MS_Aero_Thesis

INITIATION, GROWTH, AND COALESCENCE OF
SMALL FATIGUE CRACKS AT NOTCHES
A Thesis
Submitted to the Faculty
of
Purdue University
by
Eric Nielsen Forsyth
In Partial Fulfillment of the
Requirements for the Degree
of
Master of Science in Aeronautics and Astronautics
May 1993

ii
Dedicated to my parents,
George and Ardith,
and my grandparents,
Arthur and LaVerne Nielsen,
for their endless love and support.

iii
ACKNOWLEDGMENTS
This work was sponsored by the Aluminum Company of America under Project
Number TC919597TC. Special thanks are extended to Dr. A. J. Hinkle and Dr. B. J.
Shaw of Alcoa for their supervision and assistance throughout this study. In addition, the
author would like to express appreciation to the School of Materials Science at Purdue
University for the use of their specimen preparation and optical microscopy facilities.
The author would especially like to thank his major Professor, A. F. Grandt, Jr.,
for his guidance throughout this work. Professor Grandt's experience and insight were
invaluable in shaping the author's perceptions and approach to research in addition to the
course of the research itself. Thanks are also extended to Professor B. M Hillberry and
Professor H. D. Espinosa for providing their unique perspectives as members of the
author's thesis committee.
There are many other people whose support and assistance were instrumental in
the completion of this work. Thanks are due to Mark Yost, Bob Sanders, and the late
Gene Harston for technical assistance ranging from specimen fabrication to testing
equipment maintenance. Special thanks is extended to Chad Zezula for his significant
assistance with specimen testing and replica measurement. Finally, thanks are due to
Mark Doerfler and Marcus Heinimann for their advice and encouragement, as well as
Michelle Wade for her support.

iv
TABLE OF CONTENTS
Page
LIST OF TABLES............................................................................................................. vi
LIST OF FIGURES ......................................................................................................... viii
ABSTRACT..................................................................................................................... xix
CHAPTER 1 - INTRODUCTION.......................................................................................1
CHAPTER 2 - BACKGROUND.........................................................................................2
2.1 LEFM Concepts................................................................................................2
2.2 The Small Crack Problem.................................................................................4
2.3 The 7050-T7451 Aluminum Alloy...................................................................9
CHAPTER 3 - EXPERIMENTAL PROCEDURES .........................................................20
3.1 Small Crack Specimen Design and Testing Procedures.................................20
3.1.1 Specimen Testing.............................................................................22
3.1.2 The Replication Method ..................................................................23
3.2 Large Crack Testing Procedures.....................................................................25
CHAPTER 4 - EXPERIMENTAL RESULTS..................................................................35
4.1 Large Crack Growth Rate Data ......................................................................35
4.2 Small Crack Test Results................................................................................36
4.3 Small Crack Growth Rate Data.......................................................................39
CHAPTER 5 - ANALYTICAL MODELING...................................................................69
5.1 Background.....................................................................................................69
5.2 Description of Algorithm................................................................................71

v
Page
CHAPTER 6 - NUMERICAL RESULTS.........................................................................76
6.1 Back-Prediction in Specimens Used to Calculate Small Crack
da/dN-ΔK curve ...............................................................................................76
6.2 Prediction Results in Specimens Initiating Multiple Cracks ...........................79
CHAPTER 7 - CONCLUSIONS AND RECOMMENDATIONS..................................115
LIST OF REFERENCES.................................................................................................117
APPENDICES
Appendix A - Stress Intensity Factor Solutions...................................................122
Appendix B - Specimen Dimensions and Test Parameters .................................128
Appendix C - Crack Measurements for Double-Edge Notch Specimens............130

vi
LIST OF TABLES
Table Page
3.1 Parameters for fatigue test specimens........................................................27
4.1 Test matrix for the double-edge notch specimens. ....................................43
Appendix
Table
B1 Dimensions and test parameters for the double-edge notch
specimens. All tests were conducted at a stress ratio R =
0.1 and in laboratory air...........................................................................129
C1 Crack measurements for specimen 6611-a12, back notch.......................133
C2 Crack measurements for specimen 6611-a12, front notch.......................135
C3 Crack measurements for specimen 6612-b21, back notch.......................140
C10 Crack measurements for specimen 8B2, back notch ...............................152

vii
Appendix
Table Page
C11 Crack measurements for specimen 8B3, back notch ...............................153
C12 Crack measurements for specimen 8B3, front notch...............................154
C13 Crack measurements for specimen 8T3, front notch ...............................158

viii
LIST OF FIGURES
Figure Page
2.1 The two stages of fatigue...........................................................................14
2.2 The semicircular edge notch geometry and variable
definitions ..................................................................................................15
2.3 Typical fatigue crack growth rate data for large and small
cracks [11]..................................................................................................16
2.4 Stress intensity factor vs. time for constant ampltide
loading. Lower crack opening stress for small cracks
results in a larger effective stress intensity factor than a
large crack under identical loading, translating into faster
growth rates................................................................................................17
2.5 Cumulative fatigue failure distributions from 1984-1987
for the 7050-T7451 thick plate (5.5-5.9 inches = 140-150
mm)............................................................................................................18
2.6 Large crack da/dN-ΔK data for the Al 7050-T7451 alloy,
R=0.1 [29]..................................................................................................19
3.1 Double-edge notch dogbone specimen geometry and
dimensions. ................................................................................................28
3.2 Double-edge notch specimen geometry and dimensions...........................29
3.3 Hoop Stress/Remote Stress, σhoop/σrem, vs. Theta, Θ, for
the 1.11 and 2.00 inch wide specimen geometries.
σhoop/σrem was caculated with a 2-dimensional finite
element analysis of each of the specimen geometries. Θ
is defined as the angle (in radians) from the tip of the
notch to the upper/lower location where the notch meets
the specimen edge......................................................................................30

ix
Figure Page
3.4 Illustration of the replication process ........................................................31
3.5 Definition of replica coordinate system.....................................................32
3.6 CT specimens fabricated from fractured double-edge
notch specimen...........................................................................................33
3.7 CT specimen geometry and dimensions. ...................................................34
4.1 Large crack growth rate vs. stress intensity factor range
data for Al 7050-T7451 (obtained from CT specimens)
The large crack Paris Law was obtained from Reference
[29] (see Figure 2.5)...................................................................................44
4.2 Surface crack length vs. number of elapsed cycles for
specimen 6611-a12 (old material), "back" notch.......................................45
specimen 6611-a12 (old material), "front" notch. .....................................46
specimen 6612-b21 (old material), "back" notch. .....................................47
specimen 6714-a11 (old material), "back" notch.......................................48
specimen 6714-a12 (old material), "front" notch. .....................................49
specimen 7012-a22 (new material), "back" notch.....................................50
specimen 7012-a22 (new material), "front" notch.....................................51
specimen 7111-b11 (new material), "back" notch.....................................52
specimen 7111-b12 (new material), "back" notch.....................................53

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Figure Page
specimen 8B2 (3-inch plate material), "back" notch. ................................54
specimen 8B3 (3-inch plate material), "back" notch. ................................55
specimen 8B3 (3-inch plate material), "front" notch.................................56
specimen 8T3 (3-inch plate material), "front" notch. ................................57
4.15 Replica photograph of specimen 6612-b21, "back" notch,
0 cycles (after specimen alignment loading). Crack ID # 1:
2a = 0.0016 in. ...........................................................................................58
10001 cycles. Crack ID # 1: 2a = 0.0038 in. .............................................58
10001 cycles. Crack ID # 1: 2a = 0.0038 in. .............................................59
23001 cycles. Crack ID # 1: 2a = 0.0092 in. ............................................59
4.21 Replica photograph of specimen 6611-a12, "front" notch,
42507 cycles. Crack ID # 4.1: 2a = 0.0208 in. Crack ID #
4.2: 2a = 0.0025 in. Crack ID # 8: 2a = 0.0009 in. ...................................61
4.2: 2a = 0.0036 in. Crack ID # 8: 2a = 0.0032 in. ...................................61

xi
Figure Page
4.2: 2a = 0.0047 in. Crack ID # 8: 2a = 0.0032 in. ...................................62
65008 cycles. Crack ID # 4: 2a = 0.0441 in. Crack ID #
8: 2a = 0.0032 in. .......................................................................................62
4.25 SEM fractograph of specimen 6611-a12, "front" notch
fracture surface. The larger crack on the left was
identified as Crack ID # 6 during the replica measurement
process. The smaller crack on the right was identified as
Crack ID # 5 during the replica measurement process.
The reference line on the fractograph is 1000 μm in
length. ........................................................................................................63
4.26 Initiation site of Crack ID # 6. The reference line on the
fractograph is 100 μm in length.................................................................64
4.27 Close-up of initiation site of Crack ID # 6. The reference
line on the fractograph is 10 μm in length.................................................64
4.28 Initiation site of Crack ID # 5. The reference line on the
fractograph is 100 μm in length.................................................................65
4.29 Close-up of initiation site of Crack ID # 5. The reference
line on the fractograph is 10 μm in length.................................................65
4.30 Empirical expression for crack shape vs. nondimensional
length [28]. Measured values are from Crack ID #'s 5 and
6 from specimen 6611-a12, "front" notch..................................................66
4.31 Illustration of corner crack shape based on empirical
expression for c/a vs. a/t (to scale).............................................................67
4.32 Small crack growth rate vs. stress intensity factor range
data for Al 7050-T7451 (obtained from double-edge notch
specimens)..................................................................................................68
5.1 Geometry variable definitions used in the prediction
program for a surface crack and a corner crack.........................................74

xii
Figure Page
5.2 Geometry variable definitions used in the prediction
program for a typical multiple crack configuration...................................75
6.1 Actual and predicted crack growth for specimen 6612-b21,
back notch: surface crack length vs. number of cycles.............................84
6.2 Actual and predicted crack growth for specimen 6714-a11,
6.5 Actual and predicted crack growth for specimen 8B2, back
notch: surface crack length vs. number of cycles.. ...................................88
6.6 Actual and predicted crack growth for specimen 8B3, back
notch: surface crack length vs. number of cycles. ....................................89
6.7 Actual and predicted crack growth for specimen 8T3, front
notch: surface crack length vs. number of cycles. ....................................90
6.8 Predicted crack growth for specimen 6612-b21, back
notch: c/a vs. a/t for both the predicted values and the
empirical expression assumed in calculating ΔK solutions.......................91
6.9 Predicted crack growth for specimen 6714-a11, back

xiii
Figure Page
6.12 Predicted crack growth for specimen 8B2, back notch: c/a
vs. a/t for both the predicted values and the empirical
expression assumed in calculating ΔK solutions.......................................95
6.13 Predicted crack growth for specimen 8B3, back notch: c/a
6.14 Predicted crack growth for specimen 8T3, front notch: c/a
6.15 Stress intensity factor geometry for two offset parallel
cracks in a sheet under uniform uniaxial tensile stress [42]. .....................98
back notch: surface crack length vs. number of cycles.
Note: no crack interaction is considered between the
cracks. ........................................................................................................99
6.17 Predicted crack growth for specimen 7012-a22, front
empirical expression assumed in calculating ΔK solutions.
cracks. ......................................................................................................100
cracks. The stress concentration factors were adjusted to
account for the crack initiating off the midplane of the
notch at an angle Θ (see Figure 3.3)........................................................101
cracks. The stress concentration factors were adjusted to
account for the crack initiating off the midplane of the
notch at an angle Θ (see Figure 3.3)........................................................102

xiv
Figure Page
front notch: surface crack length vs. number of cycles.
Note: the presence of Crack ID #'s 2 and 3 are ignored.. ........................103
Note: the presence of Crack ID #'s 2 and 3 are ignored.. ........................104
Note: the presence of Crack ID #'s 1.1 and 2 are ignored. ......................105
Note: the presence of Crack ID #'s 1.1 and 2 are ignored. ......................106
Note: the presence of Crack ID # 2 is ignored.........................................107
Note: the presence of Crack ID # 2 is ignored.........................................108
front notch: surface crack length vs. number of cycles. ........................109
empirical expression assumed in calculating ΔK solutions.....................110
6.28 Actual and predicted crack growth for specimen 8B3,
front notch, Crack ID # 1.22: surface crack length vs.
number of cycles. Note: the presence of other cracks are
ignored. ....................................................................................................111

xv
Figure Page
6.29 Predicted crack growth for specimen 8B3, front notch,
Crack ID # 1.22: c/a vs. a/t for both the predicted values
and the empirical expression assumed in calculating ΔK
solutions. Note: the presence of other cracks are ignored. .....................112
6.30 Actual and predicted crack growth for specimen 8B3,
front notch Crack ID #'s 1.21 and 1.22: surface crack
length vs. number of cycles. Note: the presence of other
cracks are ignored. ...................................................................................113
6.31 Predicted crack growth for specimen 8B3, front notch
Crack ID #'s 1.21 and 1.22: c/a vs. a/t for both the
predicted values and the empirical expression assumed in
calculating ΔK solutions. Note: the presence of other
cracks are ignored. ...................................................................................114
Appendix
Figure
C1 Crack tip locations for Specimen 6611-a12, back notch
(N=30,003 cycles)....................................................................................159
(N=40,007 cycles)....................................................................................159
(N=50,007 cycles)....................................................................................160
C4 Crack tip locations for Specimen 6611-a12, front notch
(N=45,007 cycles)....................................................................................161
(N=65,008 cycles)....................................................................................161
C6 Crack tip locations for Specimen 6612-b21, front notch
(N=29,005 cycles)....................................................................................162
(N=37,507 cycles)....................................................................................162
(N=47,511 cycles)....................................................................................163

xvi
Appendix
Figure Page
(N=62,516 cycles)....................................................................................163
(N=60,028 cycles)....................................................................................164
(N=75,033 cycles)....................................................................................164
(N=90,037 cycles)....................................................................................165
(N=110,045 cycles)..................................................................................165
(N=113,440 cycles)..................................................................................166
(N=130,630 cycles)..................................................................................166
(N=142,130 cycles)..................................................................................167
(N=163,630 cycles)..................................................................................167
(N=53,002 cycles)....................................................................................168
(N=65,002 cycles)....................................................................................168
(N=80,004 cycles)....................................................................................169
(N=94,006 cycles)....................................................................................169
(N=106,509 cycles)..................................................................................170

xvii
Appendix
Figure Page
(N=41,001 cycles)....................................................................................171
(N=56,002 cycles)....................................................................................171
(N=71,003 cycles)....................................................................................172
(N=89,005 cycles)....................................................................................172
(N=106,509 cycles)..................................................................................173
C28 Crack tip locations for Specimen 7111-b11, back notch
(N=231,068 cycles)..................................................................................174
(N=239,072 cycles)..................................................................................174
(N=247,075 cycles)..................................................................................175
(N=211,506 cycles)..................................................................................176
(N=221,002 cycles)..................................................................................176
(N=229,003 cycles)..................................................................................177
(N=233,003 cycles). 177
C35 Crack tip locations for Specimen 8B2, back notch
(N=45,008 cycles)....................................................................................178
(N=57,501 cycles)....................................................................................178

xviii
Appendix
Figure Page
(N=72,507 cycles)....................................................................................179
(N=42,506 cycles)....................................................................................180
(N=57,509 cycles)....................................................................................180
C40 Crack tip locations for Specimen 8B3, front notch
(N=27,503 cycles)....................................................................................181
(N=40,006 cycles)....................................................................................181
(N=50,007 cycles)....................................................................................182
(N=58,509 cycles)....................................................................................182
C44 Crack tip locations for Specimen 8T3, front notch
(N=145,002 cycles)..................................................................................183
(N=162,005 cycles)..................................................................................183
(N=182,008 cycles)..................................................................................184
(N=214,005 cycles)..................................................................................184

xix
ABSTRACT
Forsyth, Eric Nielsen. M.S.A.A., Purdue University, May 1993. Initiation, Growth, and
Coalescence of Small Fatigue Cracks at Notches. Major Professor: Dr. A. F. Grandt, Jr.
This research concerns the initiation, growth and coalescence of small fatigue
cracks at semicircular edge notches in the aluminum 7050-T7451 plate alloy. Three
versions of the alloy were provided by ALCOA, each with a varying degree of
microporosity. The objective of this study was to determine if a reduction in the amount
of microporosity resulted in improved small fatigue crack growth properties. Ten
double-edge notch specimens were tested at varying stress levels with a stress ratio of R
= 0.1. Fatigue crack growth was monitored with the replication method, providing
surface crack measurements as small as 0.0006 inches (15 microns). CT specimens for
all three versions of the alloy were fatigue tested to determine the large fatigue crack
growth properties.
Results from the CT specimen tests compared favorably with fatigue crack
growth rate vs. applied stress intensity factor range data generated previously by
ALCOA, and indicated that all three versions of the alloy had identical large fatigue
crack growth properties. Results from the double-edge notch specimen tests indicate that
after initiation, small fatigue cracks grow at faster rates than large fatigue cracks under
identical ΔK loading. All three versions of the alloy demonstrated similar small fatigue

xx
crack growth rate properties after initiation. However, the versions of the alloy with
reducedmicroporosity demonstrated longer fatigue lives to initiation than the version with
the most microporosity.
The small fatigue crack da/dN-ΔK curve was incorporated in a program to back-
predict the fatigue crack growth after initiation in double-edge notch specimens that
initiated a single crack. The predicted crack growth results to breakthrough showed
reasonable agreement with the data obtained from the specimen tests. However, future
tests should be conducted at different stress levels to generalize the results obtained in
this study.

1
CHAPTER 1 - INTRODUCTION
Fatigue cracks in engineering structures often originate at stress concentrations
such as fastener holes and notched components. These cracks can initiate at initial
defects such as voids and bonded inclusions within the engineering material. Research
conducted by ALCOA on the aluminum 7050-T7451 alloy [1] has demonstrated that the
fatigue life of edge-notch specimens can be improved by reducing the amount of
microporosity within the alloy. Since the majority of a fatigue crack's life in an
engineering structure can be spent in this "small" crack stage, it is of critical importance
to understand how all of these factors interact with one another to effect the crack's
subsequent growth.
The primary objective of this study is to determine how initial microporosity
effects the initiation and growth of fatigue cracks in the aluminum 7050-T7451 alloy. To
accomplish this, fatigue testing was performed on semicircular edge-notch specimens
fabricated from three versions of the alloy with varying levels of microporosity. Fatigue
crack growth was monitored from the point of initiation, enabling crack growth rate
information to be obtained for physically small cracks. Finally, an existing program was
modified to predict the growth of these cracks from the point of initiation in the
semicircular edge notch geometry.

2
CHAPTER 2 - BACKGROUND
Fatigue, the failure mode associated with cyclic loading, is often separated into
two stages: crack initiation and crack growth (Figure 2.1). Different methodologies have
been developed to treat the life of a crack through these two stages. Stress-life and
strain-life approaches are often used to quantify crack initiation life, while linear elastic
fracture mechanics (LEFM) can be used to quantify the growth of a crack with an initial
size, ao. The damage tolerance design philosophy lends itself particularly well to the
crack growth portion of fatigue life. Using this approach, engineers assume the pre-
existence of flaws in their design. Thus, it is desirable from an analysis point of view to
treat fatigue as primarily crack growth, i. e., have a single analysis method applicable to
all crack sizes. Unfortunately, there is no strict "boundary" where the LEFM assumption
breaks down. The purpose of this research is to monitor crack growth in an aluminum
alloy from the point of initiation, and apply LEFM principles into the "gray" area
between initiation and growth in an effort to predict crack behavior.
2.1. LEFM Concepts
LEFM assumes that crack growth is controlled by the stress intensity factor, K
[2]. This term, introduced by Irwin, relates loading, crack size, and specimen geometry,
and is often given in the form

3
K a a= σ π β( ) (2.1)
where σ is the remotely applied stress, a is the crack length, and β(a) is a dimensionless
function of the crack geometry. Paris, Gomez, and Anderson [3] first demonstrated that
the rate of fatigue crack growth (da/dN) is a function of the applied stress intensity factor
range (ΔK), independent of the particular loading, crack size, and specimen geometry, i.
e.,
da dN f K/ ( )= Δ (2.2)
This expression can be integrated to obtain the cyclic fatigue life
N dN
da
f K
N
a
a
o
f
= =z z0 ( )Δ
(2.3)
If the da/dN-ΔK expression is known for a particular material, these equations can be
incorporated into an algorithm to predict the cyclic fatigue life of a crack under different
loading conditions and geometries.
Although there are many K solutions available for two dimensional geometries,
there are few closed form K solutions for three dimensional geometries. Since fatigue
cracks initially start out having two dimensions (a surface length "a" and a crack depth
"c"), three dimensional K solutions are necessary to study the growth of small cracks.
Specimen geometries that incorporate a semicircular edge notch have been found to be
useful to monitor the initiation and growth of small cracks. Newman [4] has presented
approximate K solutions for corner and surface cracks in a semicircular edge notch.

4
These solutions were developed from finite element [5, 6] and weight function [7, 8]
methods for surface and corner cracks; from boundary force analyses of through cracks
at a semicircular notch [9]; and from previously developed equations for similar crack
configurations at an open hole [10]. The solutions are given in the form
K S a QF a c a t c r c w r t r wjn= π φ( / , / , / , / , / , / , ) (2.4)
The semicircular edge notch geometry and variable definitions are illustrated in Figure
2.2. It is important to note that for a corner crack in Figure 2.2, t is defined in the present
work as the specimen thickness B, whereas for a surface crack, t is defined as B/2. The
full stress intensity factor solutions used here are given in Appendix A. More
information on the actual test specimen design used in this research is given in Chapter 3.
2.2. The Small Crack Problem
Research conducted over the past two decades has shown that for certain
materials, physically small cracks (a ≤ 0.02 inches = 0.51 mm) grow at faster rates than
large cracks under the same ΔK loading. In addition, small cracks have been observed to
grow beneath the large crack threshold, ΔKth. These phenomena are known as the "small
crack effect". Schematic differences between the growth rates of small and large cracks
is illustrated in Figure 2.3 [11]. Since a significant portion of a crack's life in an
engineering structure may be spent as a small crack, any life predictions for that
component based on large crack data would be non-conservative. Thus, it is important to
determine if an engineering material exhibits this difference between the growth of small
and large cracks, and explain why it exists [11].

5
There are several factors that are believed to be involved in the small crack effect.
As mentioned earlier, LEFM assumptions are invalidated as the crack size approaches
zero due to the fact that the plastic zone size in front of the crack is on the same order of
magnitude as the crack size itself. Nonlinear and elastic-plastic fracture mechanics
concepts, such as the J-integral [12] and strain energy densities [12], have been used to
explain the short crack effect. In addition, the continuum assumption of LEFM [13] is
invalidated because grain boundaries as well as voids and inclusion particles affect the
local stresses near the small crack front. For a large crack, these metallurgical effects are
averaged out over the larger crack's long front.
However, there are LEFM concepts which, in part, help explain the small crack
effect. In particular, crack closure has been shown to play an important role in the
accelerated growth rates of small cracks. First proposed by Elber [14], crack closure is
the concept that a crack is not fully open until a "crack opening stress" is reached. This
phenomenon can be attributed to several factors, including plastically deformed material
in the wake of a crack, crack surface roughness, and oxide debris on the crack surface.
All of these factors hinder the opening of a crack, resulting in a stress level that must be
reached before the crack can be fully open and thus propagate. It is believed that small
cracks have smaller crack opening stresses than large cracks do. Therefore, small cracks
would experience a larger effective stress intensity factor range than large cracks, even
though they are experiencing identical ΔK loading. This phenomenon is illustrated in
Figure 2.4 in a K vs. time graph.
In order to study the small crack effect, researchers have developed several
methods for measuring small fatigue cracks. Perhaps the most accurate method for
measuring small cracks is with the scanning electron microscope (SEM). When used in

6
conjunction with stereo imaging, SEM photographs provide useful information in the
closure behavior of small cracks [15]. Although the SEM has both the spatial and strain
resolution for the scale involved, cost makes its use prohibitive for routine laboratory
measurements.
Sharpe [16] has developed the interferometric strain-displacement gage (ISDG)
which acts as a non-contacting extensometer for the specimen. Two indentations are
made with a Vickers hardness tester above and below a surface crack. The diffraction
patterns created by a laser impinging on the indentations can be used to determine crack
opening displacement and thus crack size. Although the ISDG can be used for computer
control and real-time measurement of small fatigue crack tests, the location of the
initiated crack must be known before measurements can be taken.
Another method which allows for computer data acquisition is the direct current
electrical potential measurement (dcEPM) of small cracks [17]. If a current is passed
through a specimen containing a crack, the voltage difference across the crack can be
correlated to the crack length. Drawbacks to the dcEPM method include cost, the
necessity for the specimen to conduct electricity, and the fact that it has only been used
on cracks artificially created with electric discharge machining.
Resch and Nelson [18] have developed an ultrasonic method for the measurement
of small cracks. The method uses surface acoustic waves on the specimen to determine
surface crack depth; in this sense, the method is similar to the SONAR employed by
naval craft to determine underwater features.

7
A relatively simple, but more time consuming method for the measurement of
small cracks is the replication method [19]. It uses an acetate tape which makes an exact
replica of the specimen surface when acetone is applied to the surface. The method can
be used for a variety of specimen geometries and crack length measurements as small as
0.0002 inches (5.1 μm) have been obtained. Unfortunately, only the surface crack length
can be measured with this method - not the crack depth. The research presented in this
thesis utilized the replication method for the measurement of small cracks. A more
thorough discussion on the specifics of the replication method and how it was used in
conjunction with this research is presented in Chapter 3.
In 1984, an AGARD Cooperative Test Program was initiated to investigate the
small crack growth behavior under various loading conditions for the aluminum alloy
2024-T3, a common material used in airframe components [20]. Twelve participants
from nine different countries monitored the growth and coalescence of nearly 950 cracks
in over 250 single edge notch specimens. The tests were conducted at three different
stress levels for both constant amplitude loading (stress ratios, R ≡ minimum/maximum
stress = -2, -1, 0, and 0.5) and spectrum loading (FALSTAFF and GAUSSIAN)
conditions. Surface crack lengths were measured with the replication technique. The
participants involved in the test program showed good agreement on the small crack
growth rates, cyclic fatigue life to crack breakthrough (surface and/or corner cracks
became a through crack), and on crack shapes. The small cracks initiated in the tests
demonstrated the small crack effect mentioned previously by growing below the large
crack ΔK threshold and growing at faster growth rates than large cracks above the
threshold.

8
A fatigue crack growth model accounting for crack closure was developed by
Newman [21] to predict the growth of small cracks from small voids and inclusion
particles on the notch surface. The initial defect size was chosen to approximate the
initiation sites of the cracks monitored in the tests. The model was based on the Dugdale
strip-yield plastic zone [22], but modified for closure by leaving plastically deformed
material in the wake of the crack. Lee and Sharpe's experimentally measured values for
the crack opening stresses (obtained from the ISDG method) [23] showed good
correlation with Newman's analytical model, increasing the confidence in the model.
There was reasonable agreement between the experimental and predicted values for the
small crack growth rates, although the model predicted slightly slower growth rates for R
= -2 loading, and slightly faster growth rates for R = 0.5 loading. However, the model
did indicate that the small crack effect was most predominant in the tests involving
significant compressive loads. This behavior was observed in the tests themselves.
In order to allow participants to test various materials and loading conditions that
were of particular interest to their laboratories, an AGARD Supplemental Test Program
on the growth of small cracks was initiated [24]. The materials tested in the
supplemental program were: 2024-T3 and 7075-T6 aluminum alloys, 2090-T8E41
aluminum-lithium alloy, Ti-6Al-4V titanium alloy, and 4340 steel. The results from the
supplemental program were similar to the first program in that all the materials exhibited
the small crack effect to some extent. However, the effect was less pronounced in some
materials (e. g., 4340 steel) than in others. Once again, the crack growth model predicted
small crack growth rates in reasonable agreement with the experimental measurements
for most loading conditions.

9
2.3. The 7050-T7451 Aluminum Alloy
In an effort to reduce both the size and frequency of potential microporosity in
their aluminum 7050-T7451 plate alloy, the Aluminum Company of America (ALCOA)
has improved their processing techniques for the material over the past decade. Smooth
axial fatigue tests of material produced in 1985 following the process improvements have
resulted in longer fatigue lifetimes than material produced prior to the improvements [1].
Post-test fractography of the specimens fabricated from both materials revealed the size
of the micropores that resulted in crack initiation and subsequent fracture. This
microporosity size distribution was subsequently used in a probabilistic crack growth
analysis, which demonstrated analytically that the reduced microporosity material should
perform better in service than the older material with larger micropores. However, since
smooth axial fatigue tests do not take into account cracks originating from machining
defects, a test program was initiated to examine whether these type of flaws obscure the
process improvements resulting in microporosity reduction.
The objective of this program was to demonstrate the effect of microporosity on
an engineering detail, specifically, a notched specimen subjected to constant amplitude
loading [1]. The material was obtained from a single lot of 5.6 inch (14.2 cm) thick
7050-T7451 plate. Specimens fabricated from the mid-plane of the plate had a higher
degree of microporosity than the specimens fabricated from the quarter-plane of the plate.
The test specimens were 0.126 in. (3.2 mm) thick, 1.00 in. (25.4 mm) wide, and 9.00 in.
(229 mm) long, with two holes of 0.187 in. (4.75 mm) diameter located 1.00 in. (25.4
mm) apart. The goals of this specimen were to provide a symmetric stress field and to
increase the chances that a micropore would be located near a stress concentration. The

10
specimens were cycled to failure at a stress ratio R = 0.1 at maximum stress levels of 10,
12, and 20 ksi (69, 83, and 138 MPa).
Results from these tests show a significant improvement in the fatigue properties
of the low microporosity (quarter-plane) material; this can be seen in the test specimens'
log-life versus log-maximum stress plot of Figure 2.5 [1]. For example, a component
designed for a lifetime of 100,000 cycles could see a maximum stress of 110 MPa in the
low microporosity material as opposed a maximum stress of 98 MPa in the high
microporosity material; this represents an improvement of 12 percent in stress level [1].
All specimen failures in this study initiated at micropores as opposed to
machining defects. The largest micropore initiating a crack in the test program was 0.030
in. (0.75 mm); the average size of a crack initiating micropore, however, was 0.012 in.
(0.31 mm). Both of these sizes fall below current nondestructive inspection (NDI)
capabilities, which can reliably detect flaw sizes of 0.04-0.08 in. (1-2 mm) [1]. Thus,
ALCOA employed destructive techniques such as SEM examination of the fracture
surfaces to quantify the microporosity distribution. This examination revealed that the
frequency of micropores that initiated cracks in the specimens to be the major difference
between the fracture surfaces of the two versions of the material. The high porosity
(mid-plane) version of the material initiated on average a greater number of cracks (2.25
per specimen) than the low porosity (quarter-plane) version (1.33 per specimen) [1].
One of the main conclusions from the test program was that initial material
quality should be considered in the design process. To accomplish this, ALCOA utilized
the United States Air Force (USAF) Advanced Durability Analysis. This method is
based on the concept of an equivalent initial flaw size (EIFS) which represents the initial

11
microporosity distribution in the material. Since all the cracks in the specimens initiated
at micropores, an EIFS distribution (calculated via LEFM principles) based on these tests
could theoretically be equated with the actual initial microporosity distribution of the
material (determined from the earlier smooth axial fatigue tests). Subsequent analysis
demonstrated this hypothesis; the two distributions were very similar, and predicted
specimen lifetimes when used as input in a probabilistic fracture mechanics analysis [1].
Out of this test program arose two objectives for further research. First, it was
desired to further develop and assess the benefits of the probabilistic approach to
durability. ALCOA, in collaboration with Wright Laboratory's Flight Dynamics
Directorate (USAF) [25, 26], has demonstrated through further testing and analysis that a
reduction in the microporosity of Al 7050-T7451 can result in the increased performance
and reduced cost of airframe components where durability is a major design factor. In
addition, they confirmed that the USAF probabilistic failure model captured this
advantage in improved material quality, whereas more conventional fatigue design
practices did not.
ALCOA's second objective was to further quantify crack growth from micropores
by studying the effect of microporosity on the growth of physically small cracks. To
accomplish this, ALCOA initiated a test program to monitor the initiation and growth of
small cracks in the low and high microporosity versions of the 7050-T7451 plate [27].
The specimen design incorporated four semicircular edge notches, two on each side, of a
0.125 in × 2.00 in ×12.00 in (3.2 mm × 51 mm × 305 mm) rectangular specimen. Crack
initiation and growth at the notches was monitored with the replication method. After
fatigue testing of the specimens, the replicas were covered with approximately 100 - 200
Angstroms of gold so that crack measurements could be made with the SEM. Both

12
actual lengths and projected lengths of the cracks were obtained from the replicas with an
automatic image analysis system (IBAS) [27, 28]. In addition, fractography was
performed on the fractured specimen surfaces to examine the crack initiation sites.
From these small crack tests, ALCOA researchers have obtained small crack
length, L, versus number of cycles N, as well as dL/dN-ΔK plots. Although data analysis
is still being performed, some qualitative observations could be made from the
preliminary results. First, the material with the low microporosity initiated cracks later
than the material with the higher microporosity. In addition, large pores in the materials
appear to have the greatest influence on crack initiation and propagation. Finally, the
dL/dN-ΔK plots show little difference between the low and high microporosity versions
of the alloy [27].
The research presented in this thesis is an extension of ALCOA's effort to
determine the effect of microporosity on the initiation and growth of small cracks in the
Al 7050-T7451 alloy. Three versions of the material were supplied for this effort. The
versions of the material shall be referred to in this thesis as "old", "new", and "three-inch
plate" material. Both the "old" and "new" materials were obtained from a six-inch plate,
and contain more microporosity than the "three-inch plate" version of the aluminum
alloy. This is due to the fact that the three-inch version of the material was rolled for a
longer period of time than the six-inch version, effectively "squeezing" out any remaining
microporosity.
Large crack da/dN-ΔK data for the Al 7050-T7451 alloy is shown in Figure 2.6
[29]. Although the large crack growth rates for all three versions of the material exhibit
the same da/dN-ΔK relationship [30], it is believed that the initial microporosity in each

13
of the three versions will effect the small crack growth rates in different ways. It was
hoped that the reduced microporosity versions of the material would delay crack
initiation, and exhibit better overall fatigue properties, thus justifying its increased cost.
The purpose of this research is determine whether this assumption is true by performing
small fatigue crack tests on all three versions of the material. In addition, an existing
crack growth prediction program was modified to analyze the growth of small fatigue
cracks from semicircular edge notches based on the experimental results.

Figure 2.1 The two stages of fatigue.

Figure 2.2 The semicircular edge notch geometry and variable definitions.

Figure 2.3 Typical fatigue crack growth rate data for large and small cracks [11].

Figure 2.4 Stress intensity factor vs. time for constant ampltide loading. Lower crack opening stress for small cracks results in
a larger effective stress intensity factor than a large crack under identical loading, translating into faster growth
rates.

Figure 2.5 Cumulative fatigue failure distributions from 1984-1987 for the 7050-T7451 thick plate (5.5-5.9 inches = 140-150
mm).

Figure 2.6 Large crack da/dN-ΔK data for the Al 7050-T7451 alloy, R=0.1 [29].

20
CHAPTER 3 - EXPERIMENTAL PROCEDURES
In this chapter, the experimental procedures for the test program are presented.
The first section covers the procedures involved with the acquisition of small crack
growth rate data, while the second section covers the procedures involved with the
acquisition of baseline data through large crack testing.
3.1. Small Crack Specimen Design and Testing Procedures
As mentioned previously, specimens which incorporate a semicircular edge notch
are useful in the procurement of small crack growth rate data. The original specimen
design used in this study was a double-edge notch dogbone specimen, and is illustrated in
Figure 3.1. The test specimen was secured to the load frame through pin-hole grips.
Two semicircular edge notches were placed on opposite sides of the specimen in order to
increase the amount of obtainable data in a single test. However, two constraints are
placed on this type of specimen design. First, the width of the specimen must be large
enough so that the growth of a small crack in one notch is not affected by the presence of
the opposite notch and/or other small cracks growing in the opposite notch. At the same
time, the width is limited by the diameter of the pins used to grip the specimen. Trial
tests would often fail in the pinhole grip area if the dog boned width was greater than the
0.75 inch (1.9 cm) diameter of the pins. Two tests were successfully performed with this
specimen design, but considering the constraints involved, a better design was required.

21
Discussions with ALCOA personnel on this problem [30] centered on the method
used to grip the specimen. To circumvent specimen failure in the grips, ALCOA
supplied this study with a grip design that "clamped" the specimen to the grips. The
normal force applied to the specimen faces generates enough friction to prevent the
specimen from slipping out of the grips. These "friction" grips allowed the specimen to
be simplified to a double-edge notch specimen with no dogbone. This rectangular-
shaped specimen is illustrated in Figure 3.2, and was the design used in this rest of the
test program. By increasing the specimen width to 2.00 inches (5.08 cm), this ensured
that the two notches of radii = 0.188 inch (4.78 mm) interacted little with each other.
A two-dimensional finite element analysis was performed on both the dogbone
and rectangular geometries to determine if the specimen width had any effect on the
stress distribution at the notch. The stress concentration factor, Kt, is defined as the ratio
of the hoop stress divided by the remote stress. In Figure 3.3, Kt's obtained from the
finite element analysis are plotted versus the angle off of the mid plane of the notch, Θ,
for both geometries. The figure reveals that both geometries exhibit essentially the same
stress distribution at the notch. Therefore, the crack initiation data obtained from both
specimen geometries were treated as equivalent in this study.
To ensure that crack initiation occurs at material inhomogeneities and not
machining marks, the notch surfaces were polished down to a 600 grit, followed by a
diamond paste. The surface is then etched with a Keller's etch for 10-30 seconds. This
removes any residual stresses generated by the machining and polishing process, and
provides a "map" of the notch surface by extracting grain boundaries.

22
3.1.1. Specimen Testing
Specimens fabricated from all three materials were tested in a servo-hydraulic test
machine with analog-based electronic controls in laboratory air under a constant stress
ratio (R = 0.1). Strain gages were placed on both sides of the specimen to measure the
difference in strain experienced during axial loading, giving an indication of the bending
present in the specimen. All tests involved applying two or three cycles to the maximum
load to ensure the specimen was aligned correctly within the grips, and that potential
bending strains were less than 5 % of the strains induced by the axial loading at the
commencement of testing. Most specimens were loaded at a maximum nominal stress
σnom =16 ksi (110 MPa), although two specimens were loaded at σnom = 15 ksi (103
MPa) and one specimen at σnom = 18 ksi (124 MPa). Table 3.1 lists the test parameters
for the ten semicircular edge notch test specimens.
"Old" material specimens are identified by the number 6•••-••, and were
obtained from blanks of the aluminum 7050-T7451 alloy with the most microporosity.
"New" material specimens are identified by the number 7•••-••, and were obtained from
blanks of the material with less microporosity. The blanks themselves were formed from
a six-inch thick plate at the ALCOA Technical Center. In addition, the numbering
system for the "old" and "new" material specimens is an abbreviation of the numbering
system ALCOA provided with the blanks. "Three-inch plate" specimens are identified
by the number 8••, and were obtained from a three inch thick plate of aluminum 7050-
T7451. This version of the material has the least amount of microporosity of all three
materials [30].

23
3.1.2. The Replication Method
As mentioned previously, surface crack initiation and growth were monitored
with the replication method [19]. Cycling was suspended periodically throughout the
test, and the specimen held under a constant tensile load while the notch surface was
replicated. The tensile load was equal to eighty percent of the mean load, ensuring that
all crack faces were open in the notch and thus making detection of the crack easier. The
notch surface was bathed with 1-2 drops of acetone from a hypodermic needle. Finally, a
0.003 inch (76 μm) thick acetate tape was placed within the notch; this is shown
schematically in Figure 3.4. The acetone softens the tape, allowing it to conform to the
notch surface and flow into the mouths of open cracks. Great care must be taken during
the replication process so that no air bubbles are trapped between the notch surface and
the tape. No information of the notch surface is transferred to the tape where a bubble is
located. After approximately 25 seconds, the tape is dry, leaving an exact replica of the
notch surface. At this point, the tape can be removed from the notch surface and testing
can recommence. Approximately 25-50 replicas were taken throughout each test to
sufficient enough data points are available for analysis.
Once the fatigue test was completed, analysis of the replicas begins. In several of
the tests, individual cracks coalesced into a single crack. To keep track of the crack
coalescence process, the following crack identification system was developed. When
measuring the cracks from the replicas, the last replica taken was examined first. This
replica would usually include a through-the-thickness crack, and sometimes smaller
surface and corner cracks that did not become the dominant crack. Each of these cracks
would be given an integer identification number 1, 2, 3, etc. As these cracks were traced
back in time through earlier replicas, an initial crack, say Crack ID # 2, would "divide"
into two smaller cracks (i. e. crack coalescence). These two cracks would then be given

24
the identification numbers 2.1 and 2.2, indicating that they coalesced into Crack ID # 2 at
a later time in the test. Similarly, Crack ID # 2.1 could "divide" into Crack ID's # 2.11
and # 2.12 as they were traced through earlier replicas. This crack identification system
provides a simple means to keep track of crack coalescence history, and hopefully aids in
following this coalescence process in a single plot of crack length versus cycles for a
particular specimen notch.
Cracks were measured from the replicas via two different methods. Larger
cracks, defined as a ≥ 0.003 inches (76 μm) were measured with a low powered
(magnification ≈ 7 ×) optical microscope. Replicas were mounted on a slide viewing
stage and the crack tip coordinates were measured using two micrometers attached to the
stage. The micrometers provided resolutions of 0.0001 inches (2.5 μm). The crack tip
coordinates were then converted to the x-s coordinate system and subsequently into the
x-Θ coordinate system. The x-coordinate is the distance along the bore of the notch.
The s-coordinate is defined as the notch radius × Θ, where Θ is the angle in radians
above/below the mid plane of the notch; see Figure 3.5. This determined the spatial
location of the crack within the notch, and subsequently its length. For cracks smaller
than 0.003 inches (76 μm), a higher powered optical microscope was used. This
microscope provided magnifications up to 1120 ×, and crack lengths were measured from
a video screen connected to the microscope. Due to the limited viewing field, only crack
lengths could be obtained from this method - not crack tip coordinates. However, spatial
location of the cracks along the notch bore could be obtained from other measurements
with the other microscope once crack lengths become larger. Cracks lengths in the range
a ≈ 0.003 inches (76 μm) were measured with both methods; these lengths showed good
agreement with each other.

25
An important concern with measurement is that the replicas would shrink 5-10%
as they dried on the specimen surface. Therefore, the measurements were normalized
with a shrinking factor. This factor was simply the ratio of the known notch thickness to
the measured replica width, providing a scale for all measurements made on that replica.
Small crack growth experiment results from the semicircular edge notch
specimens are presented in Chapter 4.
3.2. Large Crack Testing Procedures
ALCOA researchers have established the large fatigue crack growth properties for
the aluminum 7050-T7451 alloy through numerous fatigue tests under various loading
conditions and specimen geometries [31]. However, for completeness it was decided to
quantify large fatigue crack growth rate properties for the alloys. In addition,
supplementary large crack testing would further substantiate ALCOA's belief that all
three versions of the alloy exhibited the same large crack growth properties [30].
To accomplish this, compact tension (CT) specimens were fabricated from
fractured semicircular edge notch specimens, as is illustrated in Figure 3.6. The CT
specimens were designed in accordance with ASTM Standard E647 [32], and its
geometry is shown in Figure 3.7. The CT specimens were fabricated from fractured
semicircular edge notch specimens to conserve the material used in this study. Although
the CT specimens consist of material that has been previously cycled, it is believed that
once a pre crack has started in the specimen, the large crack growth properties are
relatively unaffected by the previous loading.

26
Table 3.1 lists the test parameters for the four CT test specimens. CT specimen
pre cracking was conducted according to ASTM Standard E647 [32]. Traveling
microscopes were mounted on both sides of the specimen in order to obtain front and
back through-crack lengths. One of the microscopes was attached to a digital measuring
system accurate to 0.0005 inches (13 μm). However, due to equipment problems with
the second digital measuring system, a microscale accurate to 0.005 inches (130 μm) was
used to obtain crack lengths with the other microscope. Through crack lengths were
taken as the average of the front and back crack lengths, and were recorded to the nearest
0.005 inch (130 μm).
The large crack growth rate data obtained from the CT specimen tests are
presented in Chapter 4.

Table 3.1 Parameters for fatigue test specimens.
Specimen ID Material Type Specimen
Type
Max. Nominal
Stress
(ksi / MPa)
Stress Ratio Frequency
6714-a11 "old" dogbone DEN 15 / 103 0.1 5 Hz
6714-a12 "old" dogbone DEN 15 / 103 0.1 10 Hz
6612-b21 "old" friction DEN 16 / 110 0.1 10 Hz
6611-a12 "old" friction DEN 16 / 110 0.1 10 Hz
7111-b11 "new" friction DEN 15, 18 / 103, 124 0.1 8 Hz
7111-b12 "new" friction DEN 16 / 110 0.1 10 Hz
7012-a22 "new" friction DEN 16 / 110 0.1 10 Hz
8T3 "3-inch plate" friction DEN 16 / 110 0.1 10 Hz
8B3 "3-inch plate" friction DEN 16 / 110 0.1 10 Hz
8B2 "3-inch plate" friction DEN 16 / 110 0.1 10 Hz
Max. Load for
CT Specimens
(lbs / N)
6611-a12-CT2 "old" CT 350 / 1560 0.1 10 Hz
7012-a21-CT4 "new" CT 350 / 1560 0.1 10 Hz
8T3-CT3 "3-inch plate" CT 450 / 2000 0.1 10 Hz
8T3-CT4 "3-inch plate" CT 450 / 2000 0.1 10 Hz

Figure 3.1 Double-edge notch dogbone specimen geometry and dimensions.

Figure 3.2 Double-edge notch specimen geometry and dimensions.

Figure 3.4 Illustration of the replication process.

Figure 3.5 Definition of replica coordinate system.

Figure 3.6 CT specimens fabricated from fractured double-edge notch specimen.

Figure 3.7 CT specimen geometry and dimensions.

Theta vs. (Remote Stress/Hoop Stress)
Semicircular Edge Notch Geometries
Theta (radians)
Srem/Shoop
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Width = 1.11 in.
Width = 2 in.
Figure 3.3 Hoop Stress/Remote Stress, σhoop/σrem, vs. Theta, Θ, for the 1.11 and 2.00 inch wide specimen geometries.
σhoop/σrem was caculated with a 2-dimensional finite element analysis of each of the specimen geometries. Θ is
defined as the angle (in radians) from the tip of the notch to the upper/lower location where the notch meets the
specimen edge.

35
CHAPTER 4 - EXPERIMENTAL RESULTS
4.1. Large Crack Growth Rate Data
Four CT specimens were tested to obtain the large crack growth properties for all
three versions of the aluminum 7050-T7451 alloy. All CT specimens were tested at a
frequency of 10 Hz and a stress ratio of 0.1. The "old" material CT specimen was loaded
at a maximum load of 300 lbs (1330 N). The "new" material CT specimen was loaded at
a maximum load of 350 lbs (1560 N). Finally, two "3-inch plate" material CT specimens
were loaded at maximum loads of 300 lbs (1330 N) and 450 lbs (2000 N). All CT
specimen testing and data analysis was performed in accordance with ASTM Standard
E647 [32]. The stress intensity factor range solution for the CT specimen geometry is
given by
Δ
Δ
K
P
B W
=
+
−
+ − + −
( )
( )
( . . . . . )/
2
1
0 886 4 64 13 32 14 72 5 63 2
2 3 4α
α
α α α α (4.1)
where a ≡ crack length, W ≡ width, B ≡ thickness, ΔP ≡ applied load range, and α = a/W.
Fatigue crack growth rates were calculated with a seven-point polynomial technique [32].
The fatigue crack growth rates for the CT specimens are plotted against the
applied stress intensity factor range in Figure 4.1. Two important things can be discerned
from this

36
"large" crack growth rate curve. First, all three versions of the alloy exhibit essentially
the same large fatigue crack growth rate properties. Second, the CT crack growth rate
data correlates well with numerous fatigue tests performed by ALCOA under various
loading conditions and specimen geometries. The Paris law expression shown in Figure
4.1 for Al 7050-T7451 (R = 0.1) was obtained from Reference [29] for the data
reproduced here in Figure 2.5, and is given by
da
dN
K= × −
3 9 10 10 4 175
. ( ) .
Δ (4.2)
The units for ΔK in Equation 4.2 are ksi√in, while da/dN is measured in inches/cycle.
Although CT test data obtained here do not extend into either the threshold ΔK or the
fracture toughness regions of the da/dN-ΔK curve, it does correlate well with the
ALCOA generated data shown in Figure 2.5 [29]. Thus, the ALCOA Paris Law
expression (Equation 4.2) is used here for subsequent analysis of the materials' large
crack growth properties.
4.2. Small Crack Test Results
Table 4.1 summarizes the results of the double-edge notch specimen tests. Before
studying the results of individual tests, some general information should be noted first.
The terms "front" and back" identify the notch location relative to the servo-hydraulic
test machine. The table indicates that cracks initiated at an equal rate in both the front
and back notches for all the specimens tested. This provided added assurance that
potential bending was kept to a minimum in the tests, i. e., there was no bias as to which
notches caused crack initiation.

37
In addition, two different crack lengths are used here to define fatigue crack
"initiation." Although crack lengths of 2a<0.001 inch (25 μm) were obtained, several
tests had cracks of that length traceable back to "zero" cycles. In actuality, however, the
term "zero" cycles does not include specimen loading which occurred during gripping
and alignment procedures. All tests involved applying two or three cycles to the
maximum load to ensure the specimen was aligned correctly within the grips, and that
potential bending strains were less than five percent of the strains induced by the axial
loading. Therefore, a more generous initiation length of 2a=0.005 inch (127 μm) was
also included in the table. It is important to note that both definitions of "initiation" place
the crack length well within the small crack region of 2a < 0.02 inch (500 μm).
Surface crack length vs. number of elapsed cycles for the double-edge notch tests
are plotted in Figures 4.2 - 4.14. Surface crack lengths are plotted until "breakthrough",
i. e., until the surface crack has become a through crack at the notch.
In several of the tests, a single crack initiated at approximately the center of the
notch and grew into the through crack that eventually caused specimen failure. A series
of replica photographs in Figures 4.15 - 4.20 illustrates the growth of a lone crack in the
"back" notch of specimen 6612-b21. In Figure 4.15, a crack appears to be emanating
from a micropore at "0" cycles. By 10,001 cycles (Figure 4.16), the crack has grown and
established itself. Figures 4.17 - 4.20 follow the growth of the crack at a lower
magnification from 10,001 cycles to 42,509 cycles.
Some tests, however, were characterized by multiple cracks initiating at several
points along the bore of the notch. These cracks in turn coalesced into larger cracks, with
a dominant crack eventually leading to specimen failure. Specimen 6611-a12 (Figure

38
4.3) is the most prolific example of multiple crack initiation, with ten different cracks
initiating in the front notch. The most likely reason for the large number of cracks is the
high degree of microporosity in the "old" material, resulting in a greater number of
initiation sites in this specimen. An interesting phenomenon associated with multiple
crack initiation / interaction is illustrated in specimen 7012-a22 (Figures 4.7 and 4.8).
For some of the cracks, the final length measurements are smaller than measurements
taken at previous cycles. It may be possible that extension of the large dominant crack
prevents complete opening of adjacent smaller cracks, and thus makes them appear to be
smaller as life progresses. For example, in Figure 4.8, Crack ID #1 in the final
measurement is a through crack; it is fully open. However, Crack ID #'s 2 and 3 are only
small surface cracks compared to #1, and are only partially open in the final
measurements.
A series of replica photographs in Figures 4.21 - 4.24 illustrate multiple cracks
interacting with one another in the front notch of specimen 6611-a12. At 42,507 cycles
(Figure 4.21), Crack ID #'s 4.1 and 4.2 are shown in the center and upper-right hand
corner, respectively, while Crack ID # 8 is essentially a micropore to the left of # 4.1. By
50,007 cycles (Figure 4.22), # 8 has established itself, while # 4.1 is growing towards
both # 4.2 and # 8. At 60,008 cycles (Figure 4.23), however, # 4.1 has bypassed # 8, and
has almost coalesced with # 4.2. By 65,008 cycles (Figure 4.24), cracks 4.1 and 4.2
have coalesced into Crack ID # 4. Crack ID # 8 is starting to close due to its close
proximity to the larger # 4.
Examination of the fracture surfaces provides another method in determining
fatigue crack initiation. ALCOA researchers have examined the fracture surfaces of Al
7050-T7451 open hole fatigue specimens with the scanning electron microscope (SEM)

39
in order to locate crack initiation sites [1]. In that study, they determined that the fatigue
cracks initiated from micropores in the material rather than machining flaws. SEM
examination of this study's double-edge notch specimen fracture surfaces is currently
being performed by Jon Elsner on a JEOL JSM-T300 SEM [33]. The accelerating
voltage is 25 kV, and utilizes background scatter electrons as the imaging technique. An
example of Elsner's current work is presented here to illustrate the technique and initial
results.
Figure 4.25 is a fractograph of the "front" notch fracture surface for specimen
6611-a12, and shows two cracks which developed at this notch. Although catastrophic
failure initiated at the "back" notch of this specimen, the elliptical shapes of the dominant
cracks in the "front" notch were preserved. The larger crack on the left was identified as
Crack ID # 6 during the replica measurement process, while the smaller crack on the
right was identified as Crack ID # 5. Figures 4.26 - 4.29 show larger magnifications of
the initiation sites for Crack ID #'s 5 and 6. In both cases, the initiation sites appear to be
micropores in the material just beneath the notch surface.
4.3. Small Crack Growth Rate Data
Specimens where a single crack initiated in one of the notches were used here to
characterize the small fatigue crack growth rate data. As shown in Equation 2.4,
Newman has developed approximate K solutions for corner and surface cracks in a
semicircular edge notch [4]. Variable definitions are illustrated in Figure 2.3, whereas
noted previously t is defined as B for a corner crack, whereas for a surface crack, t is
defined as B/2. Newman's full stress intensity factor solutions are given here in
Appendix A.

40
It is important to note that the specimen design met all restrictions placed on the
edge notch geometry for the K solution to be valid except for the requirement that r/w =
0.0625 (see Appendix A for complete geometry restrictions). For the early dog bone
double-edge notch specimens, r/w ≈ 0.0845; for the friction grip double edge-notch
specimens, r/w ≈ 0.0469. As mentioned previously in Chapter 3, a finite element
analysis was performed to determine the stress distribution at the notch for both specimen
geometries. The stress concentration factor Kt at the mid plane of the notch (Θ = 0) was
calculated to be Kt = 3.03 for the dogbone geometry and Kt = 3.05 for the friction grip
geometry. These results are 3.5 % less than the stress concentration factor used in the
Newman ΔK solutions of Kt = 3.15 for uniform displacement [4]. Because of the close
correlation between the finite element analysis Kt's and the Kt used in Newman's stress
intensity factor solutions, the r/w restriction was considered insignificant in this study.
The replication method can only obtain the surface lengths of cracks, or "2a", and
not the crack depths, "c", defined in Figure 2.2. Since the stress intensity factor solution
for cracks at a semicircular edge notch depends on the crack aspect ratio, a/c, an
expression for a/c is required to calculate ΔK's for the double-edge notch specimen.
Swain and Newman measured crack lengths in both the a and c direction with the use of
marker loads in the 2024-T3 aluminum alloy [34]. Based on the experimental data, they
developed an empirical relationship between the crack shape, c/a, and the non
dimensional length, a/t, given by
c a = −0.9 0.25(a t)2
(4.3)

41
This expression is plotted in Figure 4.30. In addition, a representation of the crack
shapes predicted by the expression is shown to scale in Figure 4.31 for a corner crack.
As mentioned earlier, the SEM examination of the fracture surface of specimen 6611-
a12 allows for actual crack shape measurements to be obtained for Crack ID #'s 5 and 6
in the "front" notch. These measurements are also plotted in Figure 4.30, and correlate
well with Swain and Newman's empirical prediction. Although an exhaustive
examination of all specimen fracture surfaces has not been performed at this time, the
empirical expression for c/a vs. a/t should be adequate for calculating the stress intensity
factor ranges.
The small crack growth rates for double-edge notch specimens where a single
crack initiated along the bore of a notch are plotted against the applied stress intensity
factor ranges in Figure 4.32. The Paris Law expression obtained from ALCOA fatigue
tests for Al 7050-T7451 (Equation 4.2) is also plotted for comparison. The crack growth
rate data obtained from these tests demonstrates that small cracks do, in fact, grow faster
than large cracks at equivalent ΔK loading near the threshold region. However, the small
crack growth rate data merges with the large crack Paris Law at higher ΔK's. A linear
regression was performed on the data to obtain the Paris Law constants for the small
crack growth rate data. The "small" crack Paris Law expression is given by
da
dN
K= × −
8 22 10 9 2 807
. ( ) .
Δ (4.4)
The units for ΔK in Equation 4.4 are ksi√in, while da/dN is measured in inches/cycle. It
is important to note that there is greater variability in the small crack growth rate data
compared to large crack growth rate data. This is not surprising due to the fact that
LEFM principles are being pushed to the limit as well as uncertainties in the small crack

42
measurements. The Paris Law expressions for both small and large crack growth rate in
Al 7050-T7451 are incorporated in a computer program that predicts the crack growth of
surface and corner cracks in a variety of geometries. In addition, the single crack growth
rate Paris Law expressions will be used to predict the growth of multiple cracks by taking
into account interaction between the crack tips. The program's background and
implementation in this study is presented in Chapter 5.

Table 4.1 Test matrix for the double-edge notch specimens.
Specimen ID Nominal
Stress (ksi /
MPa)
Number of
Cracks (front)
Number of
Cracks (back)
Cycles to
2a ≥ 0.001 "
(25 μm)
Cycles to
2a ≥ 0.005 "
(127 μm)
Cycles to
Specimen
Failure
"Old"
Material:
6714-a11 15 / 103 0 2 0 25,002 120,523
6612-b21 16 / 110 0 1 0 20,002 76,952
6714-a12 15 / 103 2 0 30,150 65,485 174,958
6611-a12 16 / 110 10 3 2,501 12,502 75,066
"New"
Material:
7111-b11 15, 18/103,
124†
0 1 183,071 224,564 254,835
7111-b12 16 / 110 0 2 162,511 200,004 243,912
7012-a22 16 / 110 4 4 0 27,501 125,262
3 " Plate
Material:
8T3 16 / 110 1 0 120,708 128,209 245,771
8B3 16 / 110 6 1 15,001 22,502 77,798
8B2 16 / 110 0 1 15,002 22,503 86,962
† Specimen 7111-b11 was loaded at a maximum nominal stress of 15 ksi for 194,403 cycles. At that point, the maximum nominal stress
was increased to 18 ksi.

Fatigue Crack Growth Rate
Data for Al 7050-T7451 Alloy
Delta K, ksi sqrt(in.)
da/dN,in./cycle
1.00E-10
1.00E-09
1.00E-08
1.00E-07
1.00E-06
1.00E-05
1.00E-04
1.00E-03
1 10 100
8T3-CT4 (3 in. plate material)
8T3-CT3 (3 in. plate material)
6611-a12-CT2 (old material - more
microporosity)
7012-a21-CT4 (new material - less
microporosity)
Large Crack Paris Law
da/dN=3.9e(-10)*dK^(4.175)
Figure 4.1 Large crack growth rate vs. stress intensity factor range data for Al 7050-T7451 (obtained from CT specimens)
The large crack Paris Law was obtained from Reference [29] (see Figure 2.5).

Crack Length vs. Number of Cycles:
Specimen 6611-a12, Back Notch
Number of Cycles, N
CrackLength,a(inches)
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0 10000 20000 30000 40000 50000 60000 70000
Crack ID # 1
Crack ID # 1.1
Crack ID # 1.2
Crack ID # 2
Figure 4.2 Surface crack length vs. number of elapsed cycles for specimen 6611-a12 (old material), "back" notch.

Specimen 6611-a12, Front Notch
Number of Cycles, N
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0 10000 20000 30000 40000 50000 60000 70000
Crack ID # 1
Crack ID # 2
Crack ID # 3
Crack ID # 4.1
Crack ID # 4.2
Crack ID # 4
Crack ID # 5
Crack ID 6.1
Crack ID 6.2
Crack ID # 6
Crack Id # 7
Crack ID # 8
Figure 4.3 Surface crack length vs. number of elapsed cycles for specimen 6611-a12 (old material), "front" notch.

Specimen 6612-b21, Back Notch
Number of Cycles, N
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0 10000 20000 30000 40000 50000 60000 70000
Crack ID # 1
Figure 4.4 Surface crack length vs. number of elapsed cycles for specimen 6612-b21 (old material), "back" notch.

Number of Cycles, N
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0 20000 40000 60000 80000 100000 120000
Crack ID #2
Crack ID # 1
Figure 4.5 Surface crack length vs. number of elapsed cycles for specimen 6714-a11 (old material), "back" notch.

Number of Cycles, N
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0 20000 40000 60000 80000 100000 120000 140000 160000 180000
Crack ID # 1.1
Crack ID # 1.2
Crack ID # 1
Figure 4.6 Surface crack length vs. number of elapsed cycles for specimen 6714-a12 (old material), "front" notch.

Number of Cycles, N
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0 20000 40000 60000 80000 100000 120000
Crack ID # 1
Crack ID # 2
Crack ID # 3
Crack ID # 4
Figure 4.7 Surface crack length vs. number of elapsed cycles for specimen 7012-a22 (new material), "back" notch.

Number of Cycles, N
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0 20000 40000 60000 80000 100000 120000
Crack ID # 1
Crack ID # 2.1
Crack ID # 2.2
Crack ID # 2
Crack ID # 3
Figure 4.8 Surface crack length vs. number of elapsed cycles for specimen 7012-a22 (new material), "front" notch.

Number of Cycles, N
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0 50000 100000 150000 200000 250000
Crack ID # 1
Figure 4.9 Surface crack length vs. number of elapsed cycles for specimen 7111-b11 (new material), "back" notch.

Number of Cycles, N
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0 50000 100000 150000 200000 250000
Crack ID # 1.1
Crack ID # 1.2
Crack ID # 1
Figure 4.10 Surface crack length vs. number of elapsed cycles for specimen 7111-b12 (new material), "back" notch.

Specimen 8B2, Back Notch
Number of Cycles, N
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0 10000 20000 30000 40000 50000 60000 70000 80000
Crack ID # 1
Figure 4.11 Surface crack length vs. number of elapsed cycles for specimen 8B2 (3-inch plate material), "back" notch.

Specimen 8B3, Back Notch
Number of Cycles, N
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0 10000 20000 30000 40000 50000 60000
Crack ID # 1
Figure 4.12 Surface crack length vs. number of elapsed cycles for specimen 8B3 (3-inch plate material), "back" notch.

Specimen 8B3, Front Notch
Number of Cycles, N
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0 10000 20000 30000 40000 50000 60000
Crack ID # 1
Crack ID # 1.1
Crack ID # 1.2
Crack ID # 1.21
Crack ID # 1.22
Crack ID # 2
Crack ID # 3
Crack ID # 4
Crack ID # 5
Figure 4.13 Surface crack length vs. number of elapsed cycles for specimen 8B3 (3-inch plate material), "front" notch.

Specimen 8T3, Front Notch
Number of Cycles, N
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0 50000 100000 150000 200000 250000
Crack ID # 1
Figure 4.14 Surface crack length vs. number of elapsed cycles for specimen 8T3 (3-inch plate material), "front" notch.

Empirical Expression for
Crack Shape vs. Nondimensional Length
a/t
c/a
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Empirical Expression for a/t
vs. c/a
Measured Values of a/t vs. c/a
Figure 4.30 Empirical expression for crack shape vs. nondimensional length [28]. Measured values are from Crack ID #'s 5 and
6 from specimen 6611-a12, "front" notch. Definitions for c, a, and t are illustrated in Figure 4.31.

Figure 4.31 Illustration of corner crack shape based on empirical expression for c/a vs. a/t (to scale). Note: for the corner
crack, t≡B.

Small Fatigue Crack Growth Rate
Data for Al 7050-T7451 Alloy
Delta K, ksi sqrt(in.)
da/dN,in./cycle
1E-10
1E-09
1E-08
1E-07
1E-06
1E-05
0.0001
0.001
1 10 100
Large Crack Paris Law
Small Crack Paris Law
6714-a11, back notch
6612-b21, back notch
8B2, back notch
8B3, back notch
8T3, front notch
da/dN=3.9e(-10)*dK^(4.175)
da/dN=8.22e(-9)*(dK)^2.807
Figure 4.32 Small crack growth rate vs. stress intensity factor range data for Al 7050-T7451 (obtained from double-edge notch
specimens).

Figure 4.25 SEM fractograph of specimen 6611-a12, "front" notch fracture surface.
The larger crack on the left was identified as Crack ID # 6 during the
replica measurement process. The smaller crack on the right was
identified as Crack ID # 5 during the replica measurement process. The
reference line on the fractograph is 1000 μm in length.

Figure 4.26 Initiation site of Crack ID # 6. The reference line on the fractograph is
100 μm in length.
Figure 4.27 Close-up of initiation site of Crack ID # 6. The reference line on the
fractograph is 10 μm in length.

Figure 4.28 Initiation site of Crack ID # 5. The reference line on the fractograph is
100 μm in length.
Figure 4.29 Close-up of initiation site of Crack ID # 5. The reference line on the
fractograph is 10 μm in length.

Figure 4.15 Replica photograph of specimen 6612-b21, "back" notch, 0 cycles (after
specimen alignment loading). Crack ID # 1: 2a = 0.0016 in.
Figure 4.16 Replica photograph of specimen 6612-b21, "back" notch, 10001 cycles.
Crack ID # 1: 2a = 0.0038 in.

Crack ID # 1: 2a = 0.0038 in.
Crack ID # 1: 2a = 0.0092 in.

Crack ID # 1: 2a = 0.0179 in.
Crack ID # 1: 2a = 0.0337 in.

Figure 4.21 Replica photograph of specimen 6611-a12, "front" notch, 42507 cycles.
Crack ID # 4.1: 2a = 0.0208 in. Crack ID # 4.2: 2a = 0.0025 in.
Crack ID # 8: 2a = 0.0009 in.
Crack ID # 8: 2a = 0.0032 in.

Crack ID # 8: 2a = 0.0032 in.
Crack ID # 4: 2a = 0.0441 in. Crack ID # 8: 2a = 0.0032 in.

69
CHAPTER 5 - ANALYTICAL MODELING
A computer program was employed to predict both crack shape and fatigue crack
growth in the double-edge notch specimens tested in this study. A brief history of the
program and its various implementations is presented in this chapter. This is followed by
modifications made to the program to fit this study as well as a description of how the
algorithm works.
5.1. Background
The multi-degree of freedom algorithm used in this study was originally coded by
Tritsch [34] to predict the fatigue life and crack growth shapes for both single and double
cracks located along the bore of a hole loaded under remote tension; see Figures 5.1 and
5.2. The program utilized the Newman-Raju stress intensity factor solutions [36] for a
single surface or corner crack in a hole. These K solutions could subsequently be
modified with correction factors to handle various geometries as well as crack interaction
effects.
Tritsch's original algorithm utilized Bowie's two-dimensional stress intensity
factor solution for a through-cracked hole [37] to develop correction factors so that the
computer program could be used for various specimen geometries. For this study,
however, Newman's three-dimensional K solutions for corner and surface cracks in a

70
semicircular edge notch [4] were available, thus eliminating the need to use correction
factors based on two-dimensional geomtries. For the cases of two cracks along the bore
of a hole (Figure 5.2), Tritsch employed a crack interaction factor developed by Heath
and Grandt [38]. They used the Finite Element-Alternating Method (FEAM) to obtain
stress intensity factor solutions for both a single corner crack along the bore of a hole and
symmetric corner cracks on the same side of the hole. These solutions were calculated
for crack shapes of a/c = 1.11, 1.5, 2.0, and 3.0. The interaction factor, γ, is then given by
γ =
K
K
symmetric c c
gle c c
L
L
. .
sin . .
(5.1)
Ksymmetric cc is the stress intensity factor for two corner cracks along the bore of a hole
symmetric with respect to the plane at the half-thickness of the specimen, while Ksingle cc
is the stress intensity factor for the single corner crack along the bore of a hole. The
interaction factor is a function of the crack shape, a/c, and the non dimensional separation
distance between the two symmetric corner cracks, ts/a. Polynomial expressions were
subsequently fit to the FEAM results, and incorporated into the program. It is important
to note that the interaction factor was employed only on the tips of the two cracks
adjacent to one another; the crack tips next to the free surfaces were not modified. In
addition, although γ was based on the interaction between two symmetric corner cracks,
it was employed to handle various unsymmetric combinations of corner and surface
cracks.
Scheumann [39] updated the original code to take into account interaction effects
between unsymmetric cracks on opposite sides of a hole in a plate. In addition, Grandt,
Hinkle, Scheumann, and Todd [29] developed an interaction factor based on Trantina and

71
Barishpolsky's [40] effective stress intensity factor for an ellipsoidal void in a large body
with an equatorial crack. This interaction factor was employed to predict the growth of
cracks initiating from particles and micropores in Al 7050-T7451 open hole fatigue
specimens [29]. Although predictions based on the Trantina-Barishpolsky interaction
factor compared favorably with actual specimen lives in [29], the interaction factor was
not used in this study. Instead, the small crack growth rate equation given in Equation
4.4 will be utilized in a modified version of Tritsch's original program [35] to back-
predict fatigue crack growth to breakthrough and crack shapes in the double-edge notch
fatigue specimens tested in this study.
5.2. Description of Algorithm
The goal of a multi-degree of freedom algorithm is to predict both crack size and
shape as a function of the applied load. As mentioned previously, Tritsch's original
algorithm [35] was modified to utilize Newman's three-dimensional K solutions for
corner and surface cracks in a semicircular edge notch [4]. Figure 5.1 illustrates
geometry variable definitions utilized in the prediction code for the case of a single
surface crack or corner crack. The initial dimensions of the crack are defined by the
input coordinates, i. e.,
a a for surface cracks x x
c y
1 1 3 1
1 2
2( ) = −
=
(5.2), (5.3)
The crack depth, c1, was selected to grow an increment Δc1 = 0.0001×c1. The number
of cycles required for c1 to grow an increment Δc1 is
Δ ΔN c dy dN= 1 2( ) (5.4).

72
where dy2/dN, the crack growth rate in the c direction, is given by Equations 4.4 and 4.2
for small and large cracks, respectively. The stress intensity factor solutions for a surface
and corner crack in an edge notch [4] are used to calculate the crack growth rates. The
subsequent growth of the surface crack tips, xi, are then
Δ Δx dx dN Ni i= ×( ) (5.5).
Similarly, dxi/dN is given by Equations 4.4 and 4.2 for small and large cracks,
respectively. The new crack coordinates defining the surface crack length and depth are
then recalculated, and the process is repeated. This algorithm is easily extended to
handle two or more cracks; a typical multi-crack configuration is shown in Figure 5.2.
After each iteration, the program checks for crack tip free surface contact, or in
the case of the double crack configuration, crack coalescence. If a change in the crack
type occurs, e. g., a surface crack becomes a corner crack, the change is noted and all
subsequent calculations are based on the new crack type. Once the surface or corner
crack has broken through, iteration proceeds as a through crack with an initial crack
length equal to the last crack depth calculation, c1. After breakthrough, the program
utilizes the stress intensity factor solution presented by Newman [4] for a through crack
located at a semicircular edge notch; it is given in the following form
K S cF c w c r r wn= π ( / , / , / ) (5.6).

73
The full stress intensity factor solution is given in Appendix A. Iteration continues until
either the fracture toughness of the material is reached, or a specified maximum crack
growth rate is exceeded.
The prediction code was adjusted to include a correction factor for finite notch
width. The finite width correction factor was incorporated because the crack did not
always initiate at the half-thickness of the specimen, i. e., one of the crack tips was closer
to its respective specimen side than the other crack tip. However, the ΔK solution [4]
assumes that a surface crack is located at the center of the notch. The correction factor is
the ratio of stress intensity factors for a centrally located through crack, K2D center,and an
eccentrically located through crack in a sheet of finite width, K2D eccentric, [41], i. e.,
γ fnw
D eccentric
D center
K
K
= 2
2
L
L
..
.
(5.7)
The results of the back-prediction of fatigue crack shape and growth in the double edge-
notch specimen are discussed in Chapter 6.

74
Figure 5.1 Geometry variable definitions used in the prediction program for a surface
crack and a corner crack.

75
Figure 5.2 Geometry variable definitions used in the prediction program for a typical
multiple crack configuration.

76
CHAPTER 6 - NUMERICAL RESULTS
6.1 Back-Prediction in Specimens Used to Calculate Small Crack da/dN-ΔK curve
To verify the validity of the life prediction code, crack growth predictions were
first performed for the specimens that were used to generate the small-crack da/dN-ΔK
data in Chapter 4. Although these tests provided the fatigue crack growth data for the
predictive model (Equation 4.4), these calculations are independent of the input data file
in the sense that crack shape is predicted by the two-degree-of-freedom analysis
described in Chapter 5. Recall that only the crack surface dimension "2a" was measured
from the replicas, so that when computing the cyclic stress intensity factor ΔK to
generate Equation 4.4, it was necessary to assume a corresponding crack depth "c" (i.e.,
Equation 4.3, which assumes that crack shape c/a is a function of crack size a/t). Thus,
applying the crack growth analysis program to these tests provides an independent
prediction for the crack shape c/a, which can then be compared with the assumed shape
(Equation 4.3) employed to establish the da/dN relationship.
Plots are given for both the surface crack length, a, vs. number of elapsed cycles,
N, and crack aspect ratio, c/a, vs. the non dimensional surface crack length, a/t.
Definitions of these crack parameters are illustrated in Figure 2.2.
In the a vs. N plots of Figures 6.1-6.7, there is little difference between the
predicted results of the original algorithm and the predicted results with the finite width

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