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.

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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.

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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

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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

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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 C4 Crack measurements for specimen 6714-a11, back notch.......................141 C5 Crack measurements for specimen 6714-a12, front notch.......................142 C6 Crack measurements for specimen 7012-a22, back notch.......................144 C7 Crack measurements for specimen 7012-a22, front notch.......................147 C8 Crack measurements for specimen 7111-b11, back notch.......................150 C9 Crack measurements for specimen 7111-b12, back notch.......................151 C10 Crack measurements for specimen 8B2, back notch ...............................152

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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

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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

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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 4.3 Surface crack length vs. number of elapsed cycles for

specimen 6611-a12 (old material), "front" notch. .....................................46 4.4 Surface crack length vs. number of elapsed cycles for

specimen 6612-b21 (old material), "back" notch. .....................................47 4.5 Surface crack length vs. number of elapsed cycles for

specimen 6714-a11 (old material), "back" notch.......................................48 4.6 Surface crack length vs. number of elapsed cycles for

specimen 6714-a12 (old material), "front" notch. .....................................49 4.7 Surface crack length vs. number of elapsed cycles for

specimen 7012-a22 (new material), "back" notch. ....................................50 4.8 Surface crack length vs. number of elapsed cycles for

specimen 7012-a22 (new material), "front" notch.....................................51 4.9 Surface crack length vs. number of elapsed cycles for

specimen 7111-b11 (new material), "back" notch.....................................52 4.10 Surface crack length vs. number of elapsed cycles for

specimen 7111-b12 (new material), "back" notch.....................................53

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Figure Page 4.11 Surface crack length vs. number of elapsed cycles for

specimen 8B2 (3-inch plate material), "back" notch. ................................54 4.12 Surface crack length vs. number of elapsed cycles for

specimen 8B3 (3-inch plate material), "back" notch. ................................55 4.13 Surface crack length vs. number of elapsed cycles for

specimen 8B3 (3-inch plate material), "front" notch. ................................56 4.14 Surface crack length vs. number of elapsed cycles for

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

4.16 Replica photograph of specimen 6612-b21, "back" notch,

10001 cycles. Crack ID # 1: 2a = 0.0038 in. .............................................58 4.17 Replica photograph of specimen 6612-b21, "back" notch,

10001 cycles. Crack ID # 1: 2a = 0.0038 in. .............................................59 4.18 Replica photograph of specimen 6612-b21, "back" notch,

23001 cycles. Crack ID # 1: 2a = 0.0092 in. ............................................59 4.19 Replica photograph of specimen 6612-b21, "back" notch,

33506 cycles. Crack ID # 1: 2a = 0.0179 in. ............................................60 4.20 Replica photograph of specimen 6612-b21, "back" notch,

42509 cycles. Crack ID # 1: 2a = 0.0337 in. ............................................60 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

4.22 Replica photograph of specimen 6611-a12, "front" notch,


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Figure Page 4.23 Replica photograph of specimen 6611-a12, "front" notch,


4.24 Replica photograph of specimen 6611-a12, "front" notch,

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

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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,

back notch: surface crack length vs. number of cycles.............................85 6.3 Actual and predicted crack growth for specimen 7111-b11,

back notch: surface crack length vs. number of cycles.............................86 6.4 Actual and predicted crack growth for specimen 7111-b12,

back notch: surface crack length vs. number of cycles.............................87 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


6.10 Predicted crack growth for specimen 7111-b11, back


6.11 Predicted crack growth for specimen 7111-b12, back


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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

vs. a/t for both the predicted values and the empirical expression assumed in calculating ΔK solutions. ......................................96

6.14 Predicted crack growth for specimen 8T3, front notch: c/a

vs. a/t for both the predicted values and the empirical expression assumed in calculating ΔK solutions. ......................................97

6.15 Stress intensity factor geometry for two offset parallel

cracks in a sheet under uniform uniaxial tensile stress [42]. .....................98 6.16 Actual and predicted crack growth for specimen 7012-a22,

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

notch: c/a vs. a/t for both the predicted values and the empirical expression assumed in calculating ΔK solutions. Note: no crack interaction is considered between the cracks. ......................................................................................................100

6.18 Actual and predicted crack growth for specimen 7012-a22,

back notch: surface crack length vs. number of cycles. Note: no crack interaction is considered between the 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


notch: c/a vs. a/t for both the predicted values and the empirical expression assumed in calculating ΔK solutions. Note: no crack interaction is considered between the 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

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Figure Page 6.20 Actual and predicted crack growth for specimen 7012-a22,

front notch: surface crack length vs. number of cycles. Note: the presence of Crack ID #'s 2 and 3 are ignored.. ........................103

6.21 Predicted crack growth for specimen 7012-a22, front

notch: c/a vs. a/t for both the predicted values and the empirical expression assumed in calculating ΔK solutions. Note: the presence of Crack ID #'s 2 and 3 are ignored.. ........................104


back notch: surface crack length vs. number of cycles. Note: the presence of Crack ID #'s 1.1 and 2 are ignored. ......................105


notch: c/a vs. a/t for both the predicted values and the empirical expression assumed in calculating ΔK solutions. Note: the presence of Crack ID #'s 1.1 and 2 are ignored. ......................106


back notch: surface crack length vs. number of cycles. Note: the presence of Crack ID # 2 is ignored.........................................107


notch: c/a vs. a/t for both the predicted values and the empirical expression assumed in calculating ΔK solutions. Note: the presence of Crack ID # 2 is ignored.........................................108


front notch: surface crack length vs. number of cycles. ........................109 6.27 Predicted crack growth for specimen 6714-a12, front

notch: c/a vs. a/t for both the predicted values and the 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

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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 C2 Crack tip locations for Specimen 6611-a12, back notch


(N=50,007 cycles)....................................................................................160 C4 Crack tip locations for Specimen 6611-a12, front notch


(N=65,008 cycles)....................................................................................161 C6 Crack tip locations for Specimen 6612-b21, front notch



(N=47,511 cycles)....................................................................................163

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Appendix Figure Page C9 Crack tip locations for Specimen 6612-b21, front notch





(N=110,045 cycles)..................................................................................165 C14 Crack tip locations for Specimen 6714-a12, front notch




(N=163,630 cycles)..................................................................................167 C18 Crack tip locations for Specimen 7012-a22, back notch





(N=106,509 cycles)..................................................................................170

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Appendix Figure Page C23 Crack tip locations for Specimen 7012-a22, front notch





(N=106,509 cycles)..................................................................................173 C28 Crack tip locations for Specimen 7111-b11, back notch







(N=233,003 cycles). 177 C35 Crack tip locations for Specimen 8B2, back notch

(N=45,008 cycles)....................................................................................178 C36 Crack tip locations for Specimen 8B2, back notch

(N=57,501 cycles)....................................................................................178

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Appendix Figure Page C37 Crack tip locations for Specimen 8B2, back notch



(N=57,509 cycles)....................................................................................180 C40 Crack tip locations for Specimen 8B3, front notch




(N=58,509 cycles)....................................................................................182 C44 Crack tip locations for Specimen 8T3, front notch

(N=145,002 cycles)..................................................................................183 C45 Crack tip locations for Specimen 8T3, front notch



(N=214,005 cycles)..................................................................................184

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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

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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 daf 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

/Sho

op

-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

ΔΔK P

B W=

+−

+ − + −( )

( )( . . . . . )/

21

0 886 4 64 13 32 14 72 5 63 22 3 4α

αα 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

dadN

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

dadN

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 RateData for Al 7050-T7451 Alloy

Delta K, ksi sqrt(in.)

da/d

N, i

n./c

ycle

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 - moremicroporosity)

7012-a21-CT4 (new material - lessmicroporosity)

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

Cra

ck L

engt

h, a

(inc

hes)

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.

Crack Length vs. Number of Cycles:Specimen 6611-a12, Front Notch

Number of Cycles, N

Cra

ck L

engt

h, a

(inc

hes)

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.

Crack Length vs. Number of Cycles:Specimen 6612-b21, Back Notch

Number of Cycles, N

Cra

ck L

engt

h, a

(inc

hes)

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

Cra

ck L

engt

h, a

(inc

hes)

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

Cra

ck L

engt

h, a

(inc

hes)

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

Cra

ck L

engt

h, a

(inc

hes)

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

Cra

ck L

engt

h, a

(inc

hes)

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

Cra

ck L

engt

h, a

(inc

hes)

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

Cra

ck L

engt

h, a

(inc

hes)

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.

Crack Length vs. Number of Cycles:Specimen 8B2, Back Notch

Number of Cycles, N

Cra

ck L

engt

h, a

(inc

hes)

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.

Crack Length vs. Number of Cycles:Specimen 8B3, Back Notch

Number of Cycles, N

Cra

ck L

engt

h, a

(inc

hes)

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.

Crack Length vs. Number of Cycles:Specimen 8B3, Front Notch

Number of Cycles, N

Cra

ck L

engt

h, a

(inc

hes)

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.

Crack Length vs. Number of Cycles:Specimen 8T3, Front Notch

Number of Cycles, N

Cra

ck L

engt

h, a

(inc

hes)

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 forCrack 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/tvs. 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 RateData for Al 7050-T7451 Alloy

Delta K, ksi sqrt(in.)

da/d

N, i

n./c

ycle

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.

Figure 4.17 Replica photograph of specimen 6612-b21, "back" notch, 10001 cycles.

Crack ID # 1: 2a = 0.0038 in. Figure 4.18 Replica photograph of specimen 6612-b21, "back" notch, 23001 cycles.

Crack ID # 1: 2a = 0.0092 in.

Figure 4.19 Replica photograph of specimen 6612-b21, "back" notch, 33506 cycles.

Crack ID # 1: 2a = 0.0179 in. Figure 4.20 Replica photograph of specimen 6612-b21, "back" notch, 42509 cycles.

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. Figure 4.22 Replica photograph of specimen 6611-a12, "front" notch, 50007 cycles. Crack ID # 4.1: 2a = 0.0275 in. Crack ID # 4.2: 2a = 0.0036 in. Crack ID # 8: 2a = 0.0032 in.

Figure 4.23 Replica photograph of specimen 6611-a12, "front" notch, 60008 cycles. Crack ID # 4.1: 2a = 0.0379 in. Crack ID # 4.2: 2a = 0.0047 in. Crack ID # 8: 2a = 0.0032 in. Figure 4.24 Replica photograph of specimen 6611-a12, "front" notch, 65008 cycles. 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 =KKsymmetric 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 xc 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 fnwD eccentric

D center

KK

= 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

77

corrected algorithm. However, in the crack shape vs. size plots of Figures 6.8-6.14, the

finite width corrected algorithm predicts an earlier transition from a surface crack to a

corner crack than the original algorithm. The transition between the surface crack and

the corner crack occurs at the discontinuity on the predicted c/a vs. a/t curves.

All of the prediction analyses do a reasonable job in predicting the surface crack

length to breakthrough in the a vs. N plots, especially for specimen 6714-a11 (Figure

6.2). However, for most of the tests, both the original algorithm and the finite width

correction algorithm over-predicted the life to crack breakthrough to some degree.

However, using a da/dN-ΔK expression based on the same data the life prediction

program was attempting to predict should result in an equal number of over-predictions

and under-predictions for crack growth in the specimens. Since this did not occur, the

initial problem set-up was investigated.

One thing that differed between the actual data and the prediction analysis was

the treatment of the crack length in the "c" direction. Recall that in Chapter 4, the small

crack da/dN-ΔK data was generated by assuming that the crack shape c/a was a function

of the non dimensional surface length a/t (Equation 4.3). In the prediction analysis,

however, after the initial crack "c" dimension met the condition imposed by Equation 4.3,

the crack was allowed to grow freely in the "c" direction. This is evident in the crack

shape vs. size plots of Figures 6.8-6.14, where the predicted results are different than the

curve representing Equation 4.3. It is important to note that in future research, the initial

dimension in the "c" direction could be obtained from fractographs of the initiation site

on the specimen's fracture surfaces.

78

Another important point to note about the generation of the da/dN-ΔK data in

Figure 4.32 is that the crack growth rates were calculated with a seven-point polynomial

technique [32] that dropped both the first three and last three a vs. N data points. Also, as

one traces the growth of a crack back to the point of initiation, LEFM principles no

longer govern the growth of the crack. Thus, early crack measurements obtained from

the replicas may not be cracks in the LEFM sense. These two facts indicate that it may

not be appropriate to start the crack growth prediction at the first crack measurement. A

potential way to take into account the void's effect on the initiating crack is by

incorporating a correction factor based on Trantina and Barishpolsky's [40] effective

stress intensity factor for an ellipsoidal void in a large body with an equatorial crack. As

mentioned in Section 5.1, this interaction factor was employed with some success in

predicting the growth of fatigue cracks initiating from particles and micropores in Al

7050-T7451 open hole fatigue specimens [29].

Accordingly, the original algorithm prediction analysis was performed for the

same specimens, but starting at a later point in the test where the initial crack length was

on the order of 2a = 0.005 in. (127 mm). For comparison, the results were plotted on

Figures 6.1-6.14 with the earlier original algorithm and finite width correction

predictions. It is evident in the a vs. N plots of Figures 6.1-6.8 that the predictions based

on the larger initial crack lengths correlate quite well with the experimental data. In

addition, there were slight under-predictions as well as over-predictions. Thus, the

surface crack length to breakthrough predictions on the specimens used in calculating the

small crack da/dN-ΔK curve verify that the prediction code is working. In addition, the

crack shape vs. size predictions (Figures 6.9-6.14) indicate that although the prediction

program closely follows the assumed empirical expression (Equation 4.3), the predicted

crack shape falls short of the empirical assumption, particularly prior to and following

79

transition. The next step was to apply the code to predict crack growth in specimens

initiating multiple cracks.

6.2 Prediction Results in Specimens Initiating Multiple Cracks

Since there was relatively little difference between the crack growth back-

predictions of the original algorithm and the algorithm incorporating the finite width

correction, no finite width correction factor was used in the prediction analysis of

specimens initiating multiple cracks. As discussed in Section 5.1, the original algorithm

can utilize the Heath interaction factor [38] to model the growth of two collinear surface

cracks in a semicircular edge notch. However, in some of the specimens initiating

multiple cracks, crack interaction may not be a factor due to the size and distance

between the cracks involved.

For example, consider the cracks initiated in the back notch of specimen 7012-

a22. Although four cracks initiated in the back notch, no crack interaction may take

place due to the distance between the cracks. Since there are no stress intensity factor

solutions available that adequately model this particular situation, a two-dimensional

stress intensity factor solution for two offset parallel cracks in a sheet under uniform

uniaxial tensile stress [42] was used to give an estimate of the interaction between the

cracks. The geometry for this stress intensity factor model is illustrated in Figure 6.15.

The cases examined were in the back notch of specimen 7012-a22 at 80,004 cycles; the

location of the Crack ID #'s 1, 3, 4 in the back notch is illustrated in Figure C20 of

Appendix C. Interaction was considered between crack #'s 1 and 4 as well as crack #'s 3

and 4. Since the model only considers interaction between two offset parallel cracks of

80

equal length, the two cases utilized crack lengths, 2a, that were the average of the two

cracks being considered.

For the case between crack #'s 1 and 4, the ratio of the average crack length to the

distance between the furthermost crack tips, a/b, was 0.37. In addition, the ratio between

the offset distance and the average crack length, h/a, was 39.7. Similarly, for the case

between crack #'s 3 and 4, a/b=0.43 and h/a=14.4. For both cases, this translates into

virtually no interaction between either the inner or outer crack tips. For the sake of

comparison, an offset distance to average crack length ratio of h/a=0.1 would translate

into only six percent increase in the stress intensity factors for the crack #'s 1 and 4 case

and a four percent increase in the crack #'s 3 and 4 case [42]. From this analysis, it was

concluded that there was no interaction between small cracks initiating off the mid plane

of the notch and the dominate crack initiating at the mid plane of the notch.

The prediction analysis was applied individually to the four cracks initiating in

the back notch of specimen 7012-a22. The experimental measurements as well as the

predicted growth is plotted in Figure 6.16 for all four cracks. In addition, the c/a vs. a/t

predictions are plotted in Figure 6.17. For the majority of the cracks, the analysis

predicted faster growth than the experimental results. This may be due to the fact that all

cracks initiated a significant distance off the mid plane of the notch, as illustrated in

Figures C18-C22 of Appendix C. In the stress intensity factor solutions used in the

prediction analysis [4], the elastic stress concentration factor, Kt, was equal to 3.15 for

uniform displacement. As discussed in Section 3.1, however, a finite element analysis

demonstrated that Kt decreases significantly as the angle off the notch mid plane, Θ, is

increased (see Figure 3.3). Thus, the cracks in this notch may be experiencing a lower

hoop stress than the prediction analysis assumes, resulting in a slower growth rate.

81

To investigate this possibility, the Kt's were adjusted in the life prediction

program for the two cracks that initiated the farthest off the mid plane of the back notch

in specimen 7012-a22, namely, Crack ID #'s 1 and 4. The predicted surface lengths for

Crack ID #'s 1 and 4 are plotted in Figure 6.18 along with the experimental

measurements. It is evident from this figure that adjusting the Kt's in the prediction

analysis slow the crack growth in the notch too much (compared to the original

prediction results plotted in Figure 6.16). This is not altogether surprising, since cracks

initiating off the mid plane of the notch experience radial shear stresses as well as normal

hoop stresses, resulting in a mixed-mode crack propagation problem not taken into

account in the life prediction program. The crack shape vs. non dimensional length

predictions are plotted in Figure 6.19. It is apparent from this plot that the adjustment in

the stress concentration factor has little effect on the crack shape predictions.

Figure 6.20 shows the predicted crack growth for Crack ID #1 in the front notch

of specimen 7012-a22. The influence of the smaller cracks (Crack ID #'s 2 and 3) were

ignored in the prediction analysis. It is evident from Figure 6.20 that the original

computer algorithm did an excellent job in predicting the life to breakthrough of Crack

ID # 1. The crack shape vs. non dimensional length of this crack is plotted in Figure

6.21. As in the back predictions discussed in Section 6.1 and plotted in Figures 6.8-6.14,

the predicted crack shape closely followed the assumed empirical expression for c/a vs.

a/t, with a discontinuity occurring when the surface crack transitioned to a corner crack.

In Figure 6.22, the predicted growth of Crack ID #1.2 is plotted with the

experimental measurements of cracks in the back notch of specimen 6611-a12. Although

Crack ID # 1.2 would later coalesce with Crack ID # 1.1 in the test, the presence of Crack

82

ID #'s 1.1 and 2 was ignored in the analysis. As is evident in Figure 6.22, the analysis

predicts the growth to breakthrough of the dominant crack within 24 %. However, when

a prediction analysis incorporating the Heath interaction factor [38] was performed

including both Crack ID #'s 1.1 and 1.2, the predicted growth and coalescence of the two

cracks was within 12 % of the experimental measurements. Recall that breakthrough is

defined as the point in the test when the surface or corner crack in the notch transitions to

a through crack. These growth predictions are plotted in Figure 6.24. The crack shape

vs. non dimensional size predictions are plotted for both the single crack and double

crack cases in Figures 6.23 and 6.25. It is interesting to note that in the double crack

prediction analysis, the two cracks show a different trend in the crack shape predictions

prior to coalescence. Crack ID # 1.1 shows a faster growth in the "c" direction than the

"a" direction, whereas the opposite is true for Crack ID # 1.2. However, after the cracks

coalesce, the shape trends are similar to those observed in previous predictions.

In Figure 6.26, the predicted growth of Crack ID #'s 1.1 and 1.2 are plotted with

the experimental crack measurements located in the front notch of specimen 6714-a12.

The prediction analysis did a fair job in terms of predicting the number of cycles until

crack breakthrough. However, the actual growth of the two cracks did not follow the

predicted growth. This is due to the fact that the two cracks were offset from one another

in the notch, and continued to grow separately even after their tips had passed over one

another. This is illustrated in Figures C14-C17 in Appendix C. Therefore, it is not

surprising that the prediction analysis, which assumed the two cracks were collinear, did

not yield good results.

Figures 6.28-6.31 show two sets of crack growth predictions for the front notch of

specimen 8B3. The first case predicts the growth to breakthrough of Crack ID # 1.22.

83

This case ignores the presence of other cracks, including the fact that Crack ID # 1.22

coalesced with Crack ID #'s 1.21 and 1.1. As can be seen in Figure 6.28, this assumption

results in a reasonable prediction of the surface crack length to breakthrough of the

dominant crack. In the second case, the growth of both Crack ID #'s 1.21 and 1.22 were

predicted utilizing the Heath interaction factor [38]. Unfortunately, this analysis

predicted a longer life to breakthrough of the resulting dominant crack than the earlier

prediction assuming the presence of a single crack (compare Figures 6.28 and 6.30). The

reason behind this is unsure.

In general, the application of Tritsch's original algorithm [35] using the Heath

interaction factor [38] to the growth of small cracks at a notch in the Al 7050-T7451

proves to be promising. More testing is required, however, to further validate this

approach. Such future testing should be conducted at maximum nominal stresses both

above and below 16 ksi (110 MPa) for all three versions of the alloy. The results from

these tests should verify the small fatigue crack growth rate data gathered in this study, as

well as provide more cases for the prediction analysis.

Predicted Crack Growth for Specimen 6612-b21, back notch:N vs. total surface length

Number of Cycles, N

Tota

l Sur

face

Len

gth

(inch

es)

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

Original Algorithm(diamond)

Finite Width Correction(cross)

Original Algorithm - largerinitial crack (circle)

Crack ID # 1(experimental)

Figure 6.1 Actual and predicted crack growth for specimen 6612-b21, back notch: surface crack length vs. number of cycles.

Predicted Crack Growth for Specimen 6714-a11, back notch:N vs. total surface length

Number of Cycles, N

Tota

l Sur

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Len

gth

(inch

es)

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

Crack ID #2(experimental)




Figure 6.2 Actual and predicted crack growth for specimen 6714-a11, back notch: surface crack length vs. number of cycles.

Predicted Crack Growth for Specimen 7111-b11, back notch: total surface length vs. N

Number of Cycles, N

Tota

l Sur

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Len

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(inch

es)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

160000 180000 200000 220000 240000 260000 280000




Original Algorithm - largerinitial crack (triangle)


Predicted Crack Growth for Specimen 7111-b12, back notch:N vs. total surface length

Number of Cycles, N

Tota

l Sur

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Len

gth

(inch

es)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

140000 160000 180000 200000 220000 240000 260000 280000 300000 320000


Crack ID #1.1(experimental)

Crack ID # 1.2(experimental)





Predicted Crack Growth for Specimen 8B2, back notch: N vs. total surface length

Number of Cycles, N

Tota

l Sur

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Len

gth

(inch

es)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

5000 15000 25000 35000 45000 55000 65000 75000 85000 95000





Figure 6.5 Actual and predicted crack growth for specimen 8B2, back notch: surface crack length vs. number of cycles.

Predicted Crack Growth for Specimen 8B3, back notch:N vs. total surface length

Number of Cycles, N

Tota

l Sur

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Len

gth

(inch

es)

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0 10000 20000 30000 40000 50000 60000


Original Algorithm - largerinitial crack (diamond)

Figure 6.6 Actual and predicted crack growth for specimen 8B3, back notch: surface crack length vs. number of cycles.

Predicted Crack Growth for Specimen 8T3, front notch: N vs. total surface length

Number of Cycles, N

Tota

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Len

gth

(inch

es)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

100000 120000 140000 160000 180000 200000 220000 240000




Original Algorithm - largerinitial crack (circle)

Figure 6.7 Actual and predicted crack growth for specimen 8T3, front notch: surface crack length vs. number of cycles.

Predicted Crack Growth forSpecimen 6612-b21, back notch: a/t vs. c/a

a/t

c/a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

c/a=0.9-0.25(a/t)^2

Finite Width Correction(circle)


Original Algorithm - largerinital crack (triangle)

Figure 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.

Predicted Crack Growth forSpecimen 6714-a11, back notch: a/t vs. c/a

a/t

c/a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

c/a=0.9-0.25(a/t)^2



Figure 6.9 Predicted crack growth for specimen 6714-a11, back notch: c/a vs. a/t for both the predicted values and the



a/t

c/a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

c/a=0.9-0.25(a/t)^2Original Algorithm

(diamond)






a/t

c/a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

c/a=0.9-0.25(a/t)^2






Predicted Crack Growth forSpecimen 8B2, back notch: a/t vs. c/a

a/t

c/a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

c/a=0.9-0.25(a/t)^2




Figure 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.

Predicted Crack Growth forSpecimen 8B3, back notch: a/t vs. c/a

a/t

c/a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.02 0.04 0.06 0.08 0.1

c/a=0.9-0.25(a/t)^2

Original Algorithm - largerinitial crack (diamond)

Figure 6.13 Predicted crack growth for specimen 8B3, back notch: c/a vs. a/t for both the predicted values and the empirical


Predicted Crack Growth forSpecimen 8T3, front notch: a/t vs. c/a

a/t

c/a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

c/a=0.9-0.25(a/t)^2




Figure 6.14 Predicted crack growth for specimen 8T3, front notch: c/a vs. a/t for both the predicted values and the empirical



Number of Cycles, N

Tota

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Len

gth

(inch

es)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 20000 40000 60000 80000 100000 120000

Crack ID # 1, exp.

Crack ID # 1, prediction(no influence from other cracks)


Crack ID # 2, exp.Crack ID # 3, exp.


Crack ID # 4, exp.



Note: no crack interaction is considered between the cracks.

Predicted Crack Growth for Specimen 7012-a22, back notch: a/tvs. c/a

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

Assumed Empirical Expression for a/tvs. c/a

Crack ID # 1 (no interaction with othercracks)

Crack ID # 2 (")

Crack ID # 3 (")

Crack ID # 4 (")

c/a=0.9-0.25(a/t)^2


empirical expression assumed in calculating ΔK solutions. Note: no crack interaction is considered between the cracks.

Predicted Crack Growth for Specimen 7012-a22, back notch:N vs. total surface length. (Note: Kt's adjusted to reflect initiation off

midplane of notch)

Number of Cycles, N

Tota

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Len

gth

(inch

es)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 20000 40000 60000 80000 100000 120000

Crack ID # 1, exp.



Crack ID # 4, exp.


Note: no crack interaction is considered between the 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).

Predicted Crack Growth for Specimen 7012-a22, back notch: a/t vs. c/a(Note: Kt's adjusted to reflect initiation off midplane of notch)

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 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Assumed Empirical Expression for a/tvs. c/a

Crack ID # 1 (no interaction with othercracks)

Crack ID # 4 (")

c/a=0.9-0.25(a/t)^2


empirical expression assumed in calculating ΔK solutions. Note: no crack interaction is considered between the 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).

Predicted Crack Growth for Specimen 7012-a22, front notch:N vs. total surface length

Number of Cycles, N

Tota

l Sur

face

Len

gth

(inch

es)

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 (exp.)

Original algorithm prediction for # 1(ignores influence of other cracks)

Crack ID # 1 (exp.)

Crack ID # 3 (exp.)

Figure 6.20 Actual and predicted crack growth for specimen 7012-a22, front notch: surface crack length vs. number of cycles.

Note: the presence of Crack ID #'s 2 and 3 are ignored.

Predicted Crack Growth forSpecimen 7012-a22, front notch: a/t vs. c/a

a/t

c/a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

c/a=0.9-0.25(a/t)^2

Original algorithm prediction for # 1(ignores influence of other cracks)

Figure 6.21 Predicted crack growth for specimen 7012-a22, front notch: c/a vs. a/t for both the predicted values and the

empirical expression assumed in calculating ΔK solutions. Note: the presence of Crack ID #'s 2 and 3 are ignored.


Number of Cycles, N

Tota

l Sur

face

Len

gth

(inch

es)

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 90000

Crack ID # 1.2 (exp.)Crack ID # 1.1 (exp.)

Crack ID # 2 (exp.)

Crack ID # 1 (exp.)

Prediction for Crack ID #1.2-1 (ignores presence of #'s 1.1, 2)


Note: the presence of Crack ID #'s 1.1 and 2 are ignored.

Predicted Crack Growth for Specimen 6611-a12, back notch: a/tvs. c/a (Crack ID #1.2)

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

c/a=0.9-0.25(a/t)^2

Original algorithm prediction for Crack ID #1.2-1 (ignores presence of #'s 1.1, 2)


empirical expression assumed in calculating ΔK solutions. Note: the presence of Crack ID #'s 1.1 and 2 are ignored.


Number of Cycles, N

Tota

l Sur

face

Len

gth

(inch

es)

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 90000

Crack ID # 1.2 (exp.)Crack ID # 1.1 (exp.)

Prediction for Crack ID #1.2(ignores presence of # 2)

Crack ID # 1 (exp.)

Prediction for Crack ID #1(ignores presence of # 2)

Prediction for Crack ID #1.1(ignores presence of # 2)


Note: the presence of Crack ID # 2 is ignored.

Predicted Crack Growth for Specimen 6611-a12, back notch: a/tvs. c/a

a/t

c/a

0

0.2

0.4

0.6

0.8

1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

c/a=0.9-0.25(a/t)^2

Original algorithm prediction for Crack ID #1 (ignores presence of # 2)

Original algorithm prediction for Crack ID #1.1 (ignores presence of # 2)

Original algorithm prediction for Crack ID #1.2(ignores presence of # 2)


empirical expression assumed in calculating ΔK solutions. Note: the presence of Crack ID # 2 is ignored.

Predicted Crack Growth for Specimen 6714-a12, front notch:N vs. total surface length

Number of Cycles, N

Tota

l Sur

face

Len

gth

(inch

es)

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 # 1.2, exp.

Original algorithm prediction, Crack ID # 1

Original algorithm prediction, Crack ID # 1.2

Crack ID # 1.1, exp.

Crack ID # 1, exp.

Original algorithm prediction, Crack ID # 1.1

Figure 6.26 Actual and predicted crack growth for specimen 6714-a12, front notch: surface crack length vs. number of cycles.

Predicted Crack Growth for Specimen 6714-a12, front notch: a/t vs. c/a

a/t

c/a

-0.1

0.1

0.3

0.5

0.7

0.9

1.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

c/a=0.9-0.25(a/t)^2

Original Algorithm, Crack ID # 1.2

Original Algorithm, Crack ID # 1.1

Original Algorithm, Crack ID # 1

Figure 6.27 Predicted crack growth for specimen 6714-a12, front notch: c/a vs. a/t for both the predicted values and the


Predicted Crack Growth for Specimen 8B3, front notch:N vs. total surface length

Number of Cycles, N

Tota

l Sur

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Len

gth

(inch

es)

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

Prediction for Crack ID #1.22-1.2-1 (ignores presence of other cracks)

Crack ID # 1.2 (exp.)


Crack ID # 1 (exp.)

Figure 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.

Predicted Crack Growth forSpecimen 8B3, front notch: a/t vs. c/a

a/t

c/a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

c/a=0.9-0.25(a/t)^2

Original algorithm prediction for Crack ID #1.22-1.2-1 (ignores presence of other cracks)

Figure 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.

Predicted Crack Growth for Specimen 8B3, front notch:N vs. total surface length

Number of Cycles, N

Tota

l Sur

face

Len

gth

(inch

es)

0.00E+00

2.00E-02

4.00E-02

6.00E-02

8.00E-02

1.00E-01

1.20E-01

1.40E-01

1.60E-01

1.80E-01

2.00E-01

0 10000 20000 30000 40000 50000 60000 70000 80000 90000

Prediction for Crack ID #1.2-1 (ignores presence of other cracks)



Crack ID # 1 (exp.)

Prediction for Crack ID #1.22(ignores presence of other cracks)

Prediction for Crack ID #1.21 (ignores presence of other cracks)

Figure 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.

Predicted Crack Growth forSpecimen 8B3, front notch: a/t vs. c/a

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

c/a=0.9-0.25(a/t)^2

Original algorithm prediction for Crack ID # 1.2-1 (ignores presence of other cracks)

Original algorithm prediction for Crack ID #1.22(ignores presence of other cracks)

Original algorithm prediction for Crack ID #1.21 (ignores presence of other cracks)

Figure 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.

Figure 6.15 Stress intensity factor geometry for two offset parallel cracks in a sheet

under uniform uniaxial tensile stress [42].

115

CHAPTER 7 - CONCLUSIONS AND RECOMMENDATIONS

The experimental results in this study indicate that once crack initiation has

occurred, all three materials exhibit the same small crack growth behavior; this is

illustrated in the da/dN-ΔK plot of Figure 4.32. However, the increased specimen life of

the "new" material over the "old" material suggests that a reduction in microporosity does

improve the alloy's crack initiation properties. For this study, only three fatigue

specimens fabricated from the "3-inch plate" material were tested at maximum nominal

stress of 16 ksi (110 MPa); the average specimen life was slightly shorter than the "new"

material, but still longer than the "old" material. Although this may seem to indicate that

the "3-inch plate" material does not have improved fatigue properties over the earlier

versions, more testing at other maximum nominal stress levels is required before a

definitive conclusion can be made. In fact, recent fatigue testing of the three materials at

a maximum nominal stress of 18 ksi (124 MPa) [43] demonstrates that the "3-inch plate"

material has the longest average specimen life of all three materials.

Future experimental work in this study should include continued fatigue testing at

stress levels both above and below a maximum nominal stress of 16 ksi (110 MPa). This

testing would verify the da/dN-ΔK plot of Figure 4.32 at other stress levels, as well as

yield more information on the life to initiation for all three materials. These fatigue tests

(two or three tests for each stress level / material) would also provide cases for the

prediction analysis. In addition, the work currently being performed by Elsner [33] on

116

the fracture surfaces of the specimens will determine the exact nature of the initiation site

in the three materials. Finally, information on the mean size and frequency distribution

of the micropores in all three materials must be obtained to determine what effect that has

on the growth behavior of small cracks.

The analytical model for crack growth does a good job of predicting the growth of

small cracks after initiation. In addition, the original prediction algorithm demonstrated

that in the majority of the tests smaller cracks had little effect on the growth of the

dominant crack in the notch. For the few tests where crack interaction was deemed

important and crack coalescence occurred, the Heath interaction factor [38] modeled the

growth of two cracks reasonably well . However, future versions of the model should

consider the effect of cracks initiating off the mid plane of the notch.

The results from this study indicate that the initiation and growth of small cracks

may be important in the design decisions made by the engineer. Since the majority of the

cracks' lives were spent as small cracks, these results would be an important driver in a

design methodology that doesn't assume the pre-existence of large cracks in an

engineering structure. However, since all three versions of the alloy demonstrated the

same specimen life after crack initiation, i. e., 2a > 0.005 inches (127 μm), these results

would be of little use in a design methodology that assumes the existence of cracks in the

structure. Nevertheless, as inspection methods continue to improve, the initiation and

growth of small cracks in engineering materials will become of greater importance during

the design decision-making process.

LIST OF REFERENCES

117

LIST OF REFERENCES [1] Hinkle, A. J., Magnusen, P. E., Rolf, R. L., and Bucci, R. J., "Effect of

Microporosity on Notched Specimen Fatigue Life," ALCOA Division Report 57-89-02, January 1990.

[2] Irwin, G. R., "Analysis of Stresses and Strains Near the End of a Crack

Traversing a Plate," ASME, Journal of Applied Mechanics, Vol. 24, 1957, pp. 361.

[3] Paris, P. C., Gomez, M. P., and Anderson, W. E., The Trend in Engineering,

Washington State University, Vol. 13, No. 1, 1961, pp. 9-14. [4] Newman, J. C., Jr., "Fracture Mechanics Parameters for Small Fatigue Cracks,"

ASTM Symposium on Small Crack Test Methods, San Antonio, TX, 14 Nov 1990, pp. 35-39.

[5] Tan, P. W., Raju, I. S., Shivakumar, K. N., and Newman, J. C., Jr., "Evaluation of

Finite-Element Models and Stress-Intensity Factors for Surface Cracks Emanating from Stress Concentrations," Surface-Crack Growth: Models, Experiments and Structures, ASTM STP 1060, W. G. Reuter, J. H. Underwood and J. C. Newman, Jr., eds., American Society for Testing and Materials, Philadelphia, PA, 1990, pp. 34-48.

[6] Shivakumar, K. N., and Newman, J. C., Jr., "Stress-Intesity Factors for Large

Aspect Ratio Surface and Corner Cracks at a Semi-Circular Notch in a Tension Specimen," Engineering Fracture Mechanics, Vol. 38, No. 6, 1991, pp. 467-473.

[7] Zhao, W. and Wu, X. R., "Stress Intensity Factor Evaluation by Weight Function

for Surface Crack in Edge Notch," Theoretical and Applied Fracture Mechanics, Vol. 13, 1990, pp. 225-238.

[8] Zhao, W. and Wu, X. R., "Stress Intensity Factors for Corner Cracks at a Semi-

Circular Notch Under Stress Gradients," Fatigue and Fracture of Engineering Materials and Structures, Vol. 13, No. 4, 1990, pp. 347-360.

118

[9] Tan, P. W.: The Boundary Force Method for Stress Analysis of Arbitrarily Shaped Plates with Notches and Cracks, Ph.D. Thesis, George Washington University, 1986.

[10] Pickard, A. C., "Stress-Intensity Factors for Cracks with Circular and Elliptical

Crack Fronts, Determined by 3D Finite Elemental Methods," Numerical Methods in Fracture Mechanics, D. R. J. Owen and A. R. Luxmoore, eds., Pineridge Press, Swansea, U. K., 1980, pp. 599-619.

[11] Phillips, E. P., and Newman, J. C., Jr., "Impact of Small-Crack Effects on Design-

Life Calculations," Experimental Mechanics, Vol. 29, No. 2, June 1989, pp. 221-225.

[12] Broek, D., Elementary Engineering Fracture Mechanics, Third Revised Edition,

Martinus Nijhoff Publishers, 1982. [13] Leis, B. N., Kanninen, M. F., Hopper, A. T., Ahmad, J., and Broek, D., "A

Critical Review of the Short Crack Problem in Fatigue," AFWAL-TR-83-4019, 1983.

[14] Elber, W. "Fatigue Crack Closure Under Cyclic Tension," Engineering Fracture

Mechanics, Vol. 1, No. 4, 1970, pp. 705-718. [15] Davidson, D. L., "Techniques for Experimental Microcrack Mechanics," ASTM

Symposium on Small Crack Test Methods, San Antonio, TX, 13 Nov 1990. [16] Sharpe, W. N., Jira, J. R., and Larsen, J. M., "Real-Time Measurement of Small

Crack Opening Behavior Using an Interferometric Strain/Displacement Gage," ASTM Symposium on Small Crack Test Methods, San Antonio, TX, 13 Nov 1990.

[17] Gangloff, R. P., Slavick, D. C., Piascik, R. S., and Van Stone, R. H., "Direct

Current Electrical Potential Measurement of Small Fatigue Cracks," Small Crack Test Methods, ASTM STP, 1991.

[18] Resch, M. T. and Nelson, D. V., "An Ultrasonic Method for Measurement of Size

and Opening Behavior of Small Fatigue Cracks," ASTM Symposium on Small Crack Test Methods, San Antonio, TX, 14 Nov 1990.

[19] Swain, M. H., "Monitoring Small Crack Growth by the Replication Method,"

ASTM Symposium on Small Crack Test Methods, San Antonio, TX, 14 Nov 1990.

119

[20] Edwards, P. R. and Newman, J. C., Jr., "Short-Crack Growth Behaviour in an Aluminum Alloy - An AGARD Cooperative Test Programme," AGARD R-732, 1988.

[21] Newman, J. C., Jr., "A Crack Closure Model for Predicting Fatigure Crack

Growth Under Aircraft Spectrum Loading," Methods and Models for Predicting Fatigure Crack Growth Under Random Loading, J. B. Chang and C. M. Hudson, eds., American Society for Testing and Materials, ASTM STP 748, 1981, pp. 53-84.

[22] Dugdale, D. S., Journal of Mechanics and Physics of Solids, Vol. 8, No. 2, 1960,

pp. 100-104. [23] Sharpe, W. N., Jr., and Su, X., "Closure Measurements of Naturally Initiating

Small Cracks," Engineering Fracture Mechanics, Vol. 30, No. 3, 1988, pp. 275-294.

[24] Edwards, P. R. and Newman, J. C., Jr., "An AGARD Supplemental Test

Programme on the Behaviour of Short Cracks Under Constant Amplitude and Aircraft Spectrum Loading," AGARD R-767, 1990.

[25] Burns, J. G., Rudd, J. L., Harter, J. A., Magnusen, P. E., Hinkle, A. J., Bucci, R.

J., "Probabilistic Durability Evaluation of ALCOA 7050 Aluminum," 1991 USAF Structural Integrity Program Conference.

[26] Burns, J. G., Rudd, J. L., Harter, J. A., Magnusen, P. E., Hinkle, A. J., Bucci, R.

J., "Effect of Microporosity on Fatigue Durability of Thick 7050 Aluminum Plate," 1992 USAF Structural Integrity Program Conference, San Antonio, Texas, 1-3 December 1992.

[27] Shaw, B. J., "Small Fatigue Crack Growth in an Aluminum Alloy Containing

Porosity Defects - An Interim Report", ALCOA Report No. 56-91-KF33, August 1991.

[28] Shaw , B. J., Petri, R. A., Johnson, B. J., "An Analysis of the Quality of Data

Generated Using the IBAS to Measure Replicated Crack Lengths," Memorandum, ALCOA Technology Division, 23 April 1991.

[29] Grandt, A. F., Hinkle, A. J., Scheumann, T. D., and Todd, R. E., "Modeling the

Influence of Initial Material Inhomogeneites on the Fatigue Life of Notched Components," Fatigue and Fracture of Engineering Materials and Structures, Vol. 16, No. 2, 1993, pp. 199-213.

[30] Personnel communications with Dr. A. J. Hinkle, Staff Engineer, ALCOA

Technical Center, March 1991.

120

[31] Meuller, L. N. "ALCOA Aluminum Alloy 7050," Green Letter No. 220,

Aluminum Company of America, October, 1985. [32] "Standard Test Method for Constant-Load-Amplitude Fatigue Crack Growth

Rates Above 10-8 m/cycle," ASTM E647-86, American Society for Testing and Materials, 1986.

[33] Grandt, A. F., Forsyth, E. N., Zezula, C. E., and Elsner, J., "Initiation, Growth,

and Coalescence of Small Fatigue Cracks at Notches," Progress Report prepared for Alcoa Technical Center Project TC919597TC, 18 Dec 1992.

[34] Swain, M. H., and Newman, J. C. , Jr., "On the Use of Marker Loads and Replicas

for Measuring Growth Rates for Small Cracks," Fatigue Crack Topography (AGARD Conference Proceedings No. 376, 1984), 12.1-12.17.

[35] Tritsch, D. E., "Prediction of Fatigue Crack Lives and Shapes," M. S. Thesis,

School of Aeronautics and Astronautics, Purdue University, August 1983. [36] Newman, J. C., Jr., and Raju, I. S., "Stress Intensity Factor Equations for Cracks

in Three-Dimensional Finite Bodies," NASA Technical Memorandum 83200, Langley Research Center, Hampton, Virginia, August 1981.

[37] Bowie, O. L., "Analysis of an Infinite Plate Containing Radial Cracks Originating

at the Boundary of an Internal Circular Hole," Journal of Mathematics and Physics, Vol. 35, 1956, pp. 60-71.

[38] Heath, B. J. and Grandt, A. F., "Stress Intensity Factors for Coalescing and Single

Corner Flaws Along a Hole Bore in a Plate," Engineering Fracture Mechanics, Vol. 19, No. 4, 1984, pp. 665-673.

[39] Scheumann, T. D., "A Numerical and Experimental Investigation of the Effects of

Notches on Fatigue Behavior," M. S. Thesis, School of Aeronautics and Astronautics, Purdue University, August 1991.

[40] Trantina, G. G. and Barishpolsky, M., "Elastic-Plastic Analysis of Small Defects--

Voids and Inclusions," Engineering Fracture Mechanics, Vol. 20, No. 1, 1984, pp. 1-10.

[41] Partl, O. and Schijve, J., "Multiple-site-damage in 2024-T3 alloy sheet," Report

LR-660, Delft University of Technology, January 1992, pp. 17. [42] Rooke, D. P., and Cartwright, D., J., Compendium of Stress Intensity Factors,

Hillington Press, Uxbridge, Middx. England, 1976.

121

[43] Personnel communications with Mr. C. E. Zezula, Graduate Research Assistant, Purdue University, January - March 1993.

APPENDICES

122

Appendix A - Stress Intensity Factor Solutions

123

Approximate stress intensity factors for a semi-elliptical surface crack, a quarter

elliptical corner crack, and a through crack located at a semicircular edge notch are

presented here. The equations were presented by Newman in [4]. The semicircular edge

notch geometry and variable definitions for surface and corner cracks are given in Figure

2.3.

A.1. Stress Intensity Factors for a Surface or Corner Crack at a Semicircular Edge Notch

The stress intensity factors for both a surface crack and a corner crack in a

semicircular edge notch subjected to remote uniform stress or uniform displacement are

given in the form

K S a QF a c a t c r c w r t r wjn= π φ( / , / , / , / , / , / , ) (A.1)

where Fjn is the boundary correction factor. It is important to note that t is defined as

one-half the full sheet thickness for the surface crack (j = s), whereas t is defined as the

full sheet thickness for the corner crack (j = c). The shape factor, Q, is given by

Q a c= +1 1 464 1 65. ( / ) . for a/c≤1 (A.2a)

Q c a= +1 1 464 1 65. ( / ) . for a/c>1 (A.2b)

124

A.1.1. Surface Crack at a Semicircular Edge Notch

The boundary correction factor for a surface crack located at the center of

semicircular edge notch is given by

F M M a t M a t g g g g g f fsn w= + +[ ( / ) ( / ) ]1 22

34

1 2 3 4 5 φ (A.3)

The expression is valid for 0.2 < a/c < 2, a/t < 1, 1 < r/t < 3.5, (r + c)/w < 0.5, r/w =

1/16, and -π/2 < φ < π/2. For a/c≤1:

M1 1= (A.4)

M a c23 20 05 0 11= +. / [ . ( / ) ]/ (A.5)

M a c33 20 29 0 23= +. / [ . ( / ) ]/ (A.6)

g a t a t a c14 1 21 2 6 2 1 4= − − +[ ( / ) ( . / ) / ( / )] cos/ φ (A.7)

g22 3 4 21 0 358 1 425 1 578 2 156 1 0 08= + + − + +[ . . . . ] / ( . )λ λ λ λ λ (A.8)

λ φ= +1 1 0 9/ [ ( / )cos( . ) ]c r (A.9)

g a t32 101 0 1 1 1= + − −. ( cos ) ( / )φ (A.10)

g K c rT41 20 36 0 032 1= − +[ . . / ( / ) ]/ (A.11)

where KT is the elastic stress concentration factor (KT = 3.17 for uniform stress; KT =

3.15 for uniform displacement) at the edge of the semicircular notch and

g a c r t r t

a t a c5

1 2 2 3

2 3

1 0 003 0 035 10 35 1 0 5

= + + −

− −

( / ) [ . ( / ) . ( / )( cos ) ]. ( / ) ( . / ) cos

/ φ

φ (A.12)

The finite width correction is given as

125

fw = − + − +1 0 2 9 4 19 4 27 12 3 4. . . .γ γ γ γ for uniform stress (A.13a)

fw = + − +1 2 17 3 4 3 72 4 6. . .γ γ γ for uniform displacement

(A.13b)

where

γ = +( / ) ( ) //a t c r w1 2 (A.14)

The function fφ is given by

f a cφ φ φ= +[ ( / ) cos sin ] /2 2 2 1 4 (A.15)

For a/c > 1:

M c a a c11 2 1 04 0 04= −( / ) ( . . / )/ (A.16)

The functions M2, M3, g1, g2, λ, g3, g4, g5, and fw are given by Equations (A.5) through

(A.13), respectively, and fφ is given by

f c aφ φ φ= +[ ( / ) sin cos ] /2 2 2 1 4 (A.17)

A.1.2. Corner Crack at a Semicircular Edge Notch

The boundary correction factor for a corner crack located at the center of a

semicircular edge notch is given by

126

F M M a t M a t g g g g g f fcn w= + +[ ( / ) ( / ) ]1 22

34

1 2 3 4 5 φ (A.18)

The expression is valid for 0.2 < a/c < 2, a/t < 1, 1 < r/t < 2, (r + c)/w < 0.5, r/w = 1/16,

and 0 < φ < π/2. For a/c≤1:

M a c1 1 13 0 09= −. . / (A.19)

M a c2 0 54 0 89 0 2= − + +. . / ( . / ) (A.20)

M a c3 0 5 1 0 65= − +. / ( . / ) (A.21)

g a t

a t1

2 21 0 1 0 2 10 16

= + + −−

[ . . ( / ) ]( sin ). ( / )sin cos

φφ φ

(A.22)

g22 3 4 21 0 358 1 425 1 578 2 156 1 0 13= + + − + +[ . . . . ] / ( . )λ λ λ λ λ (A.23)

λ φ= +1 1 0 8/ [ ( / )cos( . )]c r (A.24)

g a c a t32 1 41 0 04 1 0 1 1 0 97 0 03= + + − +( . / )[ . ( cos ) ][ . . ( / ) ]/φ (A.25)

The functions g4, g5, and fw are given by Equations (A.11) through (A.13), respectively,

and fφ is given by Equation A.17. For a/c > 1:

M c a c a11 2 1 0 04= +( / ) ( . / )/ (A.26)

M c a240 2= . ( / ) (A.27)

M c a340 11= − . ( / ) (A.28)

g c a a t

a t c aa c a t

12 2

2

1 0 1 0 2 10 160 07 1 1

= + + −−

+ − −

( / )[ . . ( / ) ]( sin ). ( / )( / )sin cos. ( / )( / )cos

φφ φ

φ

(A.29)

g c a a t32 1 41 13 0 09 1 0 1 1 0 97 0 03= − + − +( . . / )[ . ( cos ) ][ . . ( / ) ]/φ (A.30)

127

The expressions for g2 and λ are given by Equations (A.23) and (A.24); g4, g5, and fw are

given by Equations (A.11) through (A.13), respectively; and fφ is given by Equation

(A.17).

A.2. Stress Intensity Factor for a Through Crack at a Semicircular Edge Notch

The stress intensity factor for a through crack emanating from a semicircular edge

notch subjected to remote uniform stress or uniform displacement is given in the form

K S cF c w c r r wn= π ( / , / , / ) (A.31)

The expression is valid for r/w = 1/16 and (c + r)/w < 0.08. The boundary correction

factor, fn, is

F f g fn w= 1 4 (A.32)

where g4 and fw are given by Equations (A.11) and (A.13), respectively. The function f1

is given by

f12 3 41 0 358 1 425 1 578 2 156= + + − +. . . .λ λ λ λ (A.33)

where

λ = +1 1/ ( / )c r (A.34).

Appendix B - Specimen Dimensions and Test Parameters

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.

Specimen ID

Specimen Type

Specimen Width

(in. / cm)

Specimen Thickness (in. / mm)

"Front" Notch Radius

(in. / mm)

"Back" Notch Radius

(in. / mm)

Remote Maximum

Stress (ksi / MPa)

Frequency (Hz)

Temperature (° F)

% Humidity

6611-a12 Friction DEN

2.001 / 5.082

0.1865 / 4.737

0.095 / 2.413

0.094 / 2.388

14.5 / 100

10 71 76

6612-b21 Friction DEN

2.004 / 5.090

0.188 / 4.755

0.099 / 2.515

0.099 / 2.515

14.4 / 99.3

10 71 68

6714-a11 Dbone DEN 1.11 / 2.82

0.191 / 4.851

0.094 / 2.388

0.094 / 2.388

12.5 / 86.2

5 70 61

6714-a12 Dbone DEN 1.104 / 2.804

0.1875 / 4.763

0.095 / 2.413

0.096 / 2.438

12.4 / 85.5

10 72 74

7012-a22 Friction DEN

2.002 / 5.085

0.184 / 4.674

0.093 / 2.362

0.093 / 2.362

14.5 / 100

10 71 68


2.004 / 5.090

0.186 / 4.724

0.096 / 2.438

0.096 / 2.438

13.6, 16.3 / 93.8, 112

8 71 78


2.003 / 5.088

0.1845 / 4.686

0.099 / 2.515

0.101 / 2.565

14.4 / 99.3

10 71 72

8B2 Friction DEN

2.005 / 5.093

0.184 / 4.674

0.094 / 2.388

0.094 / 2.388

14.5 / 100

10 71 74

8B3 Friction DEN

2.005 / 5.093

0.183 / 4.648

0.094 / 2.388

0.094 / 2.388

14.5 / 100

10 71 75

8T3 Friction DEN

2.005 / 5.093

0.1935 / 4.915

0.095 / 2.413

0.093 / 2.362

14.5 / 100

10 71 76

Appendix C - Crack Measurements for Double-Edge Notch Specimens

This appendix summarizes the data obtained from the ten fatigue tests conducted

in this study. Tables C1-C13 contain the actual crack measurements made in the tests.

The tables are organized on the basis of the crack identification number described in

Section 3.1.2. An important concern with measurements 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 widths (given in the table), providing a scale for all

measurements made on that replica. The corrected crack tip coordinates are given for

larger cracks (a ≥ 0.003 inches / 76 μm). These were measured with a low powered

(magnification ≈ 7 ×) optical microscope with the replica mounted on an two-

dimensional translation stage. For smaller cracks measured with a higher powered

microscope, only the corrected surface crack length was obtainable. Definitions for the

x-Θ coordinate system and the surface crack length are illustrated in Figure 3.5 and

discussed in Section 3.1.2. Although the distance off the mid plane of the notch is given

in radians in the tables, it can be converted to the "s" coordinate by multiplying the angle,

Θ, with the notch radius. Also included in the tables are the crack types - surface or

corner cracks. Thus, for the corner crack, the corrected length "a" represents the total

surface length, while for the surface crack, the corrected length "a" represents one-half

the total surface length; this is illustrated in Figure 3.5.

Also included in this appendix are the crack locations in the notch for selected

cycle counts. The figures are based on the crack x-### coordinate system illustrated in

Figure 3.5 and discussed in Section 3.1.2. Once again, the angle can be converted into

the distance, s, by multiplying by the notch radius. Two to five figures at selected cycle

counts are included for all of the specimens tested in this study. It is important to note

that some of the figures may not include all the cracks initiated at that particular cycle

count. This is because although some small crack lengths were obtained from the high

powered microscope, they were too small for their tip coordinates to be obtained from the

lower powered microscope.

Table C1 Crack measurements for specimen 6611-a12, back notch.

N Crack ID # corrected length, a (in.)

Crack Type Replica "Width" (in.)

"Left" Tip x coordinate (in.)

"Left" Tip Θ coordinate (rad.)

"Right" Tip x coordinate (in.)

"Right" Tip Θ coordinate (rad.)

45007 1 0.03929 s 0.1711 0.11118 -0.00884 0.18977 -0.04479 47507 1 0.08106 c 0.176 0.10544 -0.00498 0.1865 -0.02414 50007 1 0.09294 c 0.1778 0.09356 -0.07551 0.1865 -0.44598 55007 1 0.11784 c 0.1741 0.06867 -0.05627 0.1865 -0.09843 60008 1 0.1483 c 0.1806 0.03821 -0.10748 0.1865 -0.14703 65008 1 0.1865 c 0.1765 0 -0.05775 0.1865 -0.08024 25002 1.1 0.00104 s 0.1792 30003 1.1 0.00248 s 0.177 0.16774 -0.01383 0.1727 -0.034 32504 1.1 0.00248 s 0.1764 0.16948 -0.05914 0.17445 -0.08726 35005 1.1 0.00296 s 0.1734 0.17166 0.03108 0.17757 0.01621 40007 1.1 0.0058 s 0.1802 0.16725 -0.0701 0.17884 -0.0646 42507 1.1 0.00948 s 0.1751 0.16754 0.00533 0.1865 -0.01959 5001 1.2 0.00122 s 0.1805 7502 1.2 0.00146 s 0.1731

12502 1.2 0.00273 s 0.1764 15002 1.2 0.00315 s 0.1738 17502 1.2 0.00294 s 0.1776 0.1279 -0.08947 0.13378 -0.085 20002 1.2 0.00362 s 0.1775 0.1265 -0.06516 0.13375 -0.06181 22502 1.2 0.00363 s 0.1747 0.12917 -0.03876 0.13643 -0.03194 25002 1.2 0.00489 s 0.1792 0.12739 -0.04291 0.13717 -0.06837 30003 1.2 0.00637 s 0.177 0.12581 -0.00262 0.13856 -0.03176 32504 1.2 0.00825 s 0.1764 0.12613 -0.0344 0.14262 -0.06139 35005 1.2 0.01016 s 0.1734 0.12337 0.03452 0.14369 -0.00324 37506 1.2 0.01193 s 0.1704 0.12444 0.03833 0.1483 0.02086 40007 1.2 0.01309 s 0.1802 0.11954 -0.07451 0.14572 -0.08992 42507 1.2 0.01757 s 0.1751 0.11631 0.01327 0.15146 -0.01393 30003 2 0.00216 s 0.177 32504 2 0.00264 s 0.1764 37506 2 0.00323 s 0.1704

Table C1, continued.



"Left" Tip x _coordinate (in.)


"Right" Tip x _coordinate (in.)

"Right" Θ coordinate (rad.)

40007 2 0.00393 s 0.1802 0.08321 0.367 0.09108 0.37471 42507 2 0.00463 s 0.1751 0.08127 0.43704 0.09053 0.44497 45007 2 0.0054 s 0.1711 0.0823 0.41904 0.09309 0.4202 47507 2 0.00604 s 0.176 0.07916 0.38619 0.09124 0.39746 50007 2 0.00629 s 0.1778 0.07762 0.32621 0.09021 0.33067 55007 2 0.00691 s 0.1741 0.06984 0.37222 0.08366 0.38931

Table C2 Crack measurements for specimen 6611-a12, front notch.







2501 1 0.00219 s 0.1778 5001 1 0.00225 s 0.176

12502 1 0.00247 s 0.1781 15002 1 0.00256 s 0.1722 27503 1 0.00333 s 0.1743 32504 1 0.00341 s 0.1754 35005 1 0.0035 s 0.1708 40007 1 0.00357 s 0.1725 0.02746 -0.38949 0.0346 -0.40087 45007 1 0.00387 s 0.176 0.02395 -0.45648 0.03168 -0.46652 47507 1 0.00408 s 0.1783 0.03598 -0.49178 0.04414 -0.50609 50007 1 0.00448 s 0.1792 0.02758 -0.51582 0.03653 -0.53882 55007 1 0.00473 s 0.1756 0.02623 -0.48972 0.03569 -0.49978 60008 1 0.00502 s 0.1726 0.0215 -0.44022 0.03155 -0.455 65008 1 0.00575 s 0.1734 0.02635 -0.52355 0.03786 -0.52695 2501 2 0.00165 s 0.1778 5001 2 0.00173 s 0.176 7502 2 0.00184 s 0.1692

10002 2 0.00185 s 0.1762 12502 2 0.00198 s 0.1781 15002 2 0.00211 s 0.1722 17502 2 0.00206 s 0.1766 20002 2 0.0022 s 0.1765 27503 2 0.00253 s 0.1743 32504 2 0.0026 s 0.1754 35005 2 0.00258 s 0.1708 40007 2 0.00255 s 0.1725 42507 2 0.00243 s 0.1811 45007 2 0.00263 s 0.176 47507 2 0.00261 s 0.1783








50007 2 0.00258 s 0.1792 55007 2 0.00263 s 0.1756 60008 2 0.00259 s 0.1726 65008 2 0.00273 s 0.1734 2501 3 0.00033 s 0.1778 5001 3 0.00042 s 0.176 7502 3 0.00041 s 0.1692

10002 3 0.00044 s 0.1762 12502 3 0.00043 s 0.1781 15002 3 0.00045 s 0.1722 27503 3 0.00088 s 0.1743 35005 3 0.00103 s 0.1708 40007 3 0.00185 s 0.1725 42507 3 0.00185 s 0.1811 0.02369 -0.27539 0.02739 -0.28298 47507 3 0.00235 s 0.1783 0.03483 -0.28808 0.03954 -0.29689 50007 3 0.00265 s 0.1792 0.02748 -0.32301 0.03278 -0.33287 55007 3 0.00329 s 0.1756 0.02464 -0.2773 0.03122 -0.28513 60008 3 0.00389 s 0.1726 0.01967 -0.22184 0.02745 -0.2298 65008 3 0.00462 s 0.1734 0.02485 -0.30052 0.03409 -0.31071 65008 4 0.02205 s 0.1734 0.07927 -0.31863 0.12337 -0.16353 2501 4.1 0.00248 s 0.1778 5001 4.1 0.00277 s 0.176 7502 4.1 0.00276 s 0.1692

12502 4.1 0.00309 s 0.1781 15002 4.1 0.00348 s 0.1722 17502 4.1 0.00412 s 0.1766 20002 4.1 0.00445 s 0.1765 22502 4.1 0.0055 s 0.1688 32504 4.1 0.00718 s 0.1754 0.0975 -0.16838 0.11186 -0.16838








35005 4.1 0.00754 s 0.1708 37506 4.1 0.00851 s 0.1754 0.09431 -0.20867 0.11133 -0.21539 40007 4.1 0.01033 s 0.1725 0.09568 -0.13115 0.11633 -0.11521 42507 4.1 0.01159 s 0.1811 0.08815 -0.26997 0.11132 -0.24071 45007 4.1 0.01224 s 0.176 0.08997 -0.22001 0.11444 -0.18432 50007 4.1 0.01374 s 0.1792 0.08971 -0.31863 0.11719 -0.2529 55007 4.1 0.01561 s 0.1756 0.08645 -0.26053 0.11768 -0.15768 60008 4.1 0.01896 s 0.1726 0.07812 -0.22525 0.11605 -0.10923 2501 4.2 0.00043 s 0.1778 5001 4.2 0.00096 s 0.176 7502 4.2 0.001 s 0.1692

12502 4.2 0.00101 s 0.1781 20002 4.2 0.00123 s 0.1765 40007 4.2 0.0013 s 0.1725 42507 4.2 0.00124 s 0.1811 45007 4.2 0.00163 s 0.176 47507 4.2 0.00167 s 0.1783 50007 4.2 0.0018 s 0.1792 55007 4.2 0.00197 s 0.1756 60008 4.2 0.00234 s 0.1726 2501 5 0.00026 s 0.1778 5001 5 0.00026 s 0.176 7502 5 0.00033 s 0.1692

12502 5 0.00068 s 0.1781 15002 5 0.00085 s 0.1722 17502 5 0.00106 s 0.1766 20002 5 0.00114 s 0.1765 22502 5 0.00174 s 0.1688 35005 5 0.00224 s 0.1708 0.04521 0.16823 0.04968 0.18547








40007 5 0.00459 s 0.1725 0.04844 0.11809 0.05763 0.16247 42507 5 0.00587 s 0.1811 0.04181 -0.00005 0.05355 0.03464 45007 5 0.00588 s 0.176 0.04249 0.06331 0.05425 0.08339 50007 5 0.00729 s 0.1792 0.0461 -0.04475 0.06067 -0.00969 55007 5 0.00897 s 0.1756 0.0428 0.01225 0.06075 0.04914 60008 5 0.01226 s 0.1726 0.03566 0.08868 0.06019 0.08526 65008 5 0.01355 s 0.1734 0.04152 -0.01408 0.06862 -0.01182 60008 6 0.02739 s 0.1726 0.10114 0.15919 0.15592 0.16602 65008 6 0.03167 s 0.1734 0.10282 0.05611 0.16617 0.06177 5001 6.1 0.00102 s 0.176 7502 6.1 0.00106 s 0.1692

10002 6.1 0.00121 s 0.1762 15002 6.1 0.0013 s 0.1722 27503 6.1 0.00156 s 0.1743 32504 6.1 0.00237 s 0.1754 40007 6.1 0.00276 s 0.1725 0.11114 0.23189 0.11666 0.22279 45007 6.1 0.00355 s 0.176 0.10597 0.14585 0.11307 0.13358 47507 6.1 0.00429 s 0.1783 0.11265 0.12591 0.12123 0.12701 50007 6.1 0.00458 s 0.1792 0.10844 0.04837 0.1176 0.04728 55007 6.1 0.00621 s 0.1756 0.10653 0.1084 0.11895 0.1084 10002 6.2 0.00096 s 0.1762 12502 6.2 0.00095 s 0.1781 17502 6.2 0.00154 s 0.1766 20002 6.2 0.00214 s 0.1765 22502 6.2 0.00244 s 0.1688 27503 6.2 0.00324 s 0.1743 30003 6.2 0.00394 s 0.1752 32504 6.2 0.00427 s 0.1754 35005 6.2 0.00505 s 0.1708








40007 6.2 0.00611 s 0.1725 0.13125 0.23531 0.14347 0.25693 42507 6.2 0.00705 s 0.1811 0.12512 0.07149 0.13923 0.07583 45007 6.2 0.00789 s 0.176 0.12599 0.13581 0.14178 0.15143 47507 6.2 0.00978 s 0.1783 0.13127 0.12371 0.15083 0.14683 50007 6.2 0.01093 s 0.1792 0.12624 0.04618 0.1481 0.05604 55007 6.2 0.01561 s 0.1756 0.1198 0.09945 0.15103 0.11175 2501 7 0.00058 s 0.1778 5001 7 0.0006 s 0.176 7502 7 0.00158 s 0.1692

10002 7 0.00202 s 0.1762 12502 7 0.00233 s 0.1781 15002 7 0.00254 s 0.1722 17502 7 0.00287 s 0.1766 20002 7 0.00301 s 0.1765 0.09721 0.54029 0.10324 0.56588 27503 7 0.0038 s 0.1743 0.09619 0.65141 0.10379 0.67844 32504 7 0.00526 s 0.1754 0.0941 0.60502 0.10463 0.64195 37506 7 0.00579 s 0.1754 0.08847 0.55241 0.10006 0.60166 40007 7 0.00665 s 0.1725 0.09547 0.64274 0.10876 0.68712 42507 7 0.00716 s 0.1811 0.0898 0.47475 0.10411 0.51919 45007 7 0.00715 s 0.176 0.09049 0.54741 0.1048 0.58645 47507 7 0.00779 s 0.1783 0.09519 0.51568 0.11077 0.56633 50007 7 0.00801 s 0.1792 0.09294 0.45042 0.10897 0.49753 55007 7 0.00823 s 0.1756 0.09176 0.51869 0.10823 0.569 60008 7 0.00902 s 0.1726 0.08547 0.59141 0.10352 0.6369 65008 7 0.00898 s 0.1734 0.09164 0.4818 0.1096 0.52369 45007 8 0.00046 s 0.176 47507 8 0.00078 s 0.1783 50007 8 0.00162 s 0.1792

Table C3 Crack measurements for specimen 6612-b21, back notch.







0 1 0.00079 s 4420 10001 1 0.00191 s 4572 20002 1 0.00287 s 4547 22003 1 0.00356 s 4572 23001 1 0.00458 s 4623 24502 1 0.00529 s 4623 27504 1 0.00643 s 4623 29005 1 0.00686 s 0.1781 0.09141 -0.12923 0.10514 -0.15055 30505 1 0.00748 s 4623 32005 1 0.00757 s 4572 33506 1 0.00895 s 0.1828 0.09153 -0.11643 0.10943 -0.15279 35006 1 0.01031 s 0.1805 0.09322 -0.1484 0.11384 -0.19469 37507 1 0.01105 s 0.1787 0.08837 -0.13938 0.11046 -0.1872 40008 1 0.01379 s 0.1799 0.08423 -0.11832 0.11182 -0.1521 42509 1 0.01683 s 0.1832 0.08312 -0.12582 0.11678 -0.1818 45010 1 0.01899 s 0.1807 0.07969 -0.13316 0.11767 -0.1815 47511 1 0.02238 s 0.1827 0.07676 -0.13642 0.12153 -0.18631 50012 1 0.02642 s 0.1832 0.06906 -0.15381 0.12191 -0.19216 52513 1 0.03384 s 0.1828 0.06109 -0.14551 0.12876 -0.18915 55013 1 0.04396 s 0.1807 0.0515 -0.14997 0.13941 -0.16784 57514 1 0.04968 s 0.1822 0.052 -0.16062 0.15137 -0.18668 60015 1 0.06009 s 0.1824 0.04195 -0.13198 0.16213 -0.18403 62516 1 0.16055 c 0.1815 0.02745 -0.11124 0.188 -0.13949 65017 1 0.188 c 0.185 0 -0.18813 0.188 -0.19121








90037 1 0.03741 c 0.1874 0 -0.25341 0.03741 -0.22414 105041 1 0.05873 c 0.1818 0 -0.25419 0.05873 -0.18824 105043 1 0.06308 c 0.1841 0 -0.19779 0.06308 -0.11612 110045 1 0.06713 c 0.1858 0 -0.25191 0.06713 -0.1349

0 2 0.00196 s 4724 5000 2 0.00215 s 4704 7500 2 0.00218 s 4633

12515 2 0.00219 s 4618 15019 2 0.00221 s 4582 25022 2 0.00345 s 4796 30023 2 0.00449 s 4745 35024 2 0.00519 s 4658 40024 2 0.0052 s 4775 45025 2 0.00628 s 4762 50026 2 0.00783 s 4707 55027 2 0.00784 s 4737 60028 2 0.00986 s 0.185 0.09168 -0.0562 0.1114 -0.04631 65029 2 0.01258 s 0.1852 0.09065 -0.05673 0.11582 -0.04467 70030 2 0.015 s 0.1866 0.08588 -0.08987 0.11587 -0.05285 75033 2 0.02139 s 0.1826 0.07803 -0.05187 0.12081 0.006 80034 2 0.02813 s 0.184 0.07225 -0.06674 0.12851 0.00725 85035 2 0.03396 s 0.1873 0.0672 -0.12904 0.13512 -0.03791 90037 2 0.04423 s 0.1874 0.05483 -0.0203 0.1433 0.07837

100038 2 0.05274 s 0.1838 0.07752 -0.02641 0.183 0.02334 105041 2 0.08142 s 0.1818 0.02816 -0.05971 0.191 0.04199 110045 2 0.1873 c 0.1858 0 -0.04631 0.1873 0.01274








168630 1 0.187 c 0.1786 0 0.3348 0.187 0.28851 80577 1.1 0.0009 s 0.173 85607 1.1 0.00087 s 0.179 90637 1.1 0.00413 s 0.1782 98260 1.1 0.00493 s 0.1754

105840 1.1 0.00709 s 0.1749 113440 1.1 0.00803 s 0.1746 0.07251 0.53968 0.08857 0.54193 121040 1.1 0.00974 s 0.167 0.0729 0.64633 0.09238 0.63926 128630 1.1 0.01132 s 0.1767 0.07408 0.52685 0.09673 0.52351 130630 1.1 0.01151 s 0.1755 0.07192 0.5367 0.09494 0.54007 132630 1.1 0.01292 s 0.1737 0.0675 0.55514 0.09334 0.53361 134630 1.1 0.01398 s 0.1766 0.07285 0.5102 0.10081 0.50128 135630 1.1 0.01488 s 0.1772 0.0744 0.50204 0.10416 0.50649 136630 1.1 0.01354 s 0.1858 0.07357 0.42729 0.10065 0.43153 137630 1.1 0.01543 s 0.1824 0.07095 0.46453 0.1018 0.4764 138630 1.1 0.01562 s 0.1712 0.06368 0.50685 0.09492 0.52295 139630 1.1 0.01621 s 0.1707 0.07482 0.57636 0.10725 0.5706 140630 1.1 0.01742 s 0.1755 0.0716 0.53783 0.10645 0.5367 142130 1.1 0.01722 s 0.1754 0.0693 0.48628 0.10373 0.51209 143630 1.1 0.01972 s 0.1745 0.06494 0.52735 0.10438 0.54201 146130 1.1 0.0207 s 0.1689 0.07695 0.56894 0.11836 0.58293 148630 1.1 0.02221 s 0.1726 0.07042 0.54246 0.11484 0.56071 151130 1.1 0.02272 s 0.1749 0.06853 0.48316 0.11397 0.52142 153630 1.1 0.0226 s 0.1742 0.06548 0.50384 0.11068 0.55921 158630 1.1 0.02437 s 0.1765 0.05944 0.45673 0.10817 0.51473 161130 1.1 0.03118 s 0.1778 0.04344 0.41091 0.10581 0.50944 163630 1.1 0.03639 s 0.1737 0.02713 0.42709 0.09991 0.54494 166130 1.1 0.0437 s 0.1748 0.01434 0.41452 0.10174 0.54514 30150 1.2 0.00075 s 0.1726








45390 1.2 0.00138 s 0.1752 55450 1.2 0.00176 s 0.1775 60480 1.2 0.00238 s 0.1746 65485 1.2 0.00314 s 0.1782 75546 1.2 0.00449 s 0.1763 80577 1.2 0.005 s 0.173 85607 1.2 0.00563 s 0.179 90637 1.2 0.00789 s 0.1782 98260 1.2 0.00936 s 0.1754

105840 1.2 0.01492 s 0.1749 113440 1.2 0.0346 c 0.1746 0.14855 0.35704 0.18314 0.31307 121040 1.2 0.03728 c 0.167 0.14971 0.44831 0.187 0.38702 128630 1.2 0.04805 c 0.1767 0.13895 0.35752 0.187 0.30071 130630 1.2 0.05061 c 0.1755 0.13639 0.3909 0.187 0.31463 135630 1.2 0.0572 c 0.1772 0.1298 0.35986 0.187 0.26432 136630 1.2 0.0626 c 0.1858 0.1244 0.29062 0.187 0.20481 137630 1.2 0.06234 c 0.1824 0.12467 0.33935 0.187 0.22064 138630 1.2 0.06761 c 0.1712 0.11939 0.38383 0.187 0.32404 142130 1.2 0.06812 c 0.1754 0.11887 0.34039 0.187 0.26071 143630 1.2 0.07748 c 0.1745 0.10952 0.41003 0.187 0.30625 151130 1.2 0.08243 c 0.1749 0.10457 0.40325 0.187 0.29521 153630 1.2 0.08964 c 0.1742 0.09736 0.427 0.187 0.29141 156130 1.2 0.09478 c 0.1744 0.09221 0.4281 0.187 0.28476 158630 1.2 0.10108 c 0.1765 0.08592 0.40989 0.187 0.27941 161130 1.2 0.11138 c 0.1778 0.07562 0.40095 0.187 0.28027 163630 1.2 0.12294 c 0.1737 0.06406 0.44182 0.187 0.31603 166130 1.2 0.12335 c 0.1748 0.06365 0.46294 0.187 0.3233








35001 1 0.00129 s 0.1629 41001 1 0.00235 s 0.1738 44002 1 0.00287 s 0.1744 47002 1 0.00339 s 0.1644 50002 1 0.0037 s 0.175 53002 1 0.00362 s 0.1704 0.06177 -0.43757 0.069 -0.43061 56002 1 0.0044 s 0.1693 0.06423 -0.45826 0.07303 -0.43605 59002 1 0.00473 s 0.1713 0.05994 -0.45623 0.06939 -0.44353 62002 1 0.00477 s 0.1659 0.06544 -0.4605 0.07498 -0.44619 65002 1 0.00589 s 0.1703 0.05856 -0.46247 0.07034 -0.44272 68003 1 0.00634 s 0.1713 0.05725 -0.46547 0.06993 -0.44353 71003 1 0.00683 s 0.1723 0.06012 -0.44203 0.07379 -0.41677 74004 1 0.00728 s 0.1668 0.05681 -0.41786 0.07137 -0.39414 77004 1 0.00819 s 0.1741 0.05527 -0.48211 0.07166 -0.46848 80004 1 0.00899 s 0.1698 0.05451 -0.46503 0.07249 -0.44639 83004 1 0.0107 s 0.1703 0.0551 -0.46014 0.0765 -0.43342 86004 1 0.01122 s 0.1713 0.05747 -0.47818 0.07992 -0.46201 89005 1 0.01339 s 0.1704 0.05215 -0.46196 0.07893 -0.43757 94006 1 0.01532 s 0.1699 0.04787 -0.48082 0.07852 -0.46917 99007 1 0.01669 s 0.167 0.04418 -0.45834 0.07757 -0.45715

104008 1 0.02108 s 0.1597 0.04977 -0.40749 0.09194 -0.42483 106509 1 0.02074 s 0.1717 0.04254 -0.47611 0.08402 -0.48879 74004 2 0.00046 s 0.1668 77004 2 0.00062 s 0.1741 80004 2 0.00149 s 0.1698 83004 2 0.00181 s 0.1703 86004 2 0.00252 s 0.1713 89005 2 0.00389 s 0.1704 0.0676 0.27417 0.07537 0.29391 94006 2 0.00596 s 0.1699 0.064 0.25165 0.07592 0.28193








99007 2 0.00716 s 0.167 0.06082 0.28093 0.07514 0.32714 104008 2 0.00778 s 0.1597 0.06913 0.36434 0.08468 0.41265 106509 2 0.00777 s 0.1717 0.06226 0.23255 0.0778 0.27864 47002 3 0.00095 s 0.1644 50002 3 0.00095 s 0.175 53002 3 0.00098 s 0.1704 56002 3 0.0012 s 0.1693 59002 3 0.00142 s 0.1713 62002 3 0.00164 s 0.1659 65002 3 0.00189 s 0.1703 68003 3 0.00199 s 0.1713 71003 3 0.00242 s 0.1723 74004 3 0.00261 s 0.1668 80004 3 0.00412 s 0.1698 0.12808 0.23642 0.13632 0.24341 83004 3 0.00459 s 0.1703 0.13041 0.24505 0.13959 0.24854 89005 3 0.00475 s 0.1704 0.12958 0.22076 0.13908 0.21728 94006 3 0.00606 s 0.1699 0.12595 0.20275 0.13808 0.20507 99007 3 0.00788 s 0.167 0.12472 0.2679 0.14048 0.2679

104008 3 0.00864 s 0.1597 0.13665 0.36062 0.15393 0.36805 106509 3 0.00707 s 0.1717 0.12827 0.2314 0.14242 0.22909 15001 4 0.00113 s 0.1728 27501 4 0.00299 s 0.1672 35001 4 0.00358 s 0.1629 38001 4 0.00389 s 0.1695 44002 4 0.00584 s 0.1744 47002 4 0.00718 s 0.1644 50002 4 0.00755 s 0.175 53002 4 0.00772 s 0.1704 0.10107 0.54935 0.11651 0.54935 56002 4 0.00891 s 0.1693 0.09944 0.5491 0.11727 0.53742








59002 4 0.00961 s 0.1713 0.09861 0.56016 0.11783 0.52782 62002 4 0.01043 s 0.1659 0.10104 0.57227 0.12189 0.55081 65002 4 0.01043 s 0.1703 0.09724 0.54479 0.11809 0.52388 68003 4 0.01117 s 0.1713 0.09485 0.53475 0.11719 0.51165 71003 4 0.01153 s 0.1723 0.09419 0.56042 0.11726 0.55468 74004 4 0.01263 s 0.1668 0.09443 0.61409 0.11969 0.60223 77004 4 0.01411 s 0.1741 0.09205 0.50202 0.12027 0.47816 80004 4 0.01577 s 0.1698 0.08972 0.52771 0.12126 0.52305 86004 4 0.0181 s 0.1713 0.09764 0.48162 0.13384 0.47122 94006 4 0.02128 s 0.1699 0.08816 0.50552 0.13072 0.50435 99007 4 0.0281 s 0.167 0.08076 0.55816 0.13695 0.57593

104008 4 0.03612 s 0.1597 0.08376 0.66539 0.156 0.66539 106509 4 0.03729 s 0.1717 0.07341 0.5114 0.14799 0.5114








0 1 0.00032 s 0.1702 5001 1 0.00059 s 0.1786

10001 1 0.00065 s 0.1682 15001 1 0.00074 s 0.1702 27501 1 0.00222 s 0.1759 35001 1 0.00301 s 0.1747 38001 1 0.00315 s 0.1749 41001 1 0.00383 s 0.1761 44002 1 0.00447 s 0.1719 47002 1 0.00503 s 0.1734 50002 1 0.00577 s 0.1733 53002 1 0.00701 s 0.172 0.07317 -0.10763 0.08719 -0.12143 56002 1 0.00775 s 0.1746 0.06787 -0.13055 0.08336 -0.14415 59002 1 0.00938 s 0.1727 0.06542 -0.12502 0.08417 -0.1422 62002 1 0.01024 s 0.1707 0.06424 -0.12315 0.08472 -0.14285 65002 1 0.01196 s 0.1731 0.06665 -0.15007 0.09057 -0.17522 68003 1 0.01491 s 0.1746 0.06249 -0.13962 0.09232 -0.17361 71003 1 0.01759 s 0.1742 0.06401 -0.13065 0.09918 -0.1863 74004 1 0.02052 s 0.1753 0.06245 -0.1521 0.10349 -0.20741 77004 1 0.02431 s 0.1767 0.058 -0.14991 0.10663 -0.20925 80004 1 0.02948 s 0.1707 0.04452 -0.10808 0.10348 -0.14517 83004 1 0.03313 s 0.173 0.04361 -0.11723 0.10987 -0.18128 86004 1 0.03807 s 0.1747 0.03412 -0.094 0.11027 -0.16988 89005 1 0.04376 s 0.1707 0.03428 -0.02926 0.1218 -0.13358 94006 1 0.05478 s 0.1703 0.02658 -0.03494 0.13614 -0.12788 99007 1 0.06424 s 0.17 0.02176 -0.06714 0.15023 -0.16723

104008 1 0.16534 c 0.1795 0 -0.1368 0.16534 -0.21616 106509 1 0.184 c 0.1755 0 -0.08044 0.184 -0.19092 41001 2 0.00261 s 0.1761 0.03918 0.22007 0.04441 0.20434


vN Crack ID # corrected length, a (in.)






44002 2 0.00289 s 0.1719 0.03083 0.25807 0.03661 0.24426 47002 2 0.00313 s 0.1734 0.02918 0.23313 0.03544 0.21487 50002 2 0.0035 s 0.1733 0.03408 0.2296 0.04109 0.22047 53002 2 0.00396 s 0.172 0.03306 0.27196 0.04097 0.25356 56002 2 0.00527 s 0.1746 0.02856 0.24112 0.0391 0.20486 59002 2 0.00666 s 0.1727 0.02482 0.25304 0.03814 0.21409 62002 2 0.00728 s 0.1707 0.02565 0.26398 0.04021 0.25123 65002 2 0.0084 s 0.1731 0.02923 0.20653 0.04603 0.18253 68003 2 0.0098 s 0.1746 0.02592 0.21393 0.04553 0.18787 71003 2 0.01077 s 0.1742 0.03063 0.24756 0.05218 0.19872 74004 2 0.01197 s 0.1753 0.02981 0.21244 0.05374 0.17407 77004 2 0.01354 s 0.1767 0.0277 0.20727 0.05477 0.17368 80004 2 0.01633 s 0.1707 0.01283 0.23848 0.04549 0.21877 83004 2 0.01877 s 0.173 0.01276 0.23501 0.05031 0.18583 86004 2 0.02059 s 0.1747 0.00632 0.24235 0.0475 0.19479 89005 2 0.02382 s 0.1707 0.00593 0.29179 0.05357 0.22805 94006 2 0.05424 c 0.1703 0 0.2892 0.05424 0.22762 99007 2 0.0591 c 0.17 0 0.24825 0.0591 0.18308

104008 2 0.05433 c 0.1795 0 0.14537 0.05433 0.10018 106509 2 0.05892 c 0.1755 0 0.22732 0.05892 0.16983 35001 2.1 0.00043 s 0.1747 41001 2.1 0.00072 s 0.1761 44002 2.1 0.00107 s 0.1719

0 2.2 0.00179 s 0.1702 5001 2.2 0.00177 s 0.1786

10001 2.2 0.00155 s 0.1682 15001 2.2 0.00147 s 0.1702 20001 2.2 0.00165 s 0.1686 27501 2.2 0.00214 s 0.1759








35001 2.2 0.00216 s 0.1747 41001 2.2 0.00214 s 0.1761 44002 2.2 0.00219 s 0.1719 5001 3 0.00071 s 0.1786

10001 3 0.00097 s 0.1682 15001 3 0.00115 s 0.1702 20001 3 0.00163 s 0.1686 27501 3 0.00214 s 0.1759 35001 3 0.00309 s 0.1747 38001 3 0.00338 s 0.1749 41001 3 0.00346 s 0.1761 44002 3 0.00421 s 0.1719 47002 3 0.00437 s 0.1734 50002 3 0.00451 s 0.1733 53002 3 0.00481 s 0.172 0.1028 0.39389 0.11243 0.39965 56002 3 0.00543 s 0.1746 0.09632 0.36577 0.10718 0.3635 59002 3 0.00607 s 0.1727 0.09418 0.38593 0.10633 0.3951 62002 3 0.00706 s 0.1707 0.0941 0.38452 0.10822 0.39379 65002 3 0.00803 s 0.1731 0.09726 0.32655 0.11331 0.33569 68003 3 0.00854 s 0.1746 0.09611 0.33064 0.11318 0.34197 71003 3 0.00924 s 0.1742 0.09971 0.32706 0.1182 0.3316 74004 3 0.01086 s 0.1753 0.09762 0.30838 0.11934 0.32192 77004 3 0.01161 s 0.1767 0.09643 0.30133 0.11965 0.30469 83004 3 0.01553 s 0.173 0.08647 0.33222 0.11753 0.36538 86004 3 0.01727 s 0.1747 0.08131 0.33861 0.11586 0.36919 89005 3 0.01687 s 0.1707 0.0872 0.38568 0.12094 0.41002 94006 3 0.01761 s 0.1703 0.08654 0.38446 0.12177 0.40305 99007 3 0.01699 s 0.17 0.09059 0.34136 0.12458 0.36347

104008 3 0.01584 s 0.1795 0.08447 0.26331 0.11614 0.29086 106509 3 0.0163 s 0.1755 0.08922 0.32991 0.12183 0.35922








173070 1 0.00013 s 0.1787 183071 1 0.00053 s 0.1773 204563 1 0.00093 s 0.1805 214563 1 0.00227 s 0.173 224564 1 0.00523 s 0.1747 227565 1 0.01037 s 0.1724 229567 1 0.01105 s 0.1803 231068 1 0.01215 s 0.1805 0.09091 -0.0729 0.1152 0.00503 233069 1 0.01566 s 0.1831 0.08619 -0.0405 0.11751 -0.0426 235070 1 0.02104 s 0.1794 0.08033 -0.04009 0.12241 -0.02183 237071 1 0.02512 s 0.1808 0.07736 -0.03489 0.1276 -0.03275 239072 1 0.03565 s 0.1793 0.06046 0.00483 0.13176 0.03063 241073 1 0.04221 s 0.1797 0.05899 -0.00511 0.14341 0.00562 243074 1 0.04973 s 0.1808 0.03776 -0.03062 0.13722 0.00242 245074 1 0.06278 s 0.1808 0.03632 0.05465 0.16188 -0.1063 247075 1 0.07634 s 0.1826 0.02421 0.00907 0.17689 -0.09013 249076 1 0.185 c 0.1809 0 0.03244 0.185 -0.07408








231003 1 0.07676 s 0.1745 0.00886 -0.11441 0.16238 -0.13321 233003 1 0.184 c 0.1777 0 0.30737 0.184 0.31352 147509 1.1 0.00026 s 0.1728 162511 1.1 0.00052 s 0.1731 185002 1.1 0.00079 s 0.173 192503 1.1 0.00162 s 0.1811 200004 1.1 0.00568 s 0.1733 207504 1.1 0.00892 s 0.1783 209505 1.1 0.00947 s 0.1774 211506 1.1 0.01234 s 0.179 0.06877 -0.18868 0.09344 -0.24466 212959 1.1 0.0153 s 0.1744 0.06879 -0.11045 0.09939 -0.19192 215000 1.1 0.01651 s 0.1778 0.06665 -0.1394 0.09966 -0.20292 217001 1.1 0.02282 s 0.1705 0.06162 -0.10375 0.10727 -0.18389 219002 1.1 0.02705 s 0.1738 0.05865 -0.13056 0.11275 -0.19869 221002 1.1 0.03447 s 0.17 0.05217 -0.10694 0.12112 -0.20017 223003 1.1 0.04021 s 0.1748 0.04274 -0.14401 0.12316 -0.25344 225003 1.1 0.04826 s 0.171 0.03938 -0.08354 0.1359 -0.15705 227003 1.1 0.05264 s 0.1795 0.03731 -0.13976 0.14259 -0.25343 229003 1.1 0.06235 s 0.1716 0.02391 -0.0983 0.14862 -0.17686 229003 1.2 0.00306 s 0.1716 0.02005 -0.10254 0.02616 -0.10573

Table C10 Crack measurements for specimen 8B2, back notch.







15002 1 0.00188 s 0.1782 22503 1 0.00298 s 0.1814 37505 1 0.00534 s 0.18 40006 1 0.00657 s 0.1772 42507 1 0.00773 s 0.1812 45008 1 0.00885 s 0.1798 0.04794 -0.28664 0.06565 -0.3267 47509 1 0.01248 s 0.1781 0.04511 -0.31373 0.07008 -0.34325 50009 1 0.01336 s 0.1795 0.04323 -0.29425 0.06994 -0.31811 52509 1 0.01681 s 0.1785 0.04193 -0.27511 0.07556 -0.32309 55000 1 0.01903 s 0.1789 0.03948 -0.28997 0.07754 -0.33024 57501 1 0.0231 s 0.181 0.03286 -0.27687 0.07906 -0.3285 60002 1 0.02966 s 0.1808 0.02581 -0.26359 0.08512 -0.32605 62503 1 0.03713 s 0.1809 0.01791 -0.18791 0.09216 -0.26109 65004 1 0.04754 s 0.1817 0.00715 -0.16721 0.10223 -0.24114 67505 1 0.11402 c 0.1812 0 -0.16334 0.11402 -0.19986 70006 1 0.12895 c 0.1764 0 -0.12062 0.12895 -0.15925 72507 1 0.14332 c 0.1748 0 -0.09064 0.14332 -0.15747 75008 1 0.16824 c 0.181 0 -0.15533 0.16824 -0.18759 76009 1 0.183 c 0.1808 0 -0.18499 0.183 -0.1936

Table C11 Crack measurements for specimen 8B3, back notch.







15001 1 0.00104 s 0.1773 22502 1 0.00162 s 0.1794 25003 1 0.00167 s 0.1739 27503 1 0.00169 s 0.1718 30004 1 0.00299 s 0.1781 32504 1 0.00318 s 0.1719 35005 1 0.0033 s 0.1733 37505 1 0.00373 s 0.1738 42506 1 0.00412 s 0.1718 0.104 -0.22861 0.11224 -0.22292 45006 1 0.00492 s 0.1775 0.09889 -0.25297 0.10874 -0.23973 47506 1 0.00544 s 0.181 0.09708 -0.28169 0.10796 -0.27304 50007 1 0.00629 s 0.1754 0.09871 -0.22268 0.1113 -0.20706 52508 1 0.00672 s 0.1752 0.10261 -0.20327 0.11605 -0.19768 57509 1 0.00771 s 0.1778 0.09769 -0.21886 0.11311 -0.19794 59913 1 0.00781 s 0.1756 0.10122 -0.21864 0.11683 -0.21195

Table C12 Crack measurements for specimen 8B3, front notch.







57509 1 0.08316 s 0.1739 0.01468 0.27833 0.181 0.16761 58509 1 0.08823 s 0.1752 0.00455 0.25362 0.181 0.19867 59913 1 0.181 c 0.1788 0 0.21904 0.181 0.16951 15001 1.1 0.00024 s 0.177 22502 1.1 0.00032 s 0.1772 25003 1.1 0.00033 s 0.1739 27503 1.1 0.00034 s 0.1787 35005 1.1 0.00034 s 0.1764 37505 1.1 0.00034 s 0.1776 40006 1.1 0.00056 s 0.1772 42506 1.1 0.00063 s 0.1765 45006 1.1 0.00107 s 0.1768 47506 1.1 0.00152 s 0.1763 50007 1.1 0.00191 s 0.1769 52508 1.1 0.00416 s 0.1764 0.02658 0.20192 0.03489 0.18008 55009 1.1 0.00639 s 0.1769 0.02548 0.18711 0.03827 0.15554 47506 1.2 0.04297 s 0.1763 0.05195 0.23896 0.13788 0.16797 50007 1.2 0.04881 s 0.1769 0.04707 0.22412 0.14468 0.18275 52508 1.2 0.06264 s 0.1764 0.03602 0.22484 0.1613 0.20737 55009 1.2 0.06768 s 0.1769 0.03489 0.18493 0.17026 0.1784 22502 1.21 0.00099 s 0.1772 25003 1.21 0.0016 s 0.1739 27503 1.21 0.00169 s 0.1787 30004 1.21 0.00208 s 0.1734 32504 1.21 0.00231 s 0.1775 35005 1.21 0.00283 s 0.1764 37505 1.21 0.00341 s 0.1776 0.06115 0.25391 0.06798 0.25065 40006 1.21 0.00429 s 0.1772 0.06456 0.28628 0.07314 0.28736 42506 1.21 0.00579 s 0.1765 0.05907 0.25982 0.07066 0.2751








45006 1.21 0.00671 s 0.1768 0.05805 0.26107 0.07146 0.26543 15001 1.22 0.00121 s 0.177 22502 1.22 0.00481 s 0.1772 25003 1.22 0.00583 s 0.1739 0.09565 0.17647 0.10731 0.18643 27503 1.22 0.00689 s 0.1787 0.09551 0.15324 0.10929 0.15755 30004 1.22 0.0084 s 0.1734 0.09791 0.19594 0.11472 0.19927 32504 1.22 0.01061 s 0.1775 0.0881 0.16272 0.10931 0.17791 35005 1.22 0.01267 s 0.1764 0.08824 0.15934 0.11359 0.1779 37505 1.22 0.01519 s 0.1776 0.08459 0.16283 0.11496 0.18126 40006 1.22 0.01925 s 0.1772 0.08406 0.20152 0.12257 0.22977 42506 1.22 0.02425 s 0.1765 0.07496 0.19655 0.12347 0.19 45006 1.22 0.03128 s 0.1768 0.0691 0.25672 0.13165 0.1881 15001 2 0.00012 s 0.177 22502 2 0.00048 s 0.1772 25003 2 0.00082 s 0.1739 27503 2 0.00086 s 0.1787 32504 2 0.00092 s 0.1775 35005 2 0.00143 s 0.1764 37505 2 0.00167 s 0.1776 40006 2 0.00147 s 0.1772 42506 2 0.00204 s 0.1765 45006 2 0.0024 s 0.1768 47506 2 0.00359 s 0.1763 0.03111 -0.06685 0.03829 -0.08979 50007 2 0.00389 s 0.1769 0.03213 -0.06869 0.0399 -0.08175 52508 2 0.00416 s 0.1764 0.03047 -0.06443 0.03879 -0.07534 55009 2 0.00409 s 0.1769 0.03243 -0.08066 0.04062 -0.09046 22502 3 0.00076 s 0.1772 25003 3 0.00109 s 0.1739 27503 3 0.00142 s 0.1787








30004 3 0.00197 s 0.1734 32504 3 0.00235 s 0.1775 0.05354 0.09764 0.05823 0.08896 35005 3 0.00303 s 0.1764 37505 3 0.00331 s 0.1776 0.05391 0.09344 0.06054 0.09453 40006 3 0.00342 s 0.1772 0.05802 0.1298 0.06486 0.1298 42506 3 0.00441 s 0.1765 0.05271 0.09836 0.06153 0.09727 45006 3 0.00532 s 0.1768 0.05272 0.10206 0.06337 0.11296 47506 3 0.0057 s 0.1763 0.05236 0.0937 0.06376 0.10462 50007 3 0.00404 s 0.1769 0.05321 0.10003 0.06129 0.09785 15001 4 0.00085 s 0.177 22502 4 0.00181 s 0.1772 25003 4 0.00197 s 0.1739 27503 4 0.00275 s 0.1787 30004 4 0.00294 s 0.1734 32504 4 0.00311 s 0.1775 0.06659 -0.16597 0.07281 -0.18224 35005 4 0.00328 s 0.1764 0.06793 -0.16049 0.07449 -0.17686 37505 4 0.00418 s 0.1776 0.06726 -0.16134 0.07562 -0.1776 40006 4 0.00465 s 0.1772 0.07048 -0.12447 0.07977 -0.14403 42506 4 0.00569 s 0.1765 0.06543 -0.16674 0.07681 -0.16892 47506 4 0.00657 s 0.1763 0.06427 -0.16952 0.07741 -0.17279 50007 4 0.0067 s 0.1769 0.06405 -0.17209 0.07745 -0.17862 15001 5 0.00193 s 0.177 22502 5 0.00302 s 0.1772 25003 5 0.00316 s 0.1739 27503 5 0.00344 s 0.1787 0.08194 -0.31979 0.08883 -0.32518 30004 5 0.00386 s 0.1734 0.08476 -0.28822 0.09248 -0.29821 32504 5 0.00469 s 0.1775 0.07729 -0.32327 0.08668 -0.32327 35005 5 0.00493 s 0.1764 0.0788 -0.32095 0.08865 -0.31986 37505 5 0.00561 s 0.1776 0.07786 -0.32506 0.08907 -0.3153








40006 5 0.00592 s 0.1772 0.08059 -0.28964 0.09244 -0.27986 42506 5 0.00631 s 0.1765 0.07732 -0.32056 0.08994 -0.30965 45006 5 0.00635 s 0.1768 0.07781 -0.31615 0.0905 -0.30853 47506 5 0.00621 s 0.1763 0.07844 -0.32461 0.09086 -0.31696 50007 5 0.00614 s 0.1769 0.07807 -0.32557 0.09035 -0.31795

Table C13 Crack measurements for specimen 8T3, front notch.







105707 1 0.00037 s 0.1873 113207 1 0.00048 s 0.1915 120708 1 0.00092 s 0.1912 128209 1 0.00291 s 0.1866 135710 1 0.00521 s 0.1928 140011 1 0.00552 s 0.1915 142512 1 0.00636 s 0.1915 145002 1 0.00687 s 0.1893 0.07491 -0.00982 0.08865 -0.00551 147503 1 0.0075 s 0.19 0.07188 -0.0145 0.08689 -0.00805 150003 1 0.00744 s 0.1891 0.07202 -0.05353 0.08689 -0.04813 153003 1 0.00829 s 0.1895 0.07064 -0.01362 0.08722 -0.00393 156004 1 0.00845 s 0.1894 0.07262 -0.02682 0.08952 -0.01927 159005 1 0.00907 s 0.1915 0.07436 -0.03095 0.09249 -0.02242 162005 1 0.01057 s 0.1899 0.06824 0.00138 0.08939 0.00245 166006 1 0.01202 s 0.1889 0.06634 0.01511 0.09038 0.00754 170006 1 0.01397 s 0.1896 0.0618 0.01679 0.08974 0.01571 174007 1 0.01562 s 0.1913 0.06328 0.00909 0.09452 -0.00159 178007 1 0.01748 s 0.192 0.06184 -0.00412 0.0968 -0.02007 182008 1 0.02055 s 0.1907 0.05768 -0.00736 0.09878 -0.03413 186009 1 0.02408 s 0.1865 0.05222 0.02456 0.10038 0.00157 194001 1 0.03317 s 0.1857 0.04722 0.03364 0.11356 0.01824 198001 1 0.0402 s 0.1863 0.0403 0.05258 0.12069 0.00655 202002 1 0.04594 s 0.1892 0.03794 0.00611 0.12981 -0.04677 206003 1 0.05331 s 0.1916 0.02734 0.01514 0.13396 -0.01897 210004 1 0.06503 s 0.1887 0.01912 0.02977 0.14918 -0.00594 214005 1 0.16052 c 0.1889 0 0.01186 0.16052 -0.04651 218006 1 0.17355 c 0.1869 0 0.01569 0.17355 -0.00507 222276 1 0.194 c 0.1932 0 -0.01385 0.194 -0.00856

159

Crack Tip Locations for Specimen 6611-a12, back notch (N = 30,003 cycles)

x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1.2, left tip

# 1.2, right tip

# 1.1, left tip

# 1.1, right tip

Figure C1 Crack tip locations for Specimen 6611-a12, back notch (N=30,003 cycles).


x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1.2, left tip

# 1.2, right tip

# 1.1, left tip

# 1.1, right tip


160

Crack Tip Locations for Specimen 6611-a12,back notch (N = 50,007 cycles)

x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1, left tip

# 1, right tip

# 2, left tip

# 2, right tip

Figure C3 Crack tip locations for Specimen 6611-a12, back notch (N=50,007

cycles).

161

Crack Tip Locations for Specimen 6611a12, front notch (N = 45,007 cycles)

x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1, left tip

# 1, right tip

# 4.1, left tip

# 4.1, right tip

# 5, left tip

# 5, right tip

# 6.1, left tip

# 6.1, right tip

# 6.2, left tip

# 6.2, right tip

# 7, left tip

# 7, right tip

Figure C4 Crack tip locations for Specimen 6611-a12, front notch (N=45,007 cycles).

Crack Tip Locations for Specimen 6611a12, front notch (N = 65,008 cycles)

x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1, left tip

# 1, right tip

# 3, left tip

# 3, right tip

# 4, left tip

# 4, right tip

# 5, left tip

# 5, right tip

# 6, left tip

# 6, right tip

# 7, left tip

# 7, right tip


162

Crack Tip Locations for Specimen 6612-b21, back notch (N = 29,005 cycles)

x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1, left tip

# 1, right tip

Figure C6 Crack tip locations for Specimen 6612-b21, front notch (N=29,005 cycles).

Crack Tip Locations for Specimen 6612-b21,back notch (N = 37,507 cycles)

x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1, left tip

# 1, right tip

Figure C7 Crack tip locations for Specimen 6612-b21, front notch (N=37,507

cycles).

163


x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1, left tip

# 1, right tip

Figure C8 Crack tip locations for Specimen 6612-b21, front notch (N=47,511 cycles).


x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1, left tip

# 1, right tip

Figure C9 Crack tip locations for Specimen 6612-b21, front notch (N=62,516

cycles).

164


x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 2, left tip

# 2, right tip



x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 2, left tip

# 2, right tip


cycles).

165


x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 2, left tip

# 2, right tip

# 1, left tip

# 1, right tip



x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 2, left tip

# 2, right tip

# 1, left tip

# 1, right tip


cycles).

166

Crack Tip Locations for Specimen 6714-a12, front notch (N = 113,440 cycles)

x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1.2, left tip

# 1.2, right tip

# 1.1, left tip

# 1.1, right tip



x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1.2, left tip

# 1.2, right tip

# 1.1, left tip

# 1.1, right tip

Figure C15 Crack tip locations for Specimen 6714-a12, front notch (N=130,630

cycles).

167


x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1.2, left tip

# 1.2, right tip

# 1.1, left tip

# 1.1, right tip



x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1.2, left tip

# 1.2, right tip

# 1.1, left tip

# 1.1, right tip


cycles).

168


x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1, left tip

# 1, right tip

# 4, left tip

# 4, right tip



x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1, left tip

# 1, right tip

# 4, left tip

# 4, right tip


cycles).

169


x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1, left tip

# 1, right tip

# 3, left tip

# 3, right tip

# 4, left tip

# 4, right tip



x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1, left tip

# 1, right tip

# 2, left tip

# 2, right tip

# 3, left tip

# 3, right tip

# 4, left tip

# 4, right tip


cycles).

170


x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1, left tip

# 1, right tip

# 2, left tip

# 2, right tip

# 3, left tip

# 3, right tip

# 4, left tip

# 4, right tip


cycles).

171


x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 2, left tip

# 2, right tip



x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1, left tip

# 1, right tip

# 2, left tip

# 2, right tip

# 3, left tip

# 3, right tip


cycles).

172


x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1, left tip

# 1, right tip

# 2, left tip

# 2, right tip

# 3, left tip

# 3, right tip



x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1, left tip

# 1, right tip

# 2, left tip

# 2, right tip

# 3, left tip

# 3, right tip


cycles).

173


x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1, left tip

# 1, right tip

# 2, left tip

# 2, right tip

# 3, left tip

# 3, right tip


cycles).

174


x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1, left tip

# 1, right tip

Figure C28 Crack tip locations for Specimen 7111-b11, back notch (N=231,068 cycles).


x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1, left tip

# 1, right tip

Figure C29 Crack tip locations for Specimen 7111-b11, back notch (N=239,072

cycles).

175


x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1, left tip

# 1, right tip


cycles).

176


x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1.1, left tip

# 1.1, right tip



x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1.1, left tip

# 1.1, right tip


cycles).

177


x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1.1, left tip

# 1.1, right tip

# 1.2, left tip

# 1.2, right tip



x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1, left tip

# 1, right tip


cycles).

178

Crack Tip Locations for Specimen 8B2, back notch (N = 45,008 cycles)

x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1, left tip

# 1, right tip

Figure C35 Crack tip locations for Specimen 8B2, back notch (N=45,008 cycles).

Crack Tip Locations for Specimen 8B2, back notch (N = 57,501 cycles)

x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1, left tip

# 1, right tip


179

Crack Tip Locations for Specimen 8B2, back notch (N =72,507 cycles)

x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1, left tip

# 1, right tip


180


x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1, left tip

# 1, right tip



x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1, left tip

# 1, right tip


181

Crack Tip Locations for Specimen 8B3, front notch (N = 27,503 cycles)

x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1.22, left tip

# 1.22, right tip

# 5, left tip

# 5, right tip

Figure C40 Crack tip locations for Specimen 8B3, front notch (N=27,503 cycles).


x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1.21, left tip

# 1.21, right tip

# 1.22, left tip

# 1.22, right tip

# 3, left tip

# 3, right tip

# 4, left tip

# 4, right tip

# 5, left tip

# 5, right tip


182


x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1.2, left tip

# 1.2, right tip

# 2, left tip

# 2, right tip

# 3, left tip

# 3, right tip

# 4, left tip

# 4, right tip

# 5, left tip

# 5, right tip



x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1, left tip

# 1, right tip


183

Crack Tip Locations for Specimen 8T3, front notch (N = 145,002 cycles)

x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1, left tip

# 1, right tip

Figure C44 Crack tip locations for Specimen 8T3, front notch (N=145,002 cycles).


x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1, left tip

# 1, right tip


184


x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1, left tip

# 1, right tip



x (inches)

thet

a (r

adia

ns)

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

# 1, left tip

# 1, right tip


ms_aero_thesis

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