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AN ABSTRACT OF A THESIS
J-INTEGRAL FINITE ELEMENT ANALYSISOF SEMI-ELLIPTICAL SURFACE
CRACKS IN FLAT PLATESWITH TENSILE
LOADING
Eric N. Quillen
Master of Science in Mechanical Engineering
Linear elastic fracture mechanics (LEFM) is used when response
to the loadis elastic, and the fracture is brittle. For LEFM, the
K-factor is the most commonlyused fracture criterion. However, high
temperatures and limited high stress cycles be-fore component
replacement are factors that can cause significant plastic
deformationand a ductile failure. In these cases, an
elastic-plastic fracture mechanics (EPFM)approach is required. The
J-integral is commonly used as an EPFM fracture param-eter.
The primary goal of this research was to develop
three-dimensional finite el-ement analysis (FEA) J-integral data
for surface crack specimen geometries andcompare to existing
solutions. The finite element models were analyzed as elas-tic, and
fully plastic using ABAQUS. The J-integral data were used to find
the loadindependent variable, h1 for comparison purposes.
There were two other goals in this research. The second goal was
to examinethe effect of various finite element modelling parameters
including mesh density, ele-ment type, symmetry, and specimen size
effects, on the resulting J-integral. The thirdgoal was to perform
elastic-plastic finite element analyses that utilize a stress vs.
plas-tic strain table based on a power law hardening material
behavior. The elastic-plasticand fully plastic results were
compared.
For the most part, the current data compared well with the data
published byother researchers. The elastic results compared more
favorably than the fully plasticand elastic-plastic data. For both
the elastic and plastic analyses, the finite elementmodels (FEMs)
produced sudden increases in the K-factor and J-integral at the
freesurface and/or depth. The plastic FEMs also exhibited an
anomaly in the J-integralat the third and fourth angles from the
surface. The anomaly could be taken as ajump at the third angle or
a dip at the fourth angle, depending on how the data weretrended.
The third angle varied with the model geometry (2.71 to 11.24).
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J-INTEGRAL FINITE ELEMENT ANALYSIS
OF SEMI-ELLIPTICAL SURFACE
CRACKS IN FLAT PLATES
WITH TENSILE
LOADING
A Thesis
Presented to
the Faculty of the Graduate School
Tennessee Technological University
by
Eric N. Quillen
In Partial Fulfillment
of the Requirements for the Degree
MASTER OF SCIENCE
Mechanical Engineering
May 2005
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STATEMENT OF PERMISSION TO USE
In presenting this thesis in partial fulfillment of the
requirements for a Master
of Science degree at Tennessee Technological University, I agree
that the University
Library shall make it available to borrowers under rules of the
Library. Brief quota-
tions from this thesis are allowable without special permission,
provided that accurate
acknowledgment of the source is made.
Permission for extensive quotation from or reproduction of this
thesis may be
granted by my major professor when the proposed use of the
material is for scholarly
purposes. Any copying or use of the material in this thesis for
financial gain shall not
be allowed without my written permission.
Signature
Date
iii
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DEDICATION
This thesis is dedicated to my wife Julie, whose encouragement
has been critical
in the completion of my graduate degree and the composition of
this thesis.
iv
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ACKNOWLEDGMENTS
I would like to thank the following people for their help with
this work: Dr.
Chris Wilson, Dr. Phillip Allen, Mike Renfro, Krishna Natarajan,
and Richard
Gregory. I would also like to thank my employer, Fleetguard,
Inc., and cowork-
ers. Without their cooperation, it would not have been possible
for me to perform
this research.
v
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TABLE OF CONTENTS
Page
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . ix
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . xiii
LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . xx
Chapter
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 1
1.1 Fracture Mechanics . . . . . . . . . . . . . . . . . . . . .
. . 1
1.2 Overview of Research . . . . . . . . . . . . . . . . . . . .
. . 2
2. TECHNICAL BACKGROUND . . . . . . . . . . . . . . . . . . . .
. . 4
2.1 J-Integral . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 4
2.2 EPRI Estimation Scheme . . . . . . . . . . . . . . . . . . .
. 8
2.3 Reference Stress Method . . . . . . . . . . . . . . . . . .
. . 17
3. RESEARCH PROCEDURE . . . . . . . . . . . . . . . . . . . . .
. . . 19
3.1 Finite Element Modeling . . . . . . . . . . . . . . . . . .
. . 19
3.1.1 Mesh Generation . . . . . . . . . . . . . . . . . . . . .
. 21
3.1.1.1 mesh3d scp . . . . . . . . . . . . . . . . . . . . . . .
21
3.1.1.2 FEA-Crack . . . . . . . . . . . . . . . . . . . . . . .
22
3.2 Analysis Procedure . . . . . . . . . . . . . . . . . . . . .
. . 24
3.3 J-Integral Convergence . . . . . . . . . . . . . . . . . . .
. . 26
vi
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vii
Chapter Page
3.3.1 Load . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 26
3.3.2 Fully Plastic Zone . . . . . . . . . . . . . . . . . . . .
. 27
3.4 Comparison to Other Work . . . . . . . . . . . . . . . . . .
. 31
3.4.1 Kirk and Dodds . . . . . . . . . . . . . . . . . . . . . .
. 31
3.4.2 McClung et al. [15] . . . . . . . . . . . . . . . . . . .
. . 35
3.4.3 Lei [17] . . . . . . . . . . . . . . . . . . . . . . . . .
. . 37
3.4.4 Nasgro Computer Program . . . . . . . . . . . . . . . .
38
3.5 Mesh Refinement . . . . . . . . . . . . . . . . . . . . . .
. . 40
3.6 Finite Size Effects . . . . . . . . . . . . . . . . . . . .
. . . . 40
3.7 Material Properties . . . . . . . . . . . . . . . . . . . .
. . . 41
3.7.1 Deformation Plasticity . . . . . . . . . . . . . . . . . .
. 41
3.7.2 Incremental Plasticity . . . . . . . . . . . . . . . . . .
. 43
4. RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 49
4.1 Fully Plastic Zone . . . . . . . . . . . . . . . . . . . . .
. . . 49
4.2 Kirk and Dodds Incremental Plasticity . . . . . . . . . . .
. 49
4.3 McClung and Lei Comparisons . . . . . . . . . . . . . . . .
. 50
4.3.1 Elastic Analysis . . . . . . . . . . . . . . . . . . . . .
. . 51
4.3.2 Fully Plastic Analysis . . . . . . . . . . . . . . . . . .
. 67
4.3.3 Incremental Elastic-Plastic Analysis . . . . . . . . . . .
. 86
4.4 Mesh Refinement . . . . . . . . . . . . . . . . . . . . . .
. . 91
4.5 Size Effects . . . . . . . . . . . . . . . . . . . . . . . .
. . . 93
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viii
Chapter Page
4.5.1 Height Effects . . . . . . . . . . . . . . . . . . . . . .
. . 93
4.5.2 Width Effects . . . . . . . . . . . . . . . . . . . . . .
. . 95
5. CONCLUSIONS AND RECOMMENDATIONS . . . . . . . . . . . . .
104
5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . .
. . . 104
5.2 Recommendations . . . . . . . . . . . . . . . . . . . . . .
. . 106
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 108
APPENDICES
A: INSTRUCTIONS FOR MESH3D SCP MODIFICATIONS . . . . . . . . .
113
B: COARSE VERSUS REFINED MESHES FOR K-FACTORS . . . . . . .
115
C: COARSE VS. REFINED MESHES FOR FULLY PLASTIC MODELS . .
120
D: HEIGHT EFFECTS . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 133
E: K-FACTOR RESULTS FOR COARSE MESHES . . . . . . . . . . . . .
. 137
F: FULLY PLASTIC RESULTS FOR COARSE MESHES . . . . . . . . . . .
152
G: INCREMENTAL PLASTICITY TABLES . . . . . . . . . . . . . . . .
. . 163
VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 170
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LIST OF TABLES
Table Page
2.1 McClung et al. h1 values in tension, n = 15 [15] . . . . . .
. . . . . . . 14
2.2 McClung et al. h1 values in tension, n = 10 [15] . . . . . .
. . . . . . . 14
2.3 McClung et al. h1 values in tension, n = 5 [15] . . . . . .
. . . . . . . . 15
2.4 Lei h1 values in tension, n = 5 [17] . . . . . . . . . . . .
. . . . . . . . 16
2.5 Lei h1 values in tension, n = 10 [17] . . . . . . . . . . .
. . . . . . . . . 16
3.1 Number of nodes and elements in the duplication of the Kirk
and Dodds[23] geometries . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 31
3.2 Incremental plasticity values for the Kirk and Dodds models
. . . . . . 33
3.3 McClung et al. fully plastic geometries . . . . . . . . . .
. . . . . . . . 36
3.4 Geometries for Nasgro comparison and width effect
investigation . . . . 39
3.5 Number of crack front nodes in the coarse and refined meshes
. . . . . 40
3.6 Stress vs. plastic strain data at n = 15, used for ABAQUS
models . . . 47
3.7 Stress vs. plastic strain data at n = 10, used for ABAQUS
models . . . 47
3.8 Stress vs. plastic strain data at n = 5, used for ABAQUS
models . . . 48
4.1 Comparison of FEM results to Kirk and Dodds values . . . . .
. . . . . 50
4.2 Surface and depth phenomenon for K-factors . . . . . . . . .
. . . . . 56
4.3 Maximum percent differences between Newman-Raju and FEM
solutions(quarter symmetry) . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 57
4.4 Maximum percent differences between McClung et al. [15] and
FEMsolutions (quarter symmetry) . . . . . . . . . . . . . . . . . .
. . . . 82
ix
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xTable Page
4.5 Maximum percent differences between McClung et al. [15] and
Lei [17]solutions (quarter symmetry) . . . . . . . . . . . . . . .
. . . . . . . 83
4.6 Model 1 (a/t=0.2 and a/c=0.2): h1 values at different
heights . . . . . 94
4.7 Comparison of Nasgro and FEM results for n = 15 . . . . . .
. . . . . 95
4.8 Comparison of Nasgro and FEM results for n = 10 . . . . . .
. . . . . 96
4.9 Comparison of Nasgro and FEM results for n = 5 . . . . . . .
. . . . . 96
D.1 Model 4 (a/t=0.5 and a/c=0.2): h1 values for at different
heights (Part 1) 134
D.2 Model 4 (a/t=0.5 and a/c=0.2): h1 values for at different
heights (Part 2) 135
D.3 Model 5 (a/t=0.5 and a/c=0.6): h1 values for at different
heights . . . 135
D.4 Model 9 (a/t=0.8 and a/c=1.0): h1 values for at different
heights . . . 136
E.5 Model 1 (a/t=0.2, a/c=0.2): K-Factor data from ABAQUS (Part
1) . 138
E.6 Model 1 (a/t=0.2, a/c=0.2): K-Factor data from ABAQUS (Part
2) . 139
E.7 Model 2 (a/t=0.2, a/c=0.6): K-Factor data from ABAQUS . . .
. . . 140
E.8 Model 3 (a/t=0.2, a/c=1.0): K-Factor data from ABAQUS . . .
. . . 141
E.9 Model 4 (a/t=0.5, a/c=0.2): K-Factor data from ABAQUS (Part
1) . 142
E.10 Model 4 (a/t=0.5, a/c=0.2): K-Factor data from ABAQUS (Part
2) . 143
E.11 Model 5 (a/t=0.5, a/c=0.6): K-Factor data from ABAQUS . . .
. . . 144
E.12 Model 6 (a/t=0.5, a/c=1.0): K-Factor data from ABAQUS . . .
. . . 145
E.13 Model 7 (a/t=0.8, a/c=0.2): K-Factor data from ABAQUS (Part
1) . 146
E.14 Model 7 (a/t=0.8, a/c=0.2): K-Factor data from ABAQUS (Part
2) . 147
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xi
Table Page
E.15 Model 7 (a/t=0.8, a/c=0.2): K-Factor data from ABAQUS (Part
3) . 148
E.16 Model 8 (a/t=0.8, a/c=0.6): K-Factor data from ABAQUS (Part
1) . 149
E.17 Model 8 (a/t=0.8, a/c=0.6): K-Factor data from ABAQUS (Part
2) . 150
E.18 Model 9 (a/t=0.8, a/c=1.0): K-Factor data from ABAQUS . . .
. . . 151
F.19 Model 1 (a/t=0.2, a/c=0.2): h1 data from ABAQUS . . . . . .
. . . . 153
F.20 Model 2 (a/t=0.2, a/c=0.6): h1 data from ABAQUS . . . . . .
. . . . 154
F.21 Model 3 (a/t=0.2, a/c=1.0): h1 data from ABAQUS . . . . . .
. . . . 155
F.22 Model 4 (a/t=0.5, a/c=0.2): h1 data from ABAQUS (Part 1) .
. . . . 156
F.23 Model 4 (a/t=0.5, a/c=0.2): h1 data from ABAQUS (Part 2) .
. . . . 157
F.24 Model 5 (a/t=0.5, a/c=0.6): h1 data from ABAQUS . . . . . .
. . . . 157
F.25 Model 6 (a/t=0.5, a/c=1.0): h1 data from ABAQUS . . . . . .
. . . . 158
F.26 Model 7 (a/t=0.8, a/c=0.2): h1 data from ABAQUS (Part 1) .
. . . . 159
F.27 Model 7 (a/t=0.8, a/c=0.2): h1 data from ABAQUS (Part 2) .
. . . . 160
F.28 Model 8 (a/t=0.8, a/c=0.6): h1 data from ABAQUS . . . . . .
. . . . 161
F.29 Model 9 (a/t=0.8, a/c=1.0): h1 data from ABAQUS . . . . . .
. . . . 162
G.30 Stress vs. strain data at n = 15, based on Equation 3.13 .
. . . . . . . 164
G.31 Stress vs. strain data at n = 10, based on Equation 3.13 .
. . . . . . . 165
G.32 Stress vs. strain data at n = 5, based on Equation 3.13 . .
. . . . . . . 166
G.33 Stress vs. plastic strain data at n = 15, used for ABAQUS
models . . . 167
G.34 Stress vs. plastic strain data at n = 10, used for ABAQUS
models . . . 168
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xii
Table Page
G.35 Stress vs. plastic strain data at n = 5, used for ABAQUS
models . . . 169
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LIST OF FIGURES
Figure Page
2.1 Contour around a crack tip [4] . . . . . . . . . . . . . . .
. . . . . . . . 5
2.2 EPRI J-Integral estimation scheme [4] . . . . . . . . . . .
. . . . . . . 9
2.3 Sample of finite element mesh used by McClung et al. [15] .
. . . . . . 12
2.4 Close up of the finite element mesh around the crack front
used byMcClung et al. [15] . . . . . . . . . . . . . . . . . . . .
. . . . . . . 12
3.1 Degeneration of elements around crack tip [4] . . . . . . .
. . . . . . . 20
3.2 Plastic singularity element [4] . . . . . . . . . . . . . .
. . . . . . . . . 20
3.3 Zones created in the mesh by mesh3d scp [20] . . . . . . . .
. . . . . . 22
3.4 Mesh created using FEA-Crack . . . . . . . . . . . . . . . .
. . . . . . 23
3.5 Close up of mesh from Figure 3.4 created using FEA-Crack . .
. . . . . 23
3.6 Contours (semi-circular rings) around the crack tip . . . .
. . . . . . . 24
3.7 Coordinate scheme for mapping crack face angles . . . . . .
. . . . . . 26
3.8 Fully plastic element set consisting of the elements around
the crack tip 28
3.9 Fully plastic element set consisting of part of layer 1 . .
. . . . . . . . . 29
3.10 Fully plastic element set consisting of layer 1 . . . . . .
. . . . . . . . . 29
3.11 Fully plastic element set consisting of partial layers 1
and 2 . . . . . . . 30
3.12 Geometries used by Kirk and Dodds for estimating the
J-Integral [23] . 32
3.13 Stress vs. strain curve for Kirk and Dodds elastic-plastic
models [23] . . 34
3.14 Refined mesh along the crack front . . . . . . . . . . . .
. . . . . . . . 41
xiii
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xiv
Figure Page
3.15 Effect of n on the stress vs. strain curve using a
Ramberg-Osgood model 42
3.16 Intersection of Ramberg-Osgood curves at o . . . . . . . .
. . . . . . . 44
3.17 Elastic, modified elastic, and Ramberg-Osgood stress vs.
strain curves forn = 10 . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 45
4.1 Model 1 (a/t=0.2, a/c=0.2): Normalized K factor vs. angle
along crackfront . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 51
4.2 Model 2 (a/t=0.2, a/c=0.6): Normalized K factor vs. angle
along crackfront . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 52
4.3 Model 3 (a/t=0.2, a/c=1.0): Normalized K factor vs. angle
along crackfront . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 52
4.4 Model 4 (a/t=0.5, a/c=0.2): Normalized K factor vs. angle
along crackfront . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 53
4.5 Model 5 (a/t=0.5, a/c=0.6): Normalized K factor vs. angle
along crackfront . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 53
4.6 Model 6 (a/t=0.5, a/c=1.0): Normalized K factor vs. angle
along crackfront . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 54
4.7 Model 7 (a/t=0.8, a/c=0.2): Normalized K factor vs. angle
along crackfront . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 54
4.8 Model 8 (a/t=0.8, a/c=0.6): Normalized K factor vs. angle
along crackfront . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 55
4.9 Model 9 (a/t=0.8, a/c=1.0): Normalized K factor vs. angle
along crackfront . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 55
4.10 Elastic singularity element [4] . . . . . . . . . . . . . .
. . . . . . . . . 58
4.11 Model 1 (a/t=0.2, a/c=0.2): Normalized K-factor vs. angle
along crackfront for untied and tied nodes . . . . . . . . . . . .
. . . . . . . . . 58
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xv
Figure Page
4.12 Model 8 (a/t=0.8, a/c=0.6): Normalized K-factor vs. angle
along crackfront for untied and tied nodes . . . . . . . . . . . .
. . . . . . . . . 59
4.13 Model 1 (a/t = 0.2, a/c = 0.2): Reduced vs. full
integration elements . 60
4.14 Model 8 (a/t = 0.8, a/c = 0.6): Reduced vs. full
integration elements . 61
4.15 K-factor results from FEA-Crack Validation Manual [26] . .
. . . . . . 63
4.16 FEM mesh for a flat plate with no symmetry exploited [26] .
. . . . . . 63
4.17 FEM mesh for a flat plate with half symmetry . . . . . . .
. . . . . . . 64
4.18 Model 6 (a/t = 0.5, a/c = 1.0): K-Factor results for half
symmetry model 64
4.19 Model 8 (a/t = 0.8, a/c = 0.6): K-Factor results for half
symmetry model 65
4.20 Model 1 (a/t=0.2, a/c=0.2): h1 vs. angle along the crack
front . . . . . 67
4.21 Model 1 (a/t=0.2, a/c=0.2): h1 vs. angle along the crack
front . . . . . 68
4.22 Model 1 (a/t=0.2, a/c=0.2): h1 vs. angle along the crack
front . . . . . 68
4.23 Model 2 (a/t=0.2, a/c=0.6): h1 vs. angle along the crack
front . . . . . 69
4.24 Model 2 (a/t=0.2, a/c=0.6): h1 vs. angle along the crack
front . . . . . 69
4.25 Model 2 (a/t=0.2, a/c=0.6): h1 vs. angle along the crack
front . . . . . 70
4.26 Model 3 (a/t=0.2, a/c=1.0): h1 vs. angle along the crack
front . . . . . 70
4.27 Model 3 (a/t=0.2, a/c=1.0): h1 vs. angle along the crack
front . . . . . 71
4.28 Model 3 (a/t=0.2, a/c=1.0): h1 vs. angle along the crack
front . . . . . 71
4.29 Model 4 (a/t=0.5, a/c=0.2): h1 vs. angle along the crack
front . . . . . 72
4.30 Model 4 (a/t=0.5, a/c=0.2): h1 vs. angle along the crack
front . . . . . 72
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xvi
Figure Page
4.31 Model 4 (a/t=0.5, a/c=0.2): h1 vs. angle along the crack
front . . . . . 73
4.32 Model 5 (a/t=0.5, a/c=0.6): h1 vs. angle along the crack
front . . . . . 73
4.33 Model 5 (a/t=0.5, a/c=0.6): h1 vs. angle along the crack
front . . . . . 74
4.34 Model 5 (a/t=0.5, a/c=0.6): h1 vs. angle along the crack
front . . . . . 74
4.35 Model 6 (a/t=0.5, a/c=1.0): h1 vs. angle along the crack
front . . . . . 75
4.36 Model 6 (a/t=0.5, a/c=1.0): h1 vs. angle along the crack
front . . . . . 75
4.37 Model 6 (a/t=0.5, a/c=1.0): h1 vs. angle along the crack
front . . . . . 76
4.38 Model 7 (a/t=0.8, a/c=0.2): h1 vs. angle along the crack
front . . . . . 76
4.39 Model 7 (a/t=0.8, a/c=0.2): h1 vs. angle along the crack
front . . . . . 77
4.40 Model 7 (a/t=0.8, a/c=0.2): h1 vs. angle along the crack
front . . . . . 77
4.41 Model 8 (a/t=0.8, a/c=0.6): h1 vs. angle along the crack
front . . . . . 78
4.42 Model 8 (a/t=0.8, a/c=0.6): h1 vs. angle along the crack
front . . . . . 78
4.43 Model 8 (a/t=0.8, a/c=0.6): h1 vs. angle along the crack
front . . . . . 79
4.44 Model 9 (a/t=0.8, a/c=1.0): h1 vs. angle along the crack
front . . . . . 79
4.45 Model 9 (a/t=0.8, a/c=1.0): h1 vs. angle along the crack
front . . . . . 80
4.46 Model 9 (a/t=0.8, a/c=1.0): h1 vs. angle along the crack
front . . . . . 80
4.47 Model 6 (a/t=0.5, a/c=1.0): h1 results for half symmetry
model at n = 15 84
4.48 Model 8 (a/t=0.8, a/c=0.6): h1 results for half symmetry
model at n = 15 85
4.49 Model 1 (a/t=0.2, a/c=0.2): h1 vs. angle for fully plastic
andelastic-plastic models at n=5 . . . . . . . . . . . . . . . . .
. . . . . 87
-
xvii
Figure Page
4.50 Model 1 (a/t=0.2, a/c=0.2): h1 vs. angle for fully plastic
andelastic-plastic models at n=10 . . . . . . . . . . . . . . . . .
. . . . . 87
4.51 Model 1 (a/t=0.2, a/c=0.2): h1 vs. angle for fully plastic
andelastic-plastic models at n=15 . . . . . . . . . . . . . . . . .
. . . . . 88
4.52 Model 2 (a/t=0.2, a/c=0.6): h1 vs. angle for fully plastic
andelastic-plastic models at n=5 . . . . . . . . . . . . . . . . .
. . . . . 88
4.53 Model 2 (a/t=0.2, a/c=0.6): h1 vs. angle for fully plastic
andelastic-plastic models at n=10 . . . . . . . . . . . . . . . . .
. . . . . 89
4.54 Model 2 (a/t=0.2, a/c=0.6): h1 vs. angle for fully plastic
andelastic-plastic models at n=15 . . . . . . . . . . . . . . . . .
. . . . . 89
4.55 Elastic, Ramberg-Osgood, modified elastic, and
modifiedRamberg-Osgood stress vs. strain curves for n = 10 . . . .
. . . . . . 90
4.56 Model 1 (a/t = 0.2, a/c = 0.2): Normalized K-factor vs.
angle alongcrack front . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 91
4.57 Model 1 (a/t = 0.2, a/c = 0.2): h1 vs. angle along the
crack front . . . 92
4.58 Model 1 (a/t = 0.2, a/c = 0.2): h1 vs. c/w for Nasgro and
FEM at n = 15 97
4.59 Model 1 (a/t = 0.2, a/c = 0.2): h1 vs. c/w for Nasgro and
FEM at n = 10 97
4.60 Model 1 (a/t = 0.2, a/c = 0.2): h1 vs. c/w for Nasgro and
FEM at n = 5 98
4.61 Model 3 (a/t = 0.2, a/c = 1.0): h1 vs. c/w for Nasgro and
FEM at n = 15 98
4.62 Model 3 (a/t = 0.2, a/c = 1.0): h1 vs. c/w for Nasgro and
FEM at n = 10 99
4.63 Model 3 (a/t = 0.2, a/c = 1.0): h1 vs. c/w for Nasgro and
FEM at n = 5 99
4.64 Model 4 (a/t = 0.5, a/c = 0.2): h1 vs. c/w for Nasgro and
FEM at n = 15 100
4.65 Model 4 (a/t = 0.5, a/c = 0.2): h1 vs. c/w for Nasgro and
FEM at n = 10 100
-
xviii
Figure Page
4.66 Model 4 (a/t = 0.5, a/c = 0.2): h1 vs. c/w for Nasgro and
FEM at n = 5 101
4.67 Model 6 (a/t = 0.5, a/c = 1.0): h1 vs. c/w for Nasgro and
FEM at n = 15 101
4.68 Model 6 (a/t = 0.5, a/c = 1.0): h1 vs. c/w for Nasgro and
FEM at n = 10 102
4.69 Model 6 (a/t = 0.5, a/c = 1.0): h1 vs. c/w for Nasgro and
FEM at n = 5 102
B.1 Model 1 (a/t=0.2, a/c=0.2): Normalized K factor vs. angle
along crackfront . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 116
B.2 Model 2 (a/t=0.2, a/c=0.6): Normalized K factor vs. angle
along crackfront . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 116
B.3 Model 3 (a/t=0.2, a/c=1.0): Normalized K factor vs. angle
along crackfront . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 117
B.4 Model 4 (a/t=0.5, a/c=0.2): Normalized K factor vs. angle
along crackfront . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 117
B.5 Model 5 (a/t=0.5, a/c=0.6): Normalized K factor vs. angle
along crackfront . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 118
B.6 Model 6 (a/t=0.5, a/c=1.0): Normalized K factor vs. angle
along crackfront . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 118
B.7 Model 8 (a/t=0.8, a/c=0.6): Normalized K factor vs. angle
along crackfront . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 119
B.8 Model 9 (a/t=0.8, a/c=1.0): Normalized K factor vs. angle
along crackfront . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 119
C.9 Model 1: h1 vs. angle along the crack front . . . . . . . .
. . . . . . . . 121
C.10 Model 1: h1 vs. angle along the crack front . . . . . . . .
. . . . . . . . 121
C.11 Model 1: h1 vs. angle along the crack front . . . . . . . .
. . . . . . . . 122
C.12 Model 2: h1 vs. angle along the crack front . . . . . . . .
. . . . . . . . 122
-
xix
Figure Page
C.13 Model 2: h1 vs. angle along the crack front . . . . . . . .
. . . . . . . . 123
C.14 Model 2: h1 vs. angle along the crack front . . . . . . . .
. . . . . . . . 123
C.15 Model 3: h1 vs. angle along the crack front . . . . . . . .
. . . . . . . . 124
C.16 Model 3: h1 vs. angle along the crack front . . . . . . . .
. . . . . . . . 124
C.17 Model 3: h1 vs. angle along the crack front . . . . . . . .
. . . . . . . . 125
C.18 Model 4: h1 vs. angle along the crack front . . . . . . . .
. . . . . . . . 125
C.19 Model 4: h1 vs. angle along the crack front . . . . . . . .
. . . . . . . . 126
C.20 Model 4: h1 vs. angle along the crack front . . . . . . . .
. . . . . . . . 126
C.21 Model 5: h1 vs. angle along the crack front . . . . . . . .
. . . . . . . . 127
C.22 Model 5: h1 vs. angle along the crack front . . . . . . . .
. . . . . . . . 127
C.23 Model 5: h1 vs. angle along the crack front . . . . . . . .
. . . . . . . . 128
C.24 Model 6: h1 vs. angle along the crack front . . . . . . . .
. . . . . . . . 128
C.25 Model 6: h1 vs. angle along the crack front . . . . . . . .
. . . . . . . . 129
C.26 Model 6: h1 vs. angle along the crack front . . . . . . . .
. . . . . . . . 129
C.27 Model 8: h1 vs. angle along the crack front . . . . . . . .
. . . . . . . . 130
C.28 Model 8: h1 vs. angle along the crack front . . . . . . . .
. . . . . . . . 130
C.29 Model 8: h1 vs. angle along the crack front . . . . . . . .
. . . . . . . . 131
C.30 Model 9: h1 vs. angle along the crack front . . . . . . . .
. . . . . . . . 131
C.31 Model 9: h1 vs. angle along the crack front . . . . . . . .
. . . . . . . . 132
C.32 Model 9: h1 vs. angle along the crack front . . . . . . . .
. . . . . . . . 132
-
LIST OF SYMBOLS
Symbol Description
a Crack depthaeff Effective crack length, includes plastic zoneb
Uncracked ligament lengthc Half crack lengthds Increment of length
along the contourG Strain energy release rateh1 Dimensionless
parameter used to calculate Jplh2 Dimensionless parameter used to
calculate CTODh3 Dimensionless parameter used to calculate pn
Strain hardening exponentnj Unit vector components normal to r
Crack tip radiusrc Radius of projected circlet Specimen thicknessw
Half specimen widthx1 Distance along x-axis for projected circlex2
Distance along x-axis for projected circley1 Distance along y-axis
for projected circley2 Distance along y-axis for projected circleA
Crack areaCTOD Crack tip opening displacementE Youngs ModulusIn
Integration constantJ Elastic-plastic fracture parameterJel Elastic
portion of the J-integralJpl Plastic portion of the
J-integralJtotal Sum of Jel and JplK Stress intensity factorKnorm
Normalized K-factorP Applied loadPo Limit loadTi Traction vectorui
Displacement vectorW Specimen width Dimensionless Ramberg-Osgood
material constant Plasticity constraint factorp Load line
displacement
xx
-
xxi
Symbol Description
Strainij Strain tensoro Yield strainref Reference strain Contour
Potential Energy Reference stress factor Poissons ratio Strain
energy density Stressij Stress tensoro Yield stressref Reference
stress Angle of crack tipEPFM Elastic plastic fracture
mechanicsEPRI Electric Power Research InstituteFEA Finite element
analysisFEM Finite element modelLEFM Linear elastic fracture
mechanicsODB Output data base
-
CHAPTER 1
INTRODUCTION
1.1 Fracture Mechanics
Fracture mechanics is the study of the effects of flaws in
materials under load.
Modern fracture mechanics was originated by Griffith [1] in the
1920s when he suc-
cessfully showed that fracture in glass occurs when the strain
energy resulting from
crack growth is greater than the surface energy. In 1948, Irwin
[2] extended Griffiths
strain energy release rate, G, to include metals by accounting
for the energy absorbed
during plastic material flow around the flaw. By 1960, the
fundamental principles of
linear elastic fracture mechanics (LEFM) were in place ([3, 4],
for example).
LEFM is used to predict material failure when response to the
load is elastic
and the fracture response is brittle. LEFM uses the strain
energy release rate G or
the stress intensity factor K as a fracture criterion. K
solutions for many geometries
have been calculated in the past and are widely available [5].
However, the design
parameters for some components violate the assumptions of LEFM.
For example,
high temperatures and limited high stress cycles before
component replacement are
factors that can cause significant plastic deformation and a
ductile failure. In these
cases, where the LEFM approach is not valid, an elastic-plastic
fracture mechanics
(EPFM) approach is required.
EPFM had its beginnings in 1961, when Wells [6] noticed that
initially sharp
cracks in high toughness materials were blunted by plastic
deformation. Wells pro-
posed that the distance between the crack faces at the deformed
tip be used as a
1
-
2measure of fracture toughness. The stretch between the crack
faces at the blunted
tip is known as the crack tip opening displacement (CTOD).
In 1968 Rice [7] developed another EPFM parameter called the
J-integral by
idealizing the elastic-plastic deformation around the crack tip
to be nonlinear elastic.
The J-integral was shown to be equivalent to G for linear
elastic deformation and to
the crack tip opening displacement for elastic-plastic
deformation. During the same
year, Hutchinson [8], Rice, and Rosengren [9] showed that J was
also a nonlinear
stress intensity parameter. The J-integral can be used as an
elastic-plastic or fully
plastic crack growth fracture parameter, much like K is used as
an elastic fracture
parameter.
The J-integral can be calculated using several experimental and
analytical
techniques. The analytical techniques include the Electric Power
Research Institute
(EPRI) estimation scheme, the reference stress method, and
finite element methods.
It should be noted that many of the analytical techniques that
do not directly require
finite element methods were established using finite element
analysis.
1.2 Overview of Research
There are three goals in this research. The primary goal is to
develop three-
dimensional finite element analysis (FEA) J-integral results
using ABAQUS. These
results will be compared to existing solutions. The second goal
is to investigate
the effect of various finite element modelling parameters on the
resulting J-integral.
These parameters include mesh density, element type, symmetry,
and specimen size
effects. The third goal is to compare incremental plasticity
FEAs that utilize a stress
vs. plastic strain table based on a power law hardening material
with the deformation
plasticity solution for a power law material. This comparison
will be made in an
-
3attempt to see if the fully plastic results using a deformation
plasticity model can be
approached by a series of increasing loads in an incremental
plasticity model.
The finite element models (FEMs) used in this research were
three-dimensional
flat plates with surface cracks. The plates contained various
surface crack, height, and
width geometries. Because of the dual symmetry, only one quarter
of each plate was
modeled. Meshes from two different mesh generation programs were
used: mesh 3d
(Faleskog, 1996) and FEA-Crack from Structural Reliability
Technology.
-
CHAPTER 2
TECHNICAL BACKGROUND
In this chapter the J-integral and different J-integral
calculation methods will
be examined. The chapter begins with a discussion of the theory
and mathematical
foundation of the J-integral. Next, two methods for calculating
the J-integral are
discussed: the EPRI Estimation Scheme and the reference stress
method. Both of
these methods can be implemented using hand calculations without
an extensive
fracture mechanics background. In addition, both of these
methods are incorporated
into Nasgro, a fracture mechanics and fatigue crack growth
program. Finally, the
FEA method is used in this research, but a review is not
included here. There are
many excellent texts on the subject of FEA (for example Cook et
al. [10]).
2.1 J-Integral
Rice [7] developed J as a path-independent contour integral by
idealizing
elastic-plastic deformation to be the same as nonlinear elastic
material behavior. In
the arbitrary path around a crack tip (Figure 2.1),
J =
(dy Tiui
xds
), (2.1)
where is the strain energy density, Ti are components of the
traction vector, ui
are the displacement vector components, and ds is an increment
of length along the
4
-
5Figure 2.1 Contour around a crack tip [4]
contour(). The strain energy density and the traction vector
components are
=
ij0
ijdij (2.2)
and
Ti = ijnj, (2.3)
where ij is the stress tensor, ij is the strain tensor, and nj
are unit vector components
normal to .
In idealizing elastic-plastic behavior to be the same as
nonlinear elastic material
behavior, Rice assumed that the material stress versus strain
curve followed a power
law relationship. The Ramberg-Osgood equation is commonly used
to describe the
stress and total strain data for this type of material
response:
o=
o+
(
o
)n, (2.4)
-
6where is the total material strain, o is the reference stress
(normally defined as the
yield strength, but not necessarily the same as the 0.2% offset
yield strength), o is
the strain at the reference stress and is defined by o = o/E.
There are two other
material constants in Equation 2.4. The first of these, , is a
dimensionless constant,
and the second, n, is the strain hardening exponent (n 1).The
J-dominated elastic-plastic stress field contains a singularity of
order
r1
n+1 . For the elastic case (n = 1), this singularity reduces to
r12 in agreement
with the K-dominated field of LEFM. The following two equations
were derived by
Hutchinson [8], Rice and Rosengren [9] and are called the HRR
singularity. The HRR
singularity describes the actual stresses and strains near the
crack tip and within the
plastic zone as
ij = o
(EJ
2oInr
) 1n+1
ij (n, ) (2.5)
and
ij =oE
(EJ
2oInr
) nn+1
ij (n, ) , (2.6)
where In is an integration constant depending on n, r is the
crack tip radius, is
the angle at a point around the contour, andij and
ij are functions of n and .
Equations 2.5 and 2.6 are important because the J-integral
determines the stress
amplitude within the plastic zone. This fact establishes J as a
fracture parameter
under conditions of plastic deformation.
-
7Rice [7] also showed that the J-integral is equivalent to the
energy release rate
in a nonlinear elastic material containing a crack:
J = ddA
(2.7)
where is the potential energy and A is the area of the crack.
For linear elastic
deformation:
Jel = G =K2
E (2.8)
where, for plane strain
E =E
(1 2) , (2.9)
and, for plane stress
E = E. (2.10)
Care should be taken when using the energy release rate with
elastic-plastic
or fully plastic deformation. In an elastic material, the
potential energy is released
as the crack grows. In an elastic-plastic material, a large
amount of strain energy is
used in forming a plastically deformed region around the crack
tip. This energy will
not be recovered when the crack grows, or when the specimen is
unloaded [4].
-
82.2 EPRI Estimation Scheme
The elastic-plastic and fully plastic J-integral estimation
scheme presented by
EPRI [11] is derived from the work of Shih [12] and Hutchinson
[13]. The purpose
of this work was to devise a simple handbook-style procedure for
calculating the J-
integral. This goal was made possible by compiling
nondimensional functions in table
form that could be used to calculate J directly. The
nondimensional functions were
based on FEA results using Ramberg-Osgood materials.
The EPRI procedure computes a total J by summing the elastic and
plastic
J s for various 2D geometries. This is expressed as
Jtotal = Jel + Jpl (2.11)
where Jtotal is the total J , Jel is the elastic portion, and
Jpl is the plastic portion. For
small loads, Jel is much larger than Jpl. For large loads with
significant deformation,
Jpl dominates. This situation is shown graphically in Figure
2.2. As discussed previ-
ously, elastic-plastic behavior is idealized to follow a
nonlinear elastic path along the
stress versus strain curve.
In the EPRI estimation scheme, Jel is calculated utilizing an
adjusted crack
length (aeff ) to compensate for the strain hardening around the
crack tip and is
expressed as
Jel = G =K2(aeff )
E , (2.12)
-
9Figure 2.2 EPRI J-Integral estimation scheme [4]
where K is the stress intensity factor as a function of aeff .
The adjusted crack length
is given by
aeff = a+1
1 + (P/Po)2
1
pi
(n 1n+ 1
)(KIo
)2, (2.13)
where a is the half crack length, P is the applied load, Po is
the limit load per unit
thickness, = 2 for plane stress and = 6 for plane strain, n is
the strain hardening
exponent specific to the material, KI is the elastic stress
intensity factor, and o is
the reference stress (typically the yield strength).
-
10
The fully plastic equations for Jpl, crack mouth opening
displacement (CTOD),
and load line displacement (p), applicable for most specimen
geometries are
Jpl = oobh1
( aW
, n)( P
Po
)n+1, (2.14)
CTOD = oah2
( aW
, n)( P
Po
)n, (2.15)
and
p = oah3
( aW
, n)( P
Po
)n, (2.16)
where and n are a material constants, b is the uncracked
ligament length, W is
the specimen width, and a is the crack length. h1, h2, and h3
are dimensionless
parameters that are a function of geometry and the hardening
exponent n.
The center-cracked and single-edge-notched specimen geometries
have a dif-
ferent form for Jpl. This form reduces the effect of the crack
length to width ratio on
the value of h1, and is
Jpl = ooba
wh1
( aw, n)( P
Po
)n+1, (2.17)
where, for a center-cracked specimen, a is the half crack length
and w is the half
width. Po is the reference or limit load, and is typically the
load at which net cross
section yielding occurs. For center-cracked plate in
tension,
Po = 4co
/3 for plane strain, (2.18)
-
11
and
Po = 2co for plane stress. (2.19)
For a single-edge-crack in tension,
Po = 1.455co for plane strain, (2.20)
and
Po = 1.072co for plane stress. (2.21)
The EPRI handbook includes tabulations of h1, h2, and h3 for
various n values
and geometries. These values were calculated using results from
a finite element pro-
gram called INFEM [11]. INFEM was developed for the specific
purpose of analyzing
fully plastic cracks and utilizes incompressible elements in the
model formulation.
Further details of the finite element formulation have been
published by Needleman
and Shih [14].
In 1999 McClung, Chell, Lee, and Orient [15] extended the
original EPRI work
to include fully plastic J solutions for 3D geometries. This
work was performed using
3D finite element models. The meshes for these models were
constructed using eight-
noded brick elements in ANSYS 5.0. A typical mesh is shown in
Figure 2.3. A close
up view of the crack front may be seen in Figure 2.4.
-
12
Figure 2.3 Sample of finite element mesh used by McClung et al.
[15]
Figure 2.4 Close up of the finite element mesh around the crack
front used byMcClung et al. [15]
-
13
Although the meshes were created in ANSYS, ABAQUS was used to
perform
the analysis of the finite element models. The version of ABAQUS
used for this work
was only capable of performing an incremental plasticity
analysis. An EPRI-type
scheme was used to separate the elastic and plastic J values.
The fully plastic values
for h1 were then calculated using
h1 =Jpl
oot(
o
)n+1 . (2.22)A combination of three different a/t (0.2, 0.5,
0.8) and a/c (0.2, 0.6, and 1.0)
ratios were tabulated. The specimen geometry ratios were kept
constant for all models
at h/c = 4 and c/w = 0.25. The values of h1 were calculated for
strain hardening
exponents of n = 5, 10, and 15, and can be found in Tables 2.1,
2.2, and 2.3.
In 2004 Lei [17] duplicated part of the work performed by
McClung et al. [15]
by performing elastic and elastic-plastic finite element
analyses for plates containing
semi-elliptical surface cracks under tension. The models
contained surface cracks with
the same a/t and a/c ratios used by McClung et al. [15]. For the
elastic analysis,
Jel results were generated and converted into K using Equation
2.8. These K results
were then compared with Newman-Raju stress-intensity factor
calculations [18]. The
elastic-plastic results for strain hardening values of n = 5 and
n = 10 were presented
in terms of h1. These h1 results are reproduced in Tables 2.4
and 2.5 and compare
well with McClung et al. for most geometries. The comparison
with McClung et
al. and the current results are presented in more detail in
Chapter 4.
-
14
Table2.1
McC
lunget
al.h1values
intension,n=15
[15]
a/t
a/c
0
9
18
27
36
45
54
63
72
81
90
0.20
0.20
0.223
0.370
0.608
0.821
1.001
1.148
1.310
1.447
1.560
1.623
1.644
0.20
0.60
0.356
0.465
0.622
0.698
0.774
0.823
0.875
0.915
0.948
0.971
0.981
0.20
1.00
0.389
0.503
0.628
0.638
0.659
0.653
0.657
0.657
0.653
0.646
0.646
0.50
0.20
4.085
7.615
11.602
14.488
17.057
18.798
20.228
21.434
22.129
22.212
22.309
0.50
0.60
3.336
4.808
6.564
7.048
7.697
7.939
8.000
8.021
8.019
7.922
7.881
0.50
1.00
2.774
3.738
4.750
4.759
4.932
4.891
4.816
4.613
4.407
4.243
4.198
0.80
0.20
37.609
63.511
82.404
91.460
99.198
92.725
90.097
88.292
89.548
95.447
98.941
0.80
0.60
17.660
25.890
32.760
30.172
37.828
34.002
31.546
28.972
28.224
29.095
30.806
0.80
1.00
12.667
17.231
20.882
19.281
23.029
21.124
18.005
16.467
14.557
15.003
15.533
Table2.2
McC
lunget
al.h1values
intension,n=10
[15]
a/t
a/c
0
9
18
27
36
45
54
63
72
81
90
0.20
0.20
0.198
0.320
0.523
0.703
0.863
0.996
1.133
1.250
1.345
1.398
1.416
0.20
0.60
0.324
0.416
0.544
0.604
0.671
0.715
0.759
0.792
0.820
0.839
0.847
0.20
1.00
0.358
0.450
0.550
0.553
0.571
0.565
0.569
0.566
0.562
0.557
0.556
0.50
0.20
2.539
4.512
6.957
8.841
10.665
11.979
13.048
13.953
14.546
14.712
14.811
0.50
0.60
2.319
3.205
4.264
4.561
4.967
5.128
5.189
5.209
5.231
5.200
5.186
0.50
1.00
2.007
2.599
3.210
3.179
3.272
3.218
3.168
3.040
2.921
2.827
2.804
0.80
0.20
17.731
29.512
39.550
43.774
49.600
46.576
44.854
43.844
43.706
45.805
47.496
0.80
0.60
9.688
13.685
16.725
15.850
19.174
17.318
15.896
14.776
14.068
14.323
14.800
0.80
1.00
7.242
9.472
11.077
10.311
11.898
11.108
9.239
8.398
7.536
7.533
7.625
-
15
Table2.3
McC
lunget
al.h1values
intension,n=5[15]
a/t
a/c
0
9
18
27
36
45
54
63
72
81
90
0.20
0.20
0.164
0.252
0.407
0.544
0.676
0.789
0.897
0.988
1.062
1.103
1.117
0.20
0.60
0.286
0.352
0.441
0.480
0.533
0.570
0.605
0.631
0.652
0.666
0.672
0.20
1.00
0.321
0.383
0.446
0.440
0.452
0.446
0.447
0.442
0.439
0.435
0.435
0.50
0.20
1.325
2.139
3.357
4.384
5.480
6.371
7.136
7.764
8.222
8.428
8.516
0.50
0.60
1.502
1.916
2.412
2.548
2.764
2.860
2.917
2.938
2.968
2.973
2.976
0.50
1.00
1.377
1.658
1.931
1.867
1.894
1.839
1.800
1.730
1.677
1.637
1.630
0.80
0.20
7.224
11.273
15.743
18.150
21.870
21.460
20.632
20.051
18.960
18.993
19.369
0.80
0.60
4.983
6.449
7.582
7.389
8.421
7.750
7.034
6.695
6.266
6.114
6.178
0.80
1.00
3.910
4.728
5.251
4.849
5.142
4.775
4.080
3.734
3.397
3.285
3.270
-
16
Table2.4
Leih1values
intension,n=5[17]
a/t
a/c
0
9
18
27
36
45
54
63
72
81
90
0.2
0.2
0.179
0.3003
0.4572
0.5949
0.7223
0.8389
0.9556
1.053
1.132
1.177
1.196
0.2
0.6
0.3151
0.3958
0.4897
0.5236
0.5777
0.6134
0.6517
0.6778
0.7007
0.7113
0.7177
0.2
10.3575
0.4368
0.5011
0.487
0.5004
0.4895
0.4912
0.4858
0.4863
0.4825
0.4839
0.5
0.2
1.343
2.327
3.495
4.524
5.455
6.265
7.039
7.616
8.089
8.338
8.466
0.5
0.6
1.564
2.03
2.524
2.633
2.829
2.897
2.974
2.98
2.993
2.972
2.981
0.5
11.44
1.78
2.042
1.95
1.97
1.878
1.837
1.762
1.722
1.676
1.672
0.8
0.2
6.723
11.78
17.25
19.57
21.2
21.43
21.16
20.35
19.57
18.92
18.86
0.8
0.6
5.388
6.938
8.317
8.145
8.185
7.656
7.178
6.633
6.396
6.263
6.29
0.8
14.119
5.075
5.678
5.187
5.014
4.482
4.106
3.701
3.505
3.396
3.406
Table2.5
Leih1values
intension,n=10
[17]
a/t
a/c
0
9
18
27
36
45
54
63
72
81
90
0.2
0.2
0.2169
0.3749
0.5714
0.7474
0.9066
1.046
1.186
1.302
1.397
1.451
1.475
0.2
0.6
0.3554
0.4508
0.5774
0.633
0.7033
0.7518
0.7987
0.8338
0.8627
0.8786
0.8862
0.2
10.3995
0.5009
0.5964
0.5995
0.6222
0.6178
0.6211
0.619
0.6193
0.6172
0.618
0.5
0.2
2.533
4.723
6.907
8.859
10.34
11.45
12.45
13.09
13.66
13.91
14.11
0.5
0.6
2.379
3.254
4.285
4.588
4.954
5.071
5.164
5.142
5.122
5.07
5.077
0.5
12.055
2.682
3.26
3.253
3.341
3.231
3.148
3.012
2.907
2.818
2.799
0.8
0.2
16.01
29.51
43.27
46.42
47.17
45.94
45.71
45.34
45.11
44.93
45.3
0.8
0.6
11.08
15.38
19.3
19.13
19.01
17.43
16.15
15.23
15.31
15.48
15.67
0.8
17.925
10.68
12.74
12.13
11.77
10.45
9.369
8.439
8.159
8.23
8.408
-
17
2.3 Reference Stress Method
As discussed previously, the EPRI J estimation scheme assumes
that the mate-
rial has a power law stress-strain curve. There are many
materials that do not exhibit
this type of response. In 1984 Ainsworth [19] devised a method
for calculating J that
did not depend on the materials behavior following a power law.
This approach is
called the reference stress method. The reference stress is
defined as
ref =
(P
Po
)o (2.23)
where P is the applied load, Po is the same limit load defined
previously in the EPRI
research [11], and o is the yield strength.
The reference strain, ref , is defined as the uniaxial strain
corresponding to
ref . By inserting ref and ref into the Ramberg-Osgood equation
2.4, it can be
modified to the following form:
refo
=refo
+
(refo
)n. (2.24)
Using Equations 2.23 and 2.24, Equation 2.14 can be altered to
the form
Jpl = refbh1
(ref refo
o
). (2.25)
Equation 2.25 still contains the variable h1, a function of n -
same h1 used in the EPRI
equations discussed in the previous section. Ainsworths approach
was to choose Po
in such a way that the dependence of h1 on n was minimized. For
certain values of
-
18
Po, he found that h1 was relatively constant for n 20. As a
result,
h1 = h1( aw, 1)
(2.26)
where h1 is the average h1 for a range of ns and h1(aw, 1)is the
h1 for n equal to one.
The fully plastic solution at n = 1 is identical to the elastic
solution using a Poissons
ratio of = 0.5,
K2 (a) = bh1
( aw, 1)2ref (2.27)
where =1 for plane stress and =0.75 for plane strain. By
substituting Equation
2.27 and using the conditions that establish Equation 2.26, the
Jpl expression becomes
Jpl =KIE
(Erefref
1). (2.28)
The previously discussed McClung et al. [15] finite element
results were used
to develop another reference stress method. This reference
stress algorithm is used
within Nasgro. Nasgro is a crack propagation and fracture
mechanics program devel-
oped by NASA and the Southwest Research Institute.
-
CHAPTER 3
RESEARCH PROCEDURE
In this chapter, the technical approach used for this thesis is
presented. The
chapter begins with a discussion of the finite element modeling
including mesh gen-
eration. Next, the analysis procedure for the FEMs is discussed.
Then, the work
duplicated by other researchers is reviewed, and any material
properties or model
parameters specific to a geometry set are looked at as well.
This duplication of other
researchers work was to validate the methodology used by
ensuring that the J-integral
analysis could be performed properly. The chapter concludes with
a discussion of the
general material properties used.
3.1 Finite Element Modeling
The finite element analysis program ABAQUS was used to calculate
the K-
factors and J-integrals for a variety of specimen geometries.
The models were created
with quarter symmetry to reduce the number of nodes and elements
(hence, the
computational time) of each model.
Unless otherwise specified, the FEMs consisted of reduced
integration, 20-
noded brick elements specified as C3D20R within ABAQUS. Reduced
integration
elements are recommended in the ABAQUS User Manuals [21] for
plastic and large
strain elastic models. Full integration elements tend to be
overly stiff and the results
may oscillate. A reduced integration element has a softening
effect on the stiffness
that improves the finite element results.
The elements around the crack tip were also of type C3D20R.
However, the
elements were modified by collapsing the brick element into a
wedge (Figure 3.1).
19
-
20
When the elements were degenerated, the mid-side nodes were not
moved, and the
collapsed nodes were left untied (Figure 3.2). This allows for
movement of the nodes
as the element is deformed and produces a 1/r strain
singularity, which duplicates
the actual crack tip strain field in the plastic zone [4].
Figure 3.1 Degeneration of elements around crack tip [4]
Figure 3.2 Plastic singularity element [4]
-
21
3.1.1 Mesh Generation
Two different programs were used to generate finite element
meshes. The
first, called mesh3d scp [20] by Faleskog, is available as
freeware. Many early finite
element meshes in this work were generated with mesh3d scp.
However, this program
has serious limitations. Therefore, a second mesh generation
program, FEA-Crack,
was also used. This software is commercially available from
Structural Reliability
Technology, Colorado.
3.1.1.1 mesh3d scp. The mesh generation program mesh3d scp
generates
a one-quarter model of a surface cracked plate. The program
assumes that both the
geometry and the load possess planes of symmetry. This program
divides the model
into three zones, as shown in Figure 3.3. The element density in
each zone is altered by
changing variables in the mesh3d scp input file. The node and
element numbering in
each zone is controlled such that the application of boundary
conditions and external
loads is simplified. The meshes used to investigate the fully
plastic volume and
location were created using mesh3d scp (Figures 3.8 - 3.11).
The program mesh3d scp requires an iterative approach. The set
of input
variables for the program input file are changed, the program
generates a mesh, the
mesh is plotted and then examined graphically. This process is
repeated until a
satisfactory mesh by appearance is created. This program is
capable of generating
good meshes for some geometries. However, this program does not
work well for other
specimen geometries. For these geometries, mesh3d scp was found
to produce a bad
mesh, no mesh, or, in the worst cases, a mesh with errors.
This program was originally written to generate meshes for an
earlier version
of ABAQUS. This makes it necessary to modify the ABAQUS input
files created by
-
22
Figure 3.3 Zones created in the mesh by mesh3d scp [20]
mesh3d scp to make them compatible with recent releases of
ABAQUS(V6.5). The
file modifications used for the models in this thesis are listed
in Appendix A.
3.1.1.2 FEA-Crack. The second mesh generation program utilized
for this
research is called FEA-Crack. FEA-Crack is more robust than
mesh3d scp and does
not require the same iterative approach on the users part. The
mesh density in the
area around the crack can be controlled by adjusting the program
settings. Also, the
generated model may be viewed immediately, and required changes
to the ABAQUS
input file are minimal. A mesh created using FEA-Crack is shown
in Figures 3.4 and
3.5.
-
23
Figure 3.4 Mesh created using FEA-Crack
Figure 3.5 Close up of mesh from Figure 3.4 created using
FEA-Crack
-
24
3.2 Analysis Procedure
Each FEM analyzed for this research contained 5 contours around
the crack
tip, as seen in Figure 3.6. The results for the first contour
are generally considered
to be less accurate than the other contours because of numerical
inaccuracy [21]. For
this reason, the K-factor and J-integral data from all of the
contours, except the first,
were averaged [17]. These average K-factor and J-integral were
used for all further
calculations and comparisons.
The FEMs contained multiple node sets along the crack front. A
node set
is a group of nodes that have been associated as a group within
ABAQUS. The
number of node sets depended on the physical size of the crack
front. Each of these
particular node sets contain a number of nodes with the same
coordinates. In the
untied condition, one node in each node set is constrained so
that it can move in only
Figure 3.6 Contours (semi-circular rings) around the crack
tip
-
25
one or two directions (it stays on the plane of symmetry). The
direction of constraint
depends on the symmetry plane. These constrained nodes are
listed in another node
set called crack front nodes, which will be significant later.
The other nodes in
each node set are not constrained.
ABAQUS generates values for the K-factor and J-integral at each
of the node
sets along the crack front. An Excel macro was written to allow
for examination of
the variation of the K-factor and J-integral values generated
along the crack front.
The program was written to calculate the angle, as projected
onto a circle, at each
crack front node. The macro first finds and records the
constrained nodes found in
the node set crack front nodes, which is located in the ABAQUS
input file. The
coordinates for each of these crack front nodes are then
retrieved from the input file.
The crack coordinates are then mapped onto a circle, as shown in
Figure 3.7. The
equation for the projection circle is shown below as
x22 + y22 = r
2c . (3.1)
Two facts should be noted from Figure 3.7. First, y1 is equal to
y2. Second, the
circle radius, rc, is equal to the crack depth, a. Both of the
previous statements are
valid as long as a/c 1, which is the case for this research.
Using this information,Equation 3.1 can now be rearranged into the
form
x2 =a2 y21. (3.2)
Once x2 is known, the angle, , may be calculated using
= tan1(x2y1
). (3.3)
-
26
Figure 3.7 Coordinate scheme for mapping crack face angles
With known, the variation of the K-factor and J-integral values
can be mapped
along the crack front contour.
3.3 J-Integral Convergence
Two quantities were initially tested to ensure that the fully
plastic FEM results
had converged. The first quantity was load. The second involved
the fully plastic
zone specified for the FEMs.
3.3.1 Load
The applied load in the FEMs was adjusted until the resulting
J-integral values
did not change with an increase in load. The final load step was
also examined for
each model to ensure that the entire load was not applied. In
cases where the entire
-
27
specified load was applied, the load was increased, and the FEM
was analyzed again.
This ensured that the specified element set became fully
plastic. The fully plastic
option in ABAQUS utilizes a Ramberg-Osgood material model and
ends the analysis
when the observed strain for the selected element set exceeds
the offset yield strain
by ten times, assuming the load or maximum number of increments
have not been
reached. Also, to ensure sufficient steps in the model, the
loads were set such that at
least 33% of the specified load was applied to the model.
3.3.2 Fully Plastic Zone
The volume and location effect of the specified fully plastic
element set was
examined for two reasons. First, it was necessary to determine
how much of the
specimen must become fully plastic before the J-integral
converged. The second
reason was to simplify the model generation. The two mesh
generation programs used
in this research, mesh3d scp and FEA-Crack, established
convenient, but different,
elements sets for use as fully plastic.
The fully plastic results were generated using the *FULLY
PLASTIC command
within ABAQUS. This command requires the specification of an
element set which
is monitored for the fully plastic condition discussed
previously. Several fully plastic
element sets, or zones, were tested and the results compared.
The fully plastic element
sets used in this research are defined as follows:
LayerCR - Contains elements around the crack tip, (Figure
3.8);
-
28
Figure 3.8 Fully plastic element set consisting of the elements
around the cracktip
Partial Layer 1 - Contains elements in the first layer of the
model, but doesnot contain the elements closest to the crack tip,
(Figure 3.9);
Layer 1 - Contains the elements in the ligament plus the
elements found inLayerCR, (Figure 3.10);
Layer 2 - Contains elements in the first and second layers of
the model, butdoes not contain the elements closest to the crack
tip (Figure 3.11).
-
29
Figure 3.9 Fully plastic element set consisting of part of layer
1
Figure 3.10 Fully plastic element set consisting of layer 1
-
30
Figure 3.11 Fully plastic element set consisting of partial
layers 1 and 2
-
31
3.4 Comparison to Other Work
A series of models with different crack ratios and specimen
sizes were generated.
These models contained geometric parameters (e. g. a/t, a/c,
etc.) identical to those
used by other researchers. The current results were compared to
previous work with
the intent of validating the FEMs and methods used for this
research.
3.4.1 Kirk and Dodds
FEMs were generated with the same geometries and material
properties used
by Kirk and Dodds in 1992 [23]. These geometries are shown in
Figure 3.12. The
mesh generation program mesh3d scp was used to generate models
for all three cracks
defined by Kirk and Dodds. The models consisted of 20-noded
brick elements with
reduced integration. The number of nodes and elements in each
model is listed in
Table 3.1.
Table 3.1 Number of nodes and elements in the duplication of the
Kirk and Dodds[23] geometries
Crack 1 Crack 2 Crack 3Nodes 16,597 12,227 12,227
Elements 3562 2593 2593
-
32
Figure 3.12 Geometries used by Kirk and Dodds for estimating the
J-Integral [23]
-
33
These FEMs were analyzed to find Jtotal using an elastic-plastic
analysis.
ABAQUS utilizes an incremental plasticity model for this type of
analysis, and re-
quires a table of true stress versus plastic strain. The
material properties for these
models were derived from Figure 3.13 and are listed below:
E = 3.00 104 kpsi = 0.3 Tangent Modulus = 3.57 102 kpsi Initial
Yield = 80 kpsi.
These properties were used to calculate the total and elastic
strains at the yield stress
and an arbitrary stress, selected to be much higher than the
applied stress. This
arbitrarily large stress was used as an input because ABAQUS
does not explicitly
allow the tangent modulus to be given. The plastic strains
required by ABAQUS
were found by subtracting the total and elastic strains. Table
3.2 shows the calculated
strains.
Table 3.2 Incremental plasticity values for the Kirk and Dodds
models
, kpsi total strain elastic strain plastic strain80 2.67E-03
2.67E-03 0.00E+00200 3.36E-01 6.67E-03 3.29E-01
-
34
Figure 3.13 Stress vs. strain curve for Kirk and Dodds
elastic-plastic models [23]
-
35
3.4.2 McClung et al. [15]
The mesh generation program FEA-Crack was used to generate
models for all
nine geometries defined in the research performed by McClung et
al. (Table 3.3). Two
sets of models were generated. The first set contained a coarse
mesh. The second set
utilized a more refined mesh around the crack front. The McClung
et al. geometries
were analyzed as elastic, fully plastic and incrementally
plastic models. The elastic
and fully plastic analyses were performed using both the coarse
and refined meshes.
The incrementally plastic models were analyzed using only the
coarse meshes.
In the elastic FEM analysis, the K factor was found in two ways.
First,
ABAQUS was used to calculate K directly. Second, ABAQUS was used
to find
the elastic J , and then Equation 2.8 was used to calculate K.
These results were
compared to K factors calculated using equations from Newman and
Raju [24]. The
Newman-Raju solution is given in Equations 3.4 - 3.9.
KI =
pi
(a
Q
)[M1 +M2
(at
)2+M3
(at
)4]gffw, (3.4)
Q = 1 + 1.464(ac
)1.65, (3.5)
-
36
Table3.3
McC
lunget
al.fullyplasticgeom
etries
Model1
Model2
Model3
Model4
Model5
Model6
Model7
Model8
Model9
a/t
0.2
0.2
0.2
0.5
0.5
0.5
0.8
0.8
0.8
a/c
0.2
0.6
10.2
0.6
10.2
0.6
1h/c
44
44
44
44
4c/w
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
t1
11
11
11
11
a0.2
0.2
0.2
0.5
0.5
0.5
0.8
0.8
0.8
c1
0.33
0.2
2.5
0.83
0.5
41.33
0.8
w4
1.33
0.8
103.33
216
5.33
3.2
h4
1.33
0.8
103.33
216
5.33
3.2
-
37
M1 = 1.13 0.09(ac
),
M2 = 0.54 + 0.890.2+(ac ) ,
M3 = 0.5 10.65+ac+ 14
(1 a
c
)24,
(3.6)
g = 1 +
[0.1 + 0.35
(at
)2](1 sin )2 , (3.7)
f =
[(ac
)2cos2 + sin2
]1/4, (3.8)
fw =
[sec
(pic
2w
a
t
)]1/2, (3.9)
where KI is the K factor at a given angle, is the applied
stress, a is the crack depth,
Q is factor applicable for ac 1, c is the half crack width, t is
the specimen thickness,
is the angle, as previously defined in Figure 3.7, along the
crack front, and w is the
half specimen width.
3.4.3 Lei [17]
In 2004, Lei performed elastic and elastic-plastic J analyses on
models with
the same crack geometries used by McClung et al. [15]. He also
maintained a spec-
imen geometry ratio of c/w = 0.25. However, Lei deviated from
the McClung et
-
38
al. geometries by fixing the ratio h/w at four to one instead of
one to one. Lei also
fixed c, therefore fixing w and h, and varied a and t.
Lei used ABAQUS to perform the analyses on his models. He used
the *CON-
TOUR INTEGRAL command within ABAQUS to generate J-integral
results for
fifteen contours around the crack tip. The averages of these
contours, excluding the
first, were presented. Lei found that the deviation of data from
any one contour is
less than 5% of the average value.
Lei used consistent material properties in his analyses. The
properties for the
elastic analyses were set at E = 500 MPa and = 0.3. The
elastic-plastic analyses
used the Ramberg-Osgood stress-strain relationship (Equation
2.4), where o = 1.0
MPa, = 1, and n = 5 and 10. For all analyses, Lei used the Mises
yield criterion
and small strain isotropic hardening.
3.4.4 Nasgro Computer Program
Current FEM results were compared with the results produced
using the crack
propagation and fracture mechanics section of Nasgro. Nasgro is
a fracture mechanics
and fatigue crack growth program developed by NASA and the
Southwest Research
Institue. The same Ramberg-Osgood material properties used for
the McClung ge-
ometries were duplicated for this comparison. The different
geometries analyzed using
Nasgro are shown in Table 3.4.
-
39
Table 3.4 Geometries for Nasgro comparison and width effect
investigation
Model a a/t c c/w w1 0.2 0.2 1.0 0.25 4.001a 0.2 0.2 1.0 0.50
2.001b 0.2 0.2 1.0 0.67 1.493 0.2 0.2 0.2 0.25 0.803a 0.2 0.2 0.2
0.50 0.403b 0.2 0.2 0.2 0.67 0.304 0.5 0.5 2.5 0.25 10.04a 0.5 0.5
2.5 0.33 7.584b 0.5 0.5 2.5 0.40 6.256 0.5 0.5 0.5 0.25 2.006a 0.5
0.5 0.5 0.33 1.526b 0.5 0.5 0.5 0.40 1.25
-
40
3.5 Mesh Refinement
Two sets of finite element models were constructed using the
McClung et
al. geometries [15] found in Table 3.3. The first set contained
a coarse mesh refinement
along the crack front. The coarse mesh refinement along the
crack front can be seen
in Figure 3.5. The second set of models had three times more
elements around the
crack front (Figure 3.14). Table 3.5 shows the number of crack
front nodes in the
coarse and refined meshes.
3.6 Finite Size Effects
FEMs were generated to test the effect of specimen height and
width on the
J-integral. The a/t ratios of 0.2 and 0.5, and the a/c ratios of
0.2 and 1.0 were
used in this analysis. The height effect models utilized the
crack ratios for Model 1
(a/t = 0.2, a/c = 0.2), Model 4 (a/t = 0.5, a/c = 0.2), and
Model 9 (a/t = 0.8, a/c =
1.0). The width effect models utilized the same model geometries
used in the Nasgro
J-comparison work (Table 3.4).
Table 3.5 Number of crack front nodes in the coarse and refined
meshesModel a/t a/c Coarse Refined1 0.2 0.2 31 912 0.2 0.6 17 493
0.2 1.0 17 494 0.5 0.2 45 1335 0.5 0.6 17 496 0.5 1.0 17 497 0.8
0.2 73 2658 0.8 0.6 31 919 0.8 1.0 17 49
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41
Figure 3.14 Refined mesh along the crack front
3.7 Material Properties
The material properties, unless otherwise specified, were based
on a structural
steel. These are the same material properties used Natarajan
[22] for some FEMs
in his thesis work involving J-integral solutions. Two different
yielding models were
used in this research. The first was the Ramberg-Osgood
deformation plasticity
model. The second was an incremental plasticity method requiring
a table of and
pl. The elastic material properties for each model depended on
the yielding scheme
used for the FEA.
3.7.1 Deformation Plasticity
The following material properties were used with the
*Deformation Plasticity
command in ABAQUS:
-
42
E = 30.0 106 psi = 0.3 o = 40.0 103 psi = 0.5 n = 5, 10, and
15
where E is Youngs modulus, is Poissons ratio, o is yield or
reference stress,
is a dimensionless constant as described in Equation 2.4, and n
is the hardening
exponent. The effect of n on the stress vs. strain curves
modelled using the Ramberg-
Osgood equation is shown in Figure 3.15. Notice that the smaller
n is, the greater
the hardening slope
Figure 3.15 Effect of n on the stress vs. strain curve using a
Ramberg-Osgoodmodel
-
43
3.7.2 Incremental Plasticity
The incremental plasticity models, with the exception of the
Kirk and Dodds
comparison work, were generated using the Ramberg-Osgood
equation,
o=
0+
(
0
)n, (3.10)
shown again for convenience. The material properties listed in
the previous section
were used to generate the a new Youngs modulus and a table of
stress vs. plastic
strain for use in ABAQUS. The Youngs modulus, E = 30.0 106 psi,
used for thefully plastic analyses was not used to derive the
stress vs. plastic strain tables for
ABAQUS. It was replaced by a secant modulus,E, as shown in
Equation 3.11:
E =
0o (1 + )
. (3.11)
This value of Youngs modulus was selected because it intersects
the Ramberg-Osgood
curve at the fully plastic reference stress, o = 40.0 103 psi
(Figure 3.16).
-
44
Figure 3.16 Intersection of Ramberg-Osgood curves at o
-
45
Using this scheme, the elastic strain, and therefore the
J-integral, will be
underestimated at low stresses (Figure 3.17). But, for
sufficiently high stresses, the
elastic strain becomes overwhelmed by the plastic strain, making
the error negligible.
The reference strain can now be expressed as
o =0E. (3.12)
Figure 3.17 Elastic, modified elastic, and Ramberg-Osgood stress
vs. strain curvesfor n = 10
-
46
Multiplying both sides of Equation 3.10 by o and substituting
Equation 3.12 yields:
=E
+ o
(
0
)n. (3.13)
Equation 3.13 can be divided into the elastic and plastic
strains as
el =E, (3.14)
and
pl = o
(
o
)n. (3.15)
The plastic strains at different stresses were then calculated
for use with the *PLAS-
TIC command in ABAQUS for incremental plasticity analyses.
In summary, elastic-plastic material properties used in this
research are based
on a modified Youngs modulus. This modification makes it
possible to generate
incremental plasticity models that exhibit the same yield stress
for all ns. The elastic
properties used for the incremental plasticity analyses areE =
20 106 and = 0.3.
The stress vs. plastic strain values used with the *Plastic
command in ABAQUS are
shown in Tables 3.6 - 3.8.
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47
Table 3.6 Stress vs. plastic strain data at n = 15, used for
ABAQUS models
Stress Plastic Strain40000 0.00066741200 0.00103942400
0.00159843600 0.00242844800 0.00364946000 0.00542547200
0.00798248400 0.01163349600 0.01679750800 0.02404252000
0.03412453200 0.04804954400 0.06714255600 0.09313956800
0.12830258000 0.17556159200 0.2386960400 0.32252561600
0.43323162800 0.57863364000 0.76861465200 1.0156
Table 3.7 Stress vs. plastic strain data at n = 10, used for
ABAQUS models
Stress Plastic Strain40000 0.044000 0.00172916248400
0.00448552800 0.01070651357200 0.02383796261600 0.05001680566000
0.09971217470400 0.19012333374800 0.34859793279200 0.61739149783600
1.060160459
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48
Table 3.8 Stress vs. plastic strain data at n = 5, used for
ABAQUS models
Stress Plastic Strain40000 0.044000 0.00107448400 0.00172952800
0.00267257200 0.00398661600 0.00577466000 0.00815370400
0.01125874800 0.01524579200 0.02028883600 0.02658588000
0.03435892400 0.0438596800 0.055333101200 0.069105105600
0.085493110000 0.104851114400 0.127567118800 0.15406123200
0.184783127600 0.220223132000 0.260903136400 0.307384140800
0.360265145200 0.420186149600 0.487828154000 0.563913158400
0.649209162800 0.744528167200 0.850727171600 0.968713176000
1.099441
-
CHAPTER 4
RESULTS
This chapter begins with a discussion of the results for various
fully plastic
element sets. Next, models generated for parameters used by Kirk
and Dodds [23]
are compared to published results. McClung, Lei, and Newman-Raju
data are then
compared to current FEM results. Finally, the effects of the
specimen size on the
J-integral are examined, and the h1 values for various specimen
widths are compared
with Nasgro results.
4.1 Fully Plastic Zone
The mesh generation program mesh3d scp was used to generate FEMs
for all
four of the fully plastic zones described in Chapter 3. The same
mesh was used for
each model. Only the specified fully plastic element set was
changed for the different
models. It was found that the J-integral was identical for all
of the described zones.
Therefore, only Partial Layer 1 was used in later fully plastic
models was used for
the FEA-Crack meshes, and LayerCR was used for any fully plastic
meshes produced
with mesh scp.
4.2 Kirk and Dodds Incremental Plasticity
The results for models generated per the Kirk and Dodds
geometries are shown
in Table 4.1. The results compared quite well to the published
data. The maximum
difference between the current results and the published data
was 2.9%. It should be
noted that this excellent agreement in results was obtained even
though the meshes
49
-
50
Table 4.1 Comparison of FEM results to Kirk and Dodds values
Crack J (in-lb) Kirk and Dodds % Differencefrom FEM J
(in-lb)
1 30.9 0.749 0.732 2.31 90 0.892 0.867 2.92 30.9 2.055 2.014
2.02 90 0.892 0.867 2.933 30.9 2.077 2.046 1.53 90 3.207 3.173
1.7
used by Kirk and Dodds contained approximately 25% the number of
nodes and
elements used in this research.
4.3 McClung and Lei Comparisons
Elastic, fully plastic, and incremental plasticity FEA results
for the McClung
et al. geometries are presented in this section. The elastic
results are compared to the
Newman-Raju [24] calculations, and graphical trends are noted in
the comparison of
Leis [17] elastic results. The fully plastic data are compared
to the tabular data of
McClung et al. [15] and Lei [17]. The effects of mesh refinement
are discussed for
both the elastic and fully plastic FEMs. Finally the incremental
plasticity and fully
plastic FEA results are compared.
-
51
Figure 4.1 Model 1 (a/t=0.2, a/c=0.2): Normalized K factor vs.
angle along crackfront
4.3.1 Elastic Analysis
The K factors obtained from the FEMs with the McClung geometries
were
normalized using
Knorm =KI
pi aQ
, (4.1)
from Newman and Raju [25]. The results of the elastic FEM models
and the Newman
and Raju [24] calculations are presented in Figures 4.1-4.9.
-
52
Figure 4.2 Model 2 (a/t=0.2, a/c=0.6): Normalized K factor vs.
angle along crackfront
Figure 4.3 Model 3 (a/t=0.2, a/c=1.0): Normalized K factor vs.
angle along crackfront
-
53
Figure 4.4 Model 4 (a/t=0.5, a/c=0.2): Normalized K factor vs.
angle along crackfront
Figure 4.5 Model 5 (a/t=0.5, a/c=0.6): Normalized K factor vs.
angle along crackfront
-
54
Figure 4.6 Model 6 (a/t=0.5, a/c=1.0): Normalized K factor vs.
angle along crackfront
Figure 4.7 Model 7 (a/t=0.8, a/c=0.2): Normalized K factor vs.
angle along crackfront
-
55
Figure 4.8 Model 8 (a/t=0.8, a/c=0.6): Normalized K factor vs.
angle along crackfront
Figure 4.9 Model 9 (a/t=0.8, a/c=1.0): Normalized K factor vs.
angle along crackfront
-
56
Significant increases in the K-factor at the surface and/or
depth were observed
in all of the models. Only Models 8 (a/t = 0.8, a/c = 0.6) and 9
(a/t = 0.8, a/c = 1.0)
do not have large increases in the K-factor at the free surface.
Models 1, 4, and 7
(all with a/c = 0.2) are the only FEMs that do not contain the
same K-factor spike
repeated in the depth (Table 4.2).
When the surface and depth spikes are disregarded, the ABAQUS
results com-
pared very reasonably to the normalized K-factors calculated per
the Newman and
Raju [24] equations. This favorable comparison occurred even
though the mid-side
nodes were not moved to the quarter points, and the nodes along
the crack tip were
left untied (two conditions which yield optimum accuracy in
K-factor calculations
using FEMs). The largest observed error, approximately six
percent, occurred with
Model 8. It should also be noted that the normalized K-factor
results from the K
and elastic J models were very close. The elastic results are
summarized in Table
4.3. This summary disregards the surface and depth results.
There is no apparent
pattern to the differences.
The current FEA results were also compared visually to the
graphical results
published by Lei [17]. Lei used a different normalizing scheme,
resulting in different
Table 4.2 Surface and depth phenomenon for K-factors
Surface DepthModel a/t a/c Jump Jump1 0.2 0.2 yes no2 0.2 0.6
yes yes3 0.2 1.0 yes yes4 0.5 0.2 yes no5 0.5 0.6 yes yes6 0.5 1.0
yes yes7 0.8 0.2 yes no8 0.8 0.6 no yes9 0.8 1.0 no yes
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57
Table 4.3 Maximum percent differences between Newman-Raju and
FEM solu-tions (quarter symmetry)
Max FEA K Max FEA K Max FEA KDirectly from Jel from tied
nodes
Model a/t a/c (% diff.) (% diff.) (% diff.)1 0.2 0.2 -5.64 -5.38
-8.52 0.2 0.6 -2.28 -2.18 -3 0.2 1.0 3.11 2.88 -4 0.5 0.2 -6.40
-5.57 -5 0.5 0.6 -2.15 -1.96 -6 0.5 1.0 3.35 3.44 -7 0.8 0.2 4.52
4.53 -8 0.8 0.6 -7.07 -7.07 -7.179 0.8 1.0 2.8 3.03 -
scales on the y-axis, but the graphs had very similar shapes.
Lei also showed some
models with the same spike at the surface that was experienced
in this research.
However, the increase was not as significant. No sudden
increases were observed at
the depth of his elastic models.
An investigation was performed to find the cause of the
previously discussed
surface and depth K-factor spikes in the current FEMs. The first
step was to explore
the potential error caused by not tying the crack tip nodes or
moving the mid-side
nodes to the quarter points. Model 1 (a/t = 0.2, a/c = 0.2) and
Model 8 (a/t = 0.8,
a/c = 0.6) meshes were recreated in FEA-Crack for elastic
analysis only. These two
models were generated with an elastic singularity, 1/r, created
by tying the crack
tip nodes and moving the mid-side nodes to the quarter points
(Figure 4.10). The
results of the two elastic models are shown graphically in
Figures 4.11 and 4.12. The
K-factors produced using these two FEMs were almost identical to
the previous
results for Model 1 and Model 8.
-
58
Figure 4.10 Elastic singularity element [4]
Figure 4.11 Model 1 (a/t=0.2, a/c=0.2): Normalized K-factor vs.
angle alongcrack front for untied and tied nodes
-
59
Figure 4.12 Model 8 (a/t=0.8, a/c=0.6): Normalized K-factor vs.
angle alongcrack front for untied and tied nodes
-
60
The second step in investigating the surface and depth K-factor
spikes in-
volved changing from reduced integration to full integration
elements, type C3D20R
to C3D20 in ABAQUS, for the FEMs. The results for Models 1 and 8
are shown in
Figures 4.13 and 4.14. As mentioned in the ABAQUS Users Manuals
[21], using a
full integration element type caused the K-factor results to
oscillate. Full integration
elements did not correct the surface and depth deviations.
Figure 4.13 Model 1 (a/t = 0.2, a/c = 0.2): Reduced vs. full
integration elements
-
61
Figure 4.14 Model 8 (a/t = 0.8, a/c = 0.6): Reduced vs. full
integration elements
-
62
The default nonlinear solver for ABAQUS was also considered as a
possible
source of error in the third attempt to reduce the surface and
depth K-factor dis-
crepencies. Since the K-factor calculation is linear, the solver
used within ABAQUS
was changed to a linear perturbation. Unfortunately, the
*CONTOUR INTEGRAL
command used to output the K-factors will not function within a
linear perturbation
step. Another attempt to force a linear solution was made by
using the *STATIC
command to force the solution to be performed in one step. The
surface and depth
results were not altered by this approach.
In a fourth attempt to solve the free surface and depth spikes,
the FEA-
Crack Validation Manual [26] was examined. It was found that the
FEA programs
WARP3D [27], ABAQUS, and ANSYS were used to produce K-factor
calculations
for validating FEA-Crack meshes. The validation data were
presented graphically as
K-factor vs. angle along the crack front (Figure 4.15 [26]). The
K-factors produced
by these three FEA programs were virtually identical, except at
the free surface. At
this location, the K-factor produced by ABAQUS was approximately
7.7% higher
than the other two FEA programs. It was also noted that there
were no irregularities
presented in the depth.
Further examination of the FEA-Crack Validation Manual [26]
revealed that
the surface crack results published in the validation manual
were for full plates with no
symmetry conditions (Figure 4.16). To investigate boundary
conditions in the depth
phenomenon, Model 6 (a/t = 0.5, a/c = 1.0) and Model 8 (a/t =
0.8, a/c = 0.6)
meshes were created with half symmetry (Figure 4.17). The
results may be seen in
Figures 4.18 and 4.19.
-
63
Figure 4.15 K-factor results from FEA-Crack Validation Manual
[26]
Figure 4.16 FEM mesh for a flat plate with no symmetry exploited
[26]
-
64
Step: Step-1, Stress Analysis, Step # 1Increment 9: Step Time =
0.3394
3-D crack mesh model generated by FEA-CrackODB: model6-hs.odb
ABAQUS/Standard 6.2-007 Sat Dec 18 11:58:57 CST 2004
1
2
3Step: Step-1, Stress Analysis, Step # 1Increment 9: Step Time =
0.3394
3-D crack mesh model generated by FEA-CrackODB: model6-hs.odb
ABAQUS/Standard 6.2-007 Sat Dec 18 11:58:57 CST 2004
Figure 4.17 FEM mesh for a flat plate with half symmetry
Figure 4.18 Model 6 (a/t = 0.5, a/c = 1.0): K-Factor results for
half symmetrymodel
-
65
Figure 4.19 Model 8 (a/t = 0.8, a/c = 0.6): K-Factor results for
half symmetrymodel
-
66
The half symmetry models produced with FEA-Crack did not exhibit
a K-
factor spike in the depth. Therefore, it was deduced that the
boundary conditions
specified in the ABAQUS input file could be an issue. Perhaps
the solver within
ABAQUS is very sensitive to the boundary conditions. Problems
with the boundary
conditions in the depth or at the free surface could account for
the observed increases.
An ABAQUS input file for an elastic model was examined to
investigate the
boundary conditions of nodes along the crack front. For the
elastic models, it was
found that there were no coincident nodes along the crack front.
The crack tip nodes
were shared by the surrounding elements. The depth node was
constrained in the
directions perpendicular and parallel to the crack front. All of
the other crack tip
nodes were constrained only in the direction perpendicular to
the crack front. This
constraint scheme is logical given a quarter symmetry model with
tied nodes along
the crack front.
A possible explanation for the surface deviation is that it is a
result of the
surface elements in the 3-D model. The lateral surface is
subjected to plane stress,
but the surface elements have some thickness in the direction
normal to the free
surface. This is a plausible, but unproven, explanation for the
source of the surface
phenomenon.
-
67
4.3.2 Fully Plastic Analysis
The fully plastic results from the McClung geometries are
presented in the
following section. It should be noted that