Analysis of crack propagation in a thick-walled cylinder under fatigue loading By Iftikhar us Salam Advisors: Dr. M. Afzaal Malik Dr. Anjum Tauqir A Thesis Submitted to the Department of Mechanical Engineering College of Electrical and Mechanical Engineering National University of Sciences and Technology (NUST) in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mechanical Engineering December 2008
Analysis of Crack Propagation in a Thick-walled Cylinder Under Fatigue Loading
Oct 24, 2014
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Analysis of crack propagation
in a thick-walled cylinder under
fatigue loading
By
Iftikhar us Salam
Advisors: Dr. M. Afzaal Malik
Dr. Anjum Tauqir
A Thesis
Submitted to the Department of Mechanical Engineering
College of Electrical and Mechanical Engineering
National University of Sciences and Technology (NUST)
in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy
in Mechanical Engineering
December 2008
ii
Name: Iftikhar us Salam
Date of Degree: December 18, 2008
Institution: College of Electrical and Mechanical Engineering, National University of
Sciences and Technology (NUST), Rawalpindi, Pakistan
Major Field: Mechanical Engineering
NUST Advisor: Dr. M. Afzaal Malik
Local Advisor: Dr. Anjum Tauqir
Title of Study: ANALYSIS OF CRACK PROPAGATION IN A THICK-
WALLED CYLINDER UNDER FATIGUE LOADING
Pages in Study: 171
ABSTRACT
Reliability of materials and structures in the form of thick-walled cylinders is of critical
importance to many industries including power, nuclear, chemical, armament, and food
processing industries. Catastrophic failure of these cylinders can put the human life and
the surroundings at very high risk. For this reason, the integrity of the cylinder should be
guaranteed. The integrity of nearly all engineering structures is threatened by the
presence of cracks. Structural failure occurs if a crack larger than a critical size exists.
Although most well designed structures initially contain no critical cracks, subcritical
cracks can grow to failure under fatigue loading, called fatigue crack propagation.
Fatigue failure that is failure under repeated or cyclic loading is a serious concern of
engineering design. Under fatigue loading the component may fail at a stress level that is
far below its yield strength. In present research the fatigue crack propagation, in a thick-
walled cylinder, is analyzed through detailed experimental work and finite element
analysis and the fatigue crack growth life of the cylinder, with crack at the bore surface,
has been predicted.
iii
Extrusion process induces microstructural anisotropy in the thick-walled cylinder. The
intensive experimental work, with the help of laboratory tests on the material under
investigation, explores the details of the material and the microstructure-properties
relationship in the longitudinal and transverse orientations. The yield and tensile strength
in two orientations are not significantly different. However, percent elongation, reduction
in area, impact strength and fracture toughness of the material are superior in the axial
direction. A marked impact of anisotropy is found on the fatigue properties and shorter
fatigue life in the transverse direction was obtained, which is 41 to 62 % lower in the
tested stress range of 129 to 47 MPa.
The theoretical part of the study includes modeling and simulation based on finite
element method and the numerical technique is employed for the simulation of fatigue
crack propagation. The finite element analysis, based on linear elastic fracture mechanics
(LEFM) combined with the Paris law, suitably predicts the fatigue life and provides the
results that are in good agreement with the experimental results.
Both the experimental and numerical results of the crack growth data at different stress
levels were found in good agreement all along the Paris regime. In the near threshold
region the predicted values are conservative. With implementation of the present scheme
of work the fatigue crack growth life of the thick-walled cylinder, with internal axial
crack, has been predicted.
iv
DEDICATION
This work is dedicated to my parents; achieving this goal would not have been possible
without their undying love and prayers
v
ACKNOWLEDGMENTS
I gratefully acknowledge the support of many people without whom I never would have
completed this thesis. First and foremost, I would like to express my sincere gratitude and
thanks to my advisor, Dr. Muhammad Afzaal Malik, for his exemplary guidance and
encouragement throughout my research. I am greatly indebted to him for all his support
through out this research study.
I would like to thank my committee members, Brig Dr. Abdul Ghafoor, Dr. Ijaz Ahmed
Malik and Dr. Anjum Tauqir for their invaluable guidance and assistance throughout this
research. The unfailing support of my colleagues had provided brilliant ideas, ever lasting
optimism and assistance. I would like to thank my fellow peers and friends, Wali
Muhammad, Noveed Ejaz, Arif, Liaqat, Wajid, Altaf, Israr and many others, for their
selfless assistance and contribution in this research work. Special thanks for Dr.
Muhammad Abid for his invaluable guidance and continuous attachment during the
simulation work.
Lastly, I would like to immensely thank my parents, for their undying love,
encouragement and support throughout my life and education. Without them and their
blessings, achieving this goal would not have been possible.
vi
Table of Contents
Page ACKNOWLEDGMENTS ………………………….............. v
LIST OF TABLES …………………………………........… xi
LIST OF FIGURES ……………………………..…………. xii
NOMENCLATURE ………………………………………... xvii
CHAPTER 1 – INTRODUCTION
1.1 FATIGUE: PROGRESSIVE FRACTURE .......................................................... 1
1.1.1 STAGES OF FATIGUE FAILURE …………………………………………………. 3
1.1.2 FATIGUE CRACK GROWTH CURVE ……………………………………………. 5
1.1.3 FATIGUE CRACK PROPAGATION ………………………………………………. 8
1.1.4 STUDY OF THE FRACTURED SURFACE (FRACTOGRAPHY) ………........... 11
1.2 THICK-WALLED CYLINDERS ……………………………………………… 12
1.3 PROBLEM DESCRIPTION AND RESEARCH STRATEGY ……………… 14
1.3.1 EXTRUDED CYLINDERS ………………………………………………………… 14
1.3.2 FATIGUE PROCESS IN THICK-WALLED CYLINDERS …………………….. 14
1.3.3 DEFINITION OF THE PROBLEM BEING STUDIED …………………………. 15
1.3.4 RESEARCH STRATEGY ………………………………………………………….. 15
1.3.5 RESEARCH OBJECTIVES ………………………………………………………... 16
1.4 OVERVIEW OF THE THESIS ……………………………………………….. 17
CHAPTER 2 – LITERATURE REVIEW
2.1 HISTORICAL OVERVIEW OF FATIGUE …………………………………. 19
2.1.1 1800 ERA …………………………………………..................................................... 19
2.1.2 EARLY 1900 ERA ………………………………………………………………….. 20
2.1.3 MID 1900 ERA ……………………………………………………………………… 22
2.1.4 END 1900 ERA ……………………………………………………………………… 26
2.2 FATIGUE CRACK PROPAGATION; EFFECT OF MICROSTRUCTURE 27
vii
2.3 FINITE ELEMENT METHOD ………………………………………………... 29
2.3.1 MODELING THROUGH FEM ...…………..…………………………………... 30
2.3.2 COMPUTING THE STRESS INTENSITY FACTOR ……………………………. 31
2.4 STRESSES IN A THICK-WALLED CYLINDER …………………………… 35
2.5 NUMERICAL SIMULATION OF FATIGUE ……………………………….. 38
CHAPTER 3 – EQUIPMENT AND EXPERIMENTAL 3.1 STANDARDS ……………………………………………………………..……. . 43
3.2 EQUIPMENT ……………………………………………………………..…….. 43
3.3 MATERIAL CHARACTERIZATION …………………………………..……. 44
3.3.1 CHEMICAL COMPOSITION …………………………………………………… 44
3.3.2 MICROSTRUCTURAL EVALUATION ……………………………………….. 44
3.3.3 HARDNESS MEASUREMENT …………………………………………………. 45
3.4 MONOTONIC TENSILE TESTING ……………………………..…………... 45
3.4.1 SAMPLE PREPARATION ………………………………………………………. 45
3.4.2 TESTING PROCEDURE ………………………………………………………… 45
3.5 IMPACT TESTING ………………………………………………………..…… 47
3.6 FATIGUE CRACK GROWTH TESTING ……………..…………………….. 47
3.6.1 TEST SAMPLES AND PREPARATION ………………………………………. 47
3.6.2 TEST CONDITIONS AND PARAMETERS …………………………………… 51
3.7 FRACTOGRAPHY IN SEM ………………………………………………...… 52
CHAPTER 4 – MATERIAL & STRUCTURAL
CHARACTERIZATION
4.1 MATERIAL CHARACTERIZATION ……………………………………….. 54
4.1.1 CHEMICAL COMPOSITION ……………………………………………………. 54
4.1.2 MICROSTRUCTURAL EVALUATION ………………………………………… 55
4.2 MECHANICAL TESTING ……………………………………………………. 60
viii
4.2.1 HARDNESS TEST …………………………………………………………………. 60
4.2.2 MONOTONIC TENSILE TEST ………………………………………………….. 60
4.2.3 IMPACT TESTING ………………………………………………………………... 61
4.3 POST-FRACTURE ANALYSIS ………………………………………………. 62
4.3.1 TENSILE SAMPLES ………………………………………………………………. 62
4.3.2 IMPACT SAMPLES ……………………………………………………………….. 63
CHAPTER 5 – EXPERIMENTAL FATIGUE
CRACK GROWTH STUDY
5.1 FATIGUE CRACK GROWTH TEST – LR SAMPLES ……………………. 68
5.1.1 CRACK EXTENSION ……………………………………………………………... 68
5.1.2 FATIGUE CRACK GROWTH CURVE ………………………………………….. 72
5.1.3 FATIGUE LIFE ANALYSIS ………………………………………………………. 73
5.2 FATIGUE CRACK GROWTH TEST – CR SAMPLES …………………….. 75
5.2.1 CRACK EXTENSION ……………………………………………………………… 75
5.2.2 FATIGUE CRACK GROWTH CURVE ………………………………………….. 80
5.2.3 FATIGUE LIFE ANALYSIS ………………………………………………………. 81
5.3 COMPARISON OF THE FATIGUE BEHAVIOR – LR VS CR …………… 82
5.4 POST-FRACTURE ANALYSIS ………………………………………………. 85
5.4.1 LR SAMPLES ………………………………………………………………………. 85
5.4.2 CR SAMPLES ………………………………………………………………………. 90
CHAPTER 6 – NUMERICAL SIMULATION OF
FATIGUE CRACK PROPAGATION 6.1 SIMULATION OF FATIGUE CRACK PROPAGATION ………………….. 95
6.1.1 MODEL GEOMETRY …………………………………………………………….. 95
6.1.2 MATERIAL PROPERTIES ……………………………………………………….. 96
6.1.3 ELEMENT SELECTION AND MESHING ……………………………………… 96
6.1.4 BOUNDARY CONDITIONS AND SOLUTION …………………………………. 98
6.2 RESULTS AND DISCUSSION ………………………………………………... 99
6.2.1 LR SAMPLES ………………………………………………………………………. 99
ix
6.2.1.1 ELEMENT SIZE OPTIMIZATION ……………………………………..… 99
6.2.1.2 CRACK GROWTH ……………………………………………………..….. 100
6.2.1.3 PREDICTED FCG RATE – EXPERIMENTAL VS FEA ………………. 106
6.2.1.4 FATIGUE LIFE ANALYSIS ……………………………………………… 107
6.2.2 CR SAMPLES ……………………………………………………………………… 109
6.2.2.1 ELEMENT SIZE OPTIMIZATION ……………………………………… 109
6.2.2.2 CRACK PROPAGATION ………………………………………………… 109
6.2.2.3 PREDICTED FCG RATE – EXPERIMENTAL VS FEA ……………… 114
6.2.2.4 FATIGUE LIFE ANALYSIS …………………………………………........ 115
CHAPTER 7 – THICK-WALLED CYLINDER;
FINITE ELEMENT ANALYSIS
7.1 THICK-WALLED CYLINDER …………………………………………… 117
7.2.1 MODEL DESCRIPTION ……………………………………………..….… 117
7.2.2 MODEL EQUATIONS ……………………………………………...……... 118
7.2.3 PARAMETER DESCRIPTION …………………………………..….…… 120
7.2.4 STRESS DESCRIPTION …………………………………………..….…… 121
7.2 STATIC LOADING OF TWC - WITHOUT CRACK …………………… 121
7.2.1 ANALYTICAL SOLUTION ……………………………………...……….. 121
7.2.1 FINITE ELEMENT MODELING ………………………………..………. 124
7.2.1.1 MODEL GEOMETRY …………………………….……………… 124
7.2.1.2 MATERIAL PROPERTIES ……………………………….….…... 125
7.2.1.3 ELEMENT SELECTION AND MESHING …………….….……. 125
7.2.1.4 BOUNDARY CONDITIONS AND SOLUTION ………….…….. 126
7.2.2 COMPARISON OF THE ANALYTICAL AND NUMERICAL RESULTS
……………………………………………………………………………..…. 127
7.3 STATIC LOADING OF TWC - WITH INTERNAL AXIAL CRACK …. 130
7.2.1 GEOMETRY OF THE MODEL ………………………………..………… 130
7.2.2 MATERIAL PROPERTIES, ELEMENT TYPE AND MESHING …….. 131
7.2.3 BOUNDARY CONDITIONS AND SOLUTION ………………..……….. 131
7.2.4 DETERMINATION OF THE STRESS INTENSITY FACTOR (KI) …... 132
7.4 FEA OF FATIGUE CRACK GROWTH IN TWC ………………………. 135
7.2.1 CRACK PROPAGATION IN TWC …………………………………..…. 136
7.2.2 PREDICTED FCG RATE – EXPERIMENTAL VS FEA …………..….. 137
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7.2.3 FATIGUE CRACK GROWTH LIFE PREDICTION OF THE CYLINDER
……………………………………………………………………………………..….. 137
CHAPTER 8 – SUMMARY AND CONCLUSIONS
8.1 SUMMARY …………………………………………………………………….. 139
8.2 CONCLUSIONS ……………………………………………………………….. 140
8.2.1 CONCLUSIONS FROM THE EXPERIMENTAL DATA …………….. 140
8.2.2 CONCLUSIONS FROM FINITE ELEMENT ANALYSIS ……………. 140
8.3 RECOMMENDED FUTURE WORK ……………………………………….. 141
REFERENCES ……………………………………………… 143
ANNEXURES
I SCHEMATIC SHOWING PRODUCTION OF THICK-WALLED
CYLINDER THROUGH EXTRUSION PROCESS ................................... 162
II FLOW CHART OF THE ALGORITHM USED FOR THE SIMULATION
OF FATIGUE CRACK PROPAGATION ................................................... 166
III SAMPLE RESULTS FILE GENERATED DURING SIMULATION OF
FATIGUE CRACK PROPAGATION .......................................................... 168
xi
LIST OF TABLES
TABLE Page
4.1 Nominal chemical composition of the material, wt. % ………………………. 54
4.2 Quantitative microstructural analysis ……………………….………………... 60
4.3 Nominal mechanical properties of the extruded alloy ……………………….. 62
5.1 Fatigue life of the samples in LR orientation ……………………….……….. 74
5.2 Paris constants obtained from the experimental data ………………………… 80
5.3 Fatigue life of the samples in CR orientation ……………………….……….. 81
5.4 Comparison of the fatigue lives of the samples in two orientations …………. 84
6.1 The Paris constants – LR sample …………………………………………….. 107
7.1 The variation of stresses and displacements with internal pressure calculated
by the model equations at inner radius, ri ………………………………………… 122
7.2 The variation of stresses and displacements along the wall thickness
calculated by the model equations at pi = 50 MPa, po = 0 MPa …………………… 123
xii
LIST OF FIGURES
FIGURE Page
1.1 Sketch showing three stages of fatigue as seen on the fracture surface ……. 4
1.2 Typical fatigue crack growth curve ......……………………………………. 6
1.3 Alternating stress cycle with zero mean stress .……………………………. 7
1.4 Crack tip stress field and crack opening modes 8
1.5 Schematic showing the nucleation (I) and propagation (II) stages
during fatigue in a polycrystalline material ………………………………...
10
1.6 Illustration of Stage II crack growth showing the formation of striations …. 10
1.7. Scanning electron micrograph depicting striations on the fractured turbine
blade a) low magnification b) high magnification ………………... 12
1.8 Section of the thick-walled cylinder with internal surface crack
along the cylinder axis ……………………………………………………... 15
2.1 Local coordinates measured from the 3-D crack front …………………….. 33
2.2 (a) a half-crack model (b) a full-crack model ……………………………… 34
3.1 Flowchart of the solid mechanics analysis chain ………………………... 43
3.2 a) Photograph and b) dimensions of the specimens (in mm) used for tensile
testing ……………………………………………………............................ 46
3.3. Dimensions of the impact sample (in mm) ………………………………... 47
3.4 Crack plane orientation code for bar and hollow cylinder (ASTM E 399) ... 48
3.5. Sample orientations in the cylinder ………………………………………... 49
3.6 Dimensional details of the M(T) sample (dimensions in mm) – CR
direction …………………………………………………………………… 49
3.7 Dimensional details of the M(T) sample (dimensions in mm) – LR
direction …………………………………………………………………… 50
3.8 Photograph of the two LR test samples ……………………………………. 50
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FIGURE Page
3.9 a) Fatigue testing machine with the sample gripped in the jaws (encircled)
and the traveling microscope ………………………………………………. 52
4.1 EDS spectrum of the alloy …………………………………………………. 55
4.2. SEM micrographs of the alloy showing the constituent particles
a) longitudinal direction b) transverse direction …………………………... 56
4.3. SEM micrograph of the alloy showing clusters of particles in longitudinal
direction …………………………………………………………………… 57
4.4 Optical micrograph revealing the microstructure of the alloy
in LR sample a) low magnification b) high magnification ………………... 58
4.5. Optical micrograph revealing the microstructure of the alloy
in CR sample a) low magnification b) high magnification ………………... 59
4.6 Stress-strain diagram of the material in two orientations ...………………... 61
4.7 SEM micrographs of the fractured tensile samples a)LD b)CR orientation .. 64
4.8 SEM micrographs of the fractured tensile samples at high magnification
a) LD b) CR orientation ……………………………………………………. 65
4.9 SEM micrographs of the fractured impact samples - LD orientation
a) low magnification b) high magnification ...……………………………... 66
4.10 SEM micrographs of the fractured impact sample - CR orientation
a) low magnification b) high magnification ………………………………. 67
5.1 The crack length versus the number of cycles – LR orientation
At stress levels in % of yield strength; a) 40 b)35 c)30 d)25 e)20 f)15 …….. 69
5.2 Plot of the crack length versus the number of cycles for LR orientation ….. 72
5.3 The crack growth rate variation with K – LR orientation ………………... 73
5.4 Stress range versus the number of cycles to failure – LR orientation …....... 75
5.5 The crack length versus the number of cycles – CR orientation
At stress levels in % of yield strength; a)40 b)35 c)30 d)25 e)20 f)15 g)10 .. 76
xiv
FIGURE Page
5.6 Plot of the crack length versus the number of cycles for CR orientation ...... 79
5.7 The crack growth rate variation with K – CR orientation ………………... 80
5.8 Stress range versus the number of cycles to failure – CR orientation ……….. 82
5.9 The crack growth rate variation with K in two orientations ………………... 83
5.10 SEM micrographs of the LR samples tested at S a)129 MPa b) 64 MPa …. 86
5.11 SEM micrographs of fatigue fractured surface of LR orientation
- S equals a)129 MPa b)64 MPa (arrow indicates crack growth direction) ... 87
5.12 SEM micrograph of fatigue fractured surface of LR orientation showing
striations - S equals 129 MPa (arrow indicates crack growth direction) ……… 88
5.13 SEM micrograph of the overload fracture surface - LR orientation,
- S equals 129 MPa …………………………………………………………… 88
5.14 Polished side surface of the overload region; SEM micrographs
a) segment of the crack path b) region A at high magnification …………… 89
5.15 SEM micrographs of the CR samples tested at S a)129 MPa b) 64 MPa .. 91
5.16 SEM micrographs of fatigue fractured surface of CR orientation
(arrow indicates crack growth direction) - S equals a)129 MPa b)64 MPa
c) higher magnification of (b) showing cluster of particles ………………..
92
5.17 SEM micrograph of fatigue fractured surface of CR orientation showing
striations - S equals 129 MPa (arrow indicates crack growth direction) ……….
94
6.1 Geometry of the quarter model (QM) ……………………………………… 96
6.2 Two dimensional, 4-node, PLANE42 element ……………………………… 97
6.3 a) Quarter model of the M(T) sample - element plot with applied boundary
conditions. b) enlarged crack tip region showing mapped meshing ………. 99
xv
FIGURE Page
6.4 Element size optimization – LR orientation. Legend indicates the finite
element analysis (FEA) using elemnt size (E) of 0.5 to 0.025 mm ………... 101
6.5 a) LR sample - Von Mises stress distribution (MPa) at the stress level of
40 % and the crack length 4 mm b) enlarged crack tip region …………….. 101
6.6 Experimental and FEA results of the crack length vs the number of cycles
in the crack growth region at stress levels in % of the yield strength.
a) 40 b) 35 c) 30 d) 25 e) 20 f) 15. Legend shows the experimental data
(EXP-1 and EXP-2) and the finite element results (FEA) ………………….
103
6.7 LR sample - Comparison of the predicted crack growth rate with the
experimental observation. Legend shows the experimental data (EXP) and
the finite element analysis (FEA) results at 15 to 40 % of the yield strength .
107
6.8 Fatigue life analysis (LR sample) - Experimental (EXP) vs FEA …………. 108
6.9 Element size optimization – CR orientation. Legend indicates the finite
element analysis (FEA) using elemnt size (E) of 0.5 to 0.025 mm ………... 109
6.10 a) CR sample - Von Mises stress distribution (MPa) at the stress level of
40 % and the crack length 4 mm b) enlarged crack tip region ……………... 110
6.11 Experimental and FEA results of the crack length vs the number of cycles
in the crack growth region at stress levels in % of the yield strength.
a) 40 b) 35 c) 30 d) 25 e) 20 f) 15 g) 10. Legend shows the experimental
data (EXP-1 and EXP-2) and the finite element results (FEA) ……………..
111
6.12 CR sample - Comparison of the predicted crack growth rate with the
experimental observation. Legend shows the experimental data (EXP) and
the finite element analysis (FEA) results at 10 % of the yield strength ……
115
6.13 Fatigue life analysis (CR sample) - Experimental (EXP) vs FEA ………... 116
7.1 Two dimensional section of the TWC showing geometric parameters ……… 118
7.2 Schematic of the TWC indicating three principal stresses …………………… 119
7.3 Half model of the cylinder section subjected to internal pressure pi ………… 122
7.4 TWC Model used for FEA …………………………………………………… 124
xvi
FIGURE Page
7.5. a) Meshed model using PLANE42 element b) magnified view of boxed area;
element size is 0.5 mm …………………………………………………………… 125
7.6 Static loading - Boundary conditions applied for analysis …………………... 126
7.7 Static loading – Nodal solution showing von Mises stress distribution at
internal pressure of a) 5 MPa b) 100 MPa ……………………………………….. 127
7.8 Stress versus internal pressure - comparison of the two results at inner radius
a) tangential b) radial …………………………………………………………….. 128
7.9 Stress variation along the wall thickness of the cylinder obtained from the
two methods at internal pressure of 100 MPa a) tangential b) radial …………… 129
7.10 Schematic of two dimensional half cylinder model with internal axial crack 131
7.11 Static loading of TWC with crack - Boundary conditions applied for
analysis …………………………………………………………………………… 132
7.12 Static loading of cylinder with crack – Nodal solution showing von Mises
stress distribution at internal pressure of a) 5 MPa b) 100 MPa …………………. 133
7.13. Magnified view of the crack region shown in Fig. 7.12a with BCs ……….. 134
7.14 Plot of KI versus internal pressure at a crack length of 3 mm ………………. 134
7.15 Plot showing KI versus internal pressure at crack length of 3, 5, 7 and 10
mm ……………………………………………………………………………….. 135
7.16 Variation of KI with the increase of crack length at different internal
pressures ………………………………………………………………………….. 136
7.17 Applied cycles versus crack length of the simulation TWC model with an
initial crack length of 3 mm ……………………………………………………… 137
7.18 The variation of fatigue crack growth rate with K – Experimental vs FEA 138
7.19 Predicted fatigue crack growth life of the thick-walled cylinder at different
internal pressures …………………………………………………………………. 138
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NOMENCLATURE
a crack length, mm
da/dN crack growth rate, m/cycle
di internal diameter, mm
do external diameter, mm
E Young’s modulus, GPa
KI stress intensity factor, MPa. m
KIC fracture toughness, MPa. m
KImax maximum stress intensity factor, MPa. m
N number of load cycles
Nf number of load cycles to failure (fatigue life)
Ng number of load cycles from the initial crack length to the final
fracture (fatigue crack growth life)
Ni number of load cycles to initiate the crack
pi internal pressure, MPa
po external pressure, MPa
tw wall thickness, mm
W sample width, mm
K stress intensity factor range, MPa. m
Kth threshold stress intensity factor range, MPa. m
Keffth effective threshold stress intensity factor range, MPa. m
S stress range, MPa
σ
u ultimate tensile stress, MPa
σy
tensile yield stress, MPa
Poisson’s ratio
1
CHAPTER 1
INTRODUCTION
Reliability of materials and structures in the form of thick-walled cylinders is of
critical importance to many industries including power, nuclear, chemical, armament,
and food processing industries. Catastrophic failure of these cylinders can put the
human life and the surroundings at very high risk. For this reason, the integrity of the
cylinder should be guaranteed. The purpose of this chapter is to present a general
overview of the title. The chapter comprises the following topics:
The basic concepts of the fatigue process
Thick-walled cylinder
The problem description, research strategy and the objective of the
present research
Overview of the thesis
1.1 Fatigue: Progressive fracture
Fatigue is defined as:
“The process of progressive localized permanent structural change occurring in a
material subjected to conditions that produce fluctuating stresses and strains at
some point or points and that may culminate in cracks or complete fracture after a
sufficient number of fluctuations” [1].
Under the cyclic application of stress below the ultimate tensile strength of a material,
localized hardening or softening occurs due to plastic deformation. The localized
plastic deformation may occur at points of stress concentrations or even in absence of
a stress raiser; for example in persistent slip bands extrusion formation can result in
2
crack initiation. This localized structural change may develop cracks in the structural
member after a certain number of cycles. The loading is called fatigue loading and the
subsequent fracture is called fatigue failure. The corresponding number of load cycles
or the time during which the member is subjected to these loads before fracture occurs
is referred to as the fatigue life of the member.
It is believed that most common failures are due to fatigue; many experts of the field
suggested that 50 to 90 percent of all mechanical failures are fatigue failures [2] and
usually these failures are unexpected. Fatigue failures occur in every field of
engineering and also in interdisciplinary engineering fields. They include thermal,
mechanical fatigue failure in electrical circuit boards involving electrical engineers,
bridges involving civil engineers, automobiles involving mechanical engineers, farm
tractors involving agricultural engineers, aircraft involving aeronautical engineers,
heart valve implants involving biomedical engineers, pressure vessels involving
chemical engineers, and nuclear piping involving nuclear engineers. Thus, all fields of
engineering are involved with fatigue design of metals. Fatigue failures encompass
problems in simple items like door springs, tooth brushes, tennis racquets, electric
light bulbs, and repeated bending of paper clips and include more complex
components and structures like pressure vessels, ground vehicles, ships, aircraft, and
human body implants. The latter include automobile steering linkage, engine
connecting rods, ship propeller shafts, pressurized airplane fuselage, landing gears,
and hip replacement prostheses.
The integrity of nearly all engineering structures is threatened by the presence of
cracks. Structural failure occurs if a crack larger than a critical size exists. Although
most well designed structures initially contain no critical cracks, subcritical cracks can
grow to failure under fatigue loading, called fatigue crack growth (FCG). Because it is
3
impossible or impractical to prevent subcritical crack growth in most applications, a
damage tolerant design philosophy was developed for crack sensitive structures.
Design engineers have taken advantage of the FCG threshold concept to design for
long fatigue lives. FCG threshold ( Kth) is a value of K (crack-tip loading), below
which no significant FCG occurs. Cracks are tolerated if K is less than Keffth.
However, FCG threshold is not constant. Many variables influence Kth including
microstructure, environment, and load ratio.
1.1.1 Stages of fatigue failure
The fatigue life of a metallic material is divided into several stages: crack nucleation,
micro-crack growth, macro-crack growth, and failure. Crack nucleation is associated
with cyclic slip and is controlled by the local stress and strain concentrations.
Although the slip-band mechanism of crack formation may be necessary in pure
metals, the presence of inclusions or voids in engineering metals will greatly affect
the crack-nucleation process. Micro-crack growth, a term now referred to as the
“small-crack growth” regime, is the growth of cracks from inclusions, voids, or slip
bands, in the range of 1 to 20 m in length. Schijve [3] has shown that for polished
surfaces of pure metals and commercial alloys, the formation of a small crack, about
100 m in size, can consume 60 to 80 % of the fatigue life. The AGARD [4, 5] and
NASA/CAE [6] studies on small-crack behavior in a variety of materials showed that
about 90% of the fatigue life is consumed in crack growth from about 10 m to
failure. This is the reason that there is so much interest in the growth behavior of
small cracks. Macro-crack growth and failure regions are the regions where fracture
mechanics parameters have been successful in correlating and predicting fatigue-
crack growth and fracture.
4
Generally, a fatigue fracture has three distinct stages; crack nucleation, its growth and
ultimate failure. Equally present in the first two stages is some kind of repetitive load.
Typically, this load is mechanical; in the case of the crank of a combustion engine, it
can be a repeating pressure; in the interior of an aircraft or fixation of wings in the
fuselage of such aircrafts, it can also be of thermal origin; e.g. in boilers, heat
exchangers, etc. Fig. 1.1 shows the sketch of the three stages of fatigue as seen on the
fracture surface.
Striations - One lineper cycle
I
II
III
Fig. 1.1 Sketch showing three stages of fatigue as seen on the fracture surface
The fatigue life of a member, i.e. the total number of cycles to failure, is the sum of
cycles at the first and the second stages:
Nf = Ni + Np (1.1)
where Nf : Number of cycles to failure
Ni : Number of cycles for crack initiation
Np : Number of cycles for crack propagation
In high cycle fatigue stresses are predominantly elastic and studies show that in this
case Ni is relatively high [3]. In low cycle fatigue, bulk plasticity is involved and
stress levels are usually above the yield strength of the material. With increasing
stress level, Ni decreases and Np dominates.
5
1.1.2 Fatigue crack growth curve
In general, the fatigue crack growth process is characterized by three distinct regions
[7] as shown in Fig.1.2. The sigmoidal shape can be divided into three major regions.
Region I is the near threshold region and exhibits a threshold value, ∆Kth, below
which there is no observable crack growth. Cracks which form below the fatigue
strength are called non-propagating cracks. This region is associated with the growth
of cracks at low stress intensity factor ranges ( K) and is commonly believed to
account for a significant proportion of the fatigue life of a component.
Region II is the stable crack growth region and has been extensively studied for its
technological importance [8 - 19]. This region, normally known as the Paris region,
shows essentially a linear relationship between log da/dN and log K. This region has
received the greatest attention as it is in this region the Paris crack growth law [20]
can be applied, viz:
da/dN = C Km
(1.2)
Here m is the slope of the line obtained from the above equation and the coefficient C
is found by extending the straight line to K=1 MPa m. Both C and m are
experimentally obtained constants. The second stage prevails for an appreciable time
until finally the material fails.
In region III the fatigue crack growth rates are very high as they approach instability,
and little fatigue crack growth life is involved. This region is controlled primarily by
fracture toughness KIC of the material.
6
Figure 1.2 Typical fatigue crack growth curve
For a given material and environment, the fatigue crack growth behavior is essentially
the same for different specimens or components because the stress intensity factor
range is the principal controlling factor in fatigue crack growth [2]. Thus the fatigue
crack growth rate (da/dN) versus K data obtained on simple specimen
configurations, under constant amplitude conditions, can be used for engineering
design. Knowing the stress intensity factor expression, K, for a given component and
loading conditions, the fatigue crack growth life of the component can be assessed by
integrating the sigmoidal curve between the limits of initial crack size and final crack
size. Schematic in Fig. 1.3 shows a constant amplitude load cycle with zero mean
stress.
Log
da
/dN
Log K Kth KIC
I II III
Paris region
da/dN = C Km
7
Fig. 1.3 Alternating stress cycle with zero mean stress
The related terminology is as under:
Stress range = max - min (1.3)
Stress amplitude u = ( max - min)/2 (1.4)
Mean stress m = ( max + min)/2 (1.5)
Stress ratio R = min / max (1.6)
In a structure with crack size „a‟, the stress intensity factor is defined as:
Stress intensity factor K = a . F(a) (1.7)
and
Stress intensity factor range K = a . F(a) (1.8)
where F(a) is the shape factor related to the geometry of the component.
Fig. 1.4 shows the crack tip stress field and the schematic of the three crack
deformation modes.
max
min
u
m
Time
Str
ess
+
-
0
8
Fig. 1.4 Crack tip stress field and crack deformation modes
Many fatigue crack growth data have been obtained under constant load amplitude
test conditions using sharp cracked specimens. Mode I fatigue crack growth has
received the greatest attention because this is the predominant mode of macroscopic
fatigue crack growth. KII and KIII usually have only second order effects on both crack
initiation and crack growth rates.
1.1.3 Fatigue crack propagation
The second stage of fatigue fracture is crack growth or propagation. A crack forms on
the slip plane of a persistent slip band and initially propagates as a stage I crack.
Beyond a grain or two it becomes a stage II crack, propagating on a plane
perpendicular to the principal tensile stress until the member breaks in a rapid tensile
fracture mode. During stage II propagation, striations or ripples are formed on
Mode I
Opening
Mode II
Sliding
Mode III
Tearing
9
portions of the fatigue crack surface perpendicular to the tensile direction. The growth
of the crack from intrusion to the stage II propagation is a rapidly accelerated process.
Hence, the fatigue crack growth process is strongly controlled by the initiation of the
intrusion.
Fig. 1.5 is a schematic showing the nucleation and propagation stages during fatigue.
In this second stage, initially the crack will grow along lines of maximal shear and
later on, when the crack has grown, along the lines of maximal tensile stress. At this
stage, the „micro-crack‟ becomes a „macro-crack‟; the growth depends solely on the
properties of the bulk material and not on the microscopic or the local properties. In
the crack growth process, the crack tip experiences a succession of tensile and
compressive stresses. As compression changes to tension the crack opens and a
system of shear stresses begins to operate at the crack tip. As the tensile stress
increases plastic deformation also increases and ductile blunting of the crack tip takes
place leading to crack growth. When the cycle enters the compressive phase the shear
stresses are reversed and the state of maximum compression closes the crack almost
completely; in the process the plastic flow reverses. The process induces a striation on
the fracture surface. Subsequent tensile stress reopens the crack. The process repeats
itself thereby generating striations on the fracture surface as shown schematically in
Fig. 1.6. Crack closure also effect the fatigue crack growth and various sources of
closure have been identified, such as plasticity, oxide or debris,
roughness/microstructure, residual stress, viscous fluid penetration, phase
transformation, etc. Plasticity-induced crack closure has been one of the most widely
studied research topics in the area of fatigue crack growth. During loading, large
tensile plastic strains are developed near the crack tip, which are not fully reversed
upon unloading. This leads to the formation of a plastic wake behind the crack tip and
10
subsequently reduces the driving force for fatigue crack growth. However, crack
closure may not be as important in stage II as in near-threshold range. The second
stage, commonly known as the „Paris regime‟ prevails for an appreciable time until
finally the material fails in stage III.
Fig. 1.5 Schematic showing the nucleation (I) and propagation (II) stages
during fatigue in a polycrystalline material
Fig. 1.6 Illustration of Stage II crack growth showing the formation of striations [21]
Striations
t
S
(a)
(b)
(c)
(a)
(d)
(d)
(b)
(c)
(e)
(e)
Loading
direction
Free
surface
Crack
Stage I
Nucleation
Stage II
Propagation
11
The fatigue life of a member is affected by many factors e.g.;
1. The type of load (uniaxial, bending, torsion)
2. The nature of the load-displacement curve (linear, nonlinear)
3. The frequency of load repetitions or cycling
4. The load history (cyclic load with constant or variable amplitude, random
load, etc)
5. The size of the member
6. The material flaws
7. The grain size and microstructure
8. The manufacturing method (surface roughness, notches).
9. The localized surface irregularities (e.g. stamping) can be the point of high
stress concentration
10. The operating temperatures (creep due to high temperature, brittleness due to
low temperature)
11. The environmental operating conditions (corrosion, vacuum)
1.1.4 Study of the fractured surface (fractography)
The information stored in specimen during fracture becomes the subject of study [22]
even after the specimen undergoes fatigue failure. Sometimes, it is possible to
determine the exact location of crack initiation and one can locate the source of crack
initiation.
A fatigue fracture surface is distinctive in appearance and consists of three regions;
these are (i) a smooth portion, often possessing the nucleation site (ii) beach markings
showing the growth of the fatigue crack up to the moment of final failure and (iii) the
cleavage or shear which is the final fracture region.
12
The initiation and propagation phase normally generates a cracked surface that is
fairly flat. In an electron microscope, it is possible to distinguish tiny ripples on the
surface; each ripple actually corresponds a loading cycle – very similar to the growth
rings of a tree. A scanning electron micrograph depicting striations on the fractured
surface of a fighter aircraft turbine blade that failed due to fatigue is shown in Fig.
1.7. A crack growth rate of ~ 0.5 x 10-6
meter per loading cycle can be obtained from
the micrograph at high magnification. Furthermore, it is possible to determine
„seasonal‟ differences, which reflect, for instance, change in the amplitude of loading.
The final fracture zone can be smooth and shiny in case of brittle materials or may
have a lustrous structure indicating ductile failure.
(a) (b)
Fig. 1.7. Scanning electron micrographs depicting striations on the fractured turbine
blade a) low magnification b) high magnification
1.2 Thick-walled cylinders
Thick-walled cylinders (TWC) in the form of boilers, gun barrels, nuclear reactors,
pipelines and high-pressure containers are essential structural members for many
industries. These include chemical, petroleum, nuclear, armament and food industries.
The general function of these cylinders is to retain the processes, gas, fluids or
2 μm 20 μm
13
machinery inside them and isolate it from the surroundings. In many applications the
cylinders are prone to cyclic stresses during their normal operation. The origin of
these stresses may be the fluctuating pressures inside the pressure vessel, the pipe
containing fluid, or instead, the vibrations of the rotating machinery inside them.
Large internal pressures produce high tension hoop stresses along the inner surface of
the cylinder; the latter may result in the nucleation of the internal surface cracks due
to cyclic action of high-pressure pulses. Cracks nucleate at the surface of the bore,
grow into arrays of longitudinal and/or radial cracks and reduce the strength of the
structure resulting in premature failure at pressures which are even lower than the
design capacity. Catastrophic service failures result in loss to human life and have
significant impact on the economy. For this reason, proper material and precision in
design is necessary to ensure integrity of the cylinder during the service life.
Defects in the thick-walled cylinders may be generated during manufacturing or
during the service; these defects are the potential site of crack initiation. In addition,
during the normal operation, thick-walled cylinders may suffer from a number of
degradation mechanisms including stress corrosion cracking, creep etc. Initiation of
cracks can never be ruled out during the normal operation and must be considered
while analyzing against fatigue failure. If the primary crack growth mechanism is
slow, the cracks will be detected during routine maintenance by non-destructive
testing (NDT) so that corrective measures can be taken before crack growth moves
into a high risk regime. It is thus quite necessary to analyze, in detail, the crack
propagation behavior in industrial applications.
14
1.3 Problem description and research strategy
1.3.1 Extruded cylinders
In present work, the component being studied is a thick-walled extruded cylinder
under fatigue loading. In industries of a wide variety, the thick-walled cylinders are
the main critical components. The cylinders are manufactured through extrusion; the
most popular extrusion procedure is based on splitting the ingot to be extruded in
three or four segments, which subsequently get welded together upon passage through
a specially designed die. The cylinder produced in the process has seams in the wall
parallel to the extrusion direction. After the final heat treatment processes, the seams
become an integral part of the cylinder. However, the probability of certain
manufacturing defects may not be entirely ruled out; for instance lack of precise
control of the processing parameters may result in incomplete welding of the
segments creating a crack usually on the inner surface. The length of the crack so
generated may be small or could extend the entire length of the cylinder as shown in
Fig. 1.8. To avoid seams the double action extrusion process is used; see the
schematic of the process in Annexure I.
1.3.2 Fatigue process in thick-walled cylinders
A fatigue crack passes through three major stages and these are crack initiation, stable
crack propagation and final rapid fracture. Normally, the first stage takes quite a long
time but in a cylinder which already contains internal surface cracks, the first stage is
assumed to be already completed and fatigue consideration in the design is of crucial
importance. Under fatigue loading conditions the cracks present on the inner surface
of the cylinder may grow outwards in the radial direction and result in the complete
failure of the cylinder. Therefore a comprehensive study of these cylinders under the
15
conditions described above is indispensable. In this work the crack growth design
problem associated with a thick-walled cylinder is being studied.
1.3.3 Definition of the problem being studied
The component which is the subject of present study is an extruded cylinder of an
aluminum alloy AA 6061 in T6 heat treatment condition. Repeated pressure applied
to the cylinder from the inner surface produces cyclic stresses in the cylinder body.
The objective of the work is to predict the life of the cylinder in the presence of
fatigue loading with internal surface cracks along the cylinder axis. The problem is
analyzed with the help of finite element method (FEM) using the commercial
structural analysis ANSYS programme. The results may be used to avoid failure of
cylinders under the conditions studied as well as to derive life extension strategies.
Fig. 1.8 Section of the thick-walled cylinder with internal surface crack
along the cylinder axis
1.3.4 Research strategy
The first, rather difficult but important, step of a fatigue analysis is the accurate
determination of the material‟s resistance to fatigue loading. Normally, a fatigue
analyst first looks for the accurate fatigue data and more than often decides to create
data relevant to his own research task [23]. In the present work, the research strategy
Axial crack
Pressure
16
was based on the development of the fatigue data of the cylinder under investigation.
Since the full-scale fatigue crack growth test of the thick-walled cylinder subjected to
internal pressure involves a significant amount of time and cost, middle tension M(T)
samples taken from the cylinder were alternatively used for the fatigue crack growth
simulation of the actual cylinder [24].
Samples from two different orientations, representing the axial and the hoop stresses
in the cylinder, were subjected to various stress levels and crack growth rates and
fatigue life of the samples thus accomplished. The experimental work was replicated
through modeling and simulation of the same geometry with the help of commercially
available structural analysis software ANSYS. Research undertaken proceeded as
follows:
1. Material‟s characterization including metallurgical aspects and mechanical testing
under uniaxial tension
2. Experimental fatigue crack growth testing of the TWC and determination of the
material fatigue properties using M(T) specimens
3. Numerical simulation of fatigue crack propagation using finite element analysis of
the experimental geometries under fatigue conditions
4. Finite element analysis of the thick-walled cylinder under static loading and
fatigue conditions using the experimental fatigue data and specified boundary
conditions
5. Interpretation of the results
1.3.5 Research objectives
Material defects due to manufacturing processes in the presence of aggressive
environment under cyclic loadings can trigger the failure of the cylinder; this may
subsequently result in safety concerns and expensive outages. This thesis deals with
17
the aspect of fatigue crack propagation in a TWC. The objective of this research work
is to carry out theoretical and experimental investigations of an aluminum alloy thick-
walled cylinder under fatigue loadings. The study is performed to comprehend the
fatigue process, from the macroscopic stress-strain relations to microscopic fatigue
crack growth mechanisms, predict the fatigue life and suggest measures to improve
the fatigue properties. The research objective was achieved through material
characterization, cyclic testing, and numeric modeling.
1.4 Overview of the thesis
Chapter 1 Introduction covers the basic concepts of the fatigue process, introduction
to thick-walled cylinders, the problem description and the objective of the present
research.
Chapter 2 provides the historical review of fatigue and a brief literature review
relevant to this dissertation.
Chapter 3 gives the details of the experimental work conducted during this study.
This includes the equipments, standard methods, material characterization technique
and the details of the tests to determine the monotonic tensile properties and fatigue
crack growth rate data of the alloy. Finally the detail of the fractographic study in
scanning electron microscope (SEM) is provided.
Chapter 4 Thick-Walled Cylinder; Material and Structural Characterization provides
metallurgical and mechanical characterization of the TWC. The chapter presents the
findings from the procedures employed (EDX analysis, light microscopy and
scanning electron microscopy) to better understand the material and the
manufacturing characteristics of the extruded TWC as well as their monotonic
18
properties. Post fracture analysis provides the micro-mechanism of the fracture
process under monotonic loading.
Chapter 5 Experimental Fatigue Crack Growth Study presents the results of the crack
growth experiments of the TWC in two orientations. The fatigue life of the specimens
and the crack growth rate data of the material at different stress levels are given. The
fractographs showing the topographical variations with parametric change are also
presented.
Chapter 6 Numerical Simulation of Fatigue Crack Propagation presents the results of
modeling and simulation of the experimental geometries under fatigue conditions
using finite element analysis.
Chapter 7 provides a detailed finite element analysis of the TWC under static and
fatigue loading.
Chapter 8 Summary and Conclusions presents the conclusions of the research study
and based on the results recommendations for further research are suggested.
19
CHAPTER 2
Literature Review
Fatigue of materials involves a very complex interaction of different metallurgical,
mechanical and technological factors and is still only partly understood [2]. These
include the material intrinsic properties, thermal and mechanical processing, surface
conditions, the geometry of the component and finally the cyclic loading conditions. Each
of these factors includes a list of variables which significantly influences the mechanical
properties of the material including the fatigue behavior.
In the following sections, a brief literature review of the fatigue and finite element
method is presented. The goal of the literature survey is to develop a practical
methodology to analyze the fatigue crack propagation in thick-walled cylinders under
fatigue loading. Different aspects related to fatigue can be best understood by the
historical overview enlightening the evolution of the process with time.
2.1 Historical overview of fatigue
2.1.1 1800 Era
The word "fatigue" was introduced in the 1840s to describe failures occurring from
repeated stresses which were below yield stress. The first major impact of failures due to
repeated stresses involved the railway industry in the 1840s. It was found that railroad
axles failed regularly at shoulders [25] and immediately elimination of sharp corners was
recommended.
During 1850s and 1860s, August Wohler introduced the concept of the fatigue limit. He
performed the first systematic laboratory investigation of fatigue tests under repeated
20
stresses. Wohler has been named the "father" of systematic fatigue testing. Using stress
versus life (S-N) data, he showed how fatigue life decreased with higher stress
amplitudes and that below a certain stress amplitude, the test specimens did not fracture.
He pointed out that for fatigue, the range of stress is more important than the maximum
stress [26].
During the 1870s and 1890s, different researchers substantiated and expanded Wohler's
classical work. Gerber, among others, investigated the influence of mean stress, and
Goodman proposed a simplified theory concerning mean stresses. Their names are still
associated with diagrams involving alternating and mean stresses. Bauschinger [27] in
1886 showed that the yield strength in tension or compression was reduced after applying
a load of the opposite sign that caused inelastic deformation. This was the first indication
that a single reversal of inelastic strain could change the stress-strain behavior of metals.
This finding was the forerunner of understanding cyclic softening and hardening of
metals.
2.1.2 Early 1900 era
In the early 1900s, Ewing and Humfrey [28] used an optical microscope to pursue the
study of fatigue mechanisms. Localized slip lines and slip bands leading to the formation
of micro cracks were observed. Basquin [29] in 1910 showed that alternating stress
versus number of cycles to failure (S-N) in the finite life region could be represented as a
log-log linear relationship. His equation, further modified by others, is currently being
used to represent finite life fatigue behavior.
In the 1920s, Gough and associates contributed greatly to the understanding of fatigue
mechanisms. They discussed the combined effects of bending and torsion (multi-axial
21
fatigue). Gough published a comprehensive book on fatigue of metals in 1924 [30].
Moore and Kommers [31] published the first comprehensive American book on fatigue
of metals in 1927.
In 1920, Griffith [32] published the results of his theoretical calculations and experiments
on brittle fracture using glass. He found that the strength of glass depended on the size of
microscopic cracks. If ‘S’ is the nominal stress at fracture and ‘a’ is the crack size at
fracture, the relation is S√a = constant. With this classical pioneering work on the
importance of cracks, Griffith developed the basis for fracture mechanics. He thus
became the "early father" of fracture mechanics.
In 1924 Palmgren [33] developed a linear cumulative damage model for variable
amplitude loading and established the use of the B10 fatigue life based upon statistical
scatter for ball bearing design. McAdam [34], in the 1920s, performed extensive
corrosion fatigue studies in which he showed significant degradation of fatigue resistance
in various water solutions. This degradation was more pronounced in higher-strength
steels. In 1929-30, Haigh [35] presented a rational explanation of the difference in the
response of high tensile strength steel and of mild steel to fatigue when notches were
present. He used the concepts of notch strain analysis and residual stresses, which were
more fully developed later by others.
During the 1930s, an important practical advance was achieved by the introduction of
shot-peening in the automobile industry. Fatigue failures of springs and axles, which had
been a very common problem, thereafter became rare. Almen [36] correctly explained the
spectacular improvements by compressive residual stresses produced in the surface layers
of peened parts, and promoted the use of peening and other processes that produce
22
beneficial residual stresses. Horger [37] showed that surface rolling could prevent the
growth of cracks.
In 1937 Neuber [38] introduced stress gradient effects at notches and the elementary
block concept, which states that the average stress over a small volume at the root of the
notch is more important than the peak stress at the notch. In 1939 Gassner [39]
emphasized the importance of variable amplitude testing and promoted the use of an
eight-step block loading spectrum for simulated testing. Block testing was prominent
until closed-loop electrohydraulic test systems became available in the late 1950s and
early 1960s.
2.1.3 Mid 1900 era
During World War II, the deliberate use of compressive residual stresses became
common in the design of aircraft engines and armored vehicles. Many brittle fractures in
welded tankers and Liberty ships motivated substantial efforts and thinking concerning
preexisting discontinuities or defects in the form of cracks and the influence of stress
concentrations. Many of these brittle fractures started at square hatch corners or square
cutouts and welds. Solutions included rounding and strengthening corners, adding riveted
crack arresters, and placing greater emphasis on material properties. In 1945 Miner [40]
formulated a linear cumulative fatigue damage criterion suggested by Palmgren [33] in
1924. This criterion is now recognized as the Palmgren-Miner linear damage rule. It has
been used extensively in fatigue design and, despite its many shortcomings, remains an
important tool in fatigue life predictions.
The formation of the American Society for Testing and Materials (ASTM) Committee E-
09 on Fatigue in 1946, with Peterson as its first chairman, provided a forum for fatigue
23
testing standards and research. Peterson emphasized that the fatigue notch factor, K was a
function of the theoretical stress concentration factor, Kt, the notch and component
geometry, and the ultimate tensile strength [41]. In 1953, he published a comprehensive
book on stress concentration factors [42] and an expanded version [43] in 1974.
The Comet, the first jet-propelled passenger airplane, started service in May 1952 after
more than 300 hours of flight tests. Four days after an inspection in January 1953, one of
the Comets crashed into the Mediterranean Sea. After much of the wreckage had been
recovered and exhaustive investigation and tests on components of the Comet had been
made, it was concluded that the accident was caused by fatigue failure of the pressurized
cabin. The small fatigue cracks originated from a corner of an opening in the fuselage.
Subsequently, one more Comet aircraft failed catastrophically. By September 1953, a test
section of the cabin had been pressurized 18000 times to 57 kPa in addition to 30 prior
cycles between 70 and 110 kPa. The design stress for 57 kPa was 40 percent of the tensile
strength of the aluminum alloy. Probably the first 30 high load levels induced sufficient
residual stresses in the test section so as to falsely enhance the fatigue life of the test
component and provide overconfidence. All Comet aircraft of this type were taken out of
service, and additional attention was focused on airframe fatigue design. Shortly after
this, the first emphasis on fail-safe rather than safe-life design for aircraft gathered
momentum in the United States. This placed much more attention on maintenance and
inspection.
Major contributions to the subject of fatigue in the 1950s included the introduction of
closed-loop servo hydraulic test systems, which allowed better simulation of load
histories on specimens, components, and total mechanical systems. Electron microscopy
24
opened new horizons to a better understanding of basic fatigue mechanisms. Irwin' [44]
introduced the stress intensity factor KI which has been accepted as the basis of linear
elastic fracture mechanics (LEFM) and of fatigue crack growth life predictions. Irwin
coined the term "fracture mechanics," and because of his many important contributions to
the subject at this time, he is considered as the “modern father of fracture mechanics."
The Weibull distribution [45] provided both a two- and a three-parameter statistical
distribution for probabilistic fatigue life testing and analysis.
In the early 1960s, low-cycle strain-controlled fatigue behavior became prominent with
the Manson-Coffin [46, 47] relationship between plastic strain amplitude and fatigue life.
These ideas were promoted by Topper and Morrow [48, 49] and, along with the
development of Neuber's rule [50] and rain flow counting by Matsuishi and Endo [51] in
1968, are the basis for current notch strain fatigue analysis. The formation of the ASTM's
Special Committee on Fracture Testing of High-Strength Steels in the early 1960s was
the starting point for the formation of ASTM Committee E-24 on Fracture Testing in
1964. This committee has contributed significantly to the field of fracture mechanics and
fatigue crack growth and was combined with ASTM Committee E-09 in 1993 to form
Committee E-08 on Fatigue and Fracture. Paris [52] in the early 1960s showed that the
fatigue crack growth rate (da/dN) could best be described using the stress intensity factor
range ΔKI. In the late 1960s, the catastrophic crashes of F-111 aircraft were attributed to
brittle fracture of members containing preexisting flaws. These failures, along with
fatigue problems in other U.S. Air Force planes, laid the groundwork for the requirement
to use fracture mechanics concepts in the B-1 bomber development program of the 1970s.
This program included fatigue crack growth life considerations based on a pre established
25
detectable initial crack size. Schijves [53] in the early 1960s emphasized variable
amplitude fatigue crack growth testing in aircraft, along with the importance of tensile
overloads in the presence of cracks that can cause significant fatigue crack growth
retardation. In 1967, the Point Pleasant Bridge at Point Pleasant, West Virginia, collapsed
without warning. An extensive investigation [54] of the collapse showed that a cleavage
fracture in an eye bar caused by the growth of a flaw to a critical size was responsible.
The initial flaw was due to fatigue, stress corrosion cracking, and/ or corrosion fatigue.
This failure has had a profound influence on subsequent design requirements established
by the American Association of State and Highway and Transportation Officials
(AASHTO).
In 1970, Elber [55] demonstrated the importance of crack closure on fatigue crack
growth. He developed a quantitative model showing that fatigue crack growth was
controlled by an effective stress intensity factor range rather than an applied stress
intensity factor range. The crack closure model is commonly used in current fatigue crack
growth calculations. In 1970, Paris [56, 57] demonstrated that a threshold stress intensity
factor could be obtained for which fatigue crack growth would not occur. During the
1970s, an international independent and cooperative effort formulated several standard
load spectra for aircraft, offshore structures, and ground vehicle usage [58, 59]. These
standard spectra have been used by many engineers in a variety of applications. In July
1974, the U.S. Air Force issued Mil A-83444, which defined damage tolerance
requirements for the design of new military aircraft. The use of fracture mechanics as a
tool for fatigue was thus thoroughly established through practice and regulations. This
practice also emphasized the increased need for an improved quantitative, nondestructive
26
inspection capability as an integral part of the damage tolerance requirements.
2.1.4 End 1900 era
During the 1980s and 1990s, many researchers were investigating the complex problem
of in-phase and out-of-phase multi-axial fatigue. The critical plane method suggested by
Brown and Miller [60] motivated a new philosophy concerning this problem, and many
additional critical plane models were developed. The small crack problem was noted
during this time, and many workers attempted to understand the behavior. The small
crack problem was complex and important, since these cracks grew faster than longer
cracks based upon the same driving force. Definitions became very confusing. Interest in
the fatigue of electronic materials increased, along with significant research on thermo-
mechanical fatigue. Composite materials based on polymer, metal, and ceramic matrices
were being developed for many different industries. The largest accomplishments and
usage involved polymer and metal matrix composites. These developments were strongly
motivated by the aerospace industry but also involved other industries. During this time,
many complex and expensive aircraft components designed using safe-life design
concepts were routinely being retired with potential additional safe usage. This created
the need to determine a retirement for cause policy. From a fatigue standpoint, this meant
significant investigation and application of nondestructive inspection and fracture
mechanics. In 1988 the nearly fatal accident of the Aloha Boeing 737, after more than
90000 flights, created tremendous concern over multi site damage (MSD) and improved
maintenance and nondestructive inspection. Corrosion, corrosion fatigue, and inadequate
inspection were heavy contributors to the MSD problem that existed in many different
airplane types. Comprehensive investigations were undertaken to understand the problem
27
better and to determine how best to cope with it and resolve it. Also during the 1980 and
1990s, significant changes in many aspects of fatigue design were attributed to advances
in computer technology. These included software for different fatigue life (durability)
models and advances in the ability to simulate real loadings under variable amplitude
conditions with specimens, components, or full-scale structures. This brought
significantly more field testing into the laboratory. Integrated CAE involving dynamic
simulation, finite element analysis, and life prediction/estimation models created the idea
of restricting testing to component durability rather than using it for development.
Increased digital prototyping with less testing has become a goal of twenty-first-century
fatigue design. Additional readings on the history of fatigue can be found in [61, 62].
2.2 Fatigue crack propagation; effect of microstructure
The mechanical behavior of materials depends strongly on their microstructure. It is
known that an aluminum alloy exhibits very different properties depending on whether it
is cold rolled or heat treated under different temper conditions. Metal fatigue is also
significantly influenced by microstructure [2]. This includes chemistry, heat treatment,
cold working, grain size, anisotropy, inclusions, voids/porosity, delaminations and other
discontinuities or imperfections. If the actual fatigue data are available, microstructural
effects are inherently accounted for and, therefore, do not have to be accounted for again
[2]. Inclusions act as stress concentration sites and are common locations for microcracks
to nucleate under cyclic loading. Anisotropy caused by cold working gives increased
fatigue resistance when loaded in the direction of the working than when loaded in the
transverse direction. This is due to the elongated grain structure in the direction of the
original cold working.
28
Investigations of the aluminum alloys under fatigue loading have been extensively done
[63 - 83] and the microstructural features including the constituent particles are found to
play an important role in the nucleation and fatigue crack growth process of these alloys.
In a study of aluminum alloy AA 2026 extrusion bars, it was observed that reducing the
density of constituent particles significantly enhanced the fatigue resistance of the alloy
[84]. Zhu et. al. [85] have studied the effect of microstructure and temperature on fatigue
behavior of a cast aluminum alloy and reported that the influence of microstructure on the
fatigue resistance is greater than the influence of the testing temperature. In a study on
aluminum alloy, Merati [86] pointed out that grain size and orientation could play crucial
roles while determining the fatigue life. Generally, it is accepted that the microstucture is
important in crack initiation, relatively unimportant in propagation where Young's
modulus is the most important factor.
Suresh et. al. [64] have highlighted the influence of the grain structure and slip
characteristics in aluminum alloys. They pointed out that microstructural effects have a
strong influence on fatigue crack growth rates near threshold. Fonte et al. [73] have
studied the effect of microstructure and environment on fatigue crack growth resistance
of aluminum alloys and concluded that these significantly affect the process of fatigue
crack growth.
An autofrettage process is commonly used in pressure vessels to enhance the fatigue life
of the component. This process produces compressive tangential residual stresses near the
bore of the pressure vessel. These residual stresses counteract the large tensile tangential
stress at the inside surface of the pressure vessel subjected to a cyclic internal pressure.
29
The compressive tangential residual stresses near the inside surface due to the
autofrettage process retard crack formation and growth.
References [7-19, 73-89] cover some recent research work on fatigue crack growth
process in metals. These research works include the effects of loading parameters,
environment, surface treatment and microstructure on fatigue crack growth of different
alloys.
2.3 Finite element method
Finite element analysis (FEA) is a useful and powerful technique for determining stresses
and strains in structures or components too complex to analyze by strictly analytical
methods. With this technique, the structure or component is broken down into many
small pieces (finite number of elements) of various types, sizes and shapes. The elements
are assumed to have a simplified pattern of deformation (linear or quadratic etc.) and are
connected at "nodes" normally located at corners or edges of the elements. The elements
are then assembled mathematically using basic rules of structural mechanics, i.e.
equilibrium of forces and continuity of displacements, resulting in a large system of
simultaneous equations. By solving these large simultaneous equations system with the
help of a computer, the deformed shape of the structure or component under load may be
obtained. Based on that, stresses and strains may be calculated [90].
The finite element method (FEM) is probably the most versatile way of calculating stress
intensity factors. This method primarily involves the evaluation of displacements at nodal
points of the body which has been idealized into a system of elements connected at the
nodal points. The FEM has become a powerful tool for the numerical solution of a wide
range of engineering problems. The FEM has been extensively used to solve problems
30
involving irregular regions and complicated cracks [91]. The application of FEM in the
solid mechanics problems was first adjusted by Turner et al [92]. In 1961-1962 [93, 94],
the extension of the FEM to three-dimensional problems occurred with the development
of a tetrahedral stiffness matrix. In 1969-1970 [95, 96], the uses of finite element
methods in fracture mechanics were initiated. In LEFM problems, the conventional
constant elements and the higher order elements did not satisfy the singularity at the
crack tip. In 1971, Tracy [97] made the earliest attempt on the development of special
crack tip elements that directly model the 1/ r near the tip elastic strain field singularity
in order to solve the LEFM problems successfully. From the early 1950s to the present,
enormous advances have been made in the application of the FEM to solve complicated
engineering problems.
2.3.1 Modeling through FEM
The finite element method is a method of approximation, in which a discrete model is
generated by a set of small continuous functions defined over a number of finite regions
called elements. The FEM approximates any continuous quantity, such as displacement,
temperature or pressure. When the FEM is used in fracture mechanics, the cracked body
of interest is broken up into a collection of pre-selected finite elements which are
connected at nodal points. According to the minimum potential energy principle, the
global nodal forces [F] and the global nodal displacements [d] are related through use of
the global stiffness matrix [K] as follows [98]:
[F] = [K] [d] (2.1)
where [F] and [K] are assembled from the corresponding element matrix.
Once the displacements [d] at the nodes are obtained, the field quantity of any point
31
within an element could then be interpolated from [d] via shape functions.
2.3.2 Computing the stress intensity factor
According to LEFM theory, the condition for brittle failure can be expressed as;
K I K IC (2.2)
where KI is called the stress intensity factor (SIF) and is dependent on loading conditions
and the flaw size in the material. KIC can be considered a material property characterizing
the crack resistance, and is therefore called the plane strain fracture toughness. Thus the
same value of KIC should be found by testing specimens of the same material with
different geometries and with critical combinations of crack size and shape and fracture
stress. Within certain limits this is indeed the case, and so knowledge of KIC obtained
under standard conditions can be used to predict failure for different combinations of
stress and crack size and for different geometries.
The stress intensity factor is usually expressed as:
K I Q a (2.3)
where Q is a geometry correction factor depending on the geometry of the structural
component and the crack geometry, is the applied stress, and 'a' denotes the crack size.
The value of the critical stress intensity KIC can be determined experimentally by
measuring the fracture stress for a large plate that contains a through-thickness crack of
known length. This value can also be measured by using other specimen geometries, or
else can be used to predict critical combinations of stress and crack length in these other
geometries. This is what makes the stress intensity approach to fracture so powerful,
since values of K for different specimen geometries can be determined from conventional
32
elastic stress analyses. There are several handbooks giving relationships between the
stress intensity factor and many types of cracked bodies with different crack sizes,
orientations and shapes, and loading conditions. Furthermore, the stress intensity factor,
K is applicable to stable crack extension and does characterize processes of subcritical
cracking like fatigue and stress corrosion. It is the use of the stress intensity factor as the
characterizing parameter for crack extension that is the fundamental principle of LEFM.
Many researchers have performed studies to determine the stress intensity factors in
different geometries and components [99 - 102].
The stress intensity factor of a crack for a linear elastic fracture mechanics analysis may
be computed in ANSYS structural software by using the KCALC command [103]. This
analysis uses a fit of the nodal displacements in the vicinity of the crack. The actual
displacements around the crack for linear elastic materials, when only mode I is
considered, are as follows:
)(]2
3cos
2cos)12[(
24rOk
r
G
Ku I
(2.4)
)(]2
3sin
2sin)12[(
24rOk
r
G
Kv I
(2.5)
)(0 rOw (2.6)
where u, v and w are displacements in a local Cartesian coordinate system, r and θ are
coordinates in a local cylindrical coordinate system as shown in Fig. 2.1, G is the shear
modulus and O(r) represents terms of order r or higher.
33
Fig. 2.1 Local coordinates measured from the 3-D crack front
The value for k is given by:
k = 43 If plane strain or axisymmetric
1
3 If plane stress
where is Poisson’s ratio.
Evaluating Equations (2.4) through (2.6) at 1800 and neglecting the higher order
terms:
)1(22
kr
G
Kv I
(2.7)
Typically, the analysis models are either a full-crack or half-crack model. A full-crack
model contains both crack faces while a half-crack model takes the advantage of
symmetry about the crack plane and contains only one face as shown in the Fig. 2.2.
34
Fig. 2.2 (a) a half-crack model (b) a full-crack model
Equation (2.7) can be rewritten as:
r
v
k
GK I
||
1
22
(For a half-crack model) (2.8)
r
v
k
GK I
||
12
(For a full-crack model) (2.9)
where v is the motion of crack face with respect to the plane of symmetry and v is the
motion of one crack face with respect to the other crack face.
In the above equations, the final factors are evaluated based on the nodal displacements
and locations as given below.
BrAr
v
|| (For a half-crack model) (2.10)
BrAr
v
|| (For a full-crack model) (2.11)
Here, A and B are determined from the displacements at nodes ( J K, ) for the half-crack
model and ( J K L M, , , ) for full-crack model.
35
By letting r approach zero ( r 0 ), then
Ar
v
r
||lim
0 or A
r
v
r
||lim
0 (2.12)
Finally, K I could be formulated as
k
GAK I
1
22 (For a half-crack model) (2.13)
k
GAK I
12 (For a full-crack model) (2.14)
2.4 Stresses in a thick-walled cylinder
Numerous analytical models for determining the state of stress in a thick-walled cylinder
have been developed. The earliest of these date back to the classical elastic solution by
Lamé for an isotropic, homogeneous, thick-walled cylinder subjected to an internal and
an external pressure. Lamé’s solution for the simplified cases of cylinder subjected to an
internal or external pressure only can be readily found in many elasticity and strengths of
materials texts [104 - 106]. Models developed by Hill et al. [107], Hodge and White
[108], and Steele [109] considered both elastic and plastic deformation and compared
different plastic stress-strain laws and yield criteria.
Bland [110] incorporated a stationary temperature field and an internal and an external
pressure, developing one of the first thermo-mechanical models. He also considered the
unloading solution with reverse yielding but did not include the Bauschinger effect,
which predicts the yield strength in compression (reverse yielding) below the same
magnitude in tension. More recently, Chen presented analytical models, which
considered reverse yielding with the Bauschinger effect during unloading [111, 112].
36
Venter et al. [113] have done recent comparative studies on three experimental methods
to measure the residual compressive hoop stresses due to the hydraulic or swage
autofrettage process. They also compared their results to isotropic and kinematic strain-
hardening material models and found considerable error near the bore surface for
materials with large Bauschinger effects and therefore more severe reverse yielding.
While both models can depict reverse yielding, the error was greater for the isotropic
hardening model, which does not adequately account for the Bauschinger effect. The
location of the elastic-plastic boundary found in the experimental measurements agreed
with the theoretical models. O’Hara and Troiano [114] have investigated and compared
three analytical autofrettage calculation methods with different yield criterion and three
finite element methods incorporating the von Mises yield criterion and different strain
hardening models for the steel. While the six methods produced varying results, there
were several apparent conclusions. First, the kinematic strain hardening model predicted
reverse yielding but the isotropic model did not. Second, the location of the elastic-plastic
boundary was in general agreement for all methods. Third, the analytical solutions could
not predict reverse yielding. Lastly, the finite-element kinematic strain hardening method
compares well with the ASME Boiler and Pressure Vessel Code, Division 3 with the
exception of the reverse yielding near the bore.
Research works on thick-walled cylinders, pressure vessels and pipes under static
pressure, under fatigue loading and with internal and external surface flaws are given in
references [115 - 124], [125 - 129] and [130 - 134], respectively.
In general, industrial materials do not have uniform composition, and generation of
defects such as holes, cavities, and cracks in their substructure is inevitable. Therefore,
37
consideration of fracture mechanics criterions in the design process of metallic structures
seems to be essential. In cylindrical bodies, such as pressure vessels, pipes, borers, and
driving shafts, a semi-elliptical crack placed on the outer surface of the circular cross-
section of these bodies is considered to model the actual defects. Lin and Smith [135]
have shown that every defect with any initial shape gets the shape of a semi-elliptical
crack in a fatigue crack growth process after a few cycles. Because of very complex
loading conditions on cylindrical structural components, all three modes of fracture take
place at the crack front. In general, most of research on the surface cracks in cylindrical
bodies focuses on Mode I [136] and attempts for determining fracture mechanics
parameters in mixed mode conditions are rarely encountered in the literature.
Shlyannikov [137] did numerical analysis to calculate the aspect ratio changes for
different values of the geometrical parameters for both cylinder and surface flaw. Thick
and thin-walled cylinders containing initial semi-elliptical internal surface flaws have
been analyzed. Crack propagation paths in the diagram of flaw aspect ratio against
relative crack depth have been determined through a fracture damage zone model. He
reported that surface crack behavior is sensitive to the change of initial flaw
configuration, nominal stress level and dimensionless wall-thickness. The comparison
between the numerical predictions and experimental data shows that the agreement is
good for the aspect ratio change, demonstrating that the modeling crack growth by
fracture damage zone model is reliable.
In a pressure vessel subjected to a cyclic internal pressure, fatigue cracks initiate and
grow from the inside surface, at which the largest tensile stress occurs [138]. Seung-Kee
has performed an elastic and elastic–plastic stress analyses of an autofrettaged pressure
38
vessel with radial holes using the finite element method. He investigated the influence of
autofrettage on the fatigue life of the pressure vessel with radial holes. He concluded that
the cyclic plastic deformation and high tensile mean stress were dominant damaging
factors in the inside and outside surfaces of the pressure vessel, respectively. He also
reported that the predicted fatigue life evaluated at the critical location increased as the
autofrettage level increased.
Hojjati [117] has used ANSYS structural software for finite element modeling of the
autofrettaged vessel. He used the element PLANE42, which has the capability of elastic
and plastic material modeling, for the analysis. Raju and Newman [139] analyzed
longitudinal cracks in pressurized cylinders, using a three-dimensional finite element
modeling. Their analysis was restricted to Mode I. They reported that the stress intensity
factor for external cracks is more than that for internal ones, and the location of
maximum stress intensity factor is placed at either deepest or corner points of the crack,
depending on loading and geometry conditions.
2.5 Numerical simulation of fatigue
In particular, fatigue tests are very expensive, since they require a lot of human and
machine time, so it is very important to find models and develop suitable software in
order to simulate fatigue analyses. Numerical simulation of fatigue behavior is a very
complex task due to the absence of suitable material models able to correctly reproduce
the material behavior under cyclic load [140]. Generally, fatigue problems are modeled
with semi-empirical laws which do not take into account the physical problem of crack
initiation and propagation: these models provide more or less simple relationships
between damage increase inside the material and fatigue testing conditions.
39
A number of models for performing broad-spectrum fatigue analyses have been
proposed. However, the major shortcoming of many of these models is that they are often
material-specific and are therefore unsuitable for incorporation into computer simulation
programs for general use [141]. Bacila et. al. have proposed a simple model for fatigue
crack propagation, using the mechanical properties of the material (the yield stress and
Paris’s parameters C and m). They have reported that the computer simulation and
experimental results for crack propagation offer a good agreement and hence the
computer simulation could be used on other materials.
When FE methods are applied to fatigue problems, the component is simulated under
static load and from the numerical results some indications on the fatigue behavior are
extracted. For example, if FE analysis evidences stress concentration in a zone, that zone
is indicated as the most susceptible to fatigue problems. Shang et al. [142] employed an
elastic-plastic FE analysis to calculate the local stress–strain field intensity parameters
predicting the fatigue crack initiation life of U-shape notched steel specimens. Lee and
Koh [143] evaluated the residual stress distribution due to auto-frettage loading for
fatigue life prediction of an externally grooved thick-walled pressure vessel. Recently, FE
analysis was used for the simulation of the S/N-curves [144], for fretting fatigue [145],
and using a volumetric approach for life prediction of notched components [146]. In all
the cases, empirical or semi-empirical laws for life prediction are drawn from FEM
results, which correctly take into account stress and strain distributions.
The three major fatigue life methods used in design and analysis are the stress-life
method, the strain-life method, and the linear-elastic fracture mechanics (LEFM) method
[147]. The numerical techniques, based on linear-elastic fracture mechanics [148], using
40
data from laboratory tests are frequently used to establish fatigue failure criteria. Many
researchers used numerical approaches for the study of fatigue crack propagation. A
general trend of the numerical approach in this field can be inferred from the Refs. [13,
83, 149 - 153]. A crack tip node-release scheme was suggested in Ref. [150], in which, a
change in the boundary condition was characterized for a crack growth. McClung and
Sehitoglu have investigated fatigue crack closure by the FEM [152]. They followed the
node-release scheme at the maximum load and the crack tip was extended one element
length per cycle. Ding et al. presented a low-fatigue life prediction model for particulate-
reinforced [154] and short fiber reinforced [155] metal matrix composites. In both cases,
the matrix was the 6061 aluminum alloy and the reinforcement Al2O3.
Gwo-Chung et. al. [156] have conducted fatigue analysis of cracked thick aluminum plate
bonded with composite patches, which are used for the repairs of metallic aircraft
structures. The stress analyses were performed for four different aluminum plates. The
stress intensity factor analysis and the fatigue life calculations were performed for the
cracked aluminum plate and the cracked plate repaired with the composite patches. Their
results show that the finite element analysis combined with the Paris law could predict
the fatigue life of cracked specimens with and without repair. They concluded that fatigue
crack propagation characteristics obtained from FEM match very well with the
experimental results.
Alam et. al. [13] have simulated the crack propagation using ANSYS. They have
analyzed a 2D Finite element model of a butt-welded joint using interface elements and
presented the fatigue crack growth rate and fatigue crack propagation life. They declare
that the proposed simulation of fatigue crack propagation using an interface element is
41
relatively simple compared to other conventional FEM methods, effective in practice,
numerically less intensive and saves computer time significantly.
42
CHAPTER 3
Equipment and Experimental
This chapter presents the details of the experimental work conducted during this
study. Fig 3.1 shows the flowchart of the solid mechanics analysis chain and indicates
the two routes to solve a physical problem. The significance of the experimental work
can be admitted by the diagram. In both the routes the experiments are mandatory to
reach a satisfactory solution.
Experimental work constitutes the execution of the following tests and examinations
of the material:
Material characterization; viz., chemical composition and microstructural
evaluation
Monotonic tensile testing
Hardness testing
Impact testing
Fatigue crack growth rate test
Fractography
All samples subjected to the above experiments were taken in two directions; i)
longitudinal /axial direction (LR) ii) transverse direction (CR). The details of each of
the tests are given in the following sections along with some details of the equipment
used for the test.
43
Fig. 3.1 Flowchart of the solid mechanics analysis chain
3.1 Standards
The following standards were used while performing the experiments:
1. Standard Methods for Tension Testing Wrought and Cast Aluminum and
Magnesium Alloy Products (ASTM B 557 M)
2. Standard Test Method for Notched Bar Impact Testing of Metallic Materials
(ASTM E 23)
3. Standard Test Method for Measurement of Fatigue Crack Growth Rates
(ASTM E 647)
3.2 Equipment
The main equipment used in the testing and examination of the material includes the
following:
1. Optical microscope
Refinement
needed
Satisfactory
agreement
Physical problem
Model
development
Numerical
methods
Mathematical
formulation
Solution
Experiments
on samples
Comparison
(1)
(2)
(4)
(3)
Data from
experiments
Experiments
on component 1
2
OR
44
2. Scanning electron microscope (SEM) equipped with energy dispersive X ray
spectrometer (EDS)
3. Universal tensile testing machine
4. Vicker’s hardness tester
5. Charpy impact tester
6. Servo hydraulic fatigue testing machine
3.3 Material characterization
3.3.1 Chemical composition
The material used in this study was in the form of extruded thick-walled cylinder. The
material was analyzed on SEM equipped with Energy Dispersive X-ray analyzer
(EDS). The material was in T6 temper [157], which included a solution heat treatment
at 530 C, water quench, followed by aging at 175 C for 8 hrs.
3.3.2 Microstructural evaluation
Optical and scanning electron microscopes were used to examine the microstructural
features of the alloy. These included grain size measurement, grain orientation in
different directions, defects, morphology and the quantitative analysis of the constituent
particles. Samples were taken in two directions and prepared for microstructural
evaluation. Sample preparation included cutting of samples from the cylinder, coarse
and fine grinding and polishing and finally etching. The constituent particles were
studied in as-polished (mirror polished) condition while etched samples were used to
study the grain structure. Etching was done in 3 percent HF solution to reveal the grain
size and their orientation. Magnification up to 1000x was available. The microscope
was equipped with an image analysis system. This system was used to determine the
size, morphology and distribution of the constituent particles in the material and the
45
grain size. The microanalysis of the constituent particles was conducted with the help of
scanning electron microscope.
3.3.3 Hardness measurement
Hardness of the material was determined with the help of Vicker’s hardness testing
machine. This test also included sample preparation before measurement. Fine
polished samples were tested at a load of 10 kg for a 30 s time. An average of about
10 readings was obtained and included in the results.
3.4 Monotonic tensile testing
3.4.1 Sample preparation
Samples from the extruded cylinder were taken in LR and CR directions. Round
tensile samples were prepared according to ASTM standard E 08. The samples were
machined with final specimens having a uniform gauge length large enough to
accommodate a 25 mm extensometer. The surfaces that were loaded were carefully
ground using waterproof SiC polishing papers. Finally the surfaces were mechanically
polished using Aluminum oxide powder. The photograph and the dimensions of the
specimens used for tensile testing are shown in Figs. 3.2a and 3.2b.
3.4.2 Testing procedure
Monotonic tensile tests were conducted to obtain stress-strain curves and to gather
monotonic properties of the material. All tests were performed following the method
described in ASTM B 557 M [158]. The tests were conducted at ambient temperature
and in air atmosphere.
All tests were run in ram displacement control and essential data was recorded during
the test. The tests were conducted at cross head speed of 0.05 mm/min. The
46
corresponding strain rate was 3.33 x 10-5
per second. An extensometer of 25 mm gauge
length was attached to the sample in the gauge section. The load and elongation data
during the tests were recorded with the help of computer and subsequently stress-strain
curves were obtained. A statistical linear regression analysis was used to determine the
elastic modulus of the material. The linear correlation coefficient ‘r’ was also
determined which provides the information about the scatter in the data.
(a)
(b)
Fig. 3.2 a) Photograph and b) dimensions of the specimens (in mm)
used for tensile testing
Polished
surfaces
47
3.5 Impact testing
Samples from the extruded cylinder were taken in LR and CR directions for impact
testing. Standard samples were prepared according to the ASTM standard E 23; the
dimensions of the sample are given in the Fig. 3.3. Five samples were tested at each
orientation at room temperature.
Fig. 3.3. Dimensions of the impact sample (in mm)
3.6 Fatigue crack growth testing
3.6.1 Test samples and preparation
The fatigue crack growth rate of the alloy was determined according to the ASTM
standard E 647 [159]. The type of sample used in the study for crack growth rate
determination was middle tension (MT) specimen. Two sets of specimens were
prepared in LR and CR directions, Fig. 3.4 (ASTM E 399) [160]. The samples were
machined with final specimens having a notch along and across the extrusion
direction, Fig.3.5. The notch was in a direction such that the crack propagation from
the notch was in the radial direction for both the specimens. Under uniaxial tensile
loading, the stresses on the samples LR and CR were corresponding to the axial and
55 10
10
2
R 0.25
48
hoop stresses in the cylinder, respectively. The dimensions of the samples in the two
orientations used are shown in Figs. 3.6 and 3.7.
The surfaces of the samples were carefully ground using waterproof SiC polishing
papers. Finally the surfaces were mechanically polished using 0.05 m alumina
powder aqueous suspension. The photograph of the polished samples is shown in Fig.
3.8.
Fig. 3.4 Crack plane orientation code for bar and hollow cylinder (ASTM E 399)
C
R
L
RL
RC
CL
LC
LR
CR
49
Fig.3.5. Sample orientations in the cylinder
Fig. 3.6 Dimensional details of the M(T) sample (dimensions in mm) – CR direction
C
L
R
CR
LR
15
2
60
45
Ø7
A Center notch 6 mm
50
Fig. 3.7 Dimensional details of the M(T) sample (dimensions in mm) – LR direction
Fig. 3.8 Photograph of the two LR test samples.
80
60
Ø7
A
20
2.5
6Detail A
49
°±0°
1
R1
Root radius=0.151
.3
51
3.6.2 Test conditions and parameters
Fatigue tests were performed on a servo-hydraulic testing machine. Fig. 3.9 shows the
setup of fatigue testing with MT sample and traveling microscope for crack
measurement. Tests were conducted in tension-tension mode under constant
amplitude loading with R ratio equal to 0.1. A sinusoidal waveform was applied at a
loading frequency of 10 Hz. Tests were conducted at maximum stress of 143, 125,
106, 89, 71, 54 and 35 MPa. During fatigue testing, the tests were interrupted for short
times to measure the crack length. Three different techniques were used to measure
the crack extension:
1. Crack length was measured by light microscopy of the polished specimen surfaces
with the help of traveling microscope at a magnification of 50 x. The resolution of
the measuring system was 0.01 mm. This technique was frequently used for
measurement and the data is presented in Chapter 5.
2. A digital camera was used to capture the photographs of the specimen with the
cracks in the notch region. Subsequently, the crack sizes were measured using a
software package. This provided an enhanced accuracy of 0.005 mm.
3. Replicas for some of the samples were taken during the fatigue testing and were
used for subsequent crack growth measurement. These replicas were observed
under optical microscope at magnification of 200 x. This technique was used to
confirm the crack length measured by the light microscopy measuring system.
During testing, the number of cycles and crack extension data were recorded until
fracture. Fracture caused a substantial load drop that resulted in hydraulic shutdown,
terminating the test. Seven samples were tested for each orientation and number of
cycles to failure (Nf) were achieved. Number of cycles to failure was taken as the
fatigue life of the samples. Testing was conducted in air at room temperature (~20° C)
52
and approximately 50 percent relative humidity. The calculations of the stress
intensity factor range ( K) and crack growth rate (da/dN) were performed according
to ASTM E-647 standard.
Fig. 3.9 a) Fatigue testing machine with the sample gripped in the jaws (encircled)
and the traveling microscope
3.7 Fractography in SEM
Following the testing, the selected fracture surfaces of the monotonic and fatigue
specimens were inspected visually and then with a stereomicroscope. Further,
macroscopic and microscopic examination was done using SEM. The fracture surfaces
of the representative specimens were separated from the main body of the specimen
using a cutting tool, reducing the height of the samples to fit into the vacuum chamber
of the SEM. The reduced height samples were rinsed with acetone and attached to the
53
SEM mounting slides with colloidal silver liquid, which permanently attached the
sample to the slide. The observations were conducted with the electron beam in
alignment with the applied axial force. The images of the important features of the
fracture surfaces were recorded.
The sections of the fatigue fractured samples were also inspected via optical and SEM
microscopes to identify the specific fracture modes. Where appropriate, the fatigue
crack propagation modes were recorded using back scattered electron imaging. The
selected samples from each type of samples and with different loading conditions were
used for fractographic analysis. These fractographic specimens were chosen to achieve
a realistic representation of the fracture modes in different test conditions and sample
orientation.
54
CHAPTER 4
Material and Structural
Characterization
This chapter provides the results of the metallurgical and mechanical characterization of
the thick-walled cylinder (TWC). The findings from the procedures employed (EDX
analysis, light microscopy and scanning electron microscopy) to better understand the
material and the detailed manufacturing characteristics of the extruded TWC, as well as,
their monotonic properties are presented. Post-fracture analysis provides the micro-
mechanism of the fracture process under monotonic loading.
4.1 Material characterization
4.1.1 Chemical composition
The chemical composition of the alloy used in this study is given in Table 4.1. The alloy
conforms to the specifications of the aluminum alloy AA 6061 [161]. The EDS spectrum
of the material is shown in Fig. 4.1. The major constituents of the Alloy AA 6061 are Al,
Mg and Si.
Table 4.1 Nominal chemical composition of the material, wt. %
Material Mg Si Fe Cu Cr Mn Al
Al Cylinder 0.94 0.64 0.20 0.19 0.10 0.06 Bal.
55
Fig. 4.1. EDS spectrum of the alloy
4.1.2 Microstructural evaluation
Figs. 4.2a and 4.2b shows the SEM micrograph of the sample revealing constituent
particles in the longitudinal (LR) and transverse (CR) directions, respectively.
Examination of the material after etching suggested that the constituent particles were
present preferably along the grain boundaries. The particles can be seen aligned along the
extrusion direction. Inspection at higher magnifications revealed that clusters of particles
were present at these locations; see Fig. 4.3. It seems that these particles were formed
during solidification and fractured in fragments and aligned along the axis of the cylinder
during extrusion. EDX analysis showed that these particles were rich in Al, Fe, Si, Cr and
Mn. These constituent particles are formed during the cooling of the liquid metal when
some of the phases rich in alloying elements solidify earlier than aluminum solid
solution. The area fraction of these inherent particles in two orientations is given in Table
4.2.
56
(a)
(b)
Fig. 4.2. SEM micrographs of the alloy showing the constituent particles
a) longitudinal direction b) transverse direction
constituent
particles
LR
constituent
particles
57
Fig. 4.3. SEM micrograph of the alloy showing clusters of particles in longitudinal
direction
Figs. 4.4 and 4.5 show the microstructure of the alloy in two orientations. These
micrographs represent the plane in which crack propagation had taken place during
fatigue testing. A marked difference of the grain structure in the two orientations can be
seen at low magnification; compare figures 4.4a and 4.5a. In LR direction the elongated
grain boundaries are present which indicate that the extrusion ratio was not sufficient to
remove the original grain structure. Higher magnification images, Figs. 4.4b and 4.5b,
reveal recrystallized grains in both the orientations. It seems that the grains were
elongated during extrusion process and recrystallized into fine almost equiaxed grains
during subsequent heat treatment. The grain size of the alloy in longitudinal and
transverse directions is given in the Table 4.2. The results show that the recrystallized
grains are almost equiaxed. The morphology and the alignment of the constituent
particles were not affected by the heat treatment process.
10 m
58
(a)
(b)
Fig. 4.4 Optical micrograph revealing the microstructure of the alloy
in LR sample a) low magnification b) high magnification
59
(a)
(b)
Fig. 4.5. Optical micrograph revealing the microstructure of the alloy
in CR sample a) low magnification b) high magnification
60
Table 4.2 Quantitative microstructural analysis
Microstructural feature Orientation
Longitudinal Transverse
Grain size, m 13 5 15 4
Particles area fraction, % 6.5 1.0 5.1 1.2
4.2 Mechanical testing
4.2.1 Hardness test
The hardness of the material was measured on the samples prepared in two directions.
The results of the test are given in Table 4.3. The hardness values do not show any
difference in the two orientations indicating that the effect of extrusion on this property,
if any, has been removed after heat treatment.
4.2.2 Monotonic tensile test
The stress-strain diagrams obtained in tensile testing in two orientations are presented in
Fig. 4.6. These graphs indicate that the behavior of the material, under tensile loading,
remains essentially same up to the yield strength. This shows that the response of grain
structure and the constituent particles is similar in both the samples up to this point.
However, after this point, difference of the two curves is visible. A smaller strain value
up to the point of failure in transverse direction indicates that the elongated grain
boundaries and the constituent particles in this orientation restricted the material to
deform much before fracture.
The results obtained from the tensile tests of the extruded alloy in two orientations are
given in Table 4.3 and references [162, 163]. The results show that the yield strength and
61
the ultimate tensile strength of the material were not different in the two orientations.
However, the difference in the deformation behavior was significant. In LR direction the
material shows higher toughness represented by the area under the stress-strain curve.
The results of the tensile testing indicate that the recrystallized grain structure has
eliminated the effect of mechanical working on the yield and tensile strengths of the
material. However, the elongated grain boundaries and the clusters of the constituent
particles are responsible for the reduction in elongation and reduction of area in the
transverse direction.
0
100
200
300
400
0 0.04 0.08 0.12 0.16
Strain
Str
ess
, M
Pa
LR
CR
Fig. 4.6 Stress-strain diagram of the material in two orientations
4.2.3 Impact testing
The results obtained from the impact test of the extruded alloy in two orientations are
given in Table 4.3. The results show that the impact strength in the LR direction was four
62
times the strength in the CR direction. The lower toughness of the material in CR
orientation was in agreement with the results obtained from the tensile testing.
Table 4.3 Nominal mechanical properties of the extruded alloy
Properties Orientation
Longitudinal Transverse
0.2 % Yield strength, MPa 324 322
Ultimate tensile strength, MPa 354 353
Elongation (25 mm GL), % 16 9
Reduction in area, % 50 24
Elastic modulus, GPa 71 71
Hardness, HV 115 115
Impact strength, Joules 44 11
4.3 Post-fracture analysis
4.3.1 Tensile samples
Fig. 4.7 shows SEM fractographs of the tensile samples in two orientations. In LR
sample, necking is visible indicating higher ductility in this orientation. The fracture is at
an angle of about 45. The micrographs in both the orientations showed typical dimple
fracture indicating deformation before failure. Clusters of particles, as were observed in
the metallographic sections, were present in the dimples; Fig. 4.8. The presence of
constituent particles in the dimples showed that these were responsible for initiation of
failure.
63
4.3.2 Impact samples
Figs. 4.9 and 4.10 show the fractographs of the impact samples tested in LR and CR
orientations, respectively. A marked difference in the fracture morphology of the two
orientations can be seen at low magnification images; Figs. 4.9a and 4.10a. In LR
orientation an irregular fracture surface was present indicating a difficult path for crack to
grow. The higher impact strength obtained in this orientation is in agreement with this
observation. In CR orientation the fracture path was relatively smooth. The fracture was
most probably along the grain boundaries and aligned constituent particles. The lower
impact strength in this orientation was due to the fact that CR orientation offered easy
path for fracture. The high magnification images are shown in Figs. 4.9b and 4.10b. Fig.
4.9b shows fine shallow dimples, in LR orientation, which were likely to be created by
the separation of the grain boundaries. High density of particles can be seen in the CR
orientation; Fig. 4.10b. The fracture was most probably along the grain boundaries in
both the orientations.
64
(a)
(b)
Fig. 4.7 SEM micrographs of the fractured tensile samples
a) LR b) CR orientation
65
(a)
(b)
Fig. 4.8 SEM micrographs of the fractured tensile samples at high magnification
a) LR b) CR orientation
particles
66
(a)
(b)
Fig. 4.9 SEM micrographs of the fractured impact samples - LR orientation
a) low magnification b) high magnification
67
(a)
(b)
Fig. 4.10 SEM micrographs of the fractured impact sample - CR orientation
a) low magnification b) high magnification
68
CHAPTER 5
Experimental Fatigue
Crack Growth Study
This chapter provides the results of the crack growth experiments on the samples from
the thick-walled cylinder (TWC). Fatigue crack growth tests on samples in two
orientations, LR and CR, under uniaxial fatigue conditions at loading ratio (R) of 0.1
were conducted. The fatigue life of the specimens and the crack growth rate data of the
material at different stress levels are given. The fractographs showing the topographical
variations with parametric change are also presented.
5.1 Fatigue crack growth test – LR samples
5.1.1 Crack extension
The crack propagation was recorded against the increasing number of cycles at different
stress levels and the results are presented in Figs. 5.1a to 5.1f. The stresses are given in
percent of yield strength of the material. The data covers the range from start of the crack
at the notch tip up to the specimen failure. A pre-crack length of 1 mm was maintained as
per requirement of the standard. Fig. 5.2 shows all the data on a single plot, using the log
scale for the number of cycles so as to provide a better comparison of crack growth at
various stress levels. The number of cycles to initiate the crack at the notch and the
maximum value of the crack length increase with the decrease in the stress level.
69
2
6
10
0 1500 3000 4500 6000
N, cycles
a, m
m
EXP-1
EXP-2
(a)
2
6
10
0 1750 3500 5250 7000
N, cycles
a, m
m
EXP-1
EXP-2
(b)
Fig. 5.1 The crack length versus the number of cycles (LR orientation),
at stress levels in % of yield strength ( y); a) 40 b) 35 c) 30 d) 25 e) 20 f) 15
The legend shows the experiments on two different samples (EXP-1, EXP-2)
y%35
y%40
70
2
6
10
0 3500 7000 10500 14000
N, cycles
a, m
m
EXP-1
EXP-2
(c)
2
6
10
0 5000 10000 15000 20000
N, cycles
a, m
m
EXP-1
EXP-2
(d)
Fig. 5.1 Contd.
y%25
y%30
71
2
6
10
0 11500 23000 34500 46000
N, cycles
a, m
m
EXP-1
EXP-2
(e)
2
6
10
0 42500 85000 127500 170000
N, cycles
a, m
m
EXP
(f)
Fig. 5.1 Contd.
y%15
y%20
72
Fig. 5.2 Plot of the crack length versus the number of cycles for LR orientation. The
legend shows the stress level in percent of yield strength (YS)
5.1.2 Fatigue crack growth curve
The variation in fatigue crack growth rates with K for the samples in LR direction is
shown in Fig. 5.3. The data has been obtained at different stress levels and plotted on log-
log scale. The best fit curves have the typical sigmoidal shape. The experimental value of
the Paris constants was obtained from the plot, as discussed in section 1.1.2, and the
results are given in the following section.
0.002
0.004
0.006
0.008
1.00E+03 1.00E+04 1.00E+05 1.00E+06
Log N, cycles
a, m
15
20
20
25
25
30
30
35
35
40
40initial notch
1 mm
%YS
N, cycles
a, m
73
Fig. 5.3 The crack growth rate variation with K – LR orientation
5.1.3 Fatigue life analysis
Table 5.1 summarizes the results of the stress range ( S) and the number of cycles to
failure (Nf) for the samples tested in LR orientation. A statistical regression analysis of
the experimental data was conducted using the Microsoft Excel software which provides
the different regression types. The plot of the SN data was obtained and the trendline
added to the curve. These trendlines are used to graphically display the data trends and
are a function of the type of data used. When the data is fitted to a trendline, Excel
calculates its R-squared value. The R-squared (R2) value is an indicator with a value from
0 to 1 revealing how close the estimated values for the trendline correspond to the actual
1.00E-09
1.00E-08
1.00E-07
1.00E-06
1.00E-05
1 10 100
K, MPa
da/
dN
, m
/cy
cle
m/c
ycl
e
m
74
data. A trendline is most reliable when its R2
value is at or near 1. This value is also
known as the coefficient of determination.
The analysis of the data given in Table 5.1 and graphically presented in Fig. 5.4 shows
that the experimental data is best fitted with the power law relationship as shown in the
figure. It provides a coefficient of determination, R2 equal to 0.9857. The SN data shows
that the fatigue lifetime increases as the stress amplitude decreases.
Table 5.1 Fatigue life of the samples in LR orientation
Sample # S, MPa NfLR,
cycles
Average
NfLR, cycles
% scatter*
1. 129
8264 7667 11
2. 7070
3. 112
9986 9635 5
4. 9283
5. 97
15507 14036 15
6. 12565
7. 82
26550 23865 16
8. 21179
9. 63
63672 64609 2
10. 65546
11. 47 264476 - -
12. 31 600000 + - -
* (Standard deviation / Average Nf) x 100
+ run out (sample did not fracture)
75
Fig. 5.4 Stress range versus the number of cycles to failure – LR orientation
5.2 Fatigue crack growth test – CR samples
5.2.1 Crack extension
Fig. 5.5 shows the crack length versus the number of cycles at different stress levels in
CR orientation. The data covers the range from the start of the crack at the notch tip up to
the specimen failure. A pre-crack length of 1mm was maintained as per requirement of
the standard. Fig. 5.6 shows the entire data on a single plot, with the number of cycles in
the log scale, to provide better comparison of crack growth at different stress levels. The
number of cycles to initiate the crack at the notch and the maximum value of the crack
length increase with the decrease in the stress level, in analogy to the case of LR
orientation.
76
2
5
8
0 650 1300 1950 2600
N, cycles
a, m
m
EXP-1
EXP-2
(a)
2
5
8
0 900 1800 2700 3600
N, cycles
a, m
m
EXP-1
EXP-2
(b)
Fig. 5.5 The crack length versus the number of cycles (CR orientation),
at stress levels in % of yield strength; a) 40 b) 35 c) 30 d) 25 e) 20 f) 15 g) 10
The legend shows the experiments on two different samples (EXP-1, EXP-2)
y%40
y%35
77
2
5
8
0 1350 2700 4050 5400
N, cycles
a, m
m
EXP-1
EXP-2
(c)
2
5
8
0 3000 6000 9000 12000
N, cycles
a, m
m
EXP-1
EXP-2
(d)
Fig. 5.5 Contd.
y%25
y%30
78
2
5
8
0 5500 11000 16500 22000
N, cycles
a, m
m
EXP-1
EXP-2
(e)
2
5
8
0 15500 31000 46500 62000
N, cycles
a, m
m
(f)
Fig. 5.5 Contd.
y%20
y%15
79
2
5
8
0 62500 125000 187500 250000
N, cycles
a, m
m
(g)
Fig. 5.5 Contd.
Fig. 5.6 Plot of the crack length versus the number of cycles for CR orientation. The
legend shows the stress level in percent of yield strength (YS)
0.002
0.004
0.006
0.008
1.00E+03 1.00E+04 1.00E+05 1.00E+06
Log N, cycles
a, m
10
15
20
20
25
25
30
30
35
35
40
40initial notch
1 mm
%YS
N, cycles
a, m
y%10
80
5.2.2 Fatigue crack growth curve
The variation in fatigue crack growth rates with K for the samples in CR directions is
shown in Fig. 5.7. The data has been obtained at different stress levels and plotted on log-
log scale. The experimental value of the Paris constants was obtained from the plot and
the results are given in Table 5.2.
Table 5.2 Paris constants obtained from the experimental data
Alloy Form Orientation C, m/cycle m
AA 6061-T6 Extrusion
LR 2 x 10-10
2.6
CR 4 x 10-11
3.4
Fig. 5.7 The crack growth rate variation with K – CR orientation
1.00E-09
1.00E-08
1.00E-07
1.00E-06
1.00E-05
1 10 100
K, MPa. .sqrt (m)
da/
dN
, m
/cycl
e
m/c
ycl
e
m
81
5.2.3 Fatigue life analysis
Table 5.3 summarizes the results of the stress range ( S) and the number of cycles to
failure (Nf) for the samples tested in CR orientation. Calculation of scatter in the data is
also given in the table. Fig. 5.8 shows the same results in graphical form. Statistical
regression analysis shows that the experimental data is best fitted with the power law
relationship as shown in the figure, and provides coefficient of determination, R2 equal to
0.9899 for this orientation. As expected, the SN data shows that the fatigue lifetime
increases as the stress amplitude decreases.
Table 5.3 Fatigue life of the samples in CR orientation
Sample # S, MPa NfCR, cycles
Average
NfCR, cycles
%
scatter
1. 129
4037 3505 21
2. 2973
3. 112
5078 5054 1
4. 5029
5. 97
8376 8305 1
6. 8233
7. 82
12195 13988 18
8. 15780
9. 63
28653 29343 3
10. 30032
11. 47 99521 - -
12. 31 480179 - -
82
Fig. 5.8 Stress range versus the number of cycles to failure – CR orientation
5.3 Comparison of the fatigue behavior – LR vs CR
The comparison of the curves in Figs. 5.2 and 5.6 for the two orientations, at a given
stress level, reveals that the crack initiation was earlier in CR orientation. The start of the
crack at comparably low number of cycles indicates that the fatigue resistance of the
material in CR orientation is inferior as compared to the LR orientation.
The crack growth rate variation with K in two orientations is shown in Fig. 5.9. The plot
shows a lower Kth value for CR orientation and also the value of K at which final
fracture occurs. In the intermediate region the crack growth rate is almost identical. This
behavior, within the same material, clearly indicates the impact of the processing
technique on fatigue properties of the extruded cylinder. It may be noted that the tensile
strength of the material in two orientations was nearly the same. The inferior fatigue
resistance in the CR direction may be attributed to the presence of the elongated grain
83
boundaries and the unfavorably oriented clusters of constituent particles. The clusters of
aligned particles in CR orientation were present in the direction perpendicular to the
loading axis and more favorably oriented for crack initiation and growth.
Fig. 5.9 The crack growth rate variation with K in two orientations
Comparison of the fatigue lives of the samples in two orientations is given in Table 5.4. It
should be noted that the difference in the fatigue life in two orientations is beyond the
limit of the scatter at any stress level. It is worth mentioning here that although the
sample size in the two orientations was a bit different, however, the loading conditions
during testing were ensured to be the same for both the orientations. As already
mentioned that for a given material and environment, the fatigue crack growth behavior is
essentially the same for different specimens because the stress intensity factor range is
the principal controlling factor in fatigue crack growth [2]. Thus the da/dN versus K
m
84
data obtained from the samples, under constant amplitude conditions, can be compared
and used for engineering design.
The data clearly shows a shorter fatigue life of the samples prepared in CR orientation.
The percent decrease in the number of cycles to failure in CR orientation, compared to
LR orientation, at different stress levels is also shown in the table 5.4. As already
discussed, the shorter fatigue life in the CR orientation can be due to the presence of the
elongated grain boundaries and the aligned constituent particles. Clusters of aligned
particles in the material were found in the CR orientation; these were present in the
direction perpendicular to the loading axis. This situation is more favorable for crack
initiation and growth. The shorter fatigue life in CR orientation is of serious concern
because in this orientation the hoop stress in the cylinder has about twice the value of the
axial stress [120].
Table 5.4 Comparison of the fatigue lives of the samples in two orientations
# S, MPa
Number of cycles to failure, Nf
NfLR NfCR NfLR - NfCR %
decrease
1. 129 7667 3505 4162 54
2. 112 9635 5054 4581 48
3. 97 14036 8305 5731 41
4. 82 23865 13988 9877 41
5. 63 64609 29343 35266 55
6. 47 264476 99521 164955 62
7. 31 600000 + 480179 119821 + >.20
85
5.4 Post-fracture analysis
5.4.1 LR samples
Fractography was conducted on representative samples tested at different stress levels;
the typical features of the LR sample tested at a stress range of 129 MPa with 8264 cycles
to failure is shown in Fig 5.10a. The fractograph of the sample tested at a stress range of
63 MPa and a fatigue life of 63672 cycles is shown in 5.10b. The starter notch is visible
and marked in both the cases. At higher stress level, the crack traveled to a shorter length
and finally failed due to overstress; as seen in Fig. 5.10a. At lower stress level the crack
growth region is markedly longer and the overload region is not within the field of vision
in Fig. 5.10b which is taken at the same magnification. The facets created during the
propagation of fatigue crack were observed at higher magnification; Figs. 5.11a and
5.11b. The crack growth was along the crystallographic planes. Furthermore, the
striations on the fatigue facets are also visible as shown in Fig. 5.12, which indicated the
mechanism of the crack growth. In the region of the final rapid fracture, the presence of
dimples indicated ductile failure; Fig. 5.13.
A noticeable observation in the fatigue samples was the formation of the secondary
cracks in the overload failure region. The polished side surface of the overload failure
region of the sample tested at 40 % yield strength is shown in Fig. 5.14. Cracks
frequently propagated along the path where aligned constituent particles were present; in
other words, the aligned constituent particles created weak planes along the LR
orientation.
86
(a)
(b)
Fig. 5.10 SEM micrographs of the LR samples tested at S
a) 129 MPa b) 63 MPa
Fatigue fracture
Notc
h
Fatigue fracture
Notc
h
87
(a)
(b)
Fig. 5.11 SEM micrographs of fatigue fractured surface of LR orientation
- S equals a) 129 MPa b) 63 MPa (arrow indicates crack growth direction)
88
Fig. 5.12 SEM micrograph of fatigue fractured surface of LR orientation showing
striations - S equals 129 MPa (arrow indicates crack growth direction)
Fig. 5.13 SEM micrograph of the overload fracture surface - LR orientation,
- S equals 129 MPa
89
(a)
(b)
Fig. 5.14 Polished side surface of the overload region; SEM micrographs
a) segment of the crack path b) region A at high magnification
A
Crack path
Secondary
cracks
90
5.4.2 CR samples
The SEM micrographs of the fractured surfaces of the fatigue samples tested at stress
range of 129 MPa and 63 MPa in CR orientation are shown in Figs. 5.15a and 5.15b,
respectively. The corresponding numbers of cycles to failure for these samples were 4037
and 28653, respectively. The starter notch in both the samples is visible at the right hand
side. At higher stress level, the crack traveled to a shorter length and the final overload
fracture can be seen in Fig. 5.15a on the left of the micrograph. At a lower stress level,
the crack propagated to a longer distance before final overload failure; this can be seen in
the Fig. 5.15b.
The higher magnification SEM micrographs of the fatigue crack growth region of the
samples presented in Fig. 5.15 are shown in Fig. 5.16. The aligned morphology of the
fractured surface perpendicular to the crack propagation direction is apparent. In both the
samples, fatigue facets created during crack propagation, were observed. In this case,
however, the mode of failure locally changed at regions where the crack interacted with
the clusters of constituent particles; as can be seen in Fig. 5.16c. Similar to the LR
samples, striations on the fatigue facets were also observed. The SEM micrographs of the
striations at intermediate and high K values are shown in Figs. 5.17a and 5.17b,
respectively. A marked difference in the fatigue crack growth rate can be seen from these
micrographs.
91
(a)
(b)
Fig. 5.15 SEM micrographs of the CR samples tested at S
a) 129 MPa b) 63 MPa
Notc
h
Fatigue fracture
overload
fracture
Fatigue fracture
Notc
h overload
fracture
92
(a)
(b)
Fig. 5.16 SEM micrographs of fatigue fractured surface of CR orientation (arrow
indicates crack growth direction) - S equals a) 129 MPa b) 63 MPa c) higher
magnification of (b) showing cluster of particles
93
(c)
Fig. 5.16 Contd.
particle cluster
94
(a)
(b)
Fig. 5.17 SEM micrographs of the fatigue fractured surface of CR sample, S equals 129
MPa a) intermediate and b) high K values (arrow indicates crack growth direction)
95
CHAPTER 6
Numerical Simulation of
Fatigue Crack Propagation
This chapter presents the numerical simulation of the fatigue crack propagation. The
experimental sample geometries were modeled and fatigue crack growth was simulated
using FEM. The structural software ANSYS was used for the simulation and a code in
ANSYS Parametric Design Language (APDL) was written to replicate the crack growth
process. The data obtained from the experiments was used as the input parameters for
simulation work. Both the orientations representing axial and hoop stress in the cylinder
were analyzed. The results of the finite element analysis (FEA) are presented and
compared with the experimental results given in the previous chapter.
6.1 Simulation of fatigue crack propagation
Fatigue crack growth analysis was performed using commercially available ANSYS 9.0
finite-element software [164] by repeatedly loading the geometry, recording the stress
intensity factor KI at the crack tip, advancing the crack by node release method and then
unloading. Two dimensional finite element analysis of the M(T) sample geometry was
conducted using 4-noded quadrilateral elements under plane-strain conditions.
6.1.1 Model geometry
The symmetry in loading and geometry of the M(T) sample was taken advantage of and a
solid model for a quarter section of the sample was created in the ANSYS pre-processor.
96
Fig. 6.1 shows the geometry of the quarter model. The quarter model has an initial crack
length ‘a’ of 3 mm and a/W = 0.3, where W is the sample half width.
Fig. 6.1 Geometry of the quarter model (QM)
6.1.2 Material properties
During FEA an isotropic material model for LEFM was employed. The modulus of
elasticity E = 71 GPa and Poisson’s ratio = 0.33 were used [147].
6.1.3 Element selection and meshing
The M(T) sample was modeled in the preprocessor using a two dimensional, linear, 4-
noded, quadrilateral PLANE42 element. The element geometry is shown in Fig. 6.2. The
element possesses two degrees of freedom at each node, i.e., translation in the nodal x
and y directions and does not have rotational degrees of freedom. It also has plasticity,
creep, swelling, stress stiffening, large deflection, and large strain capabilities [165].
Other researchers have also conducted fatigue crack growth analysis by two-dimensional
finite element analyses of CT and MT geometries. They used the four-noded quadrilateral
elements and three-noded triangular elements for the simulation [166 - 169]. In order to
QM
a
2W
97
predict the fatigue crack growth from the crack tip, the crack advancing region was
mapped meshed [169]. The spacing between the consecutive nodes allowed the crack to
advance in steps of equal size by node-release technique [170]. McClung and Sehitoglu
have investigated the plasticity-induced fatigue crack closure by the FEM. They followed
the node-release scheme and the crack tip was extended one element length per cycle.
The mesh was optimized by correlation with selected experiments prior to detailed
analysis. A mesh having higher degree of refinement and smaller element size required a
greater number of load cycles to generate a prescribed amount of crack growth. The
meshed model is shown in Fig. 6.3.
Fig. 6.2 Two dimensional, four-node, PLANE42 element
X (Radial)
Y (Axial)
V) I
J
Element Coordinate
System
1
2
3
4
K L
98
6.1.4 Boundary conditions and solution
The boundary conditions applied to the M(T) sample are shown in the Fig. 6.3. The half
width of model was constrained applying symmetry boundary conditions along the left
and the bottom edges. A 3 mm long crack was modeled by applying no constraints from
0 to 3 mm along the x direction at the bottom edge [24, 170], thus providing the crack tip
node at 3 mm. The model was loaded by applying tractions at the upper edge in the y
direction, simulating mode I loading. After applying the boundary conditions and getting
the solution, the value of KImax at the crack tip was obtained, thereby K was calculated.
This value of K was used along with the experimental data to obtain the crack growth
rate, using the Paris equation [24, 156]. The crack size was increased by releasing the
crack tip node, which was equal to the distance between the two consecutive nodes along
the line of crack advancement. The number of cycles to move to the next node (one step)
was calculated using crack growth rate, and the process was repeated. The Paris law
equation used for the calculation of crack growth rate was;
da/dN = C ( K) m
During crack propagation, K value was monitored and the process stopped when the
parameter attained the fracture toughness of the material. In order to validate the FEA
results, analysis was conducted by simulating loads similar to those applied during the
experimental tests. A flow diagram showing algorithm used for the simulation is
presented in Annexure II.
99
(a) (b)
Fig. 6.3 a) Quarter model of the M(T) sample - element plot with applied boundary
conditions (BCs) b) enlarged crack tip region showing mapped meshing
6.2 Results and discussion
In the next sections, the results of the FEA of LR and CR samples are presented. These
include the study of crack growth with number of cycles, determination of the crack
growth rate with changing K and the fatigue life analysis of the samples. Prior to
detailed FEA, the mesh was optimized by correlation with selected experiments.
6.2.1 LR samples
6.2.1.1 Element size optimization
Given the severe computational burden associated with finite element simulation of
fatigue crack growth, it is important that finite element meshes not be refined
excessively. However, if the mesh used is too coarse, inaccurate results may be obtained
[169]. The optimization of the element size was conducted by selecting its values ranging
Crack tip
Symmetry BCs
Mode I Loading
Crack
100
from 0.025 to 0.5 mm, along the line of crack propagation and the FEA solution was
obtained. The plot of crack size versus number of cycles for different element sizes in
Fig. 6.4 shows that by reducing the element size from 0.5 to 0.05 mm, the results
converge. However, a further reduction in the element size down to 0.025 mm did not
show any significant difference in the results. Thus the data achieved during optimization
showed that an element size equal to 0.05 mm yields optimum results. The conclusion
was in agreement with the earlier studies [16,169 -173], according to which, an element
size in the neighborhood of 0.05 mm yielded satisfactory stable crack growth predictions
under constant amplitude loading. Based on these findings, an element size equal to 0.05
mm was selected along the line of crack propagation for further studies.
6.2.1.2 Crack growth
Using optimized element size of 0.05 mm, a detailed FEA study was performed [174].
Fig. 6.5a shows von Mises stress distribution at a stress level of 40 % and 1 mm crack
growth. This figure provides stress solution of the crack growth model at one of the FEA
conditions, as evidence that the stress distribution in the model is in agreement with the
expected solution and the maximum stress is in the crack tip region. This can be clearly
seen in Fig. 6.5b where the crack tip region is enlarged to show that the maximum stress
is at the crack tip node. It’s worth mentioning here that the calculations of the stress
intensity factor were based on LEFM and crack tip plasticity was not considered in the
analysis of the fatigue crack growth.
101
2
5
8
0 2500 5000 7500 10000
N, cycles
a, m
m
FEA-E0.5
FEA-E0.2
FEA-E0.1
FEA-E0.05
FEA-E0.025
Fig. 6.4 Element size optimization – LR orientation. Legend indicates the finite element
analysis (FEA) using elemnt size (E) of 0.5 to 0.025 mm.
(a) (b)
Fig. 6.5 a) LR sample - von Mises stress distribution (MPa) at the stress level of 40 %
and the crack length 4 mm b) enlarged crack tip region
Crack tip node
102
The output file generated during simulation process is given in Annexure III. The plots
for the crack length versus the number of cycles in Fig. 6.6 show the data obtained from
the two techniques at different stress levels. The data covers the entire range from start of
the crack at the notch up to the specimen failure. The plots provide the data obtained
from the two techniques in the crack growth region and do not include the crack initiation
cycles.
Comparison of the experimental and the FE results shows a close agreement at stress
levels ranging from 40 to 25 % of the yield strength of the material. At stress levels of 20
and 15 % of the yield stress, conservative results were obtained using FEA and a
significant deviation was noticed at a stress level of 15 %. This behavior can be explained
in terms of K. At lower stress levels, K approaches Kth and enters region I of the
fatigue crack growth curve as shown in Fig. 1.2 (Chapter 1). In this region, the
microstructural features have a greater influence on the fatigue crack growth rate [175],
as compared to region II. There are several factors that influence the near-threshold
fatigue crack growth rate, like grain size [176, 177], dispersoids [178], precipitates [179,
180], and texture [181]. The crack growth rate equation is valid only in the stable crack
growth region i.e. the Paris region. The results presented in this study are, thus,
considered to be optimum for stress levels producing K values within the Paris regime.
At lower stress levels the model provides more conservative results.
103
2
6
10
0 1500 3000 4500 6000
N, cycles
a,
mm
EXP-1
EXP-2
FEA
(a)
2
6
10
0 1750 3500 5250 7000
N, cycles
a,
mm
EXP-1
EXP-2
FEA
(b)
Fig. 6.6 Experimental and FEA results of the crack length vs the number of cycles in the
crack growth region at stress levels in % of the yield strength ( y) a) 40 b) 35 c) 30 d)
25 e) 20 f) 15. Legend shows the experimental data (EXP-1 and EXP-2) and the finite
element results (FEA)
y%35
y%40
104
2
6
10
0 3500 7000 10500 14000
N, cycles
a, m
m
EXP-1
EXP-2
FEA
(c)
2
6
10
0 5000 10000 15000 20000
N, cycles
a, m
m
EXP-1
EXP-2
FEA
(d)
Fig. 6.6 contd.
y%25
y%30
105
2
6
10
0 11500 23000 34500 46000
N, cycles
a, m
m
EXP-1
EXP-2
FEA
(e)
2
6
10
0 42500 85000 127500 170000
N, cycles
a, m
m
EXP
FEA
(f)
Fig. 6.6 contd.
y%15
y%20
106
6.2.1.3 Predicted FCG rate – Experiment vs FEA
The variation of fatigue crack growth rate with K obtained experimentally and from the
FEA at different stress levels is shown in Fig. 6.7. The experimental data on log-log scale
indicates a typical sigmoidal shape. The experimental values of the Paris constants were
determined by curve fitting through Excel. The smooth crack growth rate achieved using
the FE analysis is based on calculation using Paris equation, after getting the stress
intensity factor during analysis. It was observed in crack growth analysis that the value of
the Paris constant ‘m’ obtained experimentally gave higher deviations when used in FEA;
a minor adjustment in this constant was, thus, carried out to optimize the modeling
results. The values of the Paris constants obtained from the experimental data and that
used for the numerical analysis are given in Table 6.1. The fatigue crack growth rate
determined by the FEA using the adjusted ‘m’ value was found within the upper and
lower bounds of the crack growth rate achieved from the experimental data. The start of
the Paris region can be approximated by comparing the experimental data in Fig. 6.7 and
the crack growth rate curve. Further, the FEA results which satisfy the Paris relation i.e.
the minimum K value at the stress level of 25 % of the yield strength is also indicative
of the onset of the Paris region. A line is marked on the plot to indicate the value of K
that approximates the start of the Paris region. This value of K is almost 9 MPa. m .
Thus, the model provides the results that are considered to be optimum for stress levels
producing K values equal to or above 9 MPa. m .
107
Fig. 6.7 LR sample - Comparison of the predicted crack growth rate with the
experimental observation. Legend shows the experimental data (EXP) and the finite
element analysis (FEA) results at 15 to 40 % of the yield strength.
Table 6.1 The Paris constants – LR sample
Technique
Paris constants
C m
Experimental 2 E-10 2.60
FE analysis 2 E-10 2.70
6.2.1.4 Fatigue life analysis
The experimental and the FE results of stress range versus number of cycles to failure are
shown in the plots in Fig. 6.8. S vs Nf relation for the two data sets was obtained
through curve fitting and are also given in the figure. The number of cycles to failure
m
108
includes the number of cycles to initiate the crack and its growth up to the specimen
failure. The number of cycles to crack initiation was incorporated in the FE results, from
the experimental data. The fatigue life analysis results obtained from both the techniques
show that the fatigue lifetime increases as the stress range decreases. As may be seen, the
results obtained from the two techniques are in good agreement up to a S value of 79
MPa. At lower S values, Nf obtained from FEA was smaller than that determined
experimentally. The experimental results were 13 and 36 % higher than the FEA results
at stress ranges of 63 and 49 MPa, respectively. The disparity can be explained in terms
of the crack growth rates near threshold stress intensity factor range, as discussed in
section 6.2.1.2.
Fig. 6.8 Fatigue life analysis (LR sample) - Experiment (EXP) vs FEA
109
6.2.2 CR samples
6.2.2.1 Element size optimization
In case of the CR samples the mesh was again optimized by correlation with selected
experiments prior to detailed FEA. The element size ranging from 0.025 to 0.5 mm was
employed along the line of crack propagation and the results were compared. The plot of
crack size versus number of cycles for different element sizes (Fig. 6.9) showed that an
element size equal to 0.05 mm yielded optimum results. The trend was similar to that
observed in case of the LR samples. Thus, an element size equal to 0.05 mm was again
selected along the line of crack propagation for detailed studies.
2
5
8
0 9000 18000 27000 36000
N, cycles
a, m
m
FEA-E0.5
FEA-E0.1
FEA-E0.05
FEA-E0.025
Fig. 6.9 Element size optimization – CR orientation. Legend indicates the finite element
analysis (FEA) using elemnt size (E) of 0.5 to 0.025 mm.
6.2.2.2 Crack propagation
Fig. 6.10a shows von Mises stress distribution at a stress level of 40 % and 1 mm crack
growth. The stress distribution in the model is in agreement with the expected solution
110
and the maximum stress is in the crack tip region. Fig. 6.10b shows the enlarged crack tip
region demonstrating the maximum stress at the crack tip node.
(a)
(b)
Fig. 6.10 a) CR sample - Von Mises stress distribution (MPa) at the stress level of 40 %
and the crack length 4 mm b) enlarged crack tip region
The plots in Fig. 6.11 show the experimental and the FE results of the crack length versus
the number of cycles at different stress levels. The data covers the entire range from the
start of the crack at the notch up to the specimen failure. As can be predicted, the crack
grows faster at higher stress levels and vice versa. Comparison of the experimental and
the FE results shows close agreement at all the stress levels analyzed. The small deviation
of the FE results from the experimental values is within the scatter that was observed in
the experimental findings. This concluded that at all the stress levels analyzed; the crack
growth rate follows the Paris equation and remains in the region II of the crack growth
curve. Hence, the results presented are concluded to be optimized for all the stress levels
analyzed.
Crack tip
111
2
5
8
0 650 1300 1950 2600
N, cycles
a, m
m
EXP-1
EXP-2
FEA
(a)
2
5
8
0 900 1800 2700 3600
N, cycles
a, m
m
EXP-1
EXP-2
FEA
(b)
Fig. 6.11 Experimental and FEA results of the crack length vs the number of cycles in the
crack growth region at stress levels in % of the yield strength ( y) a) 40 b) 35 c) 30 d)
25 e) 20 f) 15 g) 10. Legend shows the experimental data (EXP-1 and EXP-2) and the
finite element results (FEA)
y%40
y%35
112
2
5
8
0 1350 2700 4050 5400
N, cycles
a, m
m
EXP-1
EXP-2
FEA
(c)
2
5
8
0 3000 6000 9000 12000N, cycles
a, m
m
EXP-1
EXP-2
FEA
(d)
Fig. 6.11 contd.
y%25
y%30
113
2
5
8
0 5500 11000 16500 22000
N, cycles
a, m
m
EXP-1
EXP-2
FEA
(e)
2
5
8
0 15500 31000 46500 62000N, cycles
a, m
m
EXP
FEA
(f)
Fig. 6.11 contd.
y%15
y%20
114
2
5
8
0 62500 125000 187500 250000
N, cycles
a, m
m
EXP
FEA
(g)
Fig. 6.11 contd.
6.2.2.3 Predicted FCG rate – Experiment vs FEA
The fatigue crack growth rate variation with K, obtained experimentally and from the
FEA is shown in Fig. 6.12. The curve fitting of the experimental data provided an
exponential fit as shown by the dashed line in the figure and yielded a regression
coefficient R2 equal to 0.9054. The values of Paris constants C and m obtained from this
data are 4E-11 and 3.4, respectively. A smooth crack growth rate achieved from the FE
analysis is based on calculations using the Paris equation. As in the case of LR samples, it
was observed in crack growth analysis that the value of the Paris constant ‘m’ obtained
experimentally gave higher deviations when used in FEA; a minor adjustment in this
constant was, thus, carried out to optimize the modeling results. The Paris constants C
and m, used for the FEA, were 4E-11 and 3.35, respectively. The optimized crack growth
y%10
115
rate achieved during the FEA provided best fit with regression coefficient R2 equal to 1.
It can be seen from the plot that the crack growth rates obtained from the experiments and
FEA are in close proximity and the predicted results lay within the upper and lower
bounds of the crack growth rate achieved from the experimental data.
Fig. 6.12 CR sample - Comparison of the predicted crack growth rate with the
experimental observation. Legend shows the experimental data (EXP) and the finite
element analysis (FEA) results at 10 % of the yield strength.
6.2.2.4 Fatigue life analysis
Plots in Fig. 6.13 show the experimental and the FE results of the S versus Nf. The
number of cycles to failure includes the cycles to initiate the crack and its growth up to
the specimen failure. The number of cycles to initiate the crack was added in the FE
results from the experimental data. The S vs Nf relations for the two data sets were
obtained through curve fitting and are also given in the figure. It can be seen that the
m
116
results obtained from the two techniques are in good agreement at all the stress levels
analyzed.
Fig. 6.13 Fatigue life analysis (CR sample) - Experiment (EXP) vs FEA
117
CHAPTER 7
Thick-Walled Cylinder;
Finite Element Analysis
In this chapter, finite element modeling of the thick-walled cylinder (TWC) under fatigue
loading is presented. As a first step, to provide a base line the stress distribution in TWC
under static internal pressure is estimated using analytical and numerical methods. The
component is then analyzed with internal axial crack under static pressure. Finally, crack
growth analysis is conducted on the component under cyclic pressure applying the
theories of fatigue process. The data obtained from the experimental work is used as the
input for the said analysis.
7.1 Thick-walled cylinder
A thick-walled cylinder or tube is one where the thickness of the wall is greater than one-
tenth of the radius. In the following sections a model of TWC is presented and the stress
distribution under internal/external pressure is discussed.
7.1.1. Model description
Consider a thick walled cylinder with outer diameter, do and inner diameter, di (Fig. 7.1).
The thickness of the cylinder tw is the difference between the inner and outer radius
where the outer radius is always greater than the inner radius. The pressures on the inner
and outer surfaces of the cylinder are pi and po, respectively.
118
Fig. 7.1 Two dimensional section of the TWC showing geometric parameters
In case of a TWC with closed ends, the cylinder experiences three principal stresses
under static internal/external pressure, i.e. tangential (T), radial (R) and axial (A) as
shown in Fig. 7.2. However, in case the cylinder has open ends there will be no axial
component of stress. The exact elastic solution for the cylinder under stress can be
obtained using Lamé’s equations. Among these stresses the tangential or hoop stress is
the maximum.
7.1.2. Model equations
Consider the TWC subjected to an internal pressure above atmospheric pressure. The
resulting stresses and expansion of the cylinder are described by the equations from 7.1 to
pi
do
tw
po di
119
7.5. These equations display how internal/external pressure and the thickness of the
cylinder relate to the stresses. This model shows that the stresses within the thick walled
cylinder depend on the inner and outer pressures and the inner and outer radii.
Fig. 7.2 Schematic of the TWC indicating three principal stresses
Model equations
(7.1)
(7.2)
(7.3)
(7.4)
22
22222)/)((
)(io
oioiooiir
rr
rrrpprprpr
22
22
io
ooiia
rr
rprp
rrr
rrpp
Er
rr
rprp
Eu
io
oioi
io
ooii
r
1)()1()1(22
22
22
22
22
22222)/)((
)(io
oioiooiih
rr
rrrpprprpr
T R
A
Po
Pi
120
(7.5)
where
σh = tangential stress variation within the material of the cylinder
σr = stress variation in the radial direction
σa = longitudinal stress within the material of the cylinder
pi = uniform internal pressure
po = uniform external pressure
ri = inside radius
ro = outside radius
r = radius, ri ≤ r ≤ ro
E = modulus of elasticity of the material
υ = Poisson's ratio of the material
ur = displacement in the radial direction due to pressurization
dua/da = relative increase in length in the axial direction
7.1.3. Parameter description
In the above equations, all the parameters are known except for the position vector ‘r’,
which varies from the inner to the outer radius. If the inner pressure is greater than the
outer pressure, then from the equations the stresses are largest as ‘r’ approaches the inner
radius. However, if the outer pressure is greater than the inner pressure, the stresses will
be largest as ‘r’ approaches the outer radius.
The elements that are located at the same radius but different angle theta will experience
the same tangential and radial stresses; this can be easily inferred from the fact that there
22
222
io
ooiia
rr
rprp
Eda
du
121
is no angular positional variable (i.e. theta) in any of the governing equations. However,
the elements at different radial lengths experience different stresses; this can be observed
from the fact that ‘r’ is a variable in the governing equations.
7.1.4. Stress description
Tangential stress affects an element in a direction tangent to its circumference, i.e.
perpendicular to the radial vector. Radial stress affects the element in a direction that is
parallel to the radial vector. For any pressure-thickness condition the difference between
the tangential and radial stress is a constant for the entire range of ‘r’. That constant can
be arrived by subtracting the radial stress from the tangential stress; the tangential stress
being always greater, the constant will be a positive value.
7.2 Static loading of TWC - without crack
The TWC is analyzed under static loading by classical theory and the results are
compared with the numerical solution. The cylinder was analyzed as an open cylinder
with no axial component of stress. Two types of analyses were conducted; one without
crack and the other with internal axial crack.
7.2.1. Analytical solution
For a TWC with the following parameters the principal stresses calculated by the model
equations are given in Tables 7.1 and 7.2. The sketch of the cylinder half section is
shown in Fig. 7.3.
pi = 5 – 100 MPa po = 0 MPa
di = 100 mm do = 150 mm tw = 25 mm
122
Fig. 7.3 Half model of the cylinder section subjected to internal pressure pi
Table 7.1 The variation in stresses and displacements with internal pressure calculated by
the model equations at inner radius, ri
pi, MPa
Principal stress, MPa Radial
displacement,
mm Tangential Radial
5 13 -5 0.0103169
10 26 -10 0.0206338
15 39 -15 0.0309507
20 52 -20 0.04126761
25 65 -25 0.05158451
30 78 -30 0.06190141
35 91 -35 0.07221831
40 104 -40 0.08253521
45 117 -45 0.09285211
50 130 -50 0.10316901
55 143 -55 0.11348592
60 156 -60 0.12380282
65 169 -65 0.13411972
70 182 -70 0.14443662
75 195 -75 0.15475352
80 208 -80 0.16507042
85 221 -85 0.17538732
90 234 -90 0.18570423
95 247 -95 0.19602113
100 260 -100 0.20633803
ri = 50 mm
pi = 5–100 MPa ro = 75 mm
tw = 25 mm
123
Table 7.2 The variation in stresses and displacements along the wall thickness calculated
by the model equations at pi = 50 MPa, po = 0 MPa
r, mm
Principal stress, MPa Radial
displacement,
mm Tangential Radial
50 130.00 -50.00 0.103169
51 126.51 -46.51 0.101894
52 123.21 -43.21 0.100682
53 120.10 -40.10 0.099530
54 117.16 -37.16 0.098435
55 114.38 -34.38 0.097393
56 111.75 -31.75 0.096402
57 109.25 -29.25 0.095459
58 106.88 -26.88 0.094562
59 104.64 -24.64 0.093708
60 102.50 -22.50 0.092894
61 100.47 -20.47 0.092120
62 98.53 -18.53 0.091383
63 96.69 -16.69 0.090682
64 94.93 -14.93 0.090014
65 93.25 -13.25 0.089378
66 91.65 -11.65 0.088773
67 90.12 -10.12 0.088197
68 88.66 -8.66 0.087650
69 87.26 -7.26 0.087129
70 85.92 -5.92 0.086634
71 84.63 -4.63 0.086163
72 83.40 -3.40 0.085716
73 82.22 -2.22 0.085292
74 81.09 -1.09 0.084889
75 80.00 0.00 0.084507
124
7.2.2. Finite element modeling
The TWC as shown in Fig. 7.3 was numerically analyzed by finite element method and
the results were compared with the analytical solution. The commercially available
ANSYS 9.0 finite element software was used for this purpose. Two dimensional finite
element analysis (FEA) was conducted using 4-noded quadrilateral elements under plane-
strain conditions.
7.2.2.1 Model Geometry
Fig. 7.4 shows the two dimensional model geometry of the cylinder used for FEA. The
symmetry of the cylinder was taken advantage of and a solid model for a half section of
the cylinder was created in the ANYSYS pre-processor. The same symmetry conditions
can also be used in the presence of axial crack. The outer diameter, do of the cylinder is
150 mm while the inner diameter, di is 100 mm. The wall thickness, tw of the cylinder is
25 mm.
Fig. 7.4 TWC Model used for FEA
125
7.2.2.2 Material properties
During FEA, an isotropic material with modulus of elasticity E = 71 GPa and Poisson’s
ratio, = 0.33 was used [147].
7.2.2.3 Element selection and meshing
The TWC was meshed using two dimensional 4-noded, PLANE42 solid elements. The
element geometry is shown in Fig. 6.2. The parametric study was conducted to see the
effects of element size on the results. Meshed model is shown in Fig. 7.5.
(a)
(b)
Fig. 7.5. a) Meshed model using PLANE42 element b) magnified view of boxed area;
element size is 0.5 mm
126
7.2.2.4 Boundary conditions and solution
The boundary conditions (BCs) applied on the TWC are shown in Fig. 7.6. The half
section of the cylinder was constrained applying symmetry boundary conditions along the
wall thickness on both edges. The model was loaded by applying pressure on the inner
wall of the cylinder, simulating internal pressure. The pressure was varied from 5 to 100
MPa. There was no outer pressure applied. Solutions were obtained at different internal
pressures and the results were compared with the analytical one.
The von Mises stress distribution obtained after solution is shown in Fig. 7.7. This value
is normally used in both fatigue and static load design of such cylinders. The parametric
study conducted to see the effect of element size reveals that the results obtained using
element size of 1 mm and less are in good agreement with the analytical results.
Fig. 7.6 Static loading - Boundary conditions applied for analysis
Symmetry BCs
Pressure
127
(a)
(b)
Fig. 7.7 Static loading – Nodal solution showing von Mises stress distribution at internal
pressure of a) 5 MPa b) 100 MPa
7.2.3. Comparison of the analytical and numerical results
The results of the stress distribution obtained from analytical (thick-walled cylinder
theory, Lamé’s equations) and numerical techniques were compared to see the validity of
the model. Figs. 7.8a and 7.8b show the graphical presentation of the analytical and the
128
FEA results of stress versus internal pressure at inner radius. The stress variation along
the wall thickness of the cylinder obtained from the two methods is shown in Fig. 7.9.
(a)
(b)
Fig. 7.8 Stress versus internal pressure - comparison of the two results at inner radius a)
tangential b) radial
0
75
150
225
300
0 20 40 60 80 100 pi, MPa
Tan
gen
tial
str
ess,
MP
a .
. FEA
TWC theory
-100
-75
-50
-25
0
0 20 40 60 80 100 pi, MPa
Rad
ial
stre
ss, M
Pa
MP
a
FEA
TWC theory
129
(a)
(b)
Fig. 7.9 Stress variation along the wall thickness of the cylinder obtained from the two
methods at an internal pressure of 100 MPa a) tangential b) radial
150
175
200
225
250
275
50 55 60 65 70 75
r, mm
Tan
gen
tial
str
ess,
MP
a
MP
a
Analytical
FEA
-100
-75
-50
-25
0
25
50 55 60 65 70 75 r, mm
Rad
ial
stre
ss, M
Pa
MP
a
Analytical
FEA
130
In Fig. 7.8, both the tangential and radial stresses obtained from the analytical and FEA
methods change linearly with the applied internal pressure. It can be seen that the results
obtained from the two techniques are in good agreement. Fig. 7.9a shows a gradual
decrease in the tangential stress from inner to outer radius. The highest tangential (hoop)
stress is found at the inner radius i.e. at the inner wall of the cylinder. In Fig. 7.9b the
change in radial stress along the wall thickness of the cylinder is presented. A
compressive stress is found which varies from 100 MPa at the inner radius to a value of 0
MPa at the outer radius. Again the results obtained from the two techniques and
presented in Figs. 7.9a and 7.9b are in fairly good agreement. This concludes that the half
model used for the stress analysis is providing satisfactory results and can be used for the
analysis of the cylinder with internal crack.
7.3 Static loading of TWC - with internal axial crack
The TWC with internal axial crack and under static loading was analyzed at different
internal pressures. Analysis was conducted to determine the stress intensity factor (KI) at
the crack tip; (KI) is used to estimate the crack growth rate under cyclic loading. The two
dimensional model of the TWC was analyzed analytically and the values for KI obtained.
These results were used for finite element analysis of the cylinder under fatigue loading.
7.3.1. Geometry of the model
Fig. 7.10 shows the modified two dimensional model geometry of the cylinder, with
internal axial crack, used for FEA. The crack was modeled on the inner bore of the
cylinder in axial direction (perpendicular to the plane of the paper). The crack depth is ‘a’
mm.
131
Fig. 7.10 Schematic of two dimensional half cylinder model with internal axial crack
7.3.2. Material properties, element type and meshing
The cracked model was analyzed using the same material properties and employing the
same element type as was used for un-cracked model. The half model with an initial
crack length of 3 mm, a/tw = 0.12 was used in FEA due to the geometrical symmetry of
the cylinder.
7.3.3. Boundary conditions and solution
The boundary conditions applied on the TWC are shown in Fig. 7.11. The half section of
the cylinder was constrained applying symmetry boundary conditions along the wall
thickness on both sides. A 3 mm long crack was modeled by applying no constraints from
ri to 3 mm along the x direction at the right wall, thus providing the crack tip node at 3
mm from the inner wall. The model was loaded by applying tractions at the inner wall of
the cylinder, simulating internal pressure. After loading the model and obtaining the
solution, KI was obtained at the crack tip by defining the path and using KCALC
command. Solutions were obtained at internal pressures varying from 5 to 100 MPa.
ri = 50 mm
pi = 5 –100 MPa ro = 75 mm
tw = 25 mm
Crack a
132
Fig. 7.11 Static loading of TWC with crack - Boundary conditions applied for analysis
The nodal solution showing von Mises stress distribution at internal pressure of 5 and 100
MPa is shown in Fig. 7.12. The maximum stress is at the crack tip node which can be
seen more clearly in the Fig. 7.13.
7.3.4. Determination of the stress intensity factor (KI)
After obtaining the solution, the stress intensity factor was determined by defining the
path and using KCALC command. In order to see the effect on KI, element size was
varied from 2 to 0.25 mm; solutions were obtained at internal pressures varying from 5 to
100 MPa. Plot in Fig. 7.14 shows the KI versus internal pressure at a crack length of 3
mm. KI increases linearly with the pressure and the effect of element size was found
negligible. Fig. 7.15 shows the KI versus internal pressure at crack length from 3 to 10
mm. Again KI increases linearly with internal pressure for all the crack sizes analyzed.
Pressure
Symmetry BCs
crack
133
(a)
(b)
Fig. 7.12 Static loading of cylinder with crack – Nodal solution showing von Mises stress
distribution at internal pressure of a) 5 MPa b) 100 MPa
crack
134
Fig. 7.13 Magnified view of the crack region shown in Fig. 7.12a with BCs
Fig. 7.14 Plot of KI versus internal pressure at a crack length of 3 mm
a = 3 mm
0
10
20
30
0 20 40 60 80 100 pi, MPa
E2 E1
E0.5
KI, M
Pa.
sqrt
(m)
MP
a.sq
rt(m
)
135
Fig. 7.15 Plot showing KI versus internal pressure at crack length of 3, 5, 7 and 10 mm
Fig. 7.16 shows the variation of KI with the increase in crack length along the wall
thickness of the cylinder at different internal pressures. The data was obtained using an
element size of 0.5 mm. The curves obtained from the data show polynomial fits which
are used for the fatigue calculations.
7.4 FEA of fatigue crack growth in TWC
The fatigue crack growth analysis of TWC was performed based on linear elastic fracture
mechanics and using Paris law. The FEA results obtained in the previous section were
used for this purpose. The relations between the stress intensity factor KI and the crack
size were used and the fatigue calculations were performed employing the same approach
as was applied in the case of M(T) samples. The analysis was conducted using the
0
20
40
60
80
0 20 40 60 80 100 pi, MPa
KI , M
Pa.
sqrt
(m)
MP
a.sq
rt(m
)
a-3
a-5
a-7
a-10
136
experimental data obtained in the CR direction which corresponds to the hoop stress in
the cylinder. The cyclic pressure was applied with R ratio equal to 0.1 and the crack was
advanced in steps of 0.05 mm. The fatigue crack growth life (Ng) of the cylinder was
determined at different internal pressures. The fatigue crack growth life was the total
applied cycles from the initial crack length to the final fracture [24].
0
50
100
150
0 5 10 15 20a, mm
KI, M
Pa.
sqrt
(m)
P10
P20
P30
P40
P50
P60
P75
P100
Fig. 7.16 Variation of KI with the increase of crack length at different internal pressures
7.4.1. Crack propagation in TWC
Fig. 7.17 shows the plots of the applied pressure cycles versus crack length of the
simulated TWC model with an initial crack size of 3 mm.
The analysis of the results showed that the crack grows faster at higher pressures and vice
versa as was observed in the case of M(T) samples. It is also clear from the plot that the
fatigue crack growth life decreases with an increase in the internal pressure.
137
1.0
10.0
100.0
10 100 1000 10000 100000 1000000ln N, cycles
ln a
, m
m.
P20
P25
P30
P40
P50
P60
Fig. 7.17 Applied cycles versus crack length of the simulated TWC model with an initial
crack length of 3 mm
7.4.2. Predicted FCG rate – Experimental vs FEA
The variation of fatigue crack growth rate with K obtained experimentally in CR
samples and from the FEA of the cylinder at different applied pressures is shown in Fig.
7.18. The smooth crack growth rate achieved using the FE analysis is based on the
calculations using Paris equation. It can be seen that the fatigue crack growth rate
obtained by the FEA lied within the upper and lower bounds of the crack growth rate
achieved from the experimental data.
7.4.3. Fatigue crack growth life prediction of the cylinder
The fatigue crack growth life of the cylinder was predicted from the FEA and is
presented in Fig. 7.19. The plot provides variation in the internal pressure versus the total
applied cycles, starting from the initial crack length to the final fracture. The curve fitting
138
of the data provides the best fit with power relation between the two values and is given
in the figure. As expected, the fatigue crack growth life of the cylinder obtained from
FEA shows that the fatigue lifetime increases as the applied pressure decreases.
1.00E-09
1.00E-08
1.00E-07
1.00E-06
1.00E-05
1 10 100
K, MPa.sqrt(m)
da/
dN
, m
/cycl
e
EXP
FEA-P20
Fig. 7.18 The variation of fatigue crack growth rate with K – Experimental vs FEA
Fig. 7.19 Predicted fatigue crack growth life of the thick-walled cylinder at different
internal pressures
y = 360.19 x -0.23
R 2 = 0.9927
0
20
40
60
80
100 1000 10000 100000 1000000 10000000
ln Ng, cycles
pi,
MP
a
MP
a
139
CHAPTER 8
Summary and Conclusions
This chapter summarizes the work done during this research study. Separate sections are
devoted to summarize the conclusions and recommendations for future work based on the
present research.
8.1 Summary
This research study consists of the tests conducted on the material of the TWC in
longitudinal and transverse directions. These include the characterization of the material,
determination of its mechanical properties and study of the fractured surfaces. Fatigue
tests were conducted on the samples prepared from the cylinder in two different
orientations and the process was simulated through ANSYS structural software using
FEM and Paris law. The fatigue crack propagation was simulated, based on linear elastic
fracture mechanics and stress intensity factor determination. Finite element analysis of
the TWC under internal static pressure was conducted, with and without internal crack.
The model with internal axial crack was subjected to cyclic loading and the fatigue crack
growth life of the cylinder was predicted, using the experimental data combined with
Paris law, at different stress levels.
140
8.2 Conclusions
The conclusions of the research work are presented in the following subsections.
8.2.1. Conclusions from the experimental data
1. The extrusion process induced microstructural anisotropy in the material thus
resulting in poorer mechanical properties in the transverse direction.
The yield and tensile strength in two orientations were not much different but a
substantial difference in the deformation behavior of the material was evident.
The elongation, reduction in area and the impact strength of the material in the
transverse direction were found to be inferior.
2. A major impact of the anisotropy was observed on the fatigue properties. The
material shows shorter fatigue life in the transverse direction, which is 41 to 62 %
lower in the tested range of 129 to 47 MPa.
8.2.2. Conclusions from Finite Element Analysis
1. The finite element analysis combined with the Paris law is shown to predict the
fatigue life of the samples and provide results in good agreement with the
experimental data. Hence, the node release technique used in this study works
satisfactorily to predict the fatigue crack growth process.
2. For stable crack growth predictions under constant amplitude loading an element
size of 0.05 mm along the line of crack propagation is sufficient to produce
optimized results.
3. Both the experimental and numerical results of the crack growth data at different
stress levels were found in good agreement. The model works satisfactorily as
141
long as the stress levels producing K values are within the Paris regime. In the
near threshold region the model provides more conservative results. Fatigue crack
growth rates determined numerically lay within the statistical scatter of the
experimental data.
4. The SN curves showed that in the Paris region the fatigue life of the samples
obtained from the experiments and the simulation were in good agreement.
5. The results of the stress distribution under static loads found by the finite element
analysis of the TWC using two dimensional, quadrilateral PLANE42 element
show good agreement with the analytical results.
6. The LEFM technique using finite element analysis combined with the Paris law
can predict the fatigue crack growth life of the TWC. As expected, the fatigue
crack growth life of the TWC with internal axial crack decreases with increasing
the internal pressure.
8.3 Recommended future work
1. In the published literature, crack growth simulation using FEA on extruded
material which inherently creates anisotropy is not documented. The satisfactory
results of the simulation in the present work may be utilized by extending the
research on other Al-base extruded alloys of hi-tech applications (where fatigue is
critical) in various heat treatment conditions.
2. The crack growth model presented in this research can be extended to non-linear
analysis introducing crack tip plasticity into the code.
142
3. Fatigue crack growth experiments on full length TWC could be a logical
continuation of future experimental work to re-validate the FEA on actual
components.
4. Predictions of the present work would be utilized to reduce the density of the
constituent particles and increase extrusion ratios.
143
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ANNEXURE I
PRODUCTION OF SEAMLESS TWC THROUGH
EXTRUSION PROCESS
162
Schematic showing the essential parts of the extrusion process
Induction
heating
Heating system
Annular ram Online quenching
Extrusion die
Dummy block
Container
Piercer/Mandrel
PRESS
163
Billet Heating
Inserting the billet
into the container
1
2
164
Starting Piercing
Piercing complete
Removal of dummy block
2
3
4
165
End product
Extrusion started 5
Extruded pipe
Quenching
ANNEXURE II
FLOW CHART OF THE ALGORITHM USED FOR THE
SIMULATION OF FATIGUE CRACK PROPAGATION
166
Define element
type/size and enter
material constants
(E, ν, C, m, KIC)
Create model geometry and
mesh
Create/open file to store results
Apply/change boundary
conditions
Solve
Define path and determine
KImax at the crack tip node
using KCALC command
Calculate K
K = (KImax - (0.1*KImax))
167
Compute crack growth rate
(CGR) using Paris equation
da/dN = C (K) m
Compute no. of cycles to move
one element size
N = element size / CGR
Add cycles to previous no. of
cycles
Write / Append results in the
file
Check if K = KIC
Exit
YES
NO
ANNEXURE III
SAMPLE RESULTS FILE GENERATED DURING SIMULATION
OF FATIGUE CRACK PROPAGATION
168
Sample – LR, Stress 40% YS
ITR WIDTH A CRATIO PC1 PC2 PRESS KI DEL-K CGR NOC-STEP T-NOC Ni+Ng
1 10 3.000 0.300 2.00E-10 2.7 143 547.64 15.59 3.32E-07 151 151 2500
2 10 3.050 0.305 2.00E-10 2.7 143 547.64 15.59 3.32E-07 151 301 2651
3 10 3.100 0.310 2.00E-10 2.7 143 554.13 15.77 3.43E-07 146 447 2801
4 10 3.150 0.315 2.00E-10 2.7 143 560.66 15.96 3.54E-07 141 588 2947
5 10 3.200 0.320 2.00E-10 2.7 143 567.21 16.14 3.65E-07 137 725 3088
6 10 3.250 0.325 2.00E-10 2.7 143 573.8 16.33 3.77E-07 133 858 3225
7 10 3.300 0.330 2.00E-10 2.7 143 580.41 16.52 3.89E-07 129 986 3358
8 10 3.350 0.335 2.00E-10 2.7 143 587.06 16.71 4.01E-07 125 1111 3486
9 10 3.400 0.340 2.00E-10 2.7 143 593.74 16.90 4.13E-07 121 1232 3611
10 10 3.450 0.345 2.00E-10 2.7 143 600.46 17.09 4.26E-07 117 1349 3732
11 10 3.500 0.350 2.00E-10 2.7 143 607.21 17.28 4.39E-07 114 1463 3849
12 10 3.550 0.355 2.00E-10 2.7 143 613.99 17.47 4.52E-07 111 1574 3963
13 10 3.600 0.360 2.00E-10 2.7 143 620.82 17.67 4.66E-07 107 1681 4074
14 10 3.650 0.365 2.00E-10 2.7 143 627.68 17.86 4.80E-07 104 1785 4181
15 10 3.700 0.370 2.00E-10 2.7 143 634.57 18.06 4.95E-07 101 1886 4285
16 10 3.750 0.375 2.00E-10 2.7 143 641.51 18.26 5.09E-07 98 1985 4386
169
17 10 3.800 0.380 2.00E-10 2.7 143 648.49 18.46 5.24E-07 95 2080 4485
18 10 3.850 0.385 2.00E-10 2.7 143 655.51 18.66 5.40E-07 93 2172 4580
19 10 3.900 0.390 2.00E-10 2.7 143 662.57 18.86 5.56E-07 90 2262 4672
20 10 3.950 0.395 2.00E-10 2.7 143 669.67 19.06 5.72E-07 87 2350 4762
21 10 4.000 0.400 2.00E-10 2.7 143 676.82 19.26 5.89E-07 85 2435 4850
22 10 4.050 0.405 2.00E-10 2.7 143 684.01 19.47 6.06E-07 83 2517 4935
23 10 4.100 0.410 2.00E-10 2.7 143 691.24 19.67 6.23E-07 80 2598 5017
24 10 4.150 0.415 2.00E-10 2.7 143 698.52 19.88 6.41E-07 78 2676 5098
25 10 4.200 0.420 2.00E-10 2.7 143 705.85 20.09 6.59E-07 76 2752 5176
26 10 4.250 0.425 2.00E-10 2.7 143 713.23 20.30 6.78E-07 74 2825 5252
27 10 4.300 0.430 2.00E-10 2.7 143 720.65 20.51 6.97E-07 72 2897 5325
28 10 4.350 0.435 2.00E-10 2.7 143 728.13 20.72 7.17E-07 70 2967 5397
29 10 4.400 0.440 2.00E-10 2.7 143 735.65 20.94 7.37E-07 68 3035 5467
30 10 4.450 0.445 2.00E-10 2.7 143 743.22 21.15 7.58E-07 66 3101 5535
31 10 4.500 0.450 2.00E-10 2.7 143 750.85 21.37 7.79E-07 64 3165 5601
32 10 4.550 0.455 2.00E-10 2.7 143 758.53 21.59 8.01E-07 62 3227 5665
33 10 4.600 0.460 2.00E-10 2.7 143 766.26 21.81 8.23E-07 61 3288 5727
34 10 4.650 0.465 2.00E-10 2.7 143 774.04 22.03 8.46E-07 59 3347 5788
35 10 4.700 0.470 2.00E-10 2.7 143 781.89 22.25 8.69E-07 58 3405 5847
170
36 10 4.750 0.475 2.00E-10 2.7 143 789.78 22.48 8.93E-07 56 3461 5905
37 10 4.800 0.480 2.00E-10 2.7 143 797.74 22.70 9.17E-07 55 3515 5961
38 10 4.850 0.485 2.00E-10 2.7 143 805.75 22.93 9.42E-07 53 3568 6015
39 10 4.900 0.490 2.00E-10 2.7 143 813.83 23.16 9.68E-07 52 3620 6068
40 10 4.950 0.495 2.00E-10 2.7 143 821.96 23.39 9.95E-07 50 3670 6120
41 10 5.000 0.500 2.00E-10 2.7 143 830.16 23.63 1.02E-06 49 3719 6170
42 10 5.050 0.505 2.00E-10 2.7 143 838.42 23.86 1.05E-06 48 3767 6219
43 10 5.100 0.510 2.00E-10 2.7 143 846.74 24.10 1.08E-06 46 3813 6267
44 10 5.150 0.515 2.00E-10 2.7 143 855.13 24.34 1.11E-06 45 3858 6313
45 10 5.200 0.520 2.00E-10 2.7 143 863.59 24.58 1.14E-06 44 3902 6358
46 10 5.250 0.525 2.00E-10 2.7 143 872.11 24.82 1.17E-06 43 3945 6402
47 10 5.300 0.530 2.00E-10 2.7 143 880.71 25.07 1.20E-06 42 3987 6445
48 10 5.350 0.535 2.00E-10 2.7 143 889.37 25.31 1.23E-06 41 4028 6487
49 10 5.400 0.540 2.00E-10 2.7 143 898.11 25.56 1.26E-06 40 4067 6528
50 10 5.450 0.545 2.00E-10 2.7 143 906.93 25.81 1.30E-06 39 4106 6567
51 10 5.500 0.550 2.00E-10 2.7 143 915.82 26.06 1.33E-06 38 4143 6606
52 10 5.550 0.555 2.00E-10 2.7 143 924.79 26.32 1.37E-06 37 4180 6643
53 10 5.600 0.560 2.00E-10 2.7 143 933.84 26.58 1.40E-06 36 4215 6680
54 10 5.650 0.565 2.00E-10 2.7 143 942.97 26.84 1.44E-06 35 4250 6715
171
55 10 5.700 0.570 2.00E-10 2.7 143 952.19 27.10 1.48E-06 34 4284 6750
. . . . . . . . . . . . .
. . . . . . . . . . . . .
ITR Iteration Number
WIDTH Sample width, mm
A Crack length, mm
CRATIO Crack ratio, A/W
PC1 Paris constant, C
PC2 Paris constant, m
PRES Stress, MPa
KI Stress intensity factor, MPa. mm
DEL-K Stress intensity factor range, MPa. m
CGR Crack growth rate, m/cycle
NOC-STEP Number of load cycles per step
T-NOC Total number of load cycles in crack propagation
Ni+Ng Total number of load cycles for crack initiation and growth
172
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