AN EXPERIMENTAL STUDY OF CRACK INITIATION AND GROWTH FROM COLDWORKED HOLES Dissertation for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY NOPPORN CHANDAWANICH 1977 ' LIBRARY Michigan State University - This is to certify that the thesis entitled An Experimental Study of Crack Initiation and Growth From Coldworked Holes presented by Nopporn Chandawanich has been accepted towards fulfillment of the requirements for Ph . D . degree in Mechanics Major professor 3 Date July 22, 1977 0-7639 ABSTRACT AN EXPERIMENTAL STUDY OF CRACK INITIATION AND GROWTH FROM COLDWORKED HOLES BY Nopporn Chandawanich Crack initiation and crack growth behavior can be imprOved by coldworking a fastener hole. This report describes the experimental studies investigating the change of residual strain during crack initiation, the stress intensity factor for the crack emanating from a circular hole, and the strains ahead of a crack tip. The specimens were subjected to low—cycle fatigue conditions. Analytical procedures were evaluated based on correlation with the test data. These procedures included elastic/plastic analysis which was utilized to determine the stress-strain distribution surrounding the fastener holes and ahead of the crack tip. The experimental data showed that the moiré method is acceptable for measuring the strains in this investigation. The data revealed that the relation between the tOtal notch strain range and cycles to initiation is satis- factory for engineering predictions. The comparison of the Crack initiation life of base line fatigue data to the test Nopporn Chandawanich data was favorable if initiation was defined as the develop- ment of‘a 0.006 inch (0.15 mm) crack. Crack growth rate and interferometric displacement gage (IDG) techniques were used to determine the stress intensity factor (KI). The test data for the coldworked specimens showed that an analytical formula was good for the longer cracks but not for shorter ones. AN EXPERIMENTAL STUDY OF CRACK INITIATION AND GROWTH FROM COLDWORKED HOLES By Nopporn Chandawanich A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Metallurgy, Mechanics, and Materials Science 1977 ACKNOWLEDGMENTS I wish to express my sincerest appreciation and ie to Dr. William N. Sharpe, Jr., my major professor helpful information and suggestions during the course investigation. I also wish to take this opportunity to thank ;. Cloud, Dr. J. S. Frame, Dr. G. E. Mase and Dr. :Grady for serving as the Guidance Committee. I would like to give grateful thanks to the Royal : Force which sponsored my graduate program. Grateful acknowledgment is hereby extended to the :e Office of Scientific Research which provides the )r this study. I want to pay special tribute to my wife, Suvalai, wonderful patience, encouragement, and help in taking our children during my study. My special thanks to l. Martin and Mr. W. J. Cesarz, Jr., for reviewing the .pt of my thesis. ii TABLE OF CONTENTS Page TABLES . . . . . . . . . . . . . . . . . . . . Vi FIGURES . . . . . . . . . . . . . . . . . . . Vii INTRODUCTION . . . . . . . . . . . . . . . . . l 1.1 Purpose and Motivation . . . . . . . . . . 2 1.2 Organization of Dissertation . . . . . . . 4 EXPERIMENTAL PROCEDURES . . . . . . . . . . . . 6 2.1 Material Specification and Specimen Preparation . . . . . . . . . . . . . 6 2.1.1 Coldworking Procedure . . . . . . . 6 2.1. 2 Material Specifications . . . . . . 6 2.1.3 Specimen Preparation . . . . . . . 10 2.2 Fatigue Testing Procedures . . . . . . 19 2.2.1 Servocontrolled Testing Machine . . 21 2. 2. 2 Mounting of Specimen . . . . . . . 24 2.3 Moir é Technique . . . . . . . . . . . . . 24 2.3.1 Grid Production . 26 2.3.2 Printing of Grille on the Specimen. 27 2.3.3 Optical System . . . . . . . . . . 29 2.3.4 Moiré Analysis . . . . . . . . . . 35 2.3.5 Pitch Error . . ; . . . . . . . . 45 2.4 The Interferometric Displacement Gage (IDG) Technique . . . . . . . . . . . 46 2.4.1 Basics of the IDG . . . . . . . 50 2. 4. 2 Displacement Measuring Techniques . 55 {ESIDUAL STRAINS AROUND COLDWORKED HOLES . . . 57 F.l Overview . . . . . . . . . . . I-Z Nadai Theory . . . . . . . . . . . . . . i.3 Hsu-Forman Theory . . . . . . . . Page .4 PotterfiTingEGrandt Theory . . . . . . . . 69 .5 AdlersDupree Solution . . . . . . . . . . 70 .6 Experimental Results . . . . . . . . . 71 3.6.1 1/4 Inch Thick Specimen . . . . . 71 3.6.2 1/8 Inch Thick Specimen with 0.004 Inch Radial Expansion . . 73 3.6.3 1/8 Inch Thick Specimen with 0. 006 Inch Radial Expansion . . 78 .7 Discussion of Results . . . . . . . . . . 78 ZACK INITIATION . . . . . . . . . . . . . . . 82 .1 Local Strain Behavior Before and at the Initiation of Crack . . . . . . . . 83 .2 Crack Initiation in Terms of Total Strain . . . . . . . . . . . . . . . . 89 3 Experimental Results . . . . . . . . . 92 4. 3.1 Local Strain Versus Distance from Edge of Hole . . . . . . . . . 92 4.3.2 Total Strains Versus Cycles to Crack Initiation . . . . . . . . 100 4 Discussion of Results . . . . . . . . . . 100 ACK GROWTH . . . . . . . . . . . . . . . . . 103 1 Overview . . . . . . . . . . . . . . . . 103 2 Experimental Results . . . . . . . . . . 105 3 Discussion of Results . . . . . . . . . . 110 ACK CLOSURE . . . . . . . . . . . . . . . . 114 1 Load—Displacement at the Crack Tips and Discussion . . . . . . . . . . . 114 2 Crack Opening Load and Discussion . . . . 115 RESS INTENSITY FACTOR . . . . . . . . . . 123 L Theories of the Stress Intensity Factor . 123 2 Experimental Results . . . . . . . . . . 131 3 Discussion of Results . . . . . . . . . . 137 iv Page ACK SURFACE DISPLACEMENT 139 O o o o o 0 0 o o c 1 Experimental Results . . . . . . . . . . 140 2 Discussion of Results 147 o o v o o o o o I 0 'RAIN AHEAD OF CRACK TIP . . . . . . . . . . 148 1 Theoretical Method . . . . . . . . . . . 149 2 Experimental Results . . . . . . . . . . 152 3 Discussion of Results 158 INCLUSIONS . . . . . . . . . . . . . . . . . 164 INCES . . . . . . . . . . . . . . . . . . . . 169 IIX A . . . . . . . . . . . . . . . . . . . . 175 tIX B . . . . . . . . . . . . . . . . . . . . 182 LIST OF TABLES meter of holes (in inches) for the pecimens . . . . . . . . . . . . . . . . .e dimensions (in inches) of Specimens Lfter sleeve removal . . . . . . . . . :idual diametral expansion (in inches) . . efficients of the stress function . . . . . . iSt squares approximation to Bowie solution for radially cracked holes . . . . . ast squares fit of finite element data for crack mouth displacement . . . . . . vi Page 12 l8 19 85 126 131 LIST OF FIGURES Page tress-strain curve for the test specimen of 7075—T6 aluminum . . . . . . . . . . . . . . . 8 hotomicrograph of the test specimen of 7075—T6 aluminum. Magnification x 100 . . . . . . . . 9 imensions of the test specimen . . . . . . . . 11 chematic of the King coldworking process and moiré printing . . . . . . . . . . . . . . l3 *hotograph of the device for pulling the mandrel through the hole . . . . . . . . . . . 15 .oad-displacement curves for pulling a mandrel through the hole . . . . . . . . . . . . . l6 . . . . l7 'hotograph of the mandrel and sleeve ’hotographs of the deformed region around the hole for (a) 0.004 inch (0.102 mm), and (b) 0.006 inch (0.152 mm) radial expansion . . 20 verall View of the experimental setup . . . . 22 pecimen setup on the MTS . . . . . . . . . 25 ptical system used for producing submaster . . 27 000 lines per inch grille of submaster. Magnification x 100; (a) from Graticules' master, (b) from Photolastic's master . . 28 tical system for printing process . . . . . . 29 e grids around the specimen hole. Magnifi— cation x 100; (a) 1000 lines per inch, (b) 1000 dots per inch . . . . . . 30 tical system for moiré photography . . . . . . 33 -al time moiré fringe pattern with initial mismatch for residual coldworked specimen by slotted aperture technique. (Magnifica- tion 3: 1) . . . . . . . . . . . . . . 34 vii Page Photograph of Fourier data processor setup . . . 36 . . 37 Optical system for Fourier data processor . Moiré fringe pattern photographs of the residual strain field around the coldworked hole; (a) first order, (b) second order . . . 38 Moiré fringe pattern photographs of the residual strain field around the hole by mismatch technique; (a) tension, (b) compression mismatch . . . . . . . . . . . . . . . . . . 39 Construction of the intersection curve . . . . 41 Photomicrograph of a fatigue crack defining the crack tip coordinates and showing ' 48 the surface indentations . . . . . . . Schematic of the IDG . . . . . . . . . . . . . . 51 . . . 54 Typical interference fringe pattern . . The equipment used for the IDG technique . . . . 56 Geometry and coordinate system used in cold— . . . . . 59 worked hole theories . . . . . . . . Residual stresses after coldworking for 0.004 inch (0.102 mm) radial expansion . . . . . . . 65 Residual strains after coldworking for 0.004 inch (0.102 mm) radial expansion . . . . . . 66 Residual stresses after coldworking for 0.006 inch (0.152 mm) radial expansion . . . . . . . 67 Residual strains after coldworking for 0.006 . 68 inch (0.152 mm) radial expansion . . . . . Residual strains measured by the moiré technique for the 1/4 inch (6.4 mm) specimen LL. Diametral expansion of the originally 0.261 inch (7. 63 mm) hole was only 0.0069 inch (0.175 mm) . . . . . 72 ole—field moiré measurement of the residual strain of specimen LL compared with . . 74 predictions of Adler-Dupree . . . . . . . viii tomparison of residual radial strain of 0.004 inch (0.102 mm) radial expan51on with the theories 0 {omparison of residual tangential strain of 0.004 inch (0.102 mm) radial expan51on with the theories fomparison of residual radial strain of 0.006 inch (0.152 mm) radial expansion with the theories 1 o {omparison of residual tangential strain of 0.006 inch (0.152 mm) radial expan51on with the theories 0 .ocal strains versus distance from Circular notch for three theories v o biré fringe patterns of the initiation of a crack from the hole edge in test specimens; coldworked (a) non—coldworked, o c (b) o omparison of local strains versus distance from hole edge with theories for non- coldworked specimens . . o omparison of local strains versus distance from hole edge with theories for cold- worked specimens . o cal strains versus distance after a crack had initiated non-coldworked specimens cal strains versus distance after a crack had initiated coldworked specimens, cal strains versus distance after a crack had initiated coldworked specimens 0 from hole edge in the from hole edge 0 v in the medium from hole edge in the heavy o mparison of total strains versus initiation life with base line fatigue data for 7075— T6 aluminum mparison of growth curves for cracks emanating from holes in various coldworked specimens ix 0 o Page 76 77 79 8O 88 93 95 96 97 98 99 101 106 Page Photographs of fatigue fractures; (a) non- coldworked specimens, (b) coldworked specimens. . . . . . . . . . . . . . . . . . 108 Comparison of growth rates for cracks emanating from holes in various coldworked specimens . . . . . . . . . . . . . . . . . 109 Photographs of plastic deformation in the wake of cracks; (a) non- -coldworked specimen, (b) coldworked specimen . . . . . . . . . 111 Photomicrographs of plastic deformation in the wake of a crack of coldworked specimen (Magnification x 100); (a) in plastic zone, (b) in elastic zone . . . . . . . . 112 Load-displacement curves of 1 mm cracks for various coldworked specimens . . . . . . . . 116 Load—displacement curves of 3 mm cracks for various coldworked specimens . . . . . . . . 117 Load-displacement curves of 6 mm cracks for various coldworked specimens . . 118 IRelationship between crack tip opening load and crack length for non-coldworked specimens . . . . . . . . 120 Relationship between crack tip opening load and crack length for medium coldworked specimens .~ 9 u u c u o a o n . . . 121 Relationship between crack tip opening load and crack length for heavy coldworked specimens . . . . . . . . . . . 122 chematic of linear superposition method . . 125 pen hole containing a radial crack subjected to pressure p(x) . . . . . . . . 127 omparison of theoretical stress intensity factor solutions for various coldworked holes . . . 130 omparison of experimental stress intensity factor with theory for non—coldworked specimens . O o o v o o o o u o c 9 ~ 0 . . 133 X Page Comparison of experimental stress intensity factor with the theories for medium coldworked specimens . . . . . . . . . . . . 135 Comparison of experimental stress intensity factor with theories for heavy coldworked specimens . . . . . . .1. . . . . . . . . . 136 Comparison of crack mouth displacement with the theory for non—coldworked specimens . . 141 Comparison of crack mouth displacement with the theory for medium coldworked specimens . 142 Comparison of crack mouth displacement with the theory for heavy coldworked specimens . 143 Comparison of crack surface profiles with the theory for non-coldworked specimens . . 144 Comparison of crack surface profiles with the theory for medium coldworked specimens . 145 Comparison of crack surface profiles with the theory for heavy coldworked specimens . 146 Moiré fringe pattern of 3 mm crack from the hole edge in the test specimens; (a) non— coldworked, (b) coldworked . . . . . . . . 153 Moiré fringe pattern of 6 mm crack from the hole edge in the test specimens; (a) non— coldworked, (b) coldworked . . . . . . . . 154 Comparison of measured strains ahead of crack tips with two theories at a crack length of 3 mm from the hole edge for non—coldworked specimens . . . . . . . . . . 156 omparison of measured strains ahead of crack tips with two theories at a crack length of 6 mm from the hole edge for non—coldworked specimens . . . . . . . . 157 omparison of measured strains ahead of crack tips with two theories at a crack length of 3 mm from hole edge for medium coldworked specimens 0 o u o o c u . 159 Page Comparison of measured strains ahead of crack tips with two theories at a crack length of 6 mm from the hole edge for medium coldworked specimens . . . . . . . 160 Comparison of measured strains ahead of crack tips with two theories at a crack length of 3 mm from the hole edge for heavy coldworked specimens . . . . . . . . 161 Comparison of measured strains ahead of crack tips with two theories at a crack length of 6 mm from the hole edge for heavy coldworked specimens . . . . . . . 162 Crack growth data for specimen No. l (non-coldworked) . . . . . . . . . . . Crack growth data for specimen No. 2 (non—coldworked) . . . . . . . . . . . . Crack growth data for specimen No. 3 (medium coldworked) . . . . . . . . . . Crack growth data for specimen No. 4 (medium coldworked) . . . . . . . . . . Crack growth data for specimen No. 5 (heavy coldworked) . . . . . . . . . . . Crack growth data for specimen No. 6 (heavy coldworked) . . . . . . . . . . . Load-displacement curves obtained interfer— ometrically at a crack length of 3 mm for non-coldworked specimen . . . . . . Load-displacement curves obtained interfer- ometrically at a crack length of 3 mm for medium coldworked specimen . . . . Load—displacement curves obtained interfer— ometrically at a crack length of 3 mm for heavy coldworked specimen . . . . . Load—displacement curves obtained interfer- ometrically at a crack length of 6 mm for non—coldworked specimen . . . . . . Load—displacement curves obtained interfer- ometrically at a crack length of 6 mm for medium coldworked specimen . . . . . 1oad—displacement curves obtained interfer- ometrically at a crack length of 6 mm for heavy coldworked specimen . . . . . xiii Page 176 177 178 179 180 181 183 184 185 186 187 188 CHAPTER 1 INTRODUCTION The presence of cracks or flaws in structural com— :s has resulted in the catastrophic failure of a :y of engineering structures. Analyses of the failed ients of pressure vessels, storage tanks, welded ship :ures, aircraft parts, bridges, pipelines, turbine ; and housings, rocket motor casings and various heavy 1e parts, have shown that crack- or flaw—induced frac- Ls often responsible for the failures. In recent years the operational lives of many mili— Lnd commercial aircraft have been limited by flaws initiate from bolt or rivet holes of aircraft struc— and propagate to failure. It is necessary for the er to account for the presence of flaws in the design aircraft structure. One way to protect against flaws is to use materials high value of fracture toughness. However, this is 1 associated with a decrease in yield strength of the 11 which reduces the load-carrying capacity. The 2r also has to account for weight-saving which is one most important criteria for aircraft design. Another economical technique to improve the fatigue the structure is to inhibit or slow the growth of l rs emanating from the holes. This can be done by pre- zssing the metal around the hole either by coldworking 1 an oversized mandrel or by interference—fit fasteners. slower growth of flaws is attributed to compressive Ldual stresses around the edge of the hole generated by pre-stressing operation. The improvement of fasteners processing techniques requires an understanding of the idual stress state around the hole and the change in this ass state with static or fatigue loading. Purpose and Motivation The purpose of this research is to study the change :he residual strain field around coldworked holes during :iation of cracks, and the crack opening displacements strains as the crack propagates. This investigation .udes the studies of crack initiation, crack propagation avior, the stress intensity factor, and the strain ahead .he crack tip. The earliest theoretical study of the coldworked dual stress and strain behavior by Nadai (l) is still useful today. In recent years, among the various ana- cal theories considered, the Hsu—Forman (2) and the ar-Ting—Grandt (3,4) are the best. The theories assume the hole is radially loaded and a state of plane stress :5 everywhere in the sheet, and that the sheet is its in extent. The theories have difficulty predicting esidual stress/strain state near the hole edge. From a mechanics viewpoint, failure of the metal 5 at the highly strained regions where cracks initiate. local repeated plastic strain is responsible for crack ation (5), procedures that account for the crack ation due to this strain would be expected to result in accurate fatigue life estimations than procedures based ninal stresses away from the stress concentration. :ical studies for the local stresses and strains have nerformed elastically by Timoshenko—Goodier (6), and 1d (7); and plastically by Neuber (8), and Stowell (9). :he local stresses and strains have been determined, a :y of methods exists for estimating the number of the initiation cycles. The analytical study of the stress intensity factor or the non-coldworked holes was originated by Bowie(10) ter modified by Grandt (11). In the Grandt solution, ack face pressure p(x) can be defined either as the ‘tress caused by remote load for the non-coldworked r the addition of that local stress and the compres- asidual stress caused by a coldworking process for the :ked holes. If the residual stresses were accurately the Grandt solution for KI as a function of crack should be accurate. Experimental information about the nature of the field around a coldworked hole obtained by Adler- (12) and Sharpe (13) does not agree with the theories. ) work has been done experimentally on the change of the :sidual strains during crack initiation for the coldworked rles in a plane stress condition. The only existing experé :ental study employs fatigue crack growth rates to deter- ne stress intensity factor calibrations for coldworked les. This study has been conducted by Grandt— nnerichs (14). Some experimental error in their study y be introduced by differentiating the data especially r longer crack lengths, the growth rates of which are ry high. Moiré, IDG, and crack length measurements are the :hniques used for determining the strain and the stress tensity factors in this investigation. The moiré method 1 observe the fringe pattern change during uniaxial ten— >n and measure the displacement in a strained body with isfactorily accurate results. The IDG technique, which loys laser interferometry, is quite sensitive (about 0.1 ron resolution) and provides most acceptable results. ck length measurements are made in this investigation to ermine the stress intensity factor for small displace— ts at very short crack (less than 1 mm), for which the technique is not applicable. Organization of Dissertation The description of the material used in this inves- tion, its properties, coldworking procedure, and speci— preparation are given in the first part of Chapter 2. the experimental procedures for fatigue testing, the tech— niques for the moiré method and the interferometric dis- placement gage (IDG) are also described in Chapter 2. A arief review of the coldworking theories and the results of measured coldworking strains are presented in Chapter 3. Theoretical stress concentration factors, both elastic and plastic, the base line fatigue data for 7075—T6 .1uminum, the crack initiation behavior, and the number of ycles required to initiate the crack for various coldworked oles are described in Chapter 4. Chapter 5 shows the arge difference in crack growth data between non-coldworked nd coldworked specimens. The crack closure which caused he crack to close above zero load and decreased the amount E crack opening is presented in Chapter 6. Chapter 7 discusses the stress intensity factors :om the Bowie solution for the non—coldworked hole and the :andt solution for the coldworked hole compared with the aasured results from the IDG and crack growth technique. "ack mouth displacements and crack surface profiles are ven in Chapter 8. The theoretical solutions, both plastic and elastic th plastic zone correction factor, for the strains ahead the crack tips compared with the measured strains by iré method are discussed in Chapter 9. The thesis con— ides with Chapter 10 which discusses the findings of this zestigation. CHAPTER 2 EXPERIMENTAL PROCEDURES Material Specification and Specimen Preparation .1 Coldworking Procedure The coldworking procedure studied in this research is one developed by J. 0. King, Inc. 711 Trabert Avenue, N.W. Atlanta, Georgia 30318 A thin—walled (about 0.0075 inch (0.19 mm) thick) ve is first inserted into the hole. A tapered mandrel is pulled through this sleeve. After the mandrel has been ved, the sleeve may or may not be removed before the aner is inserted, but usually the sleeve is left in the The specific amounts of expansion used in this study ).0080 inch (0.20 mm) and 0.012 inch (0.30 mm) diametral ision of the 0.196 inch (4.98 mm) holes. The sleeve was 'ed for these experiments because none of the modern 'ies for plane stress condition account for the pressure sleeve. Material Specifications The material used for this study was aluminum type P6, 1/8 inch (3.20 mm) thick. The stress-strain curve photograph of the microstructure are shown in Figures 1d 2.2. The Rockwell B hardness measurement for this 6 material is An ASTM standard tension test specimen was cut from the same sheet as the specimens. Different dimensions were used in order to fit the tension test specimen to the test— ing machine. This specimen was pulled in uniaxial tension at a strain rate of 0.0267 in./in. per min. Foil gages were applied to measure the strain. The tensile strengths obtained are: Yield strength (0.2% offset) 73.0 ksi Ultimate strength 76.5 ksi A Ramberg-Osgood representation was used for repre- senting the constitutive behavior of the material in the plastic range using the stress-strain curve in Figure 2.1. The form of the Ramberg—Osgood relation is 8P 0P n —— = 0(3——) (2.1) Y8 YS where ep is the plastic strain 0 e is the tensile yield strain = —%5 where Oys is ys the tensile yield stress and E is the initial slope of the stress—strain curve, 0 is the plastic stress a is a material constant, n is a power hardening coefficient. KSI STRESS - 3O 20 STRAIN - PERCENT Sure 2.1 Stress-strain curve for the test: 5138011119n 0f 7075-T6 aluminum. Figure 2.2 Photomicrograph of the test specimen of 7075-T6 aluminum. Magnification x 100. I 10 The experimentally determined values of the power hardening coefficient n = 15, and the material constant a = 1. 2.1.3 Specimen Preparation The dimensions shown in Figure 2.3 were used for all specimens in this investigation. The specimens were pre— pared by the machine shop to obtain round, nontapered holes to a specific dimension so that one can accurately measure the amount of coldworking deformation. A certain tolerance on the holes is required if one is to compare coldworking between various specimens. Holes were prepared by first drilling them with a 0.1875 inch (4.760 mm) drill and then using a honing machine to bring the diameter up to the nominal 0.195 i0.0020 inch (4.953 10.051 mm). The honing machine pro- duced straight walls in the hole (no evidence of spiraling) and square edges of the hole. The size was determined with a pIUg gage with a "go" cylinder of 0.1948 inch (4.948 mm) and a "no—go" cylinder of 0.1952 inch (4.958 mm). Upon receipt from the machine shop, the holes in the specimens were measured with a microscope equipped with an x—y stage. The greatest uncertainty in this measurement is in locating the edges of the holes accurately. Measurements were made along the diameter at 45 degree intervals, and each mea- surement was repeated at least three times. The variation in repeated measurements was usually less than 0.0001 inch (3 microns). Typical diameters measured for the specimens '9 ' \1 — \ n \ .ssI DIAMETER 8 HOLES ' l' u .I95 1 .002 HOLE ll ’1 l8.00 —* L00 - c/ I I II II .50—— <— —— .50 II p 9.00 \ I — 4 II I 4.25 [I 3.00 -— + III ' 125 I I I I Figure 2.3 Dimensions of the test Specimen. 12 are given in Table 2.1. From this measurement it was found that the hole diameters varied from 0.195 inch (4.966 mm) to 0.1999 inch (5.078 mm). TABLE 2.1.—-Diameter of Holes (in Inches) for the Specimens Specimen 0° 45° 90° 1350 1 0.1963 0.1957 0.1961 0.1960 2 0.1994 0.1987 0.1993 0.1989 3 0.1980 0.1982 0.1982 0.1980 4 0.1995 0.1996 0.1999 0.1993 5 0.1958 0.1958 0.1954 0.1955 6 0.1956 0.1956 0.1956 0.1955 7 0.1958 0.1959 0.1958 0.1956 8 0.1958 0.1958 0.1959 0.1957 After the holes had been measured, the surface of the specimens was coated with a moiré grille by a process which will be explained in detail in Section 2.3. The holes were then coldworked by pulling a mandrel through them, as illustrated by the schematic in Figure 2.4. The tapered mandrel is inserted into the sleeve, and the mandrel and sleeve inserted into the hole. The washer of the sleeve is pressed against an anvil, and the mandrel pulled through :he sleeve. This is the same as the industrial process specified by J. 0. King, Inc. A machine incorporating a Iand—operated hydraulic cylinder was constructed to pull a 13 ' ’ LOCKBOLT HEAD ’ \ \ iiii’ \ \ (ta \ \ / \\ \ \ FASTENER SLEEVE Figure 2.4 Schematic of the King coldworking process and moireI printing. 14 mandrel in the laboratory; a photograph of it is given in Figure 2.5. The tension rod linking the mandrel to the cyl- inder piston had been instrumented with strain gages to per— mit calibration of the force in terms of the cylinder hydraulic pressure. A typical load-displacement curve for pulling a mandrel through the hole is shown in Figure 2.6. The peak forces for the 0.0040 inch (0.102 mm) radial expan— sion was 1250 pounds (5.56 KN) and for the 0.0060 inch (0.152 mm) radial expansion was 1300 pounds (5.785 KN). The sleeves inserted in the hole were part number JK 5535—C06N10L from J. 0. King, Inc. These were supplied with the mild steel washer attached (see Figure 2.7), and a dry film lubricant applied to the inside and outside. The function of the mild steel washer is simply to protect the sleeve and specimen as the mandrel is pulled through; it pops off after coldworking. Several sleeves were sectioned, and the average wall thickness was found to be 0.0075 inch (0.19 mm). The 0.188, 0.190 and 0.192 inch (4.775, 4.826 and 4.877 mm) mandrels used were J. 0. King, Inc. part numbers JK 6540—06—188 to 192; one is shown in Figure 2.7. Accord- ing to the J. 0. King, Inc. literature, the 0.192 inch (4.877 mm) diameter mandrel will give a radial expansion of 0.0070 inch (0.178 mm) to a 0.195 inch (4.953 mm) hole, but this is based on a sleeve thickness of 0.0085 inch (0.216 mm). A maximum radial expansion of 0.0060 inch (0.152 mm) would be achieved with the 0.0075 inch (0.191 mm) thick 15 Figure 2.5 Photograph of the device for pulling the mandrel through the hole. LOAD - LB x 102 16 - .. — - MEDIUM COLDWORKED I 6 -P 7 HEAVY COLDWORKED l4- -6 I2 ‘ /’ r5 I SLEEVE INSERTED '0 q E INTO HOLE - 4 8 d -3 __ 6 - I ‘- \ SLEEVE REMOVED b 2 FROM HOLE 4 _ I 2 -t' l I I DISPLACEMENT - CM Figure 2.6 Load-diSplacement curves for pulling the mandrel through the hole. 17 I Is.“ it | I ‘1. ‘1‘ “I‘m I, I. ~ ' 3 t, ‘1... “II II‘ ‘IJ‘fI' 7‘ I. .I .y' I ' .. .,‘ I I?’ ?)'V Figure 2.7 Photograph of the mandrel and sleeve. 18 sleeves. Because of the variation of the hole diameters the size of the mandrels had to be selected to obtain the right amount of coldworking. The sleeve is tightly wedged into the hole after the mandrel has been pulled through it. To remove the sleeve, the washer on the end of it was surrounded by a larger washer for the anvil to react against and the mandrel was pulled through it a second time. The force required to pull the sleeve out was approximately 650-720 pounds (2.89— 3.20 KN). The dimensions of the holes and the residual diametral expansion after the removal of the sleeve are shown in Tables 2.2 and 2.3. The deformed regions near the hole edge for 0.0040 inch (0.102 mm) and 0.0060 inch (0.152 mm) radial expansions are shown in Figure 2.8. TABLE 2.2.—-Hole dimensions (in inches) of specimens after sleeve removal. Mandrel O O Specimen Diameter 0° 45° 90 135 (inch) 1 0.188 0.2041 0.2034 0.2039 0.2035 2 0.192 0.2071 0.2069 0.2074 0 2069 3 0.190 0.2056 0.2060 0.2057 0.2056 4 0.192 0.2065 0.2066 0.2066 0.2065 5 0.192 0.2077 0.2076 0.2075 0 2075 6 0.192 0.2067 0.2062 0.2070 0.2063 7 0.192 0.2080 0.2083 0.2080 0 2077 8 0.192 0.2063 0.2062 0.2064 0.2060 19 TABLE 2.3.-—Residua1 diametral expansion (in inches). Specimen 0° 45° 90° 135° 1 0.0078 0.0077 0.0078 0.0075 2 0.0077 0.0082 0.0081 0.0080 3 0.0076 0.0078 0.0075 0.0076 4 0.0070 0.0070 0.0067 0.0072 5 0.0119 0.0118 0.0121 0.0120 6 0.0111 0.0106 0.0114 0.0108 7 0.0122 0.0124 0.0122 0.0121 8 0.0105 0.0104 0.0105 0.0103 Sharpe (13) found that the hole is not uniform through the plate thickness after coldworking; it is slightly smaller on the back side where the washer is attached to the sleeve. The nature of the coldworking oper— ation is to exert a force perpendicular to the specimen surface through the sleeve and thus constrain deformation of the hole on the back side. 2.2 Fatigue Testing Procedure This section describes the experimental methods and techniques used in fatigue testing of the aluminum speci— mens. They were tested in a servocontrolled closed-loop hYdraulic testing machine, and all tests were performed at room temperature (70—75°F). 20 coflmcmdxw Hmemu ABE NmH. cc coca oeoc.o Lev See .A56 NoH.oc coca oeoo.o Ace "new ceoc can ec Scum cowwwu coEuomwp osu mo msdmquDOLm w.N wuowwm NOON MAO: 21 A moiré grille was used to measure strains on the specimen surfaces. A load cell was used to monitor the load applied to the specimen; a traveling microscope was used to measure the lengths of the propagating cracks. Figure 2.9 shows an overall view of the experimental setup. 2.2.1 Servocontrolled Testing Machine The servocontrolled closed-loop hydraulic testing machine (series 810) was manufactured by MTS Systems Corpo- ration, Minneapolis, Minnesota. The MTS series 810 consists basically of a hydraulic power supply model 506.02, an actuator (servoram) model 204.63, a load frame model 312.21, the electronic control console model 406.11, and a function generator model 410. A load cell model 661.21A—03 connects in series with the specimen and the actuator ram senses the load applied to the specimen. The machine has a dynamic rating of :20 kips. The actuator is connected to the hydraulic power supply through a servovalve and a hydraulic accumulator. Hydraulic fluid is ported through the servovalve to the actuator‘s cylinder, causing ram movement and applied force. The magnitude and direction of fluid through the servovalve is controlled by a signal from the servocontroller. Hydraulic power is supplied by a pump rated at 6 gallons per minute with a maximum pressure of 3,000 psi. The control console consists of a servocontroller, a control unit, a function generator, a cycle counter, and a 22 Figure 2.9, Overall View of the experimental setup. 23 transducer output panel. Servoram movement is controlled through the servovalve. This servovalve opens or closes according to the control signal from the servocontroller. An input module which is plugged into the servocontroller receives a programmed signal from the function generator, and after scaling it suitably, combines it with the manual command to give a composite command. Manual command is fed through the set point and/or span control of the input module. Composite command and feedback signals are compared by the servocontroller. Thus the servocontroller acting as a comparator—controller causes the control quantity (load or strain) to follow the output of the function generator. The command signal has a full scale input amplitude of 110 VDC. The servocontroller has an error detector circuit that can open a system failsafe interlock to stop the test if an error between command and feedback exceeds a preset limit. The output signal is indicated by both a digital voltmeter (DVM) and an oscilloscope. The servocontrolled hydraulic actuator loads and thus strains the specimen. Haversine waveforms were used throughout this investigation with a testing frequency of 15 cycles per second. All specimens were fatigue loaded in increments of 5,000 cycles up to 15,000 cycles, then changed to increments of 500 cycles for the noncoldworked specimens and 2,000 cycles for the coldworked specimens until cracks were seen with the microscope. After a crack had initiated, the DOT] cho inf cia the The men efft StII mop weII the was eff. 24 the increments of fatigue loading were 200 cycles for the noncoldworked and 5,000 cycles for the coldworked specimens. 2.2.2 Mounting of Specimens The specimen dimensions shown in Figure 2.3 were chosen to suit the load capacity of the machine and factors influencing the stress diffusion from grip portion to the test section. The specimens were mounted in a pair of spe— cially made grips, the top one attached to the load cell and the bottom attached to the piston rod (see Figure 2.10). These grips had holes corresponding to those in the speci— mens (Figure 2.3). The holes were 0.531 inch (13.49 mm) in diameter to accommodate the four bolts in each grip. Figure 2.10 shows the specimen and grips in position. Care was taken to eliminate any twisting of the specimen before cycling. Because of the long distance between grip section and test section, the holes in the grip section have little effect on the stress distribution in the test section. Stresses are assumed to be uniformly distributed at the area more than two inches above and below the test section. 2.3 Moiré Techniques All of the strain measurements in this investigation were obtained using moiré techniques. The application of the moire-fringe technique to theneasurementof displacement was developed in England beginning in 1951. The moiré effect is an optical phenomenon observed when two closely 25 .16" I” Figure 2.10 Specimen setup on the MTS. spaced arrays either transm producing the and Parks (15 provides wholI ticity. Simp. oped to relatI in the SPCCiml Scianmarella I grating on the Significant mg throu‘lh the mc Strain can be 2'3-1 Grid PI The st duce 0n the 5F SUfficient lin ment purPOses . from Photol as t Graticnles Lim Britain' A Se original mas te aparallel ligj The ma hality Slibmas‘ e . xperlmentatio‘ 26 spaced arrays of lines are superimposed and viewed with either transmitted or reflected light. The basic method for producing the moire fringe pattern was introduced by Durelli and Parks (15). This technique is quite useful, since it provides whole field data similar to that of photoelas- ticity. Simple mathematical relationships have been devel— oped to relate the change in the moiré pattern to the strain in the specimen by Vinckier and Dechaene (l6), and also by Sciammarella and Durelli (17). By proper choice of the grating on the specimen surface, it is possible to achieve significant magnification of the specimen's deformation through the moiré pattern; hence measurements of very small strain can be made with good resolution. 2.3.1 Grid Production The starting point for this technique is to repro- duce on the specimen a high—quality master grating having sufficient line density to be acceptable for strain-measure- ment purposes. One 1,000 lpi master grille was purchased from Photolastic Inc., Malvern, Pennsylvania, and one from Graticules Limited, Sovereign Way, Tonbridge, Kent, Great Britain. A set of submasters had been made from the original masters by contact printing on Kodak HRP plates in a parallel light field as shown in Figure 2.11. The master grille from Graticules produced the high quality submasters that have been successfully used in the experimentation (see Figure 2.12). I 00W MERCURY AR CONDENS ING Figure 2.11. 23-2 Printin. Before miCIOn alumina the specimen W‘- (which Was pun I’dsgj, with an but the Photore edge. This eff to obtain a fir. could be SOlVed Printing Proceg The pri required the us the illuminatio I38 .1 01m) in f: We . re 1n tentact 27 COLLIMATING LENS 200w MERCURY ARC LAMP FILM HOLDER CONDENSING LENS MASTER I In \ FILM PLATE ‘\‘t I EMULSION SIDE L I 1 I7 OPTICAL BENCH Figure 2.ll._—Optical system used for producing submaster. 2.3.2 Printing of Grille on the Specimen Before printing, the specimen was polished with 0.3 micron alumina and degreased with acetone. After cleaning, the specimen was coated by spraying AZ 1350B photoresist (which was purchased from Shipley Company, Inc., Newton, Mass), with an airbrush. Spraying gave a fine uniform layer, but the photoresist formed a thicker coating around the hole edge. This effect caused it to be a little more difficult to obtain a fine grille around the edge of the hole, but could be solved by using a longer exposure time during printing process. The printing process for this aluminum specimen required the use of a 200 watt mercury arc lamp to produce the illumination. This lamp was placed about 15 inches (38.1 cm.) in front of the submaster and specimen, which Were in contact with each other (see Figure 2.13). 22225;:E: a: .. Mm' 28 .noumme m_owummH0uonm Eoum Anv .uoumme m.moanowumuo Scum ADV NH.~ enemas .00H x c0wumofiwwcwmz Ace .uoummannm mo oHHHuw nose Hod momma oooH Anv 200W MERCIJ t. Figure 2.13. The ex minutes . A dc master 900 for are ShOWD in F 2'3'3 0Ptical The ca Kreuznach With length. This angle iron bar cation of 3. response deman eXploitedI led 29 200W MERCURY ARC LAMP FRAME HOLDER mm. -————.| EMULSION SIDE SUBMASTER R SPECIMEN \ . ‘ BACKUP PLATE 1 OPTICAL BENCH Figure 2.13.—-Optical system for printing process. The exposure of the photoresist was about 3 to 3% minutes. A dot grating was produced by turning the sub— master 90° for the second exposure. The acceptable results are shown in Figure 2.14. 2.3.3 Optical System The camera used for this study was a Schneider Optik Kreuznach with an f4.5 lens of 300 mm (11.8 inches) focal length. This camera was mounted on a tripod and braced with angle iron bar. The camera Was adjusted to obtain a magnifi- cation of 3. This use of magnification reduced the frequency response demands on the PhOtO SYStem and: when prOperly exploited led to an increase of measurement sen51t1v1ty. ' I 30m 30: HOLE lflDGE 30 , iuanOHIOOII col-noo-oo... filo-conga...- oblcoootoooco oil-couch...- .Jiiltloooooo Igloo-ooo-o-oc IUIIOIIIpOOIIoia Olly-coconut...- .oqqoopoopooou IIOIIII:IOIC§OI gluco- Cloth-Iooo-IIO-c .Ifidoowootoooon Icon-cocoa..- Dillon-loco. anabolqpcoopocoo- :CIOIIIOOOIIUI loot-0.00.10. . announce-.01.. OIIQJIUDIcncocI Ion-alfitooooo loo-0000.000. loo-co out-on DID-Q. .00... (a) Figure 2.14 The grids around the specimen hole. Magnification x 100; (a) 1000 (b) 1000 dots per inch. lines per inch, Illtn from a spotl; about 30° pr< improved 5119 the camera be area causing Kodak Plates Type 6 Both worked w With the 649- very good con tendency, and this experime time and temp that some dev Vallies C0U1d I ing Stage, TWO 01 fringe Patter] ni‘llle. This 1 specimen in a Photograph Wit tion on Kodak was then remoo Photographed a In the 0n 31 Illuminating the specimen with diverging white light from a spotlight with the angle of incidence of the light about 30° produced the best results. Grating contrast was improved slightly by using lighting from only one side of the camera because the coated lines shadowed the uncoated area causing it to become darker. Kodak High Resolution Plates and Kodak Spectroscopic Plates Type 649—F were used for recording grating images. Both worked well. The results reported here were obtained with the 649-F material. Kodak D—l9 developer, which gives very good contrast, high effective speed with low fogging tendency, and high useful developing capacity was chosen for this experiment. The control of exposure and development time and temperature was somewhat critical. It was found that some deviation from optimum exposure and processing values could be compensated for in the optical data process— ing stage. Two optical systems were used to obtain the moiré fringe patterns. One method is the double—exposure tech- nique. This technique was performed by first mounting the specimen in a holder on an optical bench, and taking a photograph with the high resolution camera at 3:1 magnifica— tion on Kodak Spectroscopic Plates Type 649—F. The specimen was then removed, mandrelized, replaced in its holder, and Photographed again using the same plate. In the second method the image of the model grille on the specimen was projected onto Kodak 649-F plates with a high resolut: exposure, the reference gr: pattern. This 3, meaning tt equal to 3,00 readily intro the camera in large number analysis. Th PhOtography i The p] improved with Cloud (13). f the iris diapI tions Were ca] ties as requiz Photography tc Although the 5 Photography, t Obtained. Hig be produCed by ProceSS of an SYStem‘ It 611 t urbahee trans] —- i 32 high resolution camera at magnification 3:1. After the exposure, the specimen photo plates were superimposed on the reference grille of 1,000 lpi to obtain the moiré fringe pattern. This technique improved the multiplication factor by 3, meaning that the model grille obtained the sensitivity equal to 3,000 lpi. With this method, pitch mismatch can be readily introduced. A slight change of the magnification of the camera introduced pitch mismatch which resulted in a large number of fringes producing more data points for moiré analysis. The schematic of the optical system for moiré photography is shown in Figure 2.15. The photograph from this optical setup can be improved with the slotted apertures technique described by Cloud (18). Slotted apertures were designed to fit behind the iris diaphragm inside the lens. Slot sizes and loca— tions were calculated to tune the lens to spatial frequen— cies as required. This technique was performed during photography to improve the fringe pattern (see Figure 2.16). Although the slotted—apertures technique improves the moiré photography, the highest quality pictures still cannot be Obtained. Higher contrast of the moire fringe pattern can be produced by optical fourier data processing which is the process of an optical spatial filtering in a coherent light System. It alters the input signal (i.e., the light dis- turbance transmitted through a moiré pattern) by placing .msmmuwouofia wufiofi How Ecum%m amoebao mH.N ounwflm mzmd wZHmZMDZOO » summon ESE mzma czEaEflocI/ \ I I mead Ede m I I n u are 20 cm amass madam .5:er EH3 zmfiumnm Q0 . 42 m - (2.6) where 6m isthefringespacingfortheinitialpitchnusmatch. p is the master-grating pitch. am is the initial fictitious strain. For the deformed grating the corresponding fringe spacing is 6d — TEET (2.7) where 6d is the fringe spacing of deformed grating. Ed is the strain produced by the applied deformation. The fringe spacing resulting (6f) from the superposition of e . m and 8d 18 a = p (2.8) where the plus sign applies to the case €d€m>0, and the minus sign to the case a sm<0. d To increase the precision it is desired that 5 < (S (2.9) Phis is achieved by any (2.10) lem' > O vh en edem>o, But when eds] It is then c.‘ produce more that a misma1 at least twic The ; linear strair the nonlinear strain(15) sh The E The at grating I for <5 La . granglan f0: 43 But when edem<0, inequality (Eq. 2.9) is satisfied if >2 led! (2.11) leml It is then clear that a mismatch of the same sign will always produce more fringes and so will always be advantageous; and that a mismatch of opposite sign, to be advantageous, must be at least twice the magnitude of the strain to be analyzed. The preceding methods will be satisfactory for small linear strain in X and y axes only.Fornmreaccurateresults, the nonlinear Eulerian and Lagrangian normal strain and shear strain(15) should be used. The Eulerian formula can be expressed as follows: €X=1——/(12§—‘1+(i3—u_2%_)+( :92 (2.12) 8V 8 = l- l— 2—— + + 2. Y / 3y (3— y)2 (g— Y)2 ( 13) Bu 8V Bu Bu 8v 8v _ . 8y 5§ 3X 3y 9x 3y Exyf arcs1n (1_EX)(1_€ ) (2.14) . Y The above equations are used for undeformed specimen rating; for deformed specimen grating one has to apply the agrangian formula displayed below: e =/1+28—u+ (3— :)—2+( :2) —1 (2.15) x 3X Botl cross gratir patterns at that can be WhiCh is eXE md Shif in small St 44 3V 3V 2 Eu 2 e = l+2—— + —— + —— — l 2.16 y / 3y (3y) (3y) I ) au 3v Bu Bu av 6v __ _ (____+____. By 8x 3x 3y 3x 3y (2 l7) s = arcsin XY (1+8 )(l+€ ) X Y Both Eulerian and Lagrangian methods require moiré cross grating to obtain u and V families of moiré fringe patterns at the same time. Anothermethodforstrainanalysis that can be applied to the experiments is a shifting method which is expressed in the following form: A N N 33' = 2&- = if“ ‘2-18) S X N d A I an Xi. = 1% (2.19) S a = A}; = L (2.20) where N is the fringe number. NS is the shift to pitch ratio. N:, is the moiré line order of shifting. Au is the displacement on fringe line. Ax' is the amount of shift. Ax is the distance between two points of interest. Shifting method will not produce accurate results for small strains, but it is still good fOr approximating the stra obta unia trar smal and valI qrai are errI lpi 0.0I lem and fer Wil obt ter 45 strains. The larger the shift, the more sensitivity will be obtained. In this experiment, the specimens were loaded in uniaxial tension. Dot gratings were used to measure the transverse strains, but these strains were found to be very small and could be ignored. Therefore only equations 2.2 to 2.11 have been applied to the experimental analysis. 2.3.5 Pitch Error Basic to all moiré analysis, both for displacement and strain analysis, is the assumption of a constant known value of the pitches of both master and undeformed specimen gratings. From Durelli and Parks(15), if the two pitches are well matched but have a common error of 1 percent, the erroroccurring'in displacement, assuming a density of 1,000 Lpi which have been used in this investigation, will be only ).00001 inch (0.00025 mm) and on a 0.1 inch (2.54 mm) base -ength the error in strain will be 0.0001. In the case of mismatch, the pitch of master grating nd the pitch of the undeformed specimen grating are dif— erent, but each has to be known, and errors in these figures ill be reflected directly in any analysis. It is simple to btain an overall estimate of any mismatch by overlaying mas- er and undeformed specimen gratings and recording the total meer of fringes produced in the field. A serious systematic error will be made by assuming matched pair of grating and neglecting mismatch, since a misn stra for init titi stra This repe orde ihVI 46 mismatch of 0.1 percent will immediately show up as a large strain of 0.001. The chief precaution in moiré analysis for overall variation in pitch (mismatch) is to record any initial fringes before loading and subtract out this "fic- titious" displacement and strain in the analysis. It is estimated that the pitch lines may vary some il/lOO of the pitch spacing so that the error in pitch of a 1,000 lpi grating will be the order of i0.00001 inch (0.00025 mm). On the base length of 0.08 inch (2.03 mm) that had been used in the experiments would produce a fictitious strain as great as e = W = :0.000125. This, however, would require that the error in pitch be repeated over a number of grating lines in the region in order to effect the position of a moiré fringe. Essentially, this is the problem with periodic pitch error, which repeats itself systematically over a number of grating lines and produces herringbone effects when overlaid by a strain pattern. 2.4 The Interferometric Displacement Gage (IDG) Technique Interferometry methods have been employed by several .nvestigators (20-22) to obtain the complete crack surface iSplacement field in the transparent specimens. Examination f the interference fringe patterns obtained for various 47 .oadings revealed that the crack perimeter was closed at zero load. Although the fringe patterns provided a quanti— Lative mapping of the entire crack surface profile, the Interior displacement measurements by this method are :estricted to transparent materials. In this experiment the surface displacement measurements were measured with an IDG :echnique developed by Sharpe (22). The method is based on Laser interferometry measurementsofthe ineplanedisplacement Ln the Vicinity of the crack tip. In the crack tip displacement method, one measures :he displacement at some point near the crack tip as a function of remote load by the laser interferometry method Ind then computes K from the elastic displacement relation >y Paris and Sih (23). For Mode I loading the appropriate equations for conditions of plane stress are: K . u = I 'r 6 l‘U . 22 x ~E 2? cos f 113 + Sln 2 (2.21) K , u = I r . 0 2 _ 26 Y —G/ 27 Si“ 7 [—l+u C05 2 ' (2'22) mere uX and uy are the x and y components of displacement at a point near the crack tip located by the polar coordinator (r,0) as shown in Figure 2.22. G is the shear modulus. u is the Poisson's ratio. 48 Figure 2.22 Photomicrograph of a fatigue crack defining the crack tip coordinates and showing the surface indentations. 49 The crack tip displacement method requires a tech- nique capable of measuring small displacements very near the crack tip. As a rule of thumb, the above displacement rela- tions are assumed to be sufficiently accurate within a dis— tance a/20 from the crack tip, where 'a' is the crack length. The general stress intensity factor equation is: KI = o/Fe’lma) (2.23) For the experimental method, it is useful to rear— range the elasticity equation of the stress intensity factor and the equation of the component of displacement into a form more convenient for the experimental data by combining Equations 2.22 and 2.23. The result is as follows: NI u F(a) =§(—S)—/—331 (2.24) where F(e) is the angle dependent term in Equation 2.22. The plane stress form for uy is chosen since all of the displacements are made on the specimen surface. In addition to presenting F(a) in terms of experi— mental quantities, Equation 2.24 also enables one to account for local yielding encountered in actual experiments. In particular, plastically deformed material left in the realm of a propagating fatigue crack can cause crack closure (24), the phenomena characterized by an initial nonlinear load- Ifisplacement response. Oncetheopeningload(24)isreached, however, behavior is again elastic. Thus, when the slope 50 u I? in Equation 2.24 is taken from the linear portion of the load displacement record following the opening load, an elastic stress intensity factor is obtained. 2.4.1 Basics of the IDG The principles of the IDG have been described in detail in Reference (22); only a basic review of the tech— nique is given here. Shallow reflective indentations are pressed into the polished surface of the specimen on either side of a crack as shown in Figure 2.22. When coherent light impinges upon the indentations, it is diffracted back at an angle (do) with respect to the incident beam shown schematically in Figure 2.23. Since the indentations are placed close together, the respective diffracted beams over- lap resulting in interference fringe patterns on either side of the incident laser beam. In observing the fringe pattern from a fixed posi— tion at the angle do, fringe movement occurs as the distance ‘d' between the indentations changes. Application of a ten— sile load, causing the distance between the indentations to increase, results in positive fringe motion towards the incident beam. Conversely, the removal of the tensile load results in negative fringe motionawayfromtheincidentbeam. The relationship between the indentation spacing and the fringe order shown schematically in Figure 2.23 is: d sin do = ml (2.25) 51 .UQH osu mo oeumawsom mmd 8:me ZMMHH rp o = _o s r o s r r 1— (—rP-)2, Ge = J’— (—f—)2 (3.10) /§ /3 e =__Oy8(l+U)(:p)2, e = Eiéili2l(:2)2 (3_11) r E/3 r E/3 r 2 o (l U) r u = _X§__i__ _§_ (3.12) Iii/3 The problem in plasticity theory is then to find the stresses, etc., subject to the known boundary condition at r = a and r = rp. An important part of the problem is the 62 determination of the relation betweenloadingandriy Various theories have been developed to predict these quantities, and they fall into three classes: analytical, numerical, and finite-element. The analytical theories produce closed—form equations based on eitherau1elastic-perfectly—plasticstress— strain curve or a two—parameter plastic stress—strain curve. The numerical ones develop the solution in terms of incre- mental rings between a and b corresponding to increments on the plastic stress—strain curve. The one finite element solution uses an elastic-plastic computer code. Once the hole has been expanded to the prescribed conditions, the loading is removed at r = a. This generates, because of the elastic relaxation of material outside (and inside) rp, residual stresses. The analytical and numerical theories (except for (4)) represents this unloading by super- posing an elastic stress field: or = omI§)2, 06 = —omI§)2 (3.13) where Om is the magnitude of the radial stress generated at r = a by the loading process. This guarantees that or = 0 at r = a after unloading. Four theories are discussed in the following sections and compared in the results section. In each case the resid- ual stresses and strains for 7075—T6 aluminum subjected to: u = 0.0040 inch (0.102 mm) and 0.0060 inch (0.152 mm) at a a = 0.0980 inch (2.490 mm) are calculated. 63 3.2 Nadai Theory Nadai (l) in 1943 published a theory on plastic expansion of tubes fitted into boilers. The plate of the boiler has a tube fitted into it and in the manufacturing process these tubes were expanded by a rolling process to insure a leak-free fit. He considered both the plastic deformation in the steel plate and the plastic deformation of the copper alloy tubes. He first solved the plate prob— lem, which is the one of interest here. Hisassumptionswere: 1) Uniform pressure at the edge of the hole in the plate. 2) A linear approximation to the Mises—Hencky yield criterion. 3) Perfectly plastic material response. In the plastic zone he develops the following equations: 0 or = J’EI—l + 2 ln -§—) (3.14) 5 p G S r 66 = —-‘/——(1 + 2 1n r—) (3.15) )5 p E u a u = (3 3 aE (3.16) 2 3 _r_ 3 (7 + ln r ) P where r P r a is the radius of the plastic zone; is the radial distance of the calculated point from the center; is the outer radius of the tube. 64 The residual stresses and strains after relaxation are plotted and compared with Hsu-Forman theory in Figures 3.2, 3.3, 3.4 and 3.5. For the test case of 0.0040 inch (0.102 mm) and 0.0060 inch (0.152 mm) radial expansion, the theory computes rp = 2.069a and rp = 2.281a respectively. He also developed the theory for the stresses in a tube with a general stress-strain curve. This complete theory could be applied to coldworking procedures in which the sleeve remains in the hole. Nadai formulated the prob- lem in terms of the Mises—Hencky criterion, but he linearized this criterion to obtain a closed-form solution. 3.3 Hsu—Forman Theory The Hsu—Forman theory (2) of 1975 is basically the Nadai theory extended to account for workhardening. Their solution is based on J2 deformation theory together with a modified Ramberg—Osgood law. The sheet is orthotropic in the thickness dissection but isotropic in its plane. Only small displacements are considered. The change in hole size and the variation of the sheet thickness are neglected. The assumptions are: 1) Uniform pressure at the hole. 2) Mises—Hencky yield criterion. 3) Ramberg—Osgood representation of stress—strain curve . The material behavior is represented by 65 h200 PLASTm ] ELASTm 20 ~ 0 I l 20- \\\\ H - 200 ,’ RADIAL STRESS Q I I I 40- I a < ’ ---- umAI :3 S ’ I S , -———- HSU-FORMAN _ I eoi 4°; 1 I i/_TANGENTIAL STRESS I 80‘ - 00 100“ f 800 I20 Figure 3.2 Residual stresses after coldworking for 0.0040 inch (0.102 mm) radial expansion. 66 IO- 8‘ -'--- NADAI HSU-FORMAN E 06‘ r! m 9.. I Z H 24' [—I m TANGENTIAL STRAIN 24 0 l ' l I I f l 2 3 4 5 6 DISTANCE FROM HOLE EDGE - MM Figure 3.3 Residual strains after coldworking for 0.0040 inch (0.102 mm) radial expansion. 67 zoo PLASTIC ELASTIC 20- 0 I I 20~ \\ \\ - zoo “ / H I Q 40- ’ RADIAL STRESS / I m E / m 2: / E I m .- - ~——— 60- 4°° I mmAI I I HSU-FORMAN I I 8 _ I o I TANGENTIAL STRESS I “ no I I I I —I 00 I I I I 4 300 'ZOY Figure 3.4 Residual stresses after coldworking for 0.0060 inch (0.152 mm) radial expan51on. 68 ———— NADAI E HSU-FORMAN Ed 0 Cd [:1 CL. I Z H E [-4 U) r I l 4 5 6 DISTANCE FROM HOLE EDGE - MM Figure 3.5 Residual strains after coldworking for 0.0060 inch (0.152 mm) radial expansion. e = gl—g—In_ for Io] > o (3.17) E Oys ’ ys The test material of 7075—T6 aluminum is adequately represented by n = 15, Oys = 73 Ksi (504 MPa). The solution is developed in terms of a parameter a which varies between 90° and aa, where da corresponds to a radial expansion ua which is known by measurement. vIn our case, “a is equal to 133.30 for 0.0040 inch (0.102 mm) and 137.0(3 for 0.0060 inch (0.152 mm) radial expansion. For each case of radial expansion, the elastic- plastic boundaries are found at rp = 1.98a and rp = 2.15 a. The Stresses and strains are also plotted and compared with Nadai theory in Figures 3.2, 3.3, 3.4 and 3.5. 3.4 Potter-Ting—Grandt Theory Potter and Grandt (4) applied the general solution of Potter and Ting (3) to the coldworked holes solution with an eye towards making it useful to designers. Their assumptions were: 1) Uniform radial displacement at the hole. 2) Mises—Hencky yield criterion. 3) Elastic-plastic material response. Their work is similar to the earlier work of-Nadai (l) , eXcept that they did not use a linearized yield criterion. 70 The Potter—Grandt theory has limits to allow the radial displacement. For the 7075—T6 aluminum and specimen dimen- sion used in this investigation, the maximum radial displace- ment is 0.0037 inch (.094 mm), so the 0.0040 inch (0.102 mm) and 0 .0060 inch (0 .152 mm) radial expansion of the experiments are not permitted. 3.5 Adler—Dupree Solution Adler—Dupree(12) performed an elastic—plastic finite element analysis of the two—dimensional stress field around a coldworked hole. They assumed plane stress and accounted for plastic behavior of the material by a Ramberg—Osgood formula- tion relates the equivalent plastic strain to the nth power of the equivalent stress. The radially symmetric finite element mesh was made up of quadrilateral elements divided into two layers with two triangular elements in each layer. The specific problem they considered was a 0.006 inch (0.15 mm) radial expansion of a 0.260 inch (6.60 mm) hole in 7075-T6 aluminum and then removal of that expansion (the same as our test case). This models the coldworking process of J. 0. King, Inc., in which an oversized tapered mandrel is pulled completely through a hole. The closest they could get to the hole edge for their computation was 0.025 inch (0.64 mm). Because of the limited radial expansion and the hole size of this solution, it was only used to compare with the 71 experimental data for the 1/4 inch (6.4 mm) thick specimen in the next section. 3.6 Experimental Results In this section the results of residual strains generated by coldworking are reported. Almost all of the specimens are 1/8 inch (3.2 mm) in thickness and have dimen- sions as shown in Figure 2.3. One specimen is 1/4 inch (6.4 mm) thick and has rectangular dimensions of 2.5 inches I (6.4 cm) by 3 inches (7.6 cm), the hole is 0.260 inch (6.604 mm). The hole was radially expanded by 0.0035 inch (0.089 mm). The 1/4 inch (6.4 mm) thick specimen was tested to obtain the residual strains for comparison with the indentation technique by Sharpe (13) and the finite element method by Adler and Dupree (12). In the specimen with the thickness of 1/8 inch (3.2 mm), the holes are 0.195 inch (4.95 mm). The holes were radially expanded by 0.0040 inch (0.102 mm) and 0.0060 inch (0.152 mm) then unloaded. 3.6.1 1/4 Inch Thick Specimen Residual radial and tangential strains measured by moire are plotted to compare with the indentation procedure in Figure 3.6. Note that this specimen had been tested at the beginning of this research, therefore, one cannot make measurements right up to the edge of the hole because of lack of resolution of the fringes; the radial strain was 72' —— MOIRE' TECHNIQUE a) 1_ A INDEN‘I‘ATION TECHNIQUE FOR E6 0 INDENTATTON TECHNIQUE FOR er , 6r - PERCENT COMPRESSION «5 m L l L a - PERCENT TENSION n) 6:0 O m-I DISTANCE FROM HOLE EDGE - MM Figure 3.6 Residual strains measured by the moire'tech— nique for the 1/4 inch (6.4 mm) specimen'LL Diametral expansion of the originally 0.261 inch (7.63 mm) hole was only 0.0069 inch (0.175 mm). 0 73 measured only to within 0.5 mm in Figure 3.6. The problem was that the printed grille tended to crack in the regions of high plastic deformation at the distance of about 1 mm from the edge of the hole, causing lack of definition there. This problem was solved later by curing the coating at an elevated temperature before coldworking. To assure that the moiré technique was giving the correct results, strain was also measured on the specimen by the indentation procedure. The indentations were applied, the grid printed, moire measurements made, and then the grid dissolved off the specimen and final strain measurements made. The agreement between the strains is excellent, as shown in Figure 3.6. One of the great advantages of the moire technique is that it produces whole field strains. A whole field strain measurement (for one quadrant) is shown in Figure: 3.7. This gives the strain in terms of 8x and Cy instead of the er and 80 that are more conveniently used for theo— retical analysis. The predicted strain field of Adler and Dupree (12) is also plotted in Figure 3.7. They used the finite element method and predicted a residual diametral expansion of 0.008 inch (6.2 mm), so the experiment and theory are not exactly the same (0.0069 inch versus 0.008 inch expansion), but the strains agree reasonably well. 3.6.2 1/8 Inch Thick Specimen with 0.004 Inch Radial Expansion ' The average residual strain measurements (two on each specimen) from four specimens were obtained by the 74 oauofltonm hues tonmmaoo AA . swam E.mnfioe thHmImHonz n m on .oouanmnuoap< mo ma awEHomdm mo samuum Hmspwmou 0:“ mo ucoEoHSmmo 22 I m5? x 0201?. moz1 mTH Figure 3.10 Comparison of residual radial strain of 0.0060 inch (0.152 mm) radial expansion with the theories. 80 ---- NADAI HSU-FORMAN 1: EXPERIMENTAL DATA § 1 STANDARD DEVIAT ION \ STRAIN - PERCENT TENSION N 01 A 0| 0 DISTANCE FROM HOLE EDGE - MM Figure 3.11 Comparison of residual tangential strain of 0.0060 inch (0.152 mm) radial expansion with the theories. 81 region of interest near the hole edge. The nature of cold— work by pulling a mandrel through the hole was described by Sharpe (13). His results from the indentation technique showed that the strains were larger than the theories in the area close to the hole edge. Also by his measurement, the strains were found to be larger on the front face than on the back face. The experimental strains in this investiga- tion agree well with Sharpe's results. This particular coldworking process produces very nonhomogeneous deforma- tion. Such deformation will vary through the specimen thickness; therefore, the real experimental conditions are not the same as the conditions that the theories assume. CHAPTER 4 CRACK INITIATION In this chapter, the strain prior to fatigue crack nucleation, the strain when the crack initiated, and also the number of cycles required to initiate a crack were investigated. In general, there have been a number of theories proposed for crack initiation; some theories (i.e., 5, 27) are based on the results of a broadening of a slip band which ultimately produces a crack spreading from crystal to crystal; some theories (28—30) are based on the exhaustion of ductility by stresses being raised to some critical level during cyclic loading. Other viewpoints of crack initiation are common among many investigators; crack nucleation at inclusions as well as grain boundaries and at precipitation zones (31) have been two such Viewpoints. More recently the idea of fatigue failure involving nucleation of grain bound— ary voids by interaction between moving dislocation and stationary obstacles, with subsequent void growth by vacancy diffusion was proposed (5). Cracks originating from slip bands formed by slip in crystal planes of preferred orienta— tion is the most commonly accepted of all the nucleation mechanisms (32). 82 83 The initiation of a crack for a circular notched specimen is based on the theoretical stress concentration factor that has been discussed by several authors (33-37). Characteristically, fatigue failure in service initiates at geometric diScontinuities that produce local stresses higher than the nominal stress applied. In weight-critical applications, these local stresses may be in the plastic range instead of the elastic range. 4.1 Local Strain Behavior Before and at the Initiation of as: The theoretical maximum value of the stress concen— tration factor at the edge of a circular hole in a straight tension member of infinite width is three times the applied uniform stress. This value is approximately correct when— ever the plate width is four or more times the hole diam— eter. Timoshenko—Goodier (6) concurs with this opinion if the width is five or more times the diameter. Since the ratio of plate width to hole diameter is ten for the test specimens, a theoretical stress concentration factor of 3.00 can be accepted. The elastic solution of local stresses for infinite plates, in the limiting case of plane stress is (6): 2 4 2 _ s _ a E 3a _ 4a or _ §[l F] + 2[l + F1;- ——r‘-z]COSZG (4.1) 2 l; _ s a _ s 3a 06 _ 31:1 + E2] 2[l + —?¢]C0526 (4.2) u 2 c’re = I” ‘ 2%” + 2‘32] (4'3) where s is the applied stress, 0 is the local stress, a is the radius of the hole, and r is the radius of the point determined. For this investigation, one must consider the tan- gential stress 06 and the radial stress or to determine the corresponding strain 80 by using elastic stress-strain rela- tionship alohg the line 9 = 90°. For the finite plate, the Howland solution (7)should be used to obtain more accurate results. 61 oo 0 n=1 p2n+2 2n-2 (n+1)(2n—l)e2n -—————~———————— + n(2n—1)12np + 2n 0 (n-1) (2n+1)m2n02n}c032n0] (4.4) 1 do 0°! n(2n+1)d2n 06 = s[§(1+c0526) + 2mO + ~§ + 2E {__§E:§—_-_ + o n—1 p 2n-2 (n-1)(2n-1)e2n ——————————————— + n(2n-1)12np + 2n D + (n+1)(2n+l)m2np2n}c052n0] (4.5) 00 e _ l . ~ _ 2n—2 _ _32 are — s[351n20 + 22 {n(2n 1)(12np pzn) + n=1 2n _ d2n 2n+2 p n(2n+1)(m2np )}sin2n0] (4.6) 85 0.1. The coefficients For this experiment, A = % d, e, 1, and m will have the values as in Table 4.1. TABLE 4.l.--Coefficients of the stress function. do = 5.01x10"3 —-——————- 12 = 4.13::10“3 mo = 5.69x10'4 -5 -3 -3 -3 d2 = 2.54x10 e2 = -5.08x10 14 = 1.06x10 m2 = -l.l3x10 d4 = 3.15x10'11 e4 = —4.21x10"'9 16 = 2.83::10"4 m4 = -5.95x10'4 d6 = 1.4ox10'15 e6 = —1.68x10'13 18 = 7.24x10'5 m6 = -2.28x10-4 _ -2o _ ~18 _ -5 _ _ -5 d8 — 5.00x10 e8 — 5.72x10 110— 1.82x10 m8 — 7.65x10 1 — 4 65xlO-6 m — -2 39 10'5 12‘ ' 10’ ' x 1 — 1 15x10-6 m = —7 3o 10"6 14‘ ° 12 ° X where a is the hole radius b is the specimen width, and p is the proportion of r over b. For 'a' equal to 'b' and angle 9 = 90°, the value of the function of local stress to applied stress equals 3.03. This value is close to the value for the equation of an infinite plate as discussed before. In fact, the ultimate success or failure of a struc- ture depends on the very localized behavior at its notches and the stresses which become plastic at the area close to the hole before a propagating crack starts. The theory of plasticity to predict the stress and strain concentration factor of a circular hole in a large, wide sheet specimen in tension was originated by Neuber (8) 86 and later modified by Stowell (9). An experimental study was performed by Griffith (38). Their results showed that, as the amount of plastic flow increased, the stress concen- tration factor is appreciably reduced, and the strain concentration factor is appreciably increased. In order to introduce plasticity into the predicted strain distribution Neuber's rule (8) is applied; 1 2 Kt = (KoKe) (4.7) where Kt is the theoretical concentration factor, K0 is the stress concentration factor, and Ks is the strain concentration factor. Now, consider a notched specimen subjected to nominal cyclic stress and strain ranges of As and Ae respectively. Let the local stress and strain ranges at the notch root be Ace and Ase. Substituting these quantities in Equation 4.7 we get, A08 A66 % Kt = (As I Ae ) (4'8) KéAsAe or Age = A0 (4.9) 8 which can be written as: l 1 K (AsAeE)2 = (A0 As E)2 (4 10) t e 6 ' where E is the modulus of elasticity. Equation 4.10 is an identity in nominal and actual stress and strain values. Duplicating a quantity equivalent to the R.H.S. in a smooth specimen is the same as reproducing 87 the deformation behavior at the notch root. The equivalence of the L.§.S. and the RPH.S. implies that when the value 1 (AoeAeeE)7 in a smooth specimen is equivalent to Kt(AsAeE)§ of a notched member, then the root of this notch is subjected to the same stress and strain conditions as the smooth speci- men. Thus, the fatigue life of the notch root and the smooth specimen must be the same, when the size effecthIthenotched member is taken into consideration. In fatigue application, the fatigue notch factor Kf is used instead of Kt’ Then Equation 4.10 becomes: 1 E _ 7 Kf(AsAeE) — (AoeAeeE) (4.11) l—l As long as As is less than the proportional limit, Ae can be replaced by As/E. This gives (Kt-As)2 AoeAEe = ——E——~— (4.12) The local stress and strain at the notch in Equation 4.12 can be determined by the Stadnick-Morrow techniques(35)£ Note that for the long fatigue life, Kf is nearly equal to Kt. In this investigation the cycle to initiate the crack are quite large, therefore one can use Kt instead of Kf in Equation 4.12. Figure 4.1 compares the local strain with the distance from the circular notch for three theories in the 7075-T6 aluminum specimen at the nominal stress of 30.1 Ksi 88 |.2‘ TIMOSHENKO-GOODIER ‘ —n—-Hmmmm I —--- NEUBER \ LOT‘ STRAIN - PERCENT I I I d f 0 LC 2.0 3.0 4.0 5.0 6.0 DISTANCE FROM HOLE EDGE - MM Figure 4.1 Local strains versus distance from circular notch for three theorieswuth.a remote loading of 30.1 ksi. 89 (207.7 MPa). The Howland solution gives results almost identical with the Timoshenko—Goodier solution because the small hole in the test specimen of finite width provides results almost the same as those of the infinite plate that Timoshenko-Goodier assumed. One of these two theories can be selected to compare with the measured strains, the error due to finite width of the plate can be neglected. The Neuber solution is just good for providing informationvfiiflflxi the plastic zone, therefore the dashed line in Figure 4.1 represents the Neuber solution for the plastic zone. 4.2 Crack Initiation in Terms of Total Strain Once the stress~strain behavior at the critical location is determined, a cumulative damage analysis is made based on the calculated stresses and strains. The linear relationship between both plastic and elastic strain and the fatigue life was suggested by Smith—Hirschberg—Manson (39) as: As 0' 9 b + 8% (2Nf)c (4.13) where Ase is the total strain amplitude, and Nf is the number of fatigue cycles. Morrow (40) defined the constants b, c, 0% , and 8% by looking upon the fatigue properties of the metal as follows: 0% is the fatigue strength coefficient; 0% /E is 90 elastic strain when 2Nf = 1, b is the fatigue strength exponent; the slope of the log Ase/2 versus log 2N plot, f is the fatigue ductility coefficient; the inter- l-h— cept of the log Asp/2 versus log 2N plot at f 2N = 1, and f c is the fatigue ductility exponent; the slope of the log Asp versus log 2Nf plot. The above constants are experimentally obtained from fatigue testing and substituted into Equation 4.13 to pro— duce base line fatigue data for each type of material. The first term of the R.H.S. in Equation 4.13 repre— sents the elastic strain curve and the second term repre— sents the plastic strain curve. For long lives, the elastic curve is dominant and for short lives the plastic line dominates (41). Note that the theoretical solution is based on cycles to failure instead of cycles to initiate the crack. The reason is that the constant values in Equation 4.13 are obtained from the fatigue testing for each type of metal. The test specimens used to obtain base line fatigue data are small unnotched specimens, the initial crack being followed quickly by specimen failure. Therefore, thetheory treats the cycles to initiate the crack the same as the cycles to failure when the crack propagation stage is negligible. In this investigation one can compare the number of cycles to initiation and the total strain with 91 the theoretical solution. Several authors (35—37,40—43) have worked on the total strain—life relationship and pro— duced the base line fatigue data in cumulative damage stud- ied for 7075—T6 aluminum. Endo-Morrow (42) studied the short fatigue life while Martin (41) studied the long fatigue life. The constant-amplitude loading that was applied to the specimens in this study gave the number of the cycles to initiate the crack as about 15,000. This fairly large number of initiation cycles will better fit the Martin data which were used for the long fatigue life. Data from Martin (41) was based on the completely reversed tensile and compressive stresses or zero mean stress with constant—amplitude loading. In this study, a mean stress has to be considered to eliminate error in fatigue life calculations. Morrow (40) introduced the mean stress parameter as: A0 _ A0/2 (Tr)eff _l—0 70% (4’14) m where (A0)eff is the equivalent completely reversed stress range, A0 is the true stress range, cm is the mean stress, and 0% is the fatigue strength coefficient. For long lives, A0 can be elastically related to As, the true strain range (As) can be measured from the experi- ment by the moiré technique, 0m and 0% are known va1ues, then (A0)eff will be obtained. 92 All of the above equations apply to the non—cold- worked specimen. None of the theories had worked on the total strain-life of the coldworked specimen subjected to the mandrel hole enlargement in the plane stress condition. Rich-Impellizzeri (37) predicted the design limit stress— life of coldworked holes in aluminum under plane strain conditions. Therefore, the experimental data for the total strain-life of coldworked specimens in this investigation are also compared with the base line fatigue data of the non-coldworked specimen. 4.3 Experimental Results The experimental data in this section are obtained from 6 specimens, 2 non—coldworked, 2 medium coldworked, and 2 heavy coldworked. Constant-amplitude loading at 30.1 Ksi (207.7 MPa) maximum and 3.1 Ksi (21.4 MPa) minimum were applied throughout this investigation. The local strains and total strains were measured by moiré techniques as in Chapter 2. For coldworked specimens, only the change of residual strains were measured not including the residual strains during coldworking process. Life to crack initia— tion was determined from crack growth measurements during the initial stage of the growing crack. 4.3.1 Local Strain Versus Distance from Edge of Hole Figure 4.2 shows the moiré fringe patterns which were used for measuring the strains. The local strains were measured with both maximum and zero loads applied and they .pmxuoapaoo Abv .pmxuo3paoouco: Amv wmzoeflooam umou CH owpw macs gnu Eoum xomuo 0 mo cowumHUHcH osu mo mauouumd owcflumxwuwoz N.¢ ouswwm A3 A3 93 xo mcwmuum H0u0u mo consumaamo w.¢ wuswflm mzN .Eofifi oe dimmer/mm —I . d . _ u _ a _ . _ a _ a _ . mammozfioo Same D $080338 ELEM: 0 / ammmozoaooéoz G / - $40 $3.32 E23 $3 I..l / .. /A\ 254.5 K 0.54.5 / . ll g/ / 1 c . c . - h as.“ 238... 23.x 23800. 2 .23» / - :uOnSO OzuOAu On:OAu AAEUOQO ‘HCIHII'IdNV NIVHIS _3_ av 102 cycling. The increasing amount of the strain concentration factors are very small for this material (25) due to the cyclic hardening and the small nominal stress applied. A constant nominal stress of 30 Ksi (207 MPa) will produce a local stress at the notch root of about 90 Ksi (621 MPa), this amount of local stress is just a little larger than 73 Ksi (504 MPa) which is the yield stress of 7075-T6 aluminum that was used in the experiment. Although the measured strains obtained from cyclic loading were compared with the theoretical results obtained from monotonic loading, they are still in good agreement. Comparison of crack initiation in smooth and notched specimens for the total strain-life showed that the Smith- Hirschberg—Manson solution (39) used to obtain the base line fatigue data for 7075-T6 aluminum by Martin (41) worked well to predict the cycles to initiation. The initiation life of the heavy coldworked specimens were shorter than those of the medium coldworked ones; these results might be explained by an optimum interference level (4, 44 and 45). Potter-Grandt (4) predicted that the maximum allowable mandrel interference 'ug' for this type of specimen is about 0.0037 inches (0.094 mm) radial expansion. This value compares with the 0.0040 inches (0.102 mm) radial expansion in this study. Since u; leads to the maximum region of residual compressive stresses (4), the larger radial expansion such as 0.0060 inches (0.152 mm) will reduce the amount of these residual compressive stresses and cause shorter crack initiation life. _ CHAPTER 5 CRACK GROWTH 5.1 Overview Crack propagation occupies a major portion of the useful fatigue life in many cracked structural components. Cracks tend to be generated in components with notches and unintentional stress raisers such as tool marks or burrs on polished surfaces. The toleration of visible cracks, in a "fail-safe" structure, and their growth during the opera- tion of aircraft are factors which further established the significance of crack propagation studies. There is still no method whereby the life of a structural element can be predicted with sufficient accuracy without carrying out fatigue tests which take careful account of all the parameters involved. This is mainly because the actual mechanism of fatigue is still not under— stood and therefore empirical technical solutions must be sought. In general, crack propagation rate depends on the kind of material, the alloying elements and the grain structure . Careful observation of striations during crack prop— agation indicates crack extension during each interval of load cycles. In any case, if the final objective is to make a crack growth prediction for structure member, a knowledge of the average crack growth is sufficient. In other words, crack Propagation can be considered to be continuous. 103 104 The general form of the crack propagation laws were proposed by Christensen and Harmon (46) in the form, 93 = F[0m, 0 dN , a, di] (5.1) 00m where: a is the half crack length. N is the number of cycles. 0 and 000m are the amplitude of cyclic stress, and the mean stress respectively at points remote from the crack. di are material constants (i = 1, 2, 3, - — -). The crack propagation laws mentioned above gave only indirect quantitative consideration to crack tip stresses. A more fundamental approach for crack propagation depending on crack stresses was proposed by Paris and Erdogan (47). 3% = C(AR)m (5.2) where: c and m are empirical constants. AK is the range between maximum and minimum stress intensity factors. For the 7075—T6 aluminum, the constants c and m were found from baseline fatigue tests of crack geometries with known stress intensity calibrations by Grandt and Hinnerichs (14) to be equal to 0.945045 x 10‘19 in./cycle and 3.44371 . . . L respectively when AK is in pSl-ln.2. 105 5.2 Experimental Results Crack lengths were measured on both surfaces of the specimen, but the results used were from the front side. Crack lengths between the front and back side of the non— coldworked specimens were almost identical. For the cold— worked specimens, the crack lengths on the back side were slightly shorter than the crack lengths on the front side after the same number of cycles because of the different amount of residual strains between front and back side. These different amounts of residual strains were caused by the nature of the coldworking operation in which a force was exerted perpendicular to the specimen surface through the sleeve and thus constrained deformation of the hole on the back side (13). Crack growth curves for three different coldworking processes are given in Figure 5.1. For the non-coldworked specimens, two cracks were observed during fatigue testing, one on each side of the h01e and both cracks grew almost the same length. The cracks originated at the edge of the holes in a plane perpendicular to the surface of the test piece and perpendicular to loading direction [this is called stage II (48)]. The crack lengths at this condition were about 0.1 to 0.3 mm., then the frac- ture plane rotated around the axis in the direction of the crack propagation until it formed an angle of about 45° with the loading direction and the surface of the sheet. The transition occurred gradually and had its origin in the shear 106 .msmawomdm poxsoBCHoo msowum> ca mmaos Eoum wcwuwsmso mxomno Mom mo>nno busouw mo sowauwano H.m 0 w . a: sh moH x mmqo>o N )— .- w AV AV Aw IN 4‘ nu D AU I as n my fl. mu 4 i. no amsmozoaoo >>Hso usoamomadwwtupmod H.o madman .mswafiooam poxHoSUHoo msofifig ezmzmo<4mmHo soamo TLRHPL 0 -on ( cow 0 4| .00. 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Grandt applied the integral method of Rice (53) and obtained arbitrary crack face pressure from the addition of local stresses caused by remote load and residual stresses caused by the coldworking process. For residual stress, only Potter—Ting-Grandt (3,4) solution accounts for compressive yielding which occurs on unloading at the area close to the hole edge. This compressive yield— ing will reduce the amount of residual stresses, resulting in an increase in the amount of arbitrary crack face pres— sure at the area close to the hole edge. Unfortunately, the Hsu-Forman (2) solution does not account for compressive yielding. If one knows the approx— imate compressive yield zone by the theoretical solution there will be a better comparison with the experimentaldata. The IDG techniques have a limit of 1 mm from the hole edge to measure the stress intensity factor because accurate measurements cannot be obtained for such small dis— placements. The set of indentations will be sufficiently accurate within a distance a/20 from the crack tip, where 'a' is the crack length. The distance between each pair of indentations has to be set at least 0.70 mm apart to avoid 138 interference of fringe patterns from another set, therefore, the data from IDG techniques cannot be plotted continuously as crack growth data can. The experimental results obtained are slightly higher than those of the theory. This disagreement comes from incompleteness of model and experimental errors. CHAPTER 8 CRACK SURFACE DISPLACEMENT In recent years, crack surface displacements have been used widely to analyze fatigue crack growth and as a fracture criterion for materials. Accurate determination of crack surface displacements is often required for funda- mental investigation of crack behavior. The shapes of cracks opened by tension are approximated by conic sec- tions (54) and the conic section coefficients related to plate geometry by very simple empirical equations as in Chapter 7. Experimental studies have been performed by inter- ferometric measurement technique for crack displacement in glass specimens. Crosley-Mostovoy-Ripling (20) and Sommer (21) used this technique to determine the stress intensity factor calibrations. A new method for measuring crack surface displacements applicable to these types of problems has recently been developed by Sharpe—Grandt (50). The technique, which employs laser interferometry, was mentioned before in Chapter 6. This technique is quite sensitive (about 0.1 micron resolution), is readily adapt— able to laboratory measurements, and has the capability of obtaining the entire crack displacement profile. 139 140 8.1 Experimental Results The experimental data for crack displacements in this chapter were obtained from the IDG technique by measuring the load-displacement slope in the linear region above the crack opening load for loading and unloading condition. The dis- placement for a l Ksi (6.9 MPa) increment of stress in the linear region was compared with the theoretical solutions. The data for the crack mouth displacements was measured from a set of two indentations at a distance of 75 microns from the hole edge, the data were compared with a least square polynomial from the finite element method by Grandt (11). The crack opening displacement at any distance along the crack line was also measured by the IDG technique and com— pared with Equation 7.14. The experimental data for crack mouth displacements for the double crack are plotted against the theory in Figures 8.1-8.3. The data are the average of the cracks on R.H.S. and L.H.S. of the hole. The finite element method predicts smaller displacements for the non— coldworked specimens and there is about 10 percent difference between theoretical and experimental data (see Figure 8.1). For the coldworked specimens, the residual compressive stresses caused a smaller crack opening displacement as shown in Figures 8.2 and 8.3. Crack surface profiles determined by Orange (54) for the double crack and the measured results are compared in Figures 8.4—8.6. Note that displacements closely match the theoretical solution near the crack area close to the hole 141 <1E] , :01 80- Q s 1 13 ° V7 501 O- H 2 an 51 Lil U E 40— E -——— POLYNOMIAL EXPRESSION a $6 m e i :21 lo: 2 D. EL 33.o~ r. i=o '(Q) g A EXPERI IENTAL A g 0 v DTA E SPECIMENS 1 g 2.0 - LOADING A E] UNLOADING C) V7 Lo- 0 ' f [.0 2.0 DIMENSIONLESS CRACK LENGTH a/ro 3.0 Figure 8.1 Comparison of crack mouth displacement with the theory for non-coldworked specimens. 142 O”=IK$ 10- C) 6.0' D LO \0 S’so E Q E 8 E 40« 3 'E Q s; E E ———~ POLYNOMIAL EXPRESSION O 3.0 - < 6 i H 1 m -°= 2 0(9) Z E r0 i=0 I r g 0 Q [3 EXPERIMENTAL DATA 2.0 - SPECIMENS ; 3 E3 LOADING A: [3 Q. 6 mmmmnm O V LO‘ 0 .‘o 210 {o DIMENSIONLESS CRACK LENGTH a/n Figure 8.2 Comparison of crack mouth displacement with the theory for medium coldworked specimens. 143 O'=l KSI -——-POLYNOMIAL EXPRESSION V i ( 12 9(8) EXPERIMENTAL DATA SPECIMENS g 9 LOADING z: E] mmmmnm o V II 210 {o .o . DIMENSIONLESS CRACK LENGTH a/g Figure 8.3 Comparison of crack mouth diSplacement with the theory for heavy coldworked specimens. 144 20 L8- L6- ———OmmGE % E XPERIME 8|.4- NTAL DATA 0 E SPECIMENS l g ' LOADING z; [3 E 12 E mmmmnm o v 8 S aLo- H D LL] 3 a .8 - D (I) M 3 0.6- 9:24 .4- .2- 0 (0 i0 i0 40 50 do i3 DISTANCE FROM HOLE EDGE - MM Figure 8.4 COmparison of crack surface profiles with the theory for non-coldworked Specimens. 145 O'=|KS| L8- L6. ———OmmGE £3 E EXPERIMENTAL DATA wL4- .A 0‘ g E SPECIMENS ; 3 O E A LOADING A [3 I l. 2 ' H O mmOunNG o v E E E m U u E (I) H Q L21 0 . :5 Z :3 V) :d O . E U :24 .A l I U 1 0 L0 20 30 so . 4'.o 5'.O DISTANCE FROM HOLE EDGE - MM Figure 8.5 Comparison of crack surface profiles with the theory for medium coldworked specimens. 146 O'=lKSl LG' —-—ommcE 13 53 E3 EXPERIMENTAL DATA g "4‘ g 0‘ SPECIMENS ; g O at 3 [A LOADING A: E] 22 n'Z‘ O"7 mmomnNG O 'V E E] E1 m 0L E U) H O Ed 2. E .2) U) M. 0 § 0 r22.4 0 Lo 2?) 3.0 3.0 5'0 6'0 DISTANCE FROM HOLE EDGE - MM Figure 8.6 Comparison of crack surface profiles with the theory for heavy coldworked Specimens. 147 edge of non-coldworked specimens. Due to residual compres- sive stresses, as mentioned before, the values for the experimental displacements of coldworked specimens were smaller than the value calculated from theory. 8.2 DiscussiOn of ReSults In the experiments, the natural cracks obtained from cyclic fatigue loading were not the same lengths on both sides of the circular hole at the same cycle. This unequal crack length causes some disagreement between experimental and theoretical values. For the non-coldworked specimens, the crack on one side started and grew to length about 1.0 to 2.0 mm then the other side initiated and grew at a faster rate than the first one. This second crack obtained almost the same crack length as the first crack when the lengths were about 5.0 to 6.0 mm. For the coldworked specimens, both R.H.S. and L.H.S. cracks started almost at the same fatigue cycle and maintained the same crack length until the crack on one side reached the elastic-plastic boundary. This crack then grew quite rapidly to the final crack while the crack on another side grew slowly and stopped within or just out— side the plastic zone. CHAPTER 9 STRAIN AHEAD OF CRACK TIP When a specimen containing a crack or other similar defect is subjected to a tensile or shear stress, a very high stress concentration exists at the crack tip. If the material is assumed to remain elastic and the defect is suf— ficiently sharp, analytical linear elastic solution for the stress and strain distribution in the region of the crack tip can generally be obtained. However, virtually all metals exhibit some ability to deform plastically without fracture. This results in a region of plastic deformation around the crack tip. If the size of this zone is very much smaller than all other significant dimensions of the struc— ture and defect, the stress and strain distribution outside the plastic zone is not significantly different from the distribution predicted by the elastic solution. When the plastic zone becomes larger, as in a relatively ductile material, the applicability of the above concepts becomes questionable. Attempts have been made by Irwin (55) and Dugdale (56) to correct for the effect of this plastic zone. These attempts have been somewhat empirical and approximate in nature and, although they have been used with some suc- cess in predicting actual fracture criteria, they do not represent a true physical model of stress-strain condition 148 149 at the crack tip. Analytical solutions to the aboveelastic- plastic boundary value problem have been developed both theoretically (57,58) and experimentally (59-62) during the past few years. Theoretical solutions work by Rice- Rosengren (57) and Hutchinson (58) applies directly to the tensile—mode crack problem. The experimental techniques are required to measure the strain distribution in the region of the crack tip on a microscopic scale. In 1964, Oppel—Hill (59) published the results of a study evaluating the various techniques which could be used to conduct such an experiment. They indicated that the optical—interference technique provided the greatest sensi- tivity and accuracy. However, this method only provides a measurement of through-the-thickness strains in a sheet specimen. Later on, Kobayashi-Ingstrom-Simon (60), Liu—Ke (61),andwothers discussed several other methods which could provide a means of measuring in—plane strain. The grid- interference or moiré method appeared to be one of the best methods to provide sufficient sensitivity for the measure— ment of strains. 9.1 Theoretical Method The linear elastic solution in the stresses and strains in cracked solids was mentioned before in Chapter 7. For stress applied well below the yield stress (o<o er mm «LN __.~ ohm Q. Q. E m: B h — _ _ o DDDDDDDDDDDDDDD@@@Q U m a Q 0 4 O 446.6 o o a o o G O O -_ O Q 0 a o 00 IN G 4 00 Q 00 O .O -m G O O 0 AV 0 u¢ 4 O G O 0 U a In 6O O Q [m NW - HISNHT XDVHD 178 .ApmxH03paoo abwpmfiv m .oa cmEHomam Mom mump suBOHw xomuo m.< muswam {W .n mOH meMAUWO n O BUDDBGBBGBQGGGDEQWW 0000000000 000 999999000 I _ a m mmwwwamaaaa awe R90 1 N 0 3 W O m Fl. I n M o m 0 T ¢m o I m o 1 m 179 .Apoxuospaoo Edwpmfiv g .0: cmEHommm How mumv nuBOHw xomuo ¢.< wuswam moa x mmqoso V m N _ O BDBDDBQDD _ _ boponeropODopo>opohmmwuwumfimcwfiwfiwcwfiwmm>o>o>o>o>b>o>o>09coo cc I. . ooeooaooocco mwmu Qfldflmwmmmm 6 00 [N O 0 m o “A HI. 0 umm O m H ¢m o o aKD 4 O In 0 1w mo .prxuoapaoo s>mm£v m .og GmEHomdm How dump :uzouw xomuo m.< gunman moa x mmqo>o v n N _ f — _ _ DDDDEDDEDDDBDBDDDEEEDDDEUDDD GGQQQQGQQQMMMMMMMMMMM 000000 0 0 0 0 0 D 0 0 ‘4. fluflufliMWU twnu NH - HLDNHT HOVHD 181 .Apmxuozpaoo .0605 o .0: :oEHommw How 43% ~33on 030.5 0.4 0.“:me mOH x mmqo>o _ n _ _ o nwssnnnsnssssassessnwnswwwwwwmwtksst% ; UBBGDDBDBDDBDBUBDDDDEDGDGDDDBDBD 4 44444444MM0 444444 00 444444MMMoooooooooooo @000 O 4 O l K) NW - HlONHT XDVHD o ; 182 APPENDIX B LOAD-DISPLACEMENT DATA 183 .ameommm pmxnospaoonaoc uom ES m we suwcwa xowuo w um haamowuuwaoummuoudw tmcwmuno wm>H50 uameomHmmeupmoq H.m ouswwm mZOMUHE I FZMZHDu50 quEmomHmchupmoA 45 muswam mZOMUHZ .. 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