Mechanics Reviews, 44, pp447-461, 1991.

[6] Zhong, Y.C., "Investigation of Block Shear of Coped Beams with Welded Clip Angles

Connection", Master thesis of Macau University, 2004.

[7] Khoo, H.A., Cheng, J.J.R., and Hrudey, T.M., "Determine steel properties for large strain

from a standard tension Test", Proceedings of the 2nd Material Specialty Conference of the

Canadian Society for Civil Engineering, Montreal, Quebec, Canada. (2002)

[8] Zahoor, A., DUCTILE FRACTURE HANDBOOK, VOLUMEⅠ: CIRCUMFERENTIAL

THROUGH-WALL CRACKS. Electric Power Research Institute, Palo Alto, CA, 1984.

[9] MATLAB MATLAB_User_Manual © 1994-2009 The MathWorks, Inc. 2009.

229

CHAPTER 6: CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER

WORK

6.1 CONCLUSION

It is well known that welded structures may often have imperfection at the welded joints. In

steel structures, these imperfections may take the form of cracks or other planar defects. Under

the action of mechanisms such as cyclic loading, propagation of fatigue cracks may occur. One

result of this propagation could be the failure of a single member. In the worst case, a

catastrophic collapse of a structure is also possible. Consequently, prediction of the

propagation of an existing or postulated crack becomes an important part of a safe design and

maintenance of a structure. To predict the crack propagation life and the residual life of

structure, it is necessary to know the severity of the crack, especially in terms of the crack tip

conditions. In fracture mechanics, this severity can be measured by several parameters, of

which the most widely used is the stress intensity factor (SIF) which depends on the crack size,

geometry of cracked member and mode of loading. Therefore, with the stress intensity factor

known, prediction of crack propagation can be done.

A detailed assessment of crack behavior in a tubular member under repeated loading is

necessary, especially for long deep cracks. Solutions for stress intensity factors (SIF) for

surface cracks in tubular members available in the literature were found covering a wide range

of surface cracks. However, accuracy in part of the range actually remains questionable.

Furthermore, the SIF for part through cracks is not available at all in the literature. To predict

the detailed behavior of a long deep crack, accurate SIFs are required. In the present study,

finite element analysis has been carried out to obtain the SIF of surface cracks in the range

considered unreliable from the literature and the SIF of part through cracks and through

thickness cracks were also obtained.

In this thesis, finite element analyses have been carried out to calculate stress intensity

factors (SIF) in a wide range of long-deep circumferential semi-elliptical internal surface

230

cracks in tubular members under axial tension. Basically, for the cracked tube, the solution of

stress intensity factors for surface cracks only covers a limited range. For investigating the

behavior of long surface cracks with a very small ligament, a range of a/T<0.8 and a/c<0.05 is

necessary. Moreover, to examine the transition behavior from surface crack to part through

crack, details of stress intensity factors for part through cracks are also needed. The finite

element analyses for these two kinds of crack have been carried out in this study. In order to

examine the effect of material plasticity on the effect of crack tip deformation of long-deep

circumferential semi-elliptical internal surface cracks in tubular members, finite element

analyses were carried out to study the crack deformation as well as the corresponding

J-integral value of tubular members with long-deep circumferential semi-elliptical internal

surface cracks. Comparison of the J-integral values obtained from the Elastic-Plastic analysis

and the Elastic analysis were made. Then, neural network of MATLAB was used to process

those finite element analysis results and suitable equations for predicting the SIFs were

proposed. With the current proposed equations, the fatigue crack growth trend can be

established which can help for estimating the inspection intervals for circular tube structures.

The results obtained from this-study have led to the following conclusions:

1. The maximum and minimum SIF for surface cracks are always located at the deepest

point and surface point due to the curvature introduced by the tube.

2. The difference between the maximum and minimum SIF is very large especially in the

case of long surface crack. As a result, the long deep surface crack in a tube will break

through the wall with a very small increment of crack length in the circumferential

direction.

3. In the case of the part through crack, the SIF adjacent to the outer surface is much

higher than that at the inner surface. This difference is proportional to the ratio of outer

crack length to inner crack length. When the through thickness crack is about to be

formed, this difference drops rapidly. This implies that having broken the wall, a part

through crack will grow and become a through thickness crack very quickly especially

231

for long cracks. Moreover, during the state of existing as a part through crack, the SIF

experiences a maximum value and brittle fracture or ductile tearing may happen.

4. In the case of the through thickness crack, due to the curvature of the tube, the variation

of SIF across the wall is not uniform. For long cracks, the SIF at the outer surface is

higher than that at the inner surface. Consequently, a situation analogous to an external

part through crack would be formed. When considering the crack opening area, this

could be an important factor.

5. Elastic-Plastic Fracture Mechanics analysis indicated that, the plastic deformation is

not significant for a/T values are less than 0.5. As a/T values become larger, the

influence of material plasticity to the prediction of J-integral value becomes more

significant. It is believed that the increase of the influence of material is due to the

plastic deformation near the vicinity of the crack depth. As the crack depth increases,

the influence of the plastic deformation becomes more significant. As the crack length

becomes longer, the effect of plastic deformation to the prediction of J-integral values

becomes more significant. It is also found that J-integral values increases as R/T value

increases.

6. The maximum SIF obtained in this analysis is higher than those reported in the

literature. As a result, using the reported SIF for crack growth calculations may lead to a

prediction which is not conservative.

7. Either by curve fitting or tabular listing, a set of SIF for crack tube has been established

which is able to cover the range of practical application.

6.2 RECOMMENDATIONS FOR FURTHER WORK

In the present study, the stress intensity factors for a wide range of long-deep circumferential

semi-elliptical internal surface cracks in tubular members have been determined by the finite

element method. Further investigations are recommended, and they are summarized as

follows:

232

Based on the solutions of SIFs of surface cracks, part through cracks, through thickness

cracks, and the assumption of the Paris law, fatigue analysis should be carried out and

estimates of the flooded life available for flooded member detection (FMD), which can

be carried out in parallel with the crack growth calculation.

The estimation of fatigue life should be made from initial penetration of a crack to final

failure during which internal flooding would take place, based on the whole process of

crack growth from surface crack, part through crack and through thickness crack. In

addition, the results can be compared with the conventional leak-before-break

evaluation procedure.

Besides the flooded life, the leakage rate is also an important factor of a reliable FMD.

Knowledge of this would be an important part for assessing the reliability of FMD.

The case of tubes with a single crack has been investigated in this study. Further

research is needed to examine the possibility of the case of multiple cracks for other

geometries, crack configurations and loads by the finite element method.

The crack resistance (R) increases with the crack growth in the case of ductile tearing of

steel which may occur at the end of the fatigue crack growth. A consequence of this is

higher fracture toughness JIc and KIc which will increase the flooded life. This effect has

not been considered in this study. Further investigation is needed to take this into

account.

STRESS INTENSITY FACTORS OF CIRCUMFERENTIAL SEMI ...library.umac.mo/etheses/b2182941x_ft.pdf · STRESS INTENSITY FACTORS OF CIRCUMFERENTIAL SEMI-ELLIPTICAL INTERNAL SURFACE CRACKS

Documents