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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
Faculty of Science and Technology
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
Faculty of Science and Technology
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 4.15: Distribution of Ft along crack front of tube (R/T=15.0) ..............................107
Figure 4.16: Distribution of Ft along crack front of tube (R/T=22.5) ..............................109
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 4.19: Variation of Ft at surface and deepest point of crack (R/T=15.0) ...............113
Figure 4.20: Variation of Ft at surface and deepest point of crack (R/T=22.5) ...............114
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 4.35: Distribution of Ft for part through crack of tube (R/T=15.0) ......................136
Figure 4.36: Distribution of Ft for part through crack of tube (R/T=22.5) ......................138
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 5.6 Comparison COD in x-z direction with R/T=10.0, c/R=0.477, a/T=0.5 .....183
Figure 5.7 Comparison COD in x-z direction with R/T=10.0, c/R=0.477, a/T=0.8 .....184
Figure 5.8 Comparison COD in x-z direction with R/T=10.0, c/R=0.477, a/T=0.85 ...184
Figure 5.9 Comparison COD in x-z direction with R/T=10.0, c/R=0.477, a/T=0.9 .....185
Figure 5.10 Comparison COD in x-z direction with R/T=10.0, c/R=0.477, a/T=0.93 .185
Figure 5.11 Comparison COD in x-z direction with R/T=10.0, c/R=0.477, a/T=0.95 .186
Figure 5.12 Comparison COD in x-z direction with R/T=10.0, c/R=0.477, a/T=0.97 .186
Figure 5.13 Comparison COD in x-z direction with R/T=10.0, c/R=0.477, a/T=0.98 .187
Figure 5.14 Comparison COD in x-z direction with R/T=10.0, c/R=0.477, a/T=0.99 .187
Figure 5.15 Comparison COD in x-z direction with R/T=10.0, a/T=0.9, c/R=0.106 ...188
Figure 5.16 Comparison COD in x-z direction with R/T=10.0, a/T=0.9, c/R=0.159 ...188
Figure 5.17 Comparison COD in x-z direction with R/T=10.0, a/T=0.9, c/R=0.212 ...189
Figure 5.18 Comparison COD in x-z direction with R/T=10.0, a/T=0.9, c/R=0.265 ...189
Figure 5.19 Comparison COD in x-z direction with R/T=10.0, a/T=0.9, c/R=0.318 ...190
Figure 5.20 Comparison COD in x-z direction with R/T=10.0, a/T=0.9, c/R=0.371 ...190
Figure 5.21 Comparison COD in x-z direction with R/T=10.0, a/T=0.9, c/R=0.424 ...191
Figure 5.22 Comparison COD in x-z direction with R/T=10.0, a/T=0.9, c/R=0.477 ...191
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 5.25 Comparison COD in x-z direction with a/T=0.9, c/R=0.477, R/T=15.0 ...193
Figure 5.26 Comparison COD in x-z direction with a/T=0.9, c/R=0.477, R/T=22.5 ...193
Figure 5.27 Comparison COD in y-z direction with R/T=10.0, c/R=0.477, a/T=0.2 ...194
Figure 5.28 Comparison COD in y-z direction with R/T=10.0, c/R=0.477, a/T=0.5 ...194
Figure 5.29 Comparison COD in y-z direction with R/T=10.0, c/R=0.477, a/T=0.8 ...195
Figure 5.30 Comparison COD in y-z direction with R/T=10.0, c/R=0.477, a/T=0.85 .195
Figure 5.31 Comparison COD in y-z direction with R/T=10.0, c/R=0.477, a/T=0.9 ...196
Figure 5.32 Comparison COD in y-z direction with R/T=10.0, c/R=0.477, a/T=0.93 .196
xii
Figure 5.33 Comparison COD in y-z direction with R/T=10.0, c/R=0.477, a/T=0.95 .197
Figure 5.34 Comparison COD in y-z direction with R/T=10.0, c/R=0.477, a/T=0.97 .197
Figure 5.35 Comparison COD in y-z direction with R/T=10.0, c/R=0.477, a/T=0.98 .198
Figure 5.36 Comparison COD in y-z direction with R/T=10.0, c/R=0.477, a/T=0.99 .198
Figure 5.37 Comparison COD in y-z direction with R/T=10.0, a/T=0.9, c/R=0.106 ...199
Figure 5.38 Comparison COD in y-z direction with R/T=10.0, a/T=0.9, c/R=0.159 ...199
Figure 5.39 Comparison COD in y-z direction with R/T=10.0, a/T=0.9, c/R=0.212 ...200
Figure 5.40 Comparison COD in y-z direction with R/T=10.0, a/T=0.9, c/R=0.265 ...200
Figure 5.41 Comparison COD in y-z direction with R/T=10.0, a/T=0.9, c/R=0.318 ...201
Figure 5.42 Comparison COD in y-z direction with R/T=10.0, a/T=0.9, c/R=0.371 ...201
Figure 5.43 Comparison COD in y-z direction with R/T=10.0, a/T=0.9, c/R=0.424 ...202
Figure 5.44 Comparison COD in y-z direction with R/T=10.0, a/T=0.9, c/R=0.477 ...202
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 5.47 Comparison COD in y-z direction with a/T=0.9, c/R=0.477, R/T=15.0 ...204
Figure 5.48 Comparison COD in y-z direction with a/T=0.9, c/R=0.477, R/T=22.5 ...204
Figure 5.49 Comparison J-integral along crack front with R/T=10.0, c/R=0.477, a/T=0.2
........................................................................................................................205
Figure 5.50 Comparison J-integral along crack front with R/T=10.0, c/R=0.477, a/T=0.5
........................................................................................................................205
Figure 5.51 Comparison J-integral along crack front with R/T=10.0, c/R=0.477, a/T=0.8
........................................................................................................................206
Figure 5.52 Comparison J-integral along crack front with R/T=10.0, c/R=0.477, a/T=0.85
........................................................................................................................206
Figure 5.53 Comparison J-integral along crack front with R/T=10.0, c/R=0.477, a/T=0.9
........................................................................................................................207
Figure 5.54 Comparison J-integral along crack front with R/T=10.0, c/R=0.477, a/T=0.93
........................................................................................................................207
xiii
Figure 5.55 Comparison J-integral along crack front with R/T=10.0, c/R=0.477, a/T=0.95
........................................................................................................................208
Figure 5.56 Comparison J-integral along crack front with R/T=10.0, c/R=0.477, a/T=0.97
........................................................................................................................208
Figure 5.57 Comparison J-integral along crack front with R/T=10.0, c/R=0.477, a/T=0.98
........................................................................................................................209
Figure 5.58 Comparison J-integral along crack front with R/T=10.0, c/R=0.477, a/T=0.99
........................................................................................................................209
Figure 5.59 Comparison J-integral along crack front with R/T=10.0, a/T=0.9, c/R=0.106
........................................................................................................................210
Figure 5.60 Comparison J-integral along crack front with R/T=10.0, a/T=0.9, c/R=0.159
........................................................................................................................210
Figure 5.61 Comparison J-integral along crack front with R/T=10.0, a/T=0.9, c/R=0.212
........................................................................................................................211
Figure 5.62 Comparison J-integral along crack front with R/T=10.0, a/T=0.9, c/R=0.265
........................................................................................................................211
Figure 5.63 Comparison J-integral along crack front with R/T=10.0, a/T=0.9, c/R=0.318
........................................................................................................................212
Figure 5.64 Comparison J-integral along crack front with R/T=10.0, a/T=0.9, c/R=0.371
........................................................................................................................212
Figure 5.65 Comparison J-integral along crack front with R/T=10.0, a/T=0.9, c/R=0.424
........................................................................................................................213
Figure 5.66 Comparison J-integral along crack front with R/T=10.0, a/T=0.9, c/R=0.477
........................................................................................................................213
Figure 5.67 Comparison J-integral along crack front with a/T=0.9, c/R=0.477, R/T=4.0
........................................................................................................................214
Figure 5.68 Comparison J-integral along crack front with a/T=0.9, c/R=0.477, R/T=10.0
........................................................................................................................214
xiv
Figure 5.69 Comparison J-integral along crack front with a/T=0.9, c/R=0.477, R/T=15.0
........................................................................................................................215
Figure 5.70 Comparison J-integral along crack front with a/T=0.9, c/R=0.477, R/T=22.5
........................................................................................................................215
Figure 5.71 Comparison Ft along crack front with R/T=10.0, c/R=0.477, a/T=0.2 ......216
Figure 5.72 Comparison Ft along crack front with R/T=10.0, c/R=0.477, a/T=0.5 ......216
Figure 5.73 Comparison Ft along crack front with R/T=10.0, c/R=0.477, a/T=0.8 ......217
Figure 5.74 Comparison Ft along crack front with R/T=10.0, c/R=0.477, a/T=0.85 ....217
Figure 5.75 Comparison Ft along crack front with R/T=10.0, c/R=0.477, a/T=0.9 ......218
Figure 5.76 Comparison Ft along crack front with R/T=10.0, c/R=0.477, a/T=0.93 ....218
Figure 5.77 Comparison Ft along crack front with R/T=10.0, c/R=0.477, a/T=0.95 ....219
Figure 5.78 Comparison Ft along crack front with R/T=10.0, c/R=0.477, a/T=0.97 ....219
Figure 5.79 Comparison Ft along crack front with R/T=10.0, c/R=0.477, a/T=0.98 ....220
Figure 5.80 Comparison Ft along crack front with R/T=10.0, c/R=0.477, a/T=0.99 ....220
Figure 5.81 Comparison Ft along crack front with R/T=10.0, a/T=0.9, c/R=0.106 ......221
Figure 5.82 Comparison Ft along crack front with R/T=10.0, a/T=0.9, c/R=0.159 ......221
Figure 5.83 Comparison Ft along crack front with R/T=10.0, a/T=0.9, c/R=0.212 ......222
Figure 5.84 Comparison Ft along crack front with R/T=10.0, a/T=0.9, c/R=0.265 ......222
Figure 5.85 Comparison Ft along crack front with R/T=10.0, a/T=0.9, c/R=0.318 ......223
Figure 5.86 Comparison Ft along crack front with R/T=10.0, a/T=0.9, c/R=0.371 ......223
Figure 5.87 Comparison Ft along crack front with R/T=10.0, a/T=0.9, c/R=0.424 ......224
Figure 5.88 Comparison Ft along crack front with R/T=10.0, a/T=0.9, c/R=0.477 ......224
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 5.91 Comparison Ft along crack front with a/T=0.9, c/R=0.477, R/T=15.0 ......226
Figure 5.92 Comparison Ft along crack front with a/T=0.9, c/R=0.477, R/T=22.5 ......226
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 4.17: NSIF Ft at surface point for part through crack (R/T=15.0) ...........................90
Table 4.18: NSIF Ft at 0.99T point for part through crack (R/T=15.0) .............................91
Table 4.19: NSIF Ft at surface point for part through crack (R/T=22.5) ...........................92
Table 4.20: NSIF Ft at 0.99T point for part through crack (R/T=22.5) .............................93
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 5.5 J integral values at surface point of surface crack for EPFM (R/T=15.0) .......174
Table 5.6 J integral values at deepest point of surface crack for EPFM (R/T=15.0).......174
Table 5.7 J integral values at surface point of surface crack for EPFM (R/T=22.5) .......175
Table 5.8 J integral values at deepest point of surface crack for EPFM (R/T=22.5).......175
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 5.13 NSIF Ft at surface point of surface crack for EPFM (R/T=15.0) ..................177
Table 5.14 NSIF Ft at deepest point of surface crack for EPFM (R/T=15.0) ..................177
Table 5.15 NSIF Ft at surface point of surface crack for EPFM (R/T=22.5) ..................177
Table 5.16 NSIF Ft at deepest point of surface crack for EPFM (R/T=22.5) ..................178
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
Table 5.21 The ratio of NSIF of EPFM to LEFM at surface point (R/T=15.0)...............179
Table 5.22 The ratio of NSIF of EPFM to LEFM at deepest point (R/T=15.0) ..............180
Table 5.23 The ratio of NSIF of EPFM to LEFM at surface point (R/T=22.5)...............180
Table 5.24 The ratio of NSIF of EPFM to LEFM at deepest point (R/T=22.5) ..............180
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
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[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
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
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
Table 4.6: NSIF Ft at surface point for surface crack (R/T=10.0)
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
2ф/π=0 (surface point), R/T=10.0
84
Table 4.7: NSIF Ft at deepest point for surface crack (R/T=10.0)
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 0.2937 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
2ф/π=1 (deepest point), R/T=10.0
Table 4.8: NSIF Ft at surface point for surface crack (R/T=15.0)
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 0.0518 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
2ф/π=0 (surface point), R/T=15.0
Table 4.9: NSIF Ft at deepest point for surface crack (R/T=15.0)
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 0.2462 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
2ф/π=1 (deepest point), R/T=15.0
85
Table 4.10: NSIF Ft at surface point for surface crack (R/T=22.5)
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
2ф/π=0 (surface point), R/T=22.5
Table 4.11: NSIF Ft at deepest point for surface crack (R/T=22.5)
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
2ф/π=1 (deepest point), R/T=22.5
Table 4.12: Parameters for the analysis of tubes with part through cracks
Parameter Values assigned in current FE study
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
Table 4.15: NSIF Ft at surface point for part through crack (R/T=10.0)
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 0.6278 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
Surface point R/T=10.0
89
Table 4.16: NSIF Ft at 0.99T point for part through crack (R/T=10.0)
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 1.7673 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
Table 4.17: NSIF Ft at surface point for part through crack (R/T=15.0)
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 0.4922 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
Surface point R/T=15.0
91
Table 4.18: NSIF Ft at 0.99T point for part through crack (R/T=15.0)
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 1.8016 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
Table 4.19: NSIF Ft at surface point for part through crack (R/T=22.5)
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
Surface point R/T=22.5
93
Table 4.20: NSIF Ft at 0.99T point for part through crack (R/T=22.5)
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
Parameter Values assigned in current FE study
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
Figure 4.14: Distribution of Ft along crack front of tube (R/T=10.0)
106
Figure 4.14: (Continue) Distribution of Ft along crack front of tube (R/T=10.0)
107
Figure 4.15: Distribution of Ft along crack front of tube (R/T=15.0)
108
Figure 4.15: (Continue) Distribution of Ft along crack front of tube (R/T=15.0)
109
Figure 4.16: Distribution of Ft along crack front of tube (R/T=22.5)
110
Figure 4.16: (Continue) Distribution of Ft along crack front of tube (R/T=22.5)
111
Figure 4.17: Variation of Ft at surface and deepest point of crack (R/T=4.0)
112
Figure 4.18: Variation of Ft at surface and deepest point of crack (R/T=10.0)
113
Figure 4.19: Variation of Ft at surface and deepest point of crack (R/T=15.0)
114
Figure 4.20: Variation of Ft at surface and deepest point of crack (R/T=22.5)
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
Figure 4.34: Distribution of Ft for part through crack of tube (R/T=10.0)
135
Figure 4.34: (Continue) Distribution of Ft for part through crack of tube (R/T=10.0)
136
Figure 4.35: Distribution of Ft for part through crack of tube (R/T=15.0)
137
Figure 4.35: (Continue) Distribution of Ft for part through crack of tube (R/T=15.0)
138
Figure 4.36: Distribution of Ft for part through crack of tube (R/T=22.5)
139
Figure 4.36: (Continue) Distribution of Ft for part through crack of tube (R/T=22.5)
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.
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 5.45 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
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 5.67
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
2ф/π=0(surface point), R/T=10.0
174
Table 5.4 J integral values at deepest 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 0.7631 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
2ф/π=1(deepest point), R/T=10.0
Table 5.5 J integral values at surface point of surface crack for EPFM (R/T=15.0)
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 0.0360 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
2ф/π=0(surface point), R/T=15.0
Table 5.6 J integral values at deepest point of surface crack for EPFM (R/T=15.0)
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 0.8057 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
2ф/π=1(deepest point), R/T=15.0
175
Table 5.7 J integral values at surface point of surface crack for EPFM (R/T=22.5)
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
2ф/π=0(surface point), R/T=22.5
Table 5.8 J integral values at deepest point of surface crack for EPFM (R/T=22.5)
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
2ф/π=1(deepest point), R/T=22.5
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
2ф/π=0(surface point), R/T=4.0
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
2ф/π=1(deepest point), R/T=4.0
Table 5.11 NSIF Ft 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 0.3238 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
2ф/π=0(surface point), R/T=10.0
Table 5.12 NSIF Ft at deepest 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 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
2ф/π=1(deepest point), R/T=10.0
177
Table 5.13 NSIF Ft at surface point of surface crack for EPFM (R/T=15.0)
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 0.2602 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
2ф/π=0(surface point), R/T=15.0
Table 5.14 NSIF Ft at deepest point of surface crack for EPFM (R/T=15.0)
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 1.2309 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
2ф/π=1(deepest point), R/T=15.0
Table 5.15 NSIF Ft at surface point of surface crack for EPFM (R/T=22.5)
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
2ф/π=0(surface point), R/T=22.5
178
Table 5.16 NSIF Ft at deepest point of surface crack for EPFM (R/T=22.5)
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
2ф/π=1(deepest point), R/T=22.5
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
2ф/π=0(surface point), R/T=4.0
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
2ф/π=1(deepest point), R/T=4.0
179
Table 5.19 The ratio of NSIF of EPFM to LEFM at surface point (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 0.9990 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
2ф/π=0(surface point), R/T=10.0
Table 5.20 The ratio of NSIF of EPFM to LEFM at deepest point (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 0.9990 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
2ф/π=1(deepest point), R/T=10.0
Table 5.21 The ratio of NSIF of EPFM to LEFM at surface point (R/T=15.0)
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 1.0048 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
2ф/π=0(surface point), R/T=15.0
180
Table 5.22 The ratio of NSIF of EPFM to LEFM at deepest point (R/T=15.0)
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 0.9999 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
2ф/π=1(deepest point), R/T=15.0
Table 5.23 The ratio of NSIF of EPFM to LEFM at surface point (R/T=22.5)
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
2ф/π=0(surface point), R/T=22.5
Table 5.24 The ratio of NSIF of EPFM to LEFM at deepest point (R/T=22.5)
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
2ф/π=1(deepest point), R/T=22.5
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
Figure 5.6 Comparison COD in x-z direction with R/T=10.0, c/R=0.477, a/T=0.5
184
Figure 5.7 Comparison COD in x-z direction with R/T=10.0, c/R=0.477, a/T=0.8
Figure 5.8 Comparison COD in x-z direction with R/T=10.0, c/R=0.477, a/T=0.85
185
Figure 5.9 Comparison COD in x-z direction with R/T=10.0, c/R=0.477, a/T=0.9
Figure 5.10 Comparison COD in x-z direction with R/T=10.0, c/R=0.477, a/T=0.93
186
Figure 5.11 Comparison COD in x-z direction with R/T=10.0, c/R=0.477, a/T=0.95
Figure 5.12 Comparison COD in x-z direction with R/T=10.0, c/R=0.477, a/T=0.97
187
Figure 5.13 Comparison COD in x-z direction with R/T=10.0, c/R=0.477, a/T=0.98
Figure 5.14 Comparison COD in x-z direction with R/T=10.0, c/R=0.477, a/T=0.99
188
Figure 5.15 Comparison COD in x-z direction with R/T=10.0, a/T=0.9, c/R=0.106
Figure 5.16 Comparison COD in x-z direction with R/T=10.0, a/T=0.9, c/R=0.159
189
Figure 5.17 Comparison COD in x-z direction with R/T=10.0, a/T=0.9, c/R=0.212
Figure 5.18 Comparison COD in x-z direction with R/T=10.0, a/T=0.9, c/R=0.265
190
Figure 5.19 Comparison COD in x-z direction with R/T=10.0, a/T=0.9, c/R=0.318
Figure 5.20 Comparison COD in x-z direction with R/T=10.0, a/T=0.9, c/R=0.371
191
Figure 5.21 Comparison COD in x-z direction with R/T=10.0, a/T=0.9, c/R=0.424
Figure 5.22 Comparison COD in x-z direction with R/T=10.0, a/T=0.9, c/R=0.477
192
Figure 5.23 Comparison COD in x-z direction with a/T=0.9, c/R=0.477, R/T=4.0
Figure 5.24 Comparison COD in x-z direction with a/T=0.9, c/R=0.477, R/T=10.0
193
Figure 5.25 Comparison COD in x-z direction with a/T=0.9, c/R=0.477, R/T=15.0
Figure 5.26 Comparison COD in x-z direction with a/T=0.9, c/R=0.477, R/T=22.5
194
Figure 5.27 Comparison COD in y-z direction with R/T=10.0, c/R=0.477, a/T=0.2
Figure 5.28 Comparison COD in y-z direction with R/T=10.0, c/R=0.477, a/T=0.5
195
Figure 5.29 Comparison COD in y-z direction with R/T=10.0, c/R=0.477, a/T=0.8
Figure 5.30 Comparison COD in y-z direction with R/T=10.0, c/R=0.477, a/T=0.85
196
Figure 5.31 Comparison COD in y-z direction with R/T=10.0, c/R=0.477, a/T=0.9
Figure 5.32 Comparison COD in y-z direction with R/T=10.0, c/R=0.477, a/T=0.93
197
Figure 5.33 Comparison COD in y-z direction with R/T=10.0, c/R=0.477, a/T=0.95
Figure 5.34 Comparison COD in y-z direction with R/T=10.0, c/R=0.477, a/T=0.97
198
Figure 5.35 Comparison COD in y-z direction with R/T=10.0, c/R=0.477, a/T=0.98
Figure 5.36 Comparison COD in y-z direction with R/T=10.0, c/R=0.477, a/T=0.99
199
Figure 5.37 Comparison COD in y-z direction with R/T=10.0, a/T=0.9, c/R=0.106
Figure 5.38 Comparison COD in y-z direction with R/T=10.0, a/T=0.9, c/R=0.159
200
Figure 5.39 Comparison COD in y-z direction with R/T=10.0, a/T=0.9, c/R=0.212
Figure 5.40 Comparison COD in y-z direction with R/T=10.0, a/T=0.9, c/R=0.265
201
Figure 5.41 Comparison COD in y-z direction with R/T=10.0, a/T=0.9, c/R=0.318
Figure 5.42 Comparison COD in y-z direction with R/T=10.0, a/T=0.9, c/R=0.371
202
Figure 5.43 Comparison COD in y-z direction with R/T=10.0, a/T=0.9, c/R=0.424
Figure 5.44 Comparison COD in y-z direction with R/T=10.0, a/T=0.9, c/R=0.477
203
Figure 5.45 Comparison COD in y-z direction with a/T=0.9, c/R=0.477, R/T=4.0
Figure 5.46 Comparison COD in y-z direction with a/T=0.9, c/R=0.477, R/T=10.0
204
Figure 5.47 Comparison COD in y-z direction with a/T=0.9, c/R=0.477, R/T=15.0
Figure 5.48 Comparison COD in y-z direction with a/T=0.9, c/R=0.477, R/T=22.5
205
Figure 5.49 Comparison J-integral along crack front with R/T=10.0, c/R=0.477, a/T=0.2
Figure 5.50 Comparison J-integral along crack front with R/T=10.0, c/R=0.477, a/T=0.5
206
Figure 5.51 Comparison J-integral along crack front with R/T=10.0, c/R=0.477, a/T=0.8
Figure 5.52 Comparison J-integral along crack front with R/T=10.0, c/R=0.477, a/T=0.85
207
Figure 5.53 Comparison J-integral along crack front with R/T=10.0, c/R=0.477, a/T=0.9
Figure 5.54 Comparison J-integral along crack front with R/T=10.0, c/R=0.477, a/T=0.93
208
Figure 5.55 Comparison J-integral along crack front with R/T=10.0, c/R=0.477, a/T=0.95
Figure 5.56 Comparison J-integral along crack front with R/T=10.0, c/R=0.477, a/T=0.97
209
Figure 5.57 Comparison J-integral along crack front with R/T=10.0, c/R=0.477, a/T=0.98
Figure 5.58 Comparison J-integral along crack front with R/T=10.0, c/R=0.477, a/T=0.99
210
Figure 5.59 Comparison J-integral along crack front with R/T=10.0, a/T=0.9, c/R=0.106
Figure 5.60 Comparison J-integral along crack front with R/T=10.0, a/T=0.9, c/R=0.159
211
Figure 5.61 Comparison J-integral along crack front with R/T=10.0, a/T=0.9, c/R=0.212
Figure 5.62 Comparison J-integral along crack front with R/T=10.0, a/T=0.9, c/R=0.265
212
Figure 5.63 Comparison J-integral along crack front with R/T=10.0, a/T=0.9, c/R=0.318
Figure 5.64 Comparison J-integral along crack front with R/T=10.0, a/T=0.9, c/R=0.371
213
Figure 5.65 Comparison J-integral along crack front with R/T=10.0, a/T=0.9, c/R=0.424
Figure 5.66 Comparison J-integral along crack front with R/T=10.0, a/T=0.9, c/R=0.477
214
Figure 5.67 Comparison J-integral along crack front with a/T=0.9, c/R=0.477, R/T=4.0
Figure 5.68 Comparison J-integral along crack front with a/T=0.9, c/R=0.477, R/T=10.0
215
Figure 5.69 Comparison J-integral along crack front with a/T=0.9, c/R=0.477, R/T=15.0
Figure 5.70 Comparison J-integral along crack front with a/T=0.9, c/R=0.477, R/T=22.5
216
Figure 5.71 Comparison Ft along crack front with R/T=10.0, c/R=0.477, a/T=0.2
Figure 5.72 Comparison Ft along crack front with R/T=10.0, c/R=0.477, a/T=0.5
217
Figure 5.73 Comparison Ft along crack front with R/T=10.0, c/R=0.477, a/T=0.8
Figure 5.74 Comparison Ft along crack front with R/T=10.0, c/R=0.477, a/T=0.85
218
Figure 5.75 Comparison Ft along crack front with R/T=10.0, c/R=0.477, a/T=0.9
Figure 5.76 Comparison Ft along crack front with R/T=10.0, c/R=0.477, a/T=0.93
219
Figure 5.77 Comparison Ft along crack front with R/T=10.0, c/R=0.477, a/T=0.95
Figure 5.78 Comparison Ft along crack front with R/T=10.0, c/R=0.477, a/T=0.97
220
Figure 5.79 Comparison Ft along crack front with R/T=10.0, c/R=0.477, a/T=0.98
Figure 5.80 Comparison Ft along crack front with R/T=10.0, c/R=0.477, a/T=0.99
221
Figure 5.81 Comparison Ft along crack front with R/T=10.0, a/T=0.9, c/R=0.106
Figure 5.82 Comparison Ft along crack front with R/T=10.0, a/T=0.9, c/R=0.159
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Figure 5.83 Comparison Ft along crack front with R/T=10.0, a/T=0.9, c/R=0.212
Figure 5.84 Comparison Ft along crack front with R/T=10.0, a/T=0.9, c/R=0.265
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Figure 5.85 Comparison Ft along crack front with R/T=10.0, a/T=0.9, c/R=0.318
Figure 5.86 Comparison Ft along crack front with R/T=10.0, a/T=0.9, c/R=0.371
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Figure 5.87 Comparison Ft along crack front with R/T=10.0, a/T=0.9, c/R=0.424
Figure 5.88 Comparison Ft along crack front with R/T=10.0, a/T=0.9, c/R=0.477
225
Figure 5.89 Comparison Ft along crack front with a/T=0.9, c/R=0.477, R/T=4.0
Figure 5.90 Comparison Ft along crack front with a/T=0.9, c/R=0.477, R/T=10.0
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Figure 5.91 Comparison Ft along crack front with a/T=0.9, c/R=0.477, R/T=15.0
Figure 5.92 Comparison Ft along crack front with a/T=0.9, c/R=0.477, R/T=22.5
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Figure 5.93: Comparison of Fts and FEA data at deepest point
Figure 5.94: Comparison of Fts and FEA data at surface point
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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
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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:
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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.