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L I {sit-.EIl-lzéu2nksl 5.111.}.51.‘ I. 3.15: .. ».~.t.+...:.;$r {illicit}. 1%.... . .c ( flan . . . . . z... ...:.£¢..t.::.. i :11 2.1.2.: ............ I This is to certify that the thesis entitled AN EXPERIMENTAL STUDY OF THREE-DIMENSIONAL STRAIN AROUND COLDWORKED HOLES AND IN THICK COMPACT TENSION SPECIMENS presented by Somnuek Paleebut has been accepted towards fulfillment of the requirements for Ph.D. degreein Mechanics AAW ‘ [Major professor Gary Lee Cloud Datewz ? 0-7 639 MSU LIBRARIES m RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. , _—~——.—_.——_—._M '-'~~— AN EXPERIMENTAL STUDY OF THREE-DIMENSIONAL STRAIN AROUND COLDWORKED HOLES AND IN THICK COMPACT TENSION SPECIEMNS BY Somnuek Paleebut A DISSERTATION / g/ / ’1/76 Submitted to Michigan State University in partial fulfillment for the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Metallurgy, Mechanics and Materials Science 1982 Air-.. 5.” ~‘ .. ‘_. ABSTRACT AN EXPERIMENTAL STUDY OF THREE-DIMENSIONAL STRAIN AROUND COLDWORKED HOLES AND IN THICK COMPACT TENSION SPECIMENS BY Somnuek Paleebut This research investigated and developed a new three- dimensional multiple Embedded Grid Moire Method for measur- ing the strain field in the interior and on the surface of specimens in order to study currently important problems in mechanics and structural design. Coldwork specimens and compact tension specimens which were made from transparent material were used. Specimens were fabricated with multi— ple embedded grids. These grids were photographed for each specimen state. Moire fringe photographs were ex— tracted from the replicas by coherent optical processing. Digital analysis of the fringes gave the desired strain maps. The three-dimensional nature of radial, hoop, and transverse strains which are created by drawing a tapered mandrel through a cylindrical hole were measured. The residual radial strain after coldwork inside the specimen was found to be smaller than on the surface. The transverse strain changes from tension near the top surface where the mandrel enters to maximum compression at the mid-plane, where it begins to decrease. The hoop strain is minimum at the mid—plane. The change in this component is only Somnuek Paleebut about four percent of maximum value. Potential problems in the use of such a coldworking process might arise from the low value of the interior residual strain in comparison with the surface value and the existance of tensile trans- verse normal strain near the surface. The static strain distribution near the crack tip on the surface plane and three interior planes of polycarbonate compact tension specimens were studied. The strains results that were obtained from the moire method are in good agree- ment with strain measurement from similar specimens with embedded strain gages. Near the crack tip strain 8y (per- pendicular to the crack line) was found to be much larger than strain ex on any plane along the specimen thickness. Therefore, strain 8y is more important. Strain EY on the mid-plane was found to be larger than on the surface, and it causes the fracture to start on the mid-plane. The maximum strain 8y on the mid-plane lies along the crack plane, but on the surface it starts from the crack tip and lies along two separate lines. ACKNOWLEDGEMENTS This research was supported by the National Science Foundation under grant ENG 78-02530. The author is grate- ful to National Science Foundation and Program Director Dr. Clifford Astill. The author would like to give grateful thanks to The Royal Thai Air Force which sponsored my graduate program. The author wishes to express my sincerest appreciations and gratitude to his advisor, Professor Gary Lee Cloud, for his encouragement and many valuable guidances during the course of this work. Thanks are also extended to the other members of his guidance committee, Dr. N. Altiero, Dr. J. Martin and Dr. N. Hills. Special thanks are due to Dr. N. Altiero and Dr. R. Abeyaratne for their suggestions during the course of this investigation. Finally the author wishes to thank his wife, Arasiri, and son, Ayudh, for their understanding, help, patience and encouragement. TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . LIST OF FIGURES . . . . . . . . . . . . . . . . . . Chapter 1 INTRODUCTION . . . . . . . . . . . . . 1.1 Purpose and Motivation . . . . . . . . . 1.2 Coldworked Specimens . . . . . . . . . . 1.3 Thick Compact Tension Specimens . . . . . 1.4 Organization of the Dissertation . . . . Chapter 2 SUMMARY OF MOIRE TECHNIQUE . . . . . . . 2.1 Fundamentals of the Moire Method . . . . 2.2 Strain Analysis . . . . . . . . . . . . . 2.3 Producing Submaster Grating . . . . . . . 2.4 Specimen Grid Deposition . . . . . . . . 2.5 Specimen Grating Photographs. . . . . . . 2.6 Optical Data Processing System . . . . . 2.7 Moire Fringe Photographs. . . . . . . . 2.8 Digitizing Moire Fringe Data . . . . . . 2.9 Data Reduction and Plotting Displacement and Strains . . . . . . . . . . . . . 2.9.1 Detailed Analysis and Plot of Single Data Sets . . . . . . . 2.9.2 Analysis and Summary Plotting of Multiple Data Sets . . . . . . Chapter 3 MATERIAL SPECIFICATION . . . . . . . . ii Page vi 13 15 19 20 31 34 40 40 46 48 Page Chapter 4 RESIDUAL STRAIN AROUND COLDWORK HOLES . . . 52 4.1 COldworking . . . . . . . . . . . . . . . . . 55 4.2 Residual Strain and Transverse Strain Measurement . . . . . . . . . . . . . . . . 57 4.2.1 Specimen Preparation . . . . . . . . . 57 4.2.2 Photograph of Specimen Grating . . . . 58 4.2.3 Experimental Results and Discussion . . . . . . . . . . . . . 62 4.3 Tangential Strain Measurement on Different Planes . . . . . . . . . . . . . . . . . . . 89 4.3.1 Specimen Preparation . . . . . . . . . 89 4.3.2 Photoqraph of Specimen Grating . . . . 91 4.3.3 Results and Discussion . . . . . . . . 93 4.4 Summary of Strain Field in Coldworked Specimen . . . . . . . . . . . . . . . . . . 100 Chapter 5 STRAIN MEASUREMENT NEAR A CRACK TIP ON DIFFERENT PLANES IN THICK COMPACT TENSION SPECIMEN . . . . . . . . . . . . . . . . . 102 5.1 Fundamental of Fracture Mechanics . . . . . . 102 5.1.1 Linear Elastic Fracture Mechanics . . 102 5.1.2 The Relationship Between Stress, Strain and Displacement Near a Crack Tip . . . . . . . . . . . . . . 104 5.1.3 Crack Tip Deformation . . . . . . . . 110 5.1.4 Fracture Behavior of Thin and Thick Specimen . . . . . . . . . . . 116 5.2 Specimen Preparation . . . . . . . . . . . . 119 5.3 Experimental Procedure . . . . . . . . . . . 121 5.4 Experimental Results . . . . . . . . . . . . 125 5.4.1 First Experimental Results . . . . . . 125 iii Page 5.4.2 Correction Procedure . . . . . . . . . 151 5.4.3 Final Experimental Results . . . . . . 167 5.4.3.1 Results from Moire Method . . . . . . . . . . . 167 5.4.3.2 Results from Strain Gage . . . . . . . . . . . . 179 5.5 Discussion . . . . . . . . . . . . . . . . . 194 5.6 Summary of Strain Field in Thick Compact Tension Specimen . . . . . . . . . . . . . . 199 Chapter 6 CONCLUSIONS . . . . . . . . . . . . . . . . 201 REFERENCES . . . . . . . . . . . . . . . . . . . . . . 206 APPENDIX . . . . . . . . . . . . . . . . . . . . . . . 213 iv Table 4.1 5.1a 5.1b 5.2a 5.2b LIST OF TABLES Diameter of Hole Before and After Load The NO. The The No. The Position of Strain Gages on Specimen l . . . . . . . . . . . . . . . Strain Results From Specimen No. 1 Position of Strain Gages on Specimen 2 . . . . . . . . . . . . . . . Strain Results From Specimen No. 2 Page 62 183 183 185 185 Figure 2.1 2.9 2.10 LIST OF FIGURES Moire Fringe Patterns Formed by (a) Rotation (b) Difference in Pitch and (c) Combination of Rotation and Difference in Pitch . . . . Formation of a Moire Fringe . . . . . . . . . Progression <1f the Data Reduction Procedure RequiredixiMoire Strain Analysis. . . . . . Sketch of a set-up for Photographically Producing a Submaster Grating . . . . . . . The Copper Grating Etching Process . . . . . . A Copper Grating by the A COpper Grating by the Diffraction of a Single a Sinusoidal Amplitude Diffraction of a Single Etching Method . . Stencil Method . Beam Passing Through Grating . . . . . . . Beam Passing Through two Superimposed Sinusoidal Gratings of Nearly Equal Spatial Frequency . . . . . . . Imaging System used in the Formation of Moire Interference Patterns (Ref. 9) . . . Diffraction of a Single Beam Passing Through two Superimposed Gratings of Different Spatial Frequency . . . . . . . . Optical System for Spatial Filtering in Fourier Transform Plane and Creation of Inverse Transform of Filter Image (Ref. 9) . Schematic of the Optical Processing System used for Obtaining Moire Fringe Photograph from Specimen Grating Photoplates . . . . . vi Page 11 12 l4 17 18 18 21 23 24 27 29 32 2.16C 2.17 3.1 Schematic of the Photograph Preparation for Digitizing a) To Measure Strain e b) To Measure strain 8x . . . . . . . . . . SampleMoire FringeyPhotograph . . . . . . . The Micro Datatizer System . . . . . . . . Plot of Distance Versus Fringe Order . . . . Plot of Distance from Fiducial Mark Versus Displacement . . . . . . . . . . . . Plot of Distance from Fiducial Mark Versus Strain . . . . . . . . . . . . . . . Plot of Multiple Data Set . . . . . . . . . . ‘The Modulus of Elasticity Versus Temperature of Polycarbonate (Reference 24) . . . . The Stress-strain Curve of Polycarbonate (Reference 24) . . . . . . . . . . . . . . Stress-strain Curve as a Function of the Strain Rate for a 60:40 Mixture of Laminac Polyester Resins (Reference 26) . . A Small Hole in a Thick Plate under Uniform Internal Pressure . . . . . . . . . . . . Expanding a Hole in the Thick Plate by Using a Tapered Rod . . . . . . . . . . . . . . . Schematic of the Coldworking Process . . Specimen Dimensions Schematic of Photography Process. Diameter of the Hole Along the Thickness Diametral Expansions Along the Thickness vii Page 35 37 39 43 44 45 47 49 50 50 52 54 56 60 63 64 Figure 4.10.1 4.10.2 4.10.3 4.10.4 4.10.5 4.11.1 4.11.2 4.11.3 4.11.4 4.12.1 4.12.2 4.12.3 Photographs of the Moire Fringe Pattern Noload and Loaded in the Direction Perpendicular to the Hole of the Polycarbonate Test Specimen . . . . . . . . . . . . . . Photographs of the Moire Fringe Pattern Noload and Loaded in the Direction Parallel to the Hole of the Polycarbonate Test Specimen . . . . . . . . . . . . . . . . Photograph of Moire Fringe Pattern with No-Load on Test Specimen of Mixed Polyester 60:40 . . Photograph Step Load 60:40 . . Photograph Step Load 60:40 . . Photograph Step Load 60:40 . . Photograph Step Load 60:40 . . Radial Strain at Different Thickness Radial Strain at Different Thickness Radial Strain at Different Thickness Radial Strain at Different Thickness Radial Strain at Different Thickness Radial Strain at Different Planes Along Thickness Radial Strain at Different Planes Along ‘Thickness of Moire Fringe Pattern with lst- on Test Specimen of Mixed Polyester of Moire Fringe Pattern with 2nd- on Test Specimen of Mixed Polyester 0 C O O O O O O O O O O O O O of Moire Fringe Pattern with 3rd- on Test Specimen of Mixed Polyester of Moire Fringe Pattern with 4th- on Test Specimen of Mixed Polyester Planes Along the on Left Side of Hole at lst Step . Planes Along the on Left Side of Hole at 2nd Step Planes Along the on Left Side of Hole at 3rd Step . Planes Along the on Left Side of Hole at 4th Step the Step Planes Along on Right Side of Hole at lst the on Right Side of Hole at 2nd Step the on Right Side of Hole at 3rd Step viii Page 66 66 67 68 68 68 68 69 70 71 72 73 74 75 Figure 4.12.4 4.13.1 4.13.2 4.13.3 4.13.4 4.13.5 4.14.1 4.14.2 4.14.3 4.14.4 4.15.1 4.15.2 4.15.3 4.15.4 4.16 4.17 Radial Strain at Different Planes Along the Thickness on Right Side of Hole at 4th Step Photograph of Moire Fringe Pattern with No- Load on Test Specimen of Mixed Polyester 60:40 O O O O O I O I O O O O O O O O O O I Photograph of Moire Fringe Pattern with lst-Step Load on Test Specimen of Mixed Polyester 60:40 . . . . . . . . . . . . . Photograph of Moire Fringe Pattern with 2nd- Step Load on Test Specimen of Mixed Polyester 60:40 . . . . . . . . . . . . . of Moire Fringe Pattern with 3rd- on Test Specimen of Mixed 60:40 O O O O O O O O O O C O 0 Photograph Step Load Polyester of Moire Fringe Pattern with 4th- on Test Specimen of Mixed Photograph Step Load Polyester 60:40 . . Strain in z-direction on Left Side of Hole Strain in z-direction on Left Side of Hole Strain in z-direction on Left Side of Hole Strain in z-direction at at at at at at at Different lst Step Different 2nd Step Different 3rd Step Different Lines Lines Lines Lines on Left S Strain in on Right Strain in _on Right Strain in on Right Strain in on Right Strain in ide of Hole at 4th Step z-direction at Different Side of Hole at lst Step z-direction at Different Side of Hole at 2nd Step z-direction at Different Side of Hole at 3rd Step z-direction at Different Side of Hole at 4th Step Lines Lines Lines Lines z-direction on the Midplane Specimen Dimensions. . . . . . . . ix Page 78 79 79 79 80 81 82 83 85 86 87 88 90 92 Figure Page 4.18 Schematic of the Photographic Data Recording . 94 4.19.1 The Moire Fringe Pattern of Specimen Before and After Load on Surface-Plane . . . . . . . 95 4.19.2 The Moire Fringe Pattern of Specimen Before and After Load on Quarter-Plane . . . . . . . 96 4.19.3 The Moire Fringe Pattern of Specimen Before and After Load on Mid—Plane . . . . . . . . . 97 4.20 H00p Strain Near the Edge of the Hole on Different Planes. . . . . . . . . . . . . . . 98 4.21 Comparison of Hoop Strain Near the Edge of the Hole on Each Plane . . . . . . . . . . . 99 5.1 The Basic Modes of Loading the Crack Plate . . 103 5.2 Coordinates Measured from the Leading Edge of a Crack and the Stress Components in the Crack Tip Stress Field . . . . . . . . . . . 106 5.3 V Transverse Contraction that Occurs Near the Crack Tip in a Thick Specimen. These Conditions are Opposed by the Unyielding Faces "A" of the Notch; Consequently Transverse Tensile Stresses oz and 0x are set up Ahead of the Crack (Ref. 55) . . . . . 112 5.4 a) Variation of oz Across the Thickness, in z—Direction at x=constant. b) Variation of o and ox with x on the Surface of a Notched Plate (Z = : B/2). c) Variation of o and ox with x at Mid Thickness (2:01 of a Thick Notched Plate (Ref. 67) . . . . . . . . . . . . . 113 5.5 Stress Deformation In Front of a Crack Tip During Local Yielding for (a) Plane Stress, and (b) Plane Strain (Ref. 67) . . . . . . . . 115 5.6 Schematic Drawing of the Types of Deformation Around a Crack (Ref. 69) . . . . . . . . . . 117 5.7 ffiuaRelationship Between Stress Intensity or Load and Change in Crack Length. (Ref. 56) . . . . . . . . . . . . . . . . . . 118 Figure 5.8 5.14 5.15 Page Schematic Diagram of Crack Propagation in a Plate Under Mode I Tensile Loading. Letters a,b,c,d,e refer to Successive Positions of the Crack Part. (Ref. 56) . . . 118 Specimen Dimensions of Compact Tension Specimen . . . . . . . - . . . . . . . . . . 120 Schematic Drawing of the Specimen Set Up . . 123 Moire Fringe Patterns on the Surface of the Specimen with Fatigue Crack Obtained From Dataplate with Submaster having 270 lpi 1J1 the Direction. a) Parallel and b) Perpindicular to the Crack Line . . . . . . . . . . . . . . . . . 126 Moire Fringe Patterns on the Surface of Specimen with Fatigue Crack Obtained From Data Plate with Submaster having 230 lpi in the Direction. a) Parallel and b) Perpendicular to the Crack Line . . . . 126 The Plot of Surface Strain fiy in the Direction Perpendicular to the Crack Line for Various Distances from Crack Tip in the Fatigue Crack Specimen . . . . . . . . . . . . . . . . . . 127 The Surface Strain in the Direction Parallel to the Crack Line 5x of a Fatigue Crack Specimen . . . . . . . . . . . . . . . . . . 128 Comparison of the Plot of Surface Strain 8x and ey from the Crack Tip Along the Crack Line of the Specimen with Fatigue Crack . . . 129 5.16glMoire Fringe Pattern in the Interior (quarter plane) of Fatigue Crack Specimen in the Direction IParallel and Perpendicular to the Crack Idne . . . . . . . . . . . . ... . . . 131 5.16J2The Schematic of the Deformation Zone Near Crack Tip . . . . . . . . . . . . . . . . . . 132 5.16g3The Decohesion Enclave in a Thick Plate (after Boyd (72)) . . . . . . . . . . . . . . 132 5.17 The Moire Fringe Pattern in the Interior (quarter-plane) of a Fatigue Crack Specimen in the Direction (a) Parallel, and (b) Perpendicular to the Crack Line Obtained From a Submaster Having 230 lpi . . . . . . 134 xi Figure 5.18 5.19 5.20 5.21 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.30 5.31 Moire Fringe Pattern on the Surface of Specimen Without Fatigue Crack in the Direction a) Parallel and b) Perpendicular to the Crack Line . . . . . . . . . . . . . . . Strain e on the Surface Plane of the Specimen Without ' yFatigue Crack . . . . . . . . . . . . . The Constant Strain Contours ey Around Crack Tip on the Surface Plane . . . . . . . . . . . Strain ex on the Surface Plane of Specimen Without Fatigue Crack . . . . . . . . . . . . . The Constant Strain Contours ex Around Crack Tip on Surface Plane . . . . . . . . . . . . . Comparison of the Strains 5x and ey along the Crack Line on the Surface Plane . . . . . . . . The Moire Fringe Patterns of Loaded and Unloaded Notched Specimen on the Quarter Plane in the Direction Parallel to the Crack Line . . . . . . . . . . . . . . . . . . The Moire Patterns of Loaded and Unloaded Notched Specimen on the Quarter Plane fix: the Direction Perpendicular to the Crack Line 0 O O O C O O O O O I I O O O O C O O O O The Moire Fringe Patterns of Unloaded and Loaded Notched Specimens on the Mid-Plane in the Direction Parallel to the Crack Line . . . . . The Moire Fringe Patterns of Unloaded and Loaded Notch Specimen on the Mid-Plane in the Direction Perpendicular to the Crack Line . . . . . . . . The Strain Plot of e on the Quarter-plane of a Notched Specimen . . . . . . . . . . . . . Strain ex on the Quarter Plane of Notched Specimen . . . . . . . . . . . . . . . . . . . The Strain Plot of 5y on the Mid—Plane of a Notched Specimen . . . . . . . . Strain 5x on the Mid-plane of a Notched Specimen . . . . . . . . . . . . . . . . . . xii Page 135 136 137 138 139 141 142 142 142 142 143 144 145 146 Figure 5.32 5.33 5.35 5.36 5.39 Development of Plastic Zone Size on the Surface of Polycarbonate Crack Specimen (Ref. 42). a) Kidney-shaped Zone of Deformation Near the Tip at Small Load b) Wedge-shaped Zone at Higher Extension c) Internal Kidney Within the Wedge of Still Higher Extension . . . . . . . . . . The Moire Fringe Patterns Observed on the Surface of a Quarter Thickness Specimen (0.375 in) in the Direction Parallel to the Crack Line by Taking a Photograph a) Directly of the Grating on the Surface Plane and b) by a Pass Through the Specimen Thickness The Moire Fringe Patterns Observed on the Surface Plane of a Half Thickness Specimen (0.750 in) in the Direction Parallel to the Crack Line by Taking a Photograph a) Directly to the Grating on Surface Plane and b) Through the Specimen Thickness. . . . . ._ The Strain Observed in the Direction Perpen- dicular to a Crack Line (e ) of a Specimen Thickness of 3/8 in. by Taking a Photograph a) Directly of the Surface, and b) Through the Thickness of the Specimen . . . . . . . . . . . . . . . . . The Strain Observed in the Direction Perpen— dicular to the Crack Line (ey) of the Specimen Thickness of 3/4 in. by taking a Photograph a) Directly of the Surface of the Specimen, and b) Through the Thickness of the Specimen . . . . . . . . . . . . . . . . . The Plot of Strain Error on Different Lines (xl,x2,x3) in the Crack Plane The Plot of Strain Error on Different Lines (yl,y2,...y6) in the Plane Perpendicular to Crack Plane at x=0.050 in . . . . . . . . . Schematic Showing the‘ Position of Nickel Mesh and Strain Gage on the Quarter Plane of Specimen . . . . . . . . . . . . . . . . . . . xiii Page 150 152 152 153 154 156 157 159 Figure 5.40 5.42 5.43 5.47 5.48 Schematic of Photography Process a) Through one-quarter Thickness Side and b) Through three-quarter Thickness Side . Moire Fringe Pattern in the Interior of the Specimen with one Plane Grating by Taking a Photograph Through a Quarter Thickness Side in the Direction a) Parallel and b) Perpendicular to the Crack Line . Moire Fringe Pattern in the Interior of Specimen with One Plane Grating by Taking a Photograph Through Three-Quarter Thickness Side in the Direction a) Parallel and b) Perpendicular to the Crack Line . The Strain Plot of 8 in the Interior Obtained from the Moire Fringe Pattern by Taking a Photograph Through the One-Quarter Side of the Specimen . . . . . . . . . . . . The Strain Plot of 6x in the Interoir Obtained from the Moire Fringe Pattern by Taking a Photograph Through a Quarter Thickness Side . . . . . . . The Strain Plot of e in the Interior Obtained from the Moire Fringe Pattern by Taking a Photograph Through the Three-quarter Side of the Specimen . . . . . . . . . . The Strain Plot of Ex in the Interior Obtained from the Moire Fringe Pattern by Taking a Photograph Through Three-quarter Side of the Specimen . . . . . . . . . . . . A Plot of Strain 6y on the Quarter-plane The Constant Strain Contours ey on the Quarter-plane . . . . . . . . . . . A Plot of Strain e on the Mid—Plane Y The Constant Strain Contours ey on the Mid-plane o o o o o o o o o o o o o A Comparison of and Mid-planes A ong the Crack Line The Plot of the Strain Ex on the Quarter-plane Along the Crack Line After Correction xiv on the Surface, Quarter Page 160 161 161 162 163 164 165 169 170 171 172 173 175 Figure 5.53 5.54 5.55 5.56 5.57 5.58 5.59 5.60 5.61 The Constant Strain Contours ex on the Quarter Plane . . . . . . . . . . . . . . . . . . . . The Plot of Strain ex on the Mid-Plane Along the Crack Line After Correction . . . . The ConstantStrain Contourss: on the . x Mid-plane . . . . . . . . . A Comparison of the Strain ex on the Surface, Quarter and Mid-plane Along the Crack Line 0 I O O O I O O O O O I O O I O O O 0 Positions of the Strain Gage Installed on Each Plane of Specimen No. 1 . . . . . . . . Positions of the Strain Gage Installed on Each Plane of Specimen No. 2 . . . . . . . . Overall View of the Specimen Set Up . . . . . A Comparision of the Strain Component, ey Obtained from the Moire Method and the Strain Gage O O O O O O O O O O O O O O O O O A Comparision of the Strain Component, ex Obtained from the Moire Method and the Strain Gage O O O O O O I O O O O O O 0 O O O The Plot of 6 and 8 From the Crack Tip to the End of the Width on the Surface Plane of the Specimen with the Strain Gage . . . . XV Page 176 177 178 180 182 184 187 188 191 192 CHAPTER 1 INTRODUCTION Methods derived from linear elasticity are not in general sufficient to complex problems involving plastic deformation, fatigue and fracture. It is important to consider nonlinear elastic, plastic, viscoelastic, and viscoplastic responses as well as the nonhomogeneous and nonisotropic responses of material. Experimental and theoretical three-dimensional strain analysis can be accomplished by only a few methods. Few problems have been solved completely by the theory of el- asticity. Three-dimensional photoelasticity, on the other hand, has found widespread application in its many forms. Methods involving locking-in stress in a photoelastic model are cmmmwn one. All of these techniques are destructive in that they require slicing of the model. A nondestructive photoelastic technique uses scattered light. The photo- elastic technique, in general, suffers from the limitation that it is not sufficient in itself, and it requires com- plementary methods for the separation of stresses. Another approach is the embedded grid moire method, which might be easier to implement. The moire method is basically two-dimensional. Three dimensional analysis is accomplish- ed through successive applications of the technique to various planes of a specimen. The moire technique is no different in this respect from most other methods, includ- ing three-dimensional photoelasticity, which requires opti- cal or physical slicing of the specimen. The embedded grid moire method is similar to layered model photoelasticity in that only one plane or, in certain cases, a few planes can be analyzed for any one specimen. The application of a cross-grid to analyze a two-dimensional strain field was discussed by Post (1), while applications for three- dimension strain analyses using an embedded grid in clear epoxy-resin and urethane-rubber models were made by Durelli and Daniel (2) and by Sciammarella and Chiang (3). 1.1 Purpose and Motivation The purpose of this research are: a) to develop a new three-dimensional moire method using multiple embedded and surface grating. The method requires that interior gratings be photographed through an inter- vening surface grating. b) to measure strain around a coldworked hole through the thickness of a polymeric specimen and; C) “no measure the strain components near a crack tip on the surface and in the interior of a thick specimen. 1.2 Coldworked Specimens Crack initiation and growth are major causes of failure of high performance structural components and source of difficulty to the designer. A 1971 review of aircraft structure failure showed that cracks which began at fast- ener holes were preliminary causes of one-third of early failures (4). One class of techniques for improving the fatigue performance of fasteners is to plastically deform the hole prior to or during installation of the fastener. This deformation is accomplished by preexpansion of the holes with an oversize mandrel. The surface strains and stresses around coldworked holes have been studied by many researchers. The nature of the strain fields around a coldworked hole was studied ex- perimentally by Adler and Dupree (5) and by Sharpe (6). Sharpe and Chandawanich (7) studied the change of the re- sidual strain field around coldworked holes during the in— itiation of cracks. Sharpe and Poolsuk (8) studied the elastic-plastic boundary and thickness changes around the coldworked hole. Cloud (9) investigated and described the nature of surface strain near the coldworked hole. Cloud and Tipton (10) and Cloud and Sulaimana (11) observed the strain distributions and effects of a row of holes which were parallel to and near a straight edge after coldworking. No experimental work has been done concerning the change of residual strains along the thickness of coldworked specimens, and no theory of coldworked holes is available which accounts for interior behavior. Given the nature of the mandrelizing process, the strain field near a coldworked hole is probably three- dimensional. The degree and direction of residual strains might change from one side to the other side of the specimen. The three-dimensional nature of the radial, hoop, and transverse strains created by mandrelizing were explored to establish whether the interior value of the strain correlated well with surface measurements. The embedded grid moire method was developed to measure interior values of the strain in both vertical and radial directions on a plane of symmetry through the hole center. Atransparent specimen with a grating plane (nickel mesh) at the center was coldworked by loading in four steps. A data plate was recorded at each step. The strains in the radial and vertical directions were obtained by using the fringe data reduction method developed by Cloud (9). It was expected that the strain field would not be symmetrical on both sides of the hole and not uniform along the thick- ness. The strain field along the thickness was measured to within 0.01 inches (0.254 mm) of the hole boundary. 1.3 Thick Compact Tension Specimens Many researchers have measured surface stress and strain states in the vicinity of a crack by various methods, primarily as a step towards obtaining stress intensity fac- tors (12, 13, 14, 15), although some have been interested in exploring the fundamental assumptions and limitations of fracture mechanics (16, 17, 18). Post (19) pointed out the need for an experimental study of the three-dimension character of the crack tip stress state. Baker and Fourney (20) investigated the stress intensity factor along the crack front by using an interferometric method. Cases where yielding is involved have not been investigated. Pitoniak, Grandt, Montulli and Packman (23) observed crack retardation and crack closure iri polymethylmethacrylate, and found that the crack tip opening displacement (COD) values in the interior could be much different from the surface values. Hahn and Rosenfield (21) and Underwood and Kendall (22) found a difference in the shape of the plastic zone on the surface plane and the mid—plane of a crack specimen. In this investigation polycarbonate compact tension specimens with gratings on the surface, quarter and mid- plane were studied. Strains (ex and ey) on the surface and in the interior were measured by using the multiple embedded grid moire method; results were compared with strain measurements that were obtained from the specimens with strain gages. The results from both methods were in good agreement. Near the crack tip along the crack line strain 6y was found to be much greater than ex both on the surface and in the interior. Strain 8y was maximum on the mid-plane. On the mid-plane the maximum strain Ey starts from the crack tip and lies on the crack plane. On the surface plane, how— ever, the maximum strain 8y lies on two separate lines radiating from the crack tip. .4 Organization of the Dissertation The experimental technique for grid deposition is described in the beginning of Chapter 2. The multiple embedded grid moire method is also described in Chapter 2. Description of the material used in this investi- gation, their properties, and details on Specimen preparation are given in Chapter 3. Chapter 4 describes and discusses the results from coldworked specimens. A summary of results is given at the end of this chapter. Chapter 5 describes and discusses the results from crack specimens. The first experimental results from crack specimensanxadescribed and discussed in the beginning of this chapter. The final results are compared with the re- sults that were obtained from strain gages. A discussion and summary of the results are also given at the end of this chapter. This dissertation concludes with Chapter 6 which summarizes the findings of time investigation. CHAPTER 2 SUMMARY OF MOIRE TECHNIQUE 2.1 Fundamentals of the Moire Method Moire fringes are produced by the interference of two grids; they are formed by two grills of parallel lines that are slightly different in pitch and/or orientation (64). Figure 2.1 illustrates the basic idea. It is possible to measure displacements and strains with this method by re- cording a specimen grid twice, once in the undeformed state and once in the deformed state. Superposition of the two images formed will produce moire fringes. Displacement can then be determined directly from the fringe patterns. Dif- ferentiation of the displacement data establishes the strains. There are three simple ways to perform the superposition and obtain moire fringes. One method is to bring a reference grill, which is often on a glass plate, directly into contact with the specimen grill. The interference between these two grills gives moire fringes. In this case, intimate contact is essential to produce satisfactory results. In the second method, the image of the model grill is projected onto the reference grill on a glass plate with a precision camera. The interference fringe formed on the .. "f" ("yfin‘if’ ' V 7 \ 1' i. l “(LIX/LIWJ'M, ”“111“,,“iii!!!”lily U f). v ,‘l‘l".;l.‘4’!’l“il""l ' ,“"l W 'Mig’li -y; . ‘ .‘ ‘ ‘ ‘1 ', [I i “‘1 I] f, ‘ ‘ I 1! ‘ I N w, (C) (b) (a) in Pitch and (c) Combination of Rotation and Difference in Figure 2.1 Moire Fringe Patterns Formed by (a) Rotation, (b) Difference Pitch. glass plate are then recorded by a second camera. With this method, pitch mismatch and rotational mismatch can be readily introduced. The slight change of the magnifica- tion of the camera introduces pitch mismatch. In the third method, the image of the model grill be- fore deformation and after deformation can be recorded on photoplates. The interference of the two images of the same grill before and after deformation give moire fringes. For this project, two photOplates were used for each specimen. One plate records the undeformed grill and the other records the deformed grill. Then the photoplates were each superimposed individually with the same sub- master to form moire fringe patterns. By using an optical data processing system, the moire fringe patterns from each plate were recorded on the negative film. The moire fringe photographs were then analyzed to obtain displacements and strains. 2.2 Strain Analysis The array used to produce moire fringes for strain analysis may be a series of straight parallel lines, a series of radial lines eminating from a point, a series of concentric circles, or a pattern of dots. A moire fringe can be defined as a locus of points which have the same component of displacement in a direction perpendicular to the lines of the master grating. 10 A simple moire fringe pattern can be obtained by using transmitted light through a deformed moire grill and an undeformed master as shown in Figure 2.2. In certain areas, the light is blocked, causing a moire fringe. Exam- ination of the sketch will show that one dark band appears every time eight grill lines on the model have been stretch- ed to fill the space of nine on the undistorted master array. One may number the moire fringes consecutively starting anywhere. Then, beginning from the corresponding point on the model, the relative displacement between the model and the master is easily calculated to be U = Np where U = component of displacement N = moire fringe order p = pitch of master where Ex = strain in x-direction. From the above, it can be seen that strain in a direction perpendicular to the orientation of the specimen grating is simply the pitch of the analyzer grating multiplied by the slope of the fringe order plot at the point in question. (Hue steps from moire fringe pattern to strain distribution are illustrated in Figure 2.3. 11 DARK GRAY ‘ tJGHT V : GRAY i DARK GRAY LJGHT GRAY DARK GRAY lJGHT DEFORMED GRATINGXLUNDEFORMED GRATING INCIDENT LIGHT ll 1 ill 1 l 1 ll Figure 2.2. Formation of a Moire Fringe. FRINGE PATTERN ¥ __ II} FRINGE ORDER NO- DISPLACEMENT DISTANT FROM CRACK TIP (x-xg STRAIN DISTANCE FROM CRACK TIPIX-X.) Figure 2.3 Progression of the Data Reduction Procedure Required in Moire Strain Analysis. 13 For any displacement field, as grating frequency is increased, so also is the number of moire fringes. Increas- ing fringe density gives better definition to the displace— ment field. This results in increased efficiency in the differentiation of the experimental data from which the strains are determined. The use of high density moire gratings is sometimes not wise in that there tends to be problems in handling and application. 2.3 Producing Submaster Gratings Direct photographic reproduction was employed in the manufacturecxfthe several submaster gratings required for the optical data processing of the specimen grating photo— graphs. Several of each gratings lmndmm; spatial frequencies of 125, 215, 230, 260, 266, and 270 lines per inch were produced- These values are P and % times the fundamental spatial frequency of the master grating photographs (500 or 1000 lpi divided by magnification used) plus or minus various frequency mismatches. Figure 2.4 shows a sketch of the apparatus used. The 1000 lpi (or 500 lpi) master grating was held in a labora- tory clamp base and backlit with light from a slide pro— jector. A ground glass plate was placed about 3—5 inches behind the grating to scatter the incident light. The lens and camera were the same as those used in photograph— ing the specimen grating (see section 4.2.2). The whole set—up rested upon an optical bench. The camera to subject m so 0 exo: <¢madH mo coflpmono cam madam . Enommcmne Hmflupom ca mGHHODHHm Hafiummm How Emumwm Hoowpmo Ha m ousmflm m0<¢f om.m_002 :tz zmmmom mzia mzmn ._z_ _ u - I II _0 $93 anaemic» 3,420.53 . . 55: 35:3 30:2... to ..alv. $2.2m {i 3:5 2:2. m>m CO <¢w2<0 w2<4a SCOmw2hmnwh mH.N onsmflm — 4mx 40 2.9 Data Reduction and Plotting Displacements and Strains Three computer programs (named "Moire," "Cloud," and "HOOp") were used to reduce the data and plot the displace- ments and strainsdiscussed in this study. They were speci- fically developed by Cloud and coworkers to use in this and similar projects. Details about the computer programs are given in Reference (9). COpies Of all three programs are provided in the Appendix. Program Moire was used for single data sets. Programs Cloud and Hoop were used for composite plots. The later two programs will give the same strain results but they differ in the position Of the strain plot. For this study, Program Cloud was used for the radial strain plot for the coldworked specimen and for the strain in the direction parallel to the crack line of the compact tension specimen. Program Hoop was used for the strain plot in the direction perpendicular to the crack line of the compact tension specimen and for the tangential strain near the hole edge of the coldworked specimen. 2.9.1 Detailed Analysis and Plot of Single Data Sets Program "Moire" (Appendix) was used to obtain a detailed picture of moire fringe order, displacements and strains by printout and graphical output. This picture pro- vided a means to study the original input, the computed displacement field, and the computed strain map. The Opera- tions performed by the detailed analysis routine (Appendix) are as follows: 41 1. Read in the data containing the set designation (specimen identifiers), the moire sensitivity multiplication factor from the optical fringe data processing and the moire grating spatial frequency. Then,for an "at strain" fringe pattern, the distance between two fiducial marks, the maxi- mum fringe order tO be entered, and the distances from the fiducial mark to the intersection of each moire fringe with the y-axis under study are entered. The maximum fringe order and the fringe locations for the baseline fringe pattern are also read into the computer. 2. Generate fringe order numbers to match each fringe location entered as data. 3. Fit the baseline and the "at load" data with two continuous smooth curves by means of a cubic spline fitting and smoothing routine. 4. Interpolate the calculated curves to obtain the fringe number as a function of the distance from the fidu- cial mark at 100 points on the data and baseline curves. The maximum range of the curves is sorted out and divided by 100 to establish the nodes, which must be common to both the data and the baseline curves. 5. Subtract the baseline fringe order (unloaded speci- men) from the data fringe orders (loaded specimen) for each of the 100 points and multiply this difference by the pitch (the reciprocal of the spatial frequency) of the grating on the specimen, and then divide by the sensitivity multiplica- tion factor to obtain the displacement function uy for the 42 chosen y-axis. 6. Subtract the distance from the first fiducial mark to the hole edge or crack line from each nodal y-value, in order to have all result reported in the term of distance from the important specimen feature. 7. Compute by finite differences the first deriva— tive of displacement with respect to distance along the appropriate axis. This result is the strain at each of the 100 nodes. 8. Print, if ordered by the input control cards, all values of input fringe orders, displacements, and strains. 9. Scale the data and generate a plot of the input data and baseline curves. This graph shows fringe data points and the smooth curves. 10. Plot vertical displacement, uy, as a function of the distance along the choosen axis. 11. Plot vertical strain, ey, as a function of the distance along the axis. 12. Start over with the next complete set of data. Samples of each of three graphs produced by this rou- tine are shown in Figure 2.16. The main purpose of this analysis program was to allow a detailed study of each set of data and the results it produced. Input errors, such as faulty card punching or skipping a fringe during digitizing, were immediately evid- ent. As proficiency and accuracy increased, this routine was used only as a last resort for data sets which seemed C) O (‘J .00 DA .00 /1 ~00 43 C) C? C...) C") L.) C] L... (:3 C3 CW (_I'T (_CI C) L') If) C.) L? C) fi :04 DISPLQCEMENT z'J] l 0 --_-_-- -,___-.- ---—-..--.— l I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I p!) oh Figure 2.16b Plot of Distance from Fiducial Mark d—p--———_——-- 1 P I” I -05 IJ-‘r~ Versus Displacement. I u.) LI: DISTANCE FROM FIDUCIAL MARK (IN) I l O. k D I‘.) (D L) R -.---- 0MPR IT. .. U - ." f (- ‘. I I: {CL-DU 000 IJrILJ u-‘J C.) I”’_’" I l I I 3 . . DISTANCE FROM FIDUCIAL MARK (IN) 6 C“) C') P J D Figure 2.16c Plot of Distance from Fiducial Mark Versus Strain. 46 as if they might contain faults when processed by the less- detailed summary plotting routine. 2.9.2 Analysis and Summary Plotting of Multiple Data Sets Program "Cloud" and "Hoop" (Appendix) were used to plot many results on a single graph. These two programs are very similar to the Moire Program discussed in the previous section. They differ in the input requirements and, ob- viously, in the output. The purpose of these two programs is to generate a single plot of the calculated strain versus distance which contains several sets of the data obtained for any one specimen. Two sets of strain results can be obtained on each plane of the specimen because two-way gratings were used. For each coldworked specimen, radial strain, er, for several lines along the thickness were com— bined in a single plot, and transverse strain, 5 for 2' several lines successively further from the edge of the hole were also plotted together. For each crack specimen, strain in the direction perpendicular to the crack line for sev- eral lines at increasing distance from the crack tip were produced in a single plot; and strain in the direction par— allel to the crack line for several axes parallel to the crack line were combined in one single plot. Thus, the strain at any point on the same specimen can be seen on a single plot. An example Of such a multiple data plot is given in Figure 2.17. 47 .pmm memo mamflbasz 00 head RH.~ apnoea “.2: m2... xoH0m mcu wo wHom may 0» HwHHmumm coHuoouHo on» :H topmoq ocm pmomoHoz :sz cumuumm mmcHum mvuHor on» mo mgmmquuosm a.v oncon .cwEHome umoB wumconnmo>Hom wnu Mo wHom osu ou HMHsoHpcwouwm :oHuomuHo on» :H Umpmoq paw powwOHoz suHB :uwuumm mmCHHm wMHOZ wnu mo msmmumouosm m.v musmHm 67 .ovuoo “mummsaom omtz mo coEHoomm umma so omomoHoz squ cumbumm mmcHHm wuHoz wo nmmumouonm H.0H.v musmHm . a # A . . ...:E. . . 2.. I -~- ..m‘.‘ “M- .. *— w l“’ ...-..— - Hm -¢ “a"...‘r. r n "*rwr.~ ». . 68 .ovuoo umumo>H0m totz we :wEHome umme :0 two: mwumlcuv cuHB cumuumm wocHuh ouHo: wo Lawnmowozm m.0H.v whomHh g , . ......» .oquom “wumwsaom pmtz we :oEHowa umwfi :0 @004 amumlpum :qu cumuumn mUCHnm muHoz mo camumOuocm v.0H.v wusmHm ...; a 1. .2 __ _ ._ .ovuom uwumo>H0d UmxH: Mo :oEHoon amok so two; mmumlpcm cuHB :uouumd wmcHum wuHoz mo LamumOuorm m.oH.q whoaHL l, r , .....-- e .....o .ovnom Hmummxaom poxH: wo cwEHoQO umoe co omou mwumlumH nuHB cumuumm omCHum opHoz mo Lawanuocm N.OH.v musmHm E ... a, l ...”.wr 69 C) . i u 15 m 0 u a. 1.2.31.9...11: .... 1;... " 1rd LL...“ ...”. n 6001. . 1.118 . 01.9.3 “ D mflw m w. 1 H N 1 77;; m rrw I.“ D % m m DH E n O L n S E . .I IL L n i a u H: " VII . - Liz m U . J R " m.r m (act “ C l m m R u 9 T u r3 8 uuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuu Lnu-un-..u-u-.. ,TI 6233.030400 mmOmmm wJO: no mwom" G D m m _‘ a q q a q “T rd -0 A.u,0.0 v0 0 00.0 - N0.-0 .0 0 00.0 _0.0u ZHGKhw m>_memmzou .- . Radial Strain at Different Planes Along the Thickness on Left Side of Hole at lst Steo. Figure 4.11.1 70 mo. «I. u 10 H e ......bLL “ h 681 . t 1348 . u012...) . 0 g 0 O CO. H‘ n D fin.“ ..., -.. o N 94.74717 1 E . A G u . s E u 5 N m II. L " ..nm r... a 2 u l D u E P 01.. . .L On L2 “ O t O - - 1 u U H «on» L n -0. N a u U f . R f W F i R m 5 E D “ mmm t " 99, a " WT " .8 .m H mm m m w t ....................................................................... .u----------iun.u S 62.xm030400 axon—mm mJOI mo meow" l u a u .1 m m a "a 0 fix. w H! . M 8.5 3a 0.0 86 8.0 6.? 8.5 N z~¢mpm u>Hmmmmmzou Figure 4.11.2 Thickness on Left Side of Hole at 2nd Step. 7O 2 u 10 LLLL m 6001 . .1119, . 01233 . 0 C O. O 5‘ m .. E .u DU “ E m ...... mm It m .m no -1 m OH lLZ “ AU 1 u 0 L m rnw R m 5 . L m IO m 0 . fl -------------------wm..mmmmmwmm-.umobm 30: to moon: 0 u 5 u 3 1t . . . .m a .0 00.0 v0.0 _0.0 00.0 _0.0- memo Noun Z_EMHm m>~mmmmm200 1 (IN. DIsffiNCE FROM HOLE Radial Strain at Different Planes Along the Thickness on Left Side of Hole at 2nd Step. Figure 4.11.2 71 ..- Ho ...” TJLTHT. .. m 0 6001. . 3. «Linda “ T. CTR/”1) . 0 N . u E u 5 L O m r.” Qo to u 0 D L n on m 0 u 0 IL " .A u U m s . o. m 10 m o . 0. czimoaonoo wmohmm mac: .3 moomm o u s u a . . 4 . n. ..u .o mod .55 5.0. oo.o 5.0: moan Nome szzpm m>~mwwmmzoo 1 (IN. DISTRNCE FROM HOLE Radial Strain at Different Planes Along the Thickness on Left Side of Hole at 3rd Step. Figure 4.11.3, 72 0.06 .05 L0004L LEGEND B——‘€*-—*5 Z=0.016 G———4*-—~D Z=O.l?8l Z=O.24 7:0.381 0.04 +———+———+ .03 l 0 .02 l COMPR§SSIVE STRRIN 0-01 1 EDGE OF HOLE BEFORE COLDWORKING qP----——----r-————----_on-----------—---——-n------—-—---—-—-—-—--———-—-——~—--- ------------- -0.05 9.00 0105 SKID 0215 0120 0.25 DISTQNCE FROM HOLE [IN.} Figure 4.11.4 Radial Strain at Different Planes Along the Thickness on Left Side of Hole at 4th Step. 73 .05 RLDDDIR LEGEND LQLQNN II II II H 0000 O O O O (UIUI-‘O EDGE OF HOLE BEFORE COLDWORKING ..------_---r------_--_---_--__._---------_--------_----_-_-_--------..--------_-----_ .00 0 ,-0.01 q 0.05 0100 0105 0i10 0115 0120 0.25 OISTRNCE FROM HOLE [IN.] Figure 4.12.1 Radial Strain at Different Planes Along the Thickness on Right Side of Hole at lst Step. 74 a) a... ru 1 I 181§§ .9 PJ A... -0.016?§ 2:0013Oq ; 71={ 02”,)“ I flu r.) LEGEND ru a——4+——a . CP—-€>-—-*B Ih_+_4zw. v’IuOIII ...III- [- : RLDDDZR S I ,— J : PJ I I I I I I I I I I I 1 I "'Il..ll"'l.|lIl'-l"-I"I-|-'lI---‘Ial'""|-|"l-l-"'l--'l"-'-'-I-|'|' 02.5“.030400 wmommm MAC: .....0 meow PU .--L---------—------II-d----------Oc-o-----—---------0con—~- q .4 0 4L-------- '3 no _o.o oo.o .o.o- 2! CE 010‘ 0.03 ujo LL; :> (ON LDCD Lu 0: Q. 2: CD LJ {3. 1 (IN. 0 'E FE0M'H0LE SWNE - I D ness on Right Side of Hole at 2nd Ste f 8 Radial Strain at Different Planes Along the Thic] Figure 4.12.2 75 " H0 RRRR m 600.1. . 1348 _ 0123 n O .... . 2 w 88 m -.. E m nu “ DH E u 5 3 L J m .m. O n 0 N no “ IL " m R "2 m rflm m 0 - ............ o ............ I0 023.005.5400 wmouhwm MAC... “.0 moomm 0 m s n 0 q q q _I q A b“ 0.. .o no.0 «0.0 mo.o No.0 _o,o oo.o _o.o-. zEEm mimwwmmzou -..-...... DISTRNCE FROM HOLE ) (IN. Radial Strain at Different Planes Along the Thickness on Right Side of Hole at 3rd Step. Figure 4.12.3 76 0" C3 RL0004R 33.5.? r0001 13:48 019.3 0000 = = = = 22747: LEGEND 71'" II. 01": .I II II I. .‘l'll A w [J «L fiO .r.) .Fu flu F0 AU. 0.0 F.) to .. «rd L3 no. .. 00 Iq—--‘—-—--—-—L—-—a——-—--———-—-—-—-——---—---——-—---—‘------—--‘--u-------- I) ( _0..0I N. ..ll. DTSTRNCE FROM HOLE Figure 4.12.4 Radial Strain at Different Planes Along the Thickness on Right Side of Hole at 4th Step. 77 the radial direction was measured to within 0.01 inch (0.254 mm) of the edge of the hole. The results show that the largest strain occurred within 0.1 inch (2.54 mm) of the edge of the hole after the mandrel was pulled out. The strain is maximum at the tOp and decreases towards the bottom as shown in Figure 4.11.1 and Figure 4.12.1. The shape of the fringe line pattern obviously changes as the mandrel is pulled through. From Figures 4.10.3 and 4.10.4,one can see a modification of the fringe pattern near the top of the mandrel when the mandrel was pulled down for each step (the position of the top of the mandrel is shown by the arrow). When the tapered mandrel was pull- ed out, the strain on the bottom surface was slightly lar- ger than at the tOp, and the strains inside the specimen were smaller than the surface strains, as shown in Figure 4.11.4 and 4.12.4, because the diametral expansion was smaller than on the surface. The strain in the z-direction for the polyester speci- men was measured from moire fringe patterns to within 0.05 inch (0.13 mm) of the edge of the hole after the tapered mandrel was pulled through and the sleeve was still in the hole. The photographs of moire fringe pattern from the horizontal grill (horizontal fringes) are shown in Figure 4.13.1 to Figure 4.13.5; and the plots of strain in the z— direction on the left side and the right side of the hole are shown in Figure 4.14.1 to Figure 4.14.4 and Figure 4.15.1 .osnom smpmmsaom cmtz 00 cmEHowmm umoe co UoUmoHoz anB cs00¢mm mmcHHm mHHoz mo ammsmODOLm \ ~ 9‘10&.¢ I‘M\._~ '- H.ma.e mssmHm 79 .ovnow noumw>aom .ovuow uwumw>Hom waHz 00 :wEHome umwa co anon mmumInuv pwaz 00 coEHome umma :o coo; mwumlv:N :uHB cuwuumm wmcHum wHHoE no :mmum0uonm m.MH.¢ wusmHm AuH3 :uwuumm wmcHum muHo: mo numum0uonm m.mH.v ousmHm .. i. ......1. .. a iii 0 I $11.11..“ .1 I.-. .‘ . v u 1.! . I .0... "s. I'III. . I IYI. ....H.\... .1! ’ .ovnow Hmumwxnfiom .ovuow Hmuwoxflom vmtz mo cwEHowmm umwa :0 coca mmumlvum umxHS mo :wEHomow umwh :0 Umoq mwuqumH nuH3 :prumm mmcHum wuHoz mo :mmumouonm v.MH.v muomHm SUHB :uwuumm wmcHuh muHoz mo LamumODOQm N.MH.v wusmHh _. ... a“ .. _. . ... :..‘..9. _ . . 45.111010)... ... , 80 .mmum uma um 0H0: mo . oon ummq co mmch ucmHGMMHQ um COHuowHHclu cH :Hmuum H.vH.v.mHSCHm ..z Ema anom 20.500 20:... mozfima mm o amuo .o o no.0 ro.o-_ 4‘ 4. 0 n m m. — . . . . n r ........... L ........... I0Mw #1 m.u muw nu .nv d .d "I. ma . TL .3 23 .nv HS 1% MW -0nb .4. " .....u..1 .w mm. m. "a ma 0 . in" mm . - m.= wmm " _ ozmomh L hloqohm W _ I--I I u _ u Ir.U MIN 81 .mmum can cH mHom mo 00Hm ummq no moch ucmeMMHo um GOHuomHHUIN CH chuum 020 m0\/\<1 5 L\/\/\/\/\/\/\/\/\1[ . zmszowdm L mmmnmua (5.4) = 12.1 El €22 E /2wr COS '2' J Strain in Plane Strain [o = v(o + o )] zz xx yy - KI e 2 e 3e Exx = E/EFE cos § [(1-v-2v )—(l+v) Sln 5 Sin 7? ] + . K _I 2 ..3e Eyy E7??? cos 3 [(l-v-Zv )+(l+v) Sln 7 Sln 7T ] + ~(5.5) ZZ Displacement in Plane Strain N I -—- 6 u = 77 /§£ cos 5 [1-2\)+cos2 % ] + .. . 1T KI e 29 V = :7 Vg? Sln 5 [2—Zv — cos i] + .... (5.6) 109 KI is the stress intensity factor under tensile strain (mode I). The elastic stress equation shows that the stress intensity factor (KI) characterizes the stress fields near the crack tip; i.e. for the same value of KI, the same stress fields are obtained regardless of the crack length and/or the specimen geometry. But, the stress intensity factor itself is a function of applied stress, crack length and specimen geometry. In general the stress intensity factor has the form K = o/FE f(a/w) where o is the nominal stress 2a is the crack length f(a/w) is a function of specimen geometry and w is the specimen width. When plastic deformation takes place at a crack tip, a correction to the stress field equations is necessary. Several suggestions have been made for such a correction. Irwin (66) suggests that the 'plastic enclave' is bounded by the surface on which the stress is equal to the uniaxial yield strength, 0y of the material. The approximation of plastic zone size is r = l (33-)2 for lane stres y 2n 0 p S Y 2 r = l (is) for plane strain y 6n 0y The full width of the plastic zone is rp = Zry 110 Therefore, Irwin proposes that to take care of the effect of the plastic zone, an amount of ry be added to each end of the real crack. This corrected value can be used as the effective crack length for the calculation of the stress in- tensity factor Keff = o/n(a+ry) . 5.1.3 Crack Tip Deformation The stress and strain fields near a crack tip in a thigk plate are complicated. In a thick plate, the state of the stresses and strains change from that of plane strain in the interior to that of plane stress on the surface. This means that a triaxial stress condition exists in the interior, and biaxial stresses exist on the surface. This is true only if the plastic zone is small relative to the plate thickness (Reference 68). For a triaxial state of stress where the three principal stresses, ox, 0y, oz, are equal, there are no shearing stresses. This results in almost complete constraint against plastic flow, and the elastic stresses can be increased to extremely high values. In the case of most notched specimens, oy>ax or oz, so the normal stresses are not equal, and some shearing stresses do exist. Consider a notched thick plate loaded in tension to a low nominal stress level such that there is no local yielding at the notch root and the entire plate is elastic. The high longitudinal stresses OYY';E o/a/2r'set up at a distance, r, 111 ahead of the root (for r > p) cause the material there to extend elastically and consequently to contract because of the Poisson effect. _This contraction is greatest near the notch root (Figure 5.3) where the longitudinal stresses are highest. The area,A, that has been cut by the notch does not want to contract because there are no longitudinal stresses acting along it, all of these are concentrated a- head of the root. Since the unstressed area, A, tries to maintain its original dimensions while the material ahead of the root is contracting, transverse tensile stresses, 022' are set up in the contracting material. The stress 022 is a maximum at the center of the plate (z=0). Because the faces (xy-plane) of the plate are not loaded externally, 022 drop to zero at z : B/2, and 022 decreases with increas- ing distance, r, ahead of the root. In addition, transverse tensile stresses Oxx are also set up ahead of the notch by the constraints which prevent contraction in the width (x) direction and by the cantilever type deflection induced by the presence of the notch itself. The variations of the longitudinal and transverse (elastic) stresses Oyy and Oxx’ with distance, r, in the center of the plate (z=0), are shown in Figure 5.4. At any point 022 = v (orXX + oyy). Since oxx=0 at the root surface (r=0), local yielding will occur when the longitudinal stresses Oyy are equal to the uniaxial tensile yield stress. 112 \ 62 CONT RACTION Figure 5.3 Transverse Contraction that Occurs Near the Crack Tip in a Thick Specimen. These Conditions are Opposed by the Unyielding Faces "A" of the Notch; Consequently Transverse Tensile Stresses oz and Ox are set up Ahead of the Crack (Ref. 55). °51 (a) -n 3 ..é 2 2 0' (0'2=o,z=ia/2) (b) f"\0' oi. x 0.. —O'Z= V(O;+O;fl 2 = o (C) .61 = o _ N a; 07: x Figure 5.4 a) Variation of oz Across the Thickness, in z-direction at x=constant. b) Variation of o and Ox with x on the Surface of a Notched Piate (Z 2: 7)' c) Variation of o and 0x with x at Mid-Thickness (2:0) of a Thick Notched Plate (Ref. 67). 114 When a notched piece is stressed elastically, it is possible to produce high stresses near the notch that may locally exceed the material's yield stress to produce a small plastic zone or 'enclave'. The stress distributions within this enclave depend very much on whether the defor— mation is occurring in plane stress or plane strain. In plane stress, the smallest principal stress is that through the thickness, 022' and yielding occurs on the plane at 450 to the y and z axes. Yielding occurs when 0yy 22 ys ys where 022 = 0, Tys = yield stress in shear, and Cys = yield stress. This condition is found throughout the plas- tic zone. Thus, the maximum stress in the plastic zone is equal to the material uniaxial yield stress. In plane strain, the smallest principal stress is now Oxx and the yield is consequently in the xy-plane, with c -0 =2": =0 YY XX YS YS O.=G +0 yy YS XX Problems involving both elastic and plastic deformation around notches in plane strain becomes complicated because both elastic and plastic compatibility must be satisfied. Stress must be related to both elastic strain and to plastic strain increments. 115 (o ) " «q P‘ fir4 Figure 5.5 Stress Deformation hiFront of a Crack Tip During Local Yielding for (a) Plane Stress, and (b) Plane Strain (Ref. 67). 116 5.1.4 Fracture Behavior of Thin and Thick Specimens When plane stress conditions prevail and ry 1 B, where ry is the radius of the plastic zone and B is the specimen thickness, the fracture plane often assumes a 1’45 degree orientation with respect to the load axis and the plate thickness (Reference 57). This behavior may be rationalized in terms of failure occurring on those planes containing the maximum resolved shear stress (Figure 5.6b). (Since 02 = 0 in plane stress, a Mohr circle construction will show that the plane of maximum shear will lie along 1 45 degree lines in the y-z plane). In plane strain, where dz = v(oy + ox) and ry << B, the plane of maximum shear is found in the x-y plane (Figure 5.6a). Apparently, the fracture plane under the plane strain condition lies midway between the two maximum shear planes. This Compromise probably also reflects the tendency for the crack to remain in a plane containing the maximum stress. In very thick specimens, the plane-strain instability that occurs when K = K can cause the entire structure to IC fail so that KIC = KC as in Figure 5.7a. However, in many structures, complete failure does not occur at K = K and IC a macrosc0pic slow crack growth precedes a complete failure (Figure 5.7b). In this instance, the plane-strain fracture tunnels ahead in the central portion of the structure where the degree of plane-strain loading is greatest (Figure 5.8). The material on either side of the tunnel is then loaded in plane stress and it eventually fracture by shear rupture. 117 . $0 .03: Momuu m pcsou< coameHowmm mo momwa map 00 mafizmuo oapmamnom 0 Ammmuum mcmHmv mmme Hmmnm 3 mum tor—bum -Il-by-l/. . . . -.. --.: 2. I II _ v . h s _ ... "EO- -"\\§ \s tllIf' o\ \ \ -35... .v\ _ o Acflmupm mamamv came mcflm .m musmflm A0 ‘ (:4 ZOZbNm ’ 118 K or Loud K or Loud Fracture Kc _--------------- c K _WF actor. -----.“ . IQ 504 I—G)+— ® ——-I ®-microscopic slow growth -mocroscopic slow growth Figure 5.7 a) b) The Relationship Between Stress Intensity or Load and Change in Crack Length. (Ref. 56). Figure 5.8 $91.?“ 1.59:6} ‘ plane uralh 0",.!ll; I A—Av shear up 4 T B 4 l shear Hp Schematic Diagram of Crack Propagation in a Plate Under Mode I Tensile Loading. Letters a, b, c, d, e, refer to Successive Positions of the Crack Part. (Ref. 56). 119 Each of the several "pops" lettered in Figure 5.7b result from the jumps taken by the crack front in Figure 5.8. 5.2 Specimen Preparation Twenty rectangular shaped (3.9 in. x 4.0 in. (99 X 1021mn))pieces were machined from a single sheet of 3/8 in. (9.5 mm) Merlon Polycarbonate. COpper gratings were printed on five of them by using the stencil method (see section 2.4). Five specimen blocks were made with four pieces. The pieces were fastened together with epoxy (Epon 828 and Diethylene Triamine 100:8 by weight) by bonding a grating side on the first piece to a plain side on the se- cond piece. In addition, nickle mesh with 500 lpi (20 lpmm) and 0.008 in. (0.2 mm) in thickness was placed between the second piece and the third piece. Compact tension speci- mens were machined from these blocks. The dimensions of the compact tension specimens are shown in Figure 5.9. All specimens were polished on both sides to get smooth, clear and parallel surface which let the light pass through the specimen without scattering or creating Optical distortion of the embedded grating. Finally, copper gratings were printed on the outside surfaces near the crack tip by using the stencil method. Five thick multi-layer tension specimens were made for this study. One of them had a fatigue crack to make a very sharp crack tip. The fatigue crack specimen did not give a straight crack front through the specimen thickness, and it thus did not give a good comparision of the 120 080-75 IN- (Is-um) .1 (serum) a 51' a] E, El 0’ or 2 <3 ‘0 h—-+5fl'-—-- T’ | I | m , . 1F i s § 5 A. : : 3 3. 9 i = z a C) F\ : : E .0 12 ‘ ' ' .1 s - '3‘ 3 ; i 3 8 ”3 ; 5 5 v- 0 3 ; : l : E s i : E 3 : ii. .3 j E e 3 m 5: I-——I-5 m -—-I (75 MM) (:0 ma) Figure 5.9 Specimen Dimensions of Compact Tension Specimens. 120 DIO°75 IN- (mum) .1 .1 (sewn) & ii’ a «1’ E.’ a? or 2 <3 m k—I-SIN—II ‘1' . I . 0° . . 1F i a i E 5 ‘3 : z 3 3 a i : : a a .. ‘ ‘ 3 V z : r ‘i s - i ; i i 8’ "3 ; i 3 '_ O 3 ; 3 I : : s i i i : i E i I "F Figure 5.9 3|N00 i+——+5|N-——d (76 MM) (30 ran) M Specimen Dimensions of Compact Tension Specimens. 121 strain map around the crack tip in the interior and on the surface. For the rest of the specimens only a notch was used. In addition to the multi-layer thick specimen with embedded gratings, two similar specimens having embedded strain gages were made. Several thin specimens with gratings were also used. 5.3 Experimental Procedures In this investigation, eleven compact tension specimens made from polocarbonate with the same dimensions were studied. They are: i) one fatigue crack specimen (1.5 inches (38.1 mm) thick) with gratings on the surface-plane, the quarter-plane and the mid-plane. ii) three notched specimens (1.5 inches (38.1 mm) thick) with gratings on the surface-plane, quarter-plane and mid- plane. iii) two notched specimen (0.375 inch (9.525mm) thick) with gratings on the surface-plane only. iv) two notched specimen (0.75 inch (19.05 mm) thick) with gratings on the surface-plane only. v) one notched specimens (1.5 inches (38.1 mm) thick) with a grating on quarter-plane only. vi) two notched specimens (1.5 inches (38.1 mm) thick) with strain gages on the surface-plane, the quarter-plane and the mid-plane. All of these specimens were loaded in tension (Mode I). More details about each specimen are provided later. 122 'The compact tension specimens were set up as shown in Figure 5.10. Each specimen was loaded by static loading on the hanger. A slide projector was used as a light source for this set up. Light-diffusing glass was placed in front of the light source to get a more uniform intensity on the area around the crack tip. Photographs of the specimen gratings were obtained in the same manner as before,tflmadifference be- ing that the lens was a Carl Zeiss N., S-Planer with a focal length of 120 mm. and a maximum aperture of f5.6. The camera was set up to get a magnification factor of 2. The specimens were placed to let the light pass through them in the direction parallel to the crack front. Each grating was imaged separately by focusing on each grating plane to get the maximum sharpness and contrast over the whole area near the crack tip. The first data plate was recorded by focusing on the front surface and then the entire camera set up was moved closer to the specimen to get a focus on the quarter-plane and a second data plate was recorded. The specimen was then turned around and focused on the mid-plane and the third data plate was recorded. Finally, when the camera was focused on the back surface, the fourth data plate was recorded. For each specimen, four data plates were made for the unloaded specimen as the base line on each plane, and another four data plates were made for the load- ed specimen. .00 com cmeoomm mnu mo 0cH3MHQ oeumfiwnom .oa.m musmfim 20.02wh z. <¢wl<0 own—<04 INS-0mmm cumauuzu \/\/\/\/\/\/\/\/\lg mm<40 ozaoco . . L hzmv: >0..-~-‘------—- -----‘no-Qou- cacao---.... cun— .. — --- >zu<¢oopozs no MOM mqu xomuo may 0» HMHsoHpcmmHmm COHuomHHQ 0:» GH we QHmHum mommHsm mo HOHm one MH.m musmHm “.2: ~23 xoemo 205. 324.56 «mno eu..o m...o move oouo move- 26. euho- «new m o 00e.0nx + w l H meme; 1.. ..I . sway ....I .I ...... W O a atC( (IV I rfi LL m == vial- I 3 00-0 (NI/NI) vaals Y lO'O ...--l.__-.- n----..-. 30' O 128 .cmEHommm xomuo msmHumh o no we mCHA xomuo on» o» HmHHmumm coHuomHHo on» CH chuum womwusm one vH.m wusmHm A2: a; x0410 20m... uuzo mo “Cam aflonum one mm.m ousmflm ..z: :2... 558 20c... mozfima NnO vué 0.0 00.0 00.0 00.0.. 0.6.. vNéa N909“.U “ n b 0 . .V . . fl . n O a u ._ . o . a nil. . .. CH; of 4 . ) : may «T ..v .... mt- a m..- .. 011‘.le .3. . mm... w Jim Li I.“ J 0/ -J t - I :7. ... . ....-- ,_ .... a. _ . .1. .. _ L. NH. iwl . l. I 004) .1!vl."l..' o|«.\ C . .l: \. O: l (NI/NI) NIVHLS 201) 90-0 144 . . I my N. E. U. E. L. n . .5 6 . rUKufiUGLCJm r...) «L nunuo OCC- . .. . .00. 000 .. .. .. .. .. IIVuIIVI(. E. ...- 'QPL—EX-CTB J O .. O 0°32 oTz4 oils f O-OB DISTANCE FRO" CRACK TIP ("0” 1 0°00 0'08 00.0 cm... ‘ '00 No.6 .ziz: 232.5 00 couc- 000-. Figure 5.29 Strain 5x on the Quarter Plane of a Notched Specimen. 145 «MO .coEHoon Gong-«oz M NO ocoamlpwz onu so um mo no.3 sumo-Hum ose om.m ousmflm 1. VNO 0.0 ..z: 000 000 .. oz... 545 :2: 32355 8.0- 2.? 5.? «no -’-— — ------C- I I - 20:0- r 000 90-0 (NI/NI) NIVULS 20°C 90-0 (>32 .-.------.s-e. i - .fi 055%.... 3 E: r3 2030 _.| D. mmmmw _ u N VA 0 O .0..- H E “flea-TX. E N G. _ (. E $3-; L w m . T |-__ D- WK so,“ - x M .00 wt. 5 - , A R C m “R to F E C N A T o B . -------------------- nVD IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII " uIIUI I do - . . . 8 . O . - -. . . O 00.0 00.0 #00 N00 000 N00- ¢O0- 00.0.. .52: 225m on the Mid-plane of a Notched X Strain e 'Specimen. Figure 5.31 147 strain 8y along crack line is found to be tensile strain. At this point the results of the strain in the interior of specimen obtained from the experiment do not agree with theory. The interior strain from the experimental results needs to be considered and a correction of these results have to be made. R. C. Kerber and J. S. Whittier (37) suggested that the embedded grid measurements of internal motion in trans- parent models, in some cases, can be seriously in error owing to the internal optical refraction caused by stress gradients within the model. For this study these errors were created by both the change of index of refraction and the changed of thickness of the specimen. While the speci- men was being loaded large strains occurred near the crack tip and changed the index of refraction of the specimen. Furthermore, due to the strain of contraction, Ez’ the thickness close to the crack tip is changed. As pointed out by R. P. Kambour (38) crack propagation in polymers do not behave as it would in the simple elastic, ideal brittle solids for which the original formulation of the theories was intended. In fact, it is known today that in these materials, crack propagation occurs by the formation of more or less substantial amounts of craze material followed by failure. R. P. Kambour (38) and D. 148 Hull (39) concluded that the main characteristics of a craze in transparent, isotropic polymers are: l. A craze is a highly localized region of plastic deformation. 2. Crazes formed in a uniaxial tensile stress field have a similar shape to a crack, and the plane of a craze is at a right angle to the stress axis. 3. The craze volume has a lower density than the surrounding material. Because the crazing creates holes smaller than the wavelength of light, the craze behaves as an optically homo- geneous medium and has a refractive index con- siderably lower than that of the normal polymer. As pointed out by Vincent (42), the development of a plastic zone in a polycarbonate is more complex than in the other polymers. While increasing the load, the develop- ment of a plastic zone at the crack tip of a polycarbonate specimen is not the same as in the other types of polyester such as PMMA. This notion was explained in an experimental work by Vincent (42). For PMMA the kidneybshape (or butter- fly-shape) that occurs at the crack tip becomes larger and larger with increasing load. This corresponds to the development of a plastic zone in almost all materials. This kind of development is seen in a polycarbonate specimen in the early stages too. While applying a small load, the local strains are below the yield strain and the zone is 149 kidney-shaped (or butterfly-shaped). In the second stage, however, the local strains are above the yield strain, and the zone becomes wedge shaped. Finally, at higher strains, there is a kidney-shaped zone within the wedge. The schematic of the development of a plastic zone of a polycarbonate crack Specimen is shown in Figure 5.32. In this investigation, while the specimen was loaded, it was possible to have a local yield strain at the crack tip higher than the material yield strain, and a wedge shaped plastic zone (the second stage of development of plastic zone in a polycarbonate specimen) formed near the crack tip. A craze occurred within this zone and caused the index of refraction to change near the crack tip. Clearly, the results that were obtained by taking a photograph of the interior grating through a portion of the specimen will be erroneous. This effect was demon- strated by making two compact tension specimens with the same dimensions from the same sheet of polycarbonate but with thicknesses a quarter (0.375 in) and a half (0.750 in) that of the test specimens (1.50 in.). Copper gratings were applied on one side by using the stencil method. Specimen gratings were recorded by taking photographs on the same surface plane with two types of set-ups. First a photograph was taken directly of the specimen surface by putting the grating side near the camera lens. Then, the specimen was turned around and the photograph of the speci- men grating was taken through the specimen thickness. 150 (c) Figure 5.32 Development of Plastic Zone Size on the Surface of Polycarbonate Crack Specimen (Ref 42). a) Kidney—shaped Zone of Deformation Near the Tip at Small Load. b) Wedge—shaped Zone at Higher Extension. c) Internal Kidney Within the Wedge of Still Higher Extension. 151 The moire fringe patterns that were obtained from these set-ups are shown in figure 5.33 and Figure 5.34. The plots of 8y are shown in Figure 5.35 and Figure 5.36. The results show that the strain, ey, on the same surface plane obtained from different sides of the specimen are totally different. The strain ey near the crack tip ob— tained by taking a photograph of a specimen grating direct— ly from the near surface is tensile strain; but the strain 6y obtained by taking a photograph through the specimen thickness is compressive strain. The magnitude of the strain 8y near crack tip obtained by taking a photograph through the specimen thickness is much larger than that obtained directly from the surface grating. The interior strains obtained by simple means are shown to be in serious error due to internal refraction. For the observation to be valid, the measured strains in the interior must be corrected. It was necessary to develop a suitable correction procedure. 5.4.2 Correction Procedure From previous sections, the measurements on the same surface plane obtained from different sides were found to be a lot different. The observed difference depends on specimen thickness. Assume the strain difference of a very thin specimen is zero. A plot of the strain difference (strain error) of the very thin (2 zero thickness) specimen, the quarter thickness (0.375 in. thick) specimen, and the half- thickness (0.750 in. thick) specimen (see Figures 5.35 .mmonxoflns noEfluoam on» nmoonne mmom m >n .n .mmochflnB noEfluomm on» nusounh mmom m >n .n can onoam can onwam oummusm no mcwumuo on» ou wauoouwn .m ooowuom onu no wnflumno onu mo xauoouflo .a nmmumouonm m mnfixma an onfiq nomuu nmmumouonm o mnwxoe an onfla xomuo on» cu on» on HoHHmumm nofluoouflo on» :fl .2“ own-on Hoaamumm coauoouwo on» an .cw mnm.o. nofiwoomw mmocxofine uaom m we onoam ououusw noEMoomm mmonxofins uouumzo o wo oomwusm on» no vo>uomno mnuopuom omcwum ouwoz one vm.m ouswfim on» no uo>uomno manouuow ownaum oufioz one mm.m ousmflm OwDuomno nfiouum one 02.. m2: x0 Til-oil}. . O nw .....- .ww W a .. ET” .0 _ Z 3-0..- -. I H O C 7. - -_ H. m u r 3 5 r.--..-.....--------.------.-... .M 1 -_ aw/fiv- .....oM-w-fl-lr A -\.... O .. ..\ . ... .. Hx. no 0 Z ....- .. , ....I ..I. o llllll KL? l-MIII-‘l ...-lulu. II I I foul-Ln p... a .10 r .1 . . u I ’ u. . _ . . . . H _ 90-0 (NI/NI) NIVULS .‘tfi‘fi' 3'“?- 5". ~.:.' ._ " mm“ 154 _N00 .noEfloomm on» 00 .o. nmoumOponm onflq noouu onu moonxoflne onp nmsoune .n. pno ooomusm on» 00 mauooufln o mnwxou an .GH «\m mo mmonxoflne noEHoomm onn mo Aha. on Hoasoflpnomnom noauooufla onp nH po>nowno nHoHum one 72.. 02.... x0<¢0 20mm woz 08.0 n s .0 l onomflm 0000 2060 000 900-0 (NI /N|) NIVBLS 60 0-0 BIO-0 0050 .CH omo.onx um onoam noouu on Hmasowcnomuom onon onu on .m»....m>.a>.monflqnunouomwfio no Houum nflouum no uon one mm.m onomflm .2: mmwzxo...... 20.2.0me m~m0 157 a}? >‘.‘ 3"?! S. -----. ---. ..-- --..--..- ----.. ---.--.--..--..l >1’ ......1b 5 00.0 n 5 .v00 :. n00 5 N00 5 .00 a. 000.0 c. 0000 u x £99;- 000 900-0 EGO-0 00-0 (NI/NI) NIVULS 600-0 ZlO-O 158 schematically in Figure 5.40. The photoplates of the specimen grating on the same plane were recorded from both sides of the specimen. One was recorded through a quarter of the specimen thickness and the other was re- corded through three-quarters of specimen thickness. While the specimen was loaded the strain measurement obtained from the strain gage wasa small tensile strain. Moire fringe patterns obtained from both sides are shown in Figure 5.41 and Figure 5.42 The strain 8y obtained from both sides is shown in Figure 5.43 and Figure 5.45. The results from each side can be expressed as: e1M 81R + A81 + As E3M 83R 3 where is a measure of the strain on the quarter- €1M plane by taking a photograph through one- quarter of the specimen thickness. 83M is a measure of the strain on the quarter- plane by taking a photograph through three- quarters of the specimen thickness. and e are the real strain on the 81R quarter-plane, 3R 81R = €3R’ since it is the same plane. A51 is the strain error on the quarter-plane obtained from a quarter thickness side. A83 is the strain error on the quarter-plane obtained from a three-quarters thickness side. firmwatfywmnzv v ~ ..7 ' 159 [STRAIN GAGE 1* I-— -u \' NICKEL MESH PLANE 0F NICKEL MESH AND STRA‘N GAGE K’ ' ° ' ' f Nickel F ure 5.39 Schematic Show1ng the POSiton 0 lg Mesh and Strain Gage on the Quarter Plane of Specimen. 160 GRATING PLANE ” / vvvvva I .A/vvvvxI; GROUND SPECIMEN LIGHT GLASS LOAD IN CAMERA DIFFUSER TENSION b)———.- E GRATING PLANE Figure 5.40 Schematic of Photography Process a) Through one-quarter Thickness Side and b) Through three-quarter Thickness Side. .0cflq xomuo mcu ou undoufipcwmuwm An can HmHHmumm Aw :ofluowufln map :H mwflm wmwcxoflne Hounmovlwmnca :msoune mmmm ammuUOuosm m mcflxma hp mcflumnu mamam mco cues cwEHowmm mo MeanwusH wcu ca :uouumm mmceum wufloz N¢.m wusmfim 2: S w 0... . a .. a . a . a I I. . 9‘ an! r .9, .g‘ 161 .a omo<04 .mcflq xomuu may ou amasoflpcwmuwm An can HwHHmumm Am cofluomuflo may :H wcfiw mmwcxofize _E V . Hmuumso m nmsouca mwmm ammuvouonm m mcflxme >9 mcflumuu wcwam wco ruw3 coEHowmm osu mo HOflHoucH 0:9 cw :uouumm wmcflum wufloz av.m wusmflm 2: 5:. at: Ya Dun—<04 .mcfiq xomuo man on amasofiwcwmuwm An can Hwaamumm Am :ofluooufln may 2% ocfim mmocxofics Mouumswlwwuce nwoouna mmmm nmmumoaonm m wcfixma >2 mcflumuw macaw wco sues coEHowmm mo HOfiumucH on» :H :umuumm mmcflum whee: N¢.m musmflm 161 .wCflq xomuo on» on HMHDOHccwmuom An 0cm Hoaamumm Am :0wuomufio on» Ca wpflm wmmcxofise _E V ._ kuumso m cadence wmmm cmmumouonm m mCflxme >2 mnflumnw mcmHm wqo sue: :wEMoomm wcu wo uoauwucH mnu :H :uwuumm museum oufloz He 2. amiwd Omo<04 m wuawflm 162 .cmeflommm on» mo mpflm HmuHMDOIoGO map £050H£B £0MH00¢0£Q m mcflmme >9 cumppmm mmCHHm mufloz mnp Eoum oocflmpno moauoucH may CH m m0 poam camnwm one ..z: uz... .658 22.... mozfiba .mq.m musmflm ~n0 , . vN.0 0.0 00.0 00.0 00.0.. 0.0- 0&0: Nn.0...0 r P P L m b D 0 . b n . O . l m o u __ lo l O z n ”I -IJJr II I PHIII . m .-““'|---'-'-‘-----'-'-----"¢"-l' "‘--'-'-"-‘-'-- - ........... IL IF _ LI IIIIIILII. ...I.II .H IIIIIITIITNITI I... I . ., a r L. .F: N4pr (I . I .l.H.uI.JflI\N-III . a»! F. ‘1 H. JIFIPI 4 P I I \fid . ..I (--....I . . . _ . . . _ HI v ..J.-. . — -3; c c. I: - -1- I--. . .. . .J ... I \ . - .L I. . (I > «NI I aP- 1.. u .H. ... ...” n > ,HIII IA»..- In. . “A. I D J .l‘ . I . ., _. > .....I- -- IETII --. . . a II" . . l I! J .II.II - II lu~ J Al) II M . i l . _ ._ n . L. _ . _ FL _ a -... I. I. _ . \ / I I . . . m 708 mv N m .I. mom 0 z #04) 162 .coEwoomm onu mo opflm HouHooOIocO onu amooune smonmoooam m mcflmme an unoppom omcflum oHHo: on» Eoum posflowoo Hoauouce onp ca w wo uoam cflmupm one ..z: oz: 555 22.... 32455 .36 33$. «3 . «To 2.0 mob Sb 8.? 2.0- 5.? 264w - p p p . . b n b P o u 0 . .9 . . . . l o . . u —_ o . v o . Z l O u . . .. -...IIthI 0 m .-.».M-w......m..+.---.+. ----..HI.-.U_...,.I . ............... u ........... o W. H .r III... I 'r \JI—y - IIIINIIfiuII . o I . r A r IJFL. r I I- \3 . N LIIIIIKMII . . n n N . / I _ I 1:3 x I... I 1-1--..r . N I I. r f I > .«vi- I ant In... “ fio ( I r- , HrII IS? If... n w _ . > Hi. {Ami I i u . .\II‘. . . .. . . ‘4] .. I ..I] - In Id I, ‘4‘; .... q .. a... . w . . III . m . r... h 4. -...L L .-.I_L.. m m \_/ .IL. _ ..I_ _ . _ 904) 162 .noEHoomm onu mo opflm HouHooOIono onu nmsoune nmoumouonm o mnflmme en nuoppom omnflum oHaoz onu EOHM ponflonno HofluounH onp n0 0 mo uoam camnum one Aézm ouomflm . ..z: oz: 065.0 :05 “02.2.0.0 «no , 3.0 0.0 00.0 .000 000- 0.0- 03. woo-mu F f P p. m n p P 0 _ .7 . m __ 0 .. u o l O z n - . n-"-‘|--'I'-"‘-‘A|"|'-'---|II.|'--| ||---"'--'-“--- _ “PI h b IIII-III‘ITIIIIIII- - ..IIII. Hr I.-I I- . - . AIL ,L \IVJIHrI' . I I AMHI I _II,I\ C . . A-fi III-H 0 r_ I .u III-\IJ. . uh. VIIIIIIIIOHIHI ..IIII - . c _ h I V J . - J _ .- .. III I .-_II I II r . I. I _ I I. r f I > «NI I, .aul . a - I, II:..I.0- II. n I. I II. ,I. . y .rI- .nnIIIII-. . . - II] .. I. - .-.I. . II. I .41. I — ”//,_ I“ _ .IIIW m n \— L III- — m ~ _ FL _ 4 IL I. ah. . . u x/ I... .I. . _ 00:0 (NI/NI) mvms 20$ 904) 163 a 9 . oI I . I-’ 3“ I’... r-\ I I :;"_It1I\‘JI.J ' .. 1 r::——--~e.:«—~-- 1‘ Izrr-rnzs 0 : :--—-—-e——-~—«=_ 1:33:5- 9. . : ...-...--- 'I':I'.‘!-.'JC‘3 c) : Tr ~I % I: SEE: - e—u-«ae——«~x (:-U:¢UG I : T F g : i I I i T" E- X 6‘ ' i I .2. I \ I z. 84 I U . | as ° = < I c I .- : 4'5. 0) - g \ 1 0I ----- ; :2?"- ' i N I 9- I I 9 : . O I ' g : ;:>_. 0 : O o‘ : 9 : o : 9 1 '-O-08 0'00 0'08 016 0°24 032 Figure 5.44. DISTANCE FROM CRACK TIP (IN'I The Strain Plot of ex in the Interior Obtained from the Moire Fringe Pattern by Taking a Photograph Through a Quarter-Thickness Side. 164 .noEHoomm on» No opwm HouuoovIooHne on» nmsonne nmmumouonm o mnemoe en nuouumm omnflum onfloz on» Eoum conflmuno HoauonnH onu ca 0 00 uon nwmuum one .2: wz... x032 I. .m n H - — I a. ......“ " .. r3200 a... nu .. I . nUFJF-u .. . I. ..ILW __ : : : : H . _ \ . 4 N JI «III-«III . FIJI. \‘L. " O O 2 ...w 4 .. . A..I . v . \.I E. j... j. . , 2..., u on B. M w n u it” .m _ I”. . . X W rIM AM? n P . . m-..I.._ . ,l . ..I w. . "...-01?} ~=.. " 1”WK I . u om .. x. u a I A \. n no \ .. m w -K). ” .Wnfl u . u 1 . F N EN . o L . E . c - . A Z /a. _ T o S I .......................... .0 D .----....--------.. I” - O . . . . . . .8 i ., o I . . I I. . . o 00.0 00.0 ¢0.0 No.0 00.0 Av0.0- 00.0- .ziz: 54F;- N0...0- . The Strain Plot of 5x in the Interior Obtained from the Moire Fringe Pattern by Taking a Photograph Through Threeéquarter Side of the Specimen. Figure 5.46 166 To find the strain error, the difference of strain measure- ment from both sides is given by - A51 At this point, assume that the strain error, As, is a fun- ction of material property, 6, and the thickness of the specimen. The observation that the error is proportional to the square of thickness leads to the expression, d 2 A8 = (E) (S where 6 is the material property that affects the index of the refraction while the specimen is loaded . d is the distance from the grating plane to the surface plane on the camera side. t is the specimen thickness. The strain errors from both sides are 2 [3.1: (if) a — (x) 5 2 A53 =(Qé33—925 = (3/4) a 83M - 81M = (3/4)25 - (1/4)26 = 1/26 SO' 5 = 2(€3M ' 81M) Finally the real strain on the quarter-plane is mud (9 e - e 8 :8 = 1M 1R 3R The real strain on the mid-plane can be obtained in terms of the correction factor found from the quarter-plane obser— vations by extending this idea. 167 - Ae €2M 2 2 =€2M-(g_é—2—}t_)6 1 82M - 5 (83M - 81M) ..... (II) The real strain in the interior of specimen is calculated by using eq. (I) and eq. (II) to correct the data from moire fringe patterns. Tfimastrains ELM and 82M were obtain- ed from the test specimen with two gratings in the interior, but a was obtained from the specimen with one grating in 3M the quarter plane of the specimen. 5.4.3 Final Experimental Results 5.4.3.1 Experimental Results from the Moire Method The results given in Section 5.4.1 for the specimen which gave the best results on the surface-plane, quarter- plane and mid-plane were corrected by using eq (I) and (II) in Section 5.4.2. For this Specimen, the data plate of the quarter-plane grating obtained by taking a photograph through a three-quarter thickness sidecthinot record a good grating, due to the camera lens having to focus through the copper grating on the surface plane and the nickle mesh grating on the mid-plane before getting to the grating on the quarter- plane (three-quarter thickness side). COnsequently, the moire fringe patterns and the strain plot from the three- quarter thickness side could not be obtained from this specimen. The final resultsjknrthe specimen with gratings on the quar- ter and mid-plane were corrected by using the results from 168 two different specimens. The quarter-plane (e and the m), mid-plane, 22m, results obtained from the first experiment, and the quarter-plane, 53 results obtained from the three- M' quarter thickness side from the specimen with one plane grat- ing (quarter-plane) in the interior were substituted in eq. (I) and eq. (II) in Section 5.4.2. The corrected strain plots and constant strain contours on the quarter-plane and the mid-plane are shown in Figure 5.47 to Figure 5.50. The results show that the strain 6y around the crack tip on the quarter-plane and mid-plane is tensile strain and the magnitude of the strain 5 on the mid-plane near the Y crack tip is larger than on the quarter-plane. The plots of the strain 6y along crack line on the surface-plane, quarter-plane and mid-plane are shown in Figure 5.51. The results show that the strain ey in the interior along the crack line near the crack tip is larger than the surface strain. The difference of the strain 8y on the quarter-plane and the mid-plane is small. At the same distance from the crack tip through the specimen thickness, the strain 8y near the crack tip is maximum on the mid-plane and decreases to a minimum on the surface-plane. Consider the strain plot and the constant strain contours of 8y in the interior and on the surface. The peak of the maximum strain in the in— terior is on the crack line. The peak of the maximum strain on the surface (see Figure 5.20), however, does not lie on the crack line. There are, instead, two symmetrically located strain peaks above and below the crack line along 169 D (O ‘? oi ; G) x: 0050 m. : (3) x =o-07s m- 3 ; 6) x =o-noo m- 9. 3 Q) x =o-I25 m- 0) : Io O) . 94 3 ° s 2 N \ a 8 ~— 6i 5. i 5 n- 94 Ohio 0 E O . 0- _ _- - - o I _ i o i C) 2 o 1 I 1. . . I -o-os -o-04 -o-02 o-oo o-oz 0-04 0-06 DISTANCE FROM CRACK LINE (IN) Figure 5.47 A Plot of Strain 8y on the Quarter—plane. 170 CD . o-oozoo lnIIn 3?. , ® . o-oono In/In ° ® - oooouao Iva/In ‘ © - o-oouza Win 6) - o-oouoo in/In 2 6‘ \ \ \ o \ \ \. “ “ \ \\| ‘ \ \ ‘. ‘ 0 \‘ o. > .‘ \ \ u ‘) o I o I 0°00 d.---—----o-—--.-o—..-o- DISTANCE FROM CRACK 'I’IP (IN) é f - . . '-0-I0 -0°05 0°00 0°05 0°l0 DISTANCE FRO" CRACK LINE (IN) Figure 5.48 The Constant Strain Contours CV on the Quarter-plane. ' 171 D 8 5) 5 G) Xao-oso in ; ® x=o-075 in 3 ; ® X=O°IOO In 3 3 @ x =o-125 in n . ..‘O z 9. z o E _z_ N g 8. r- o (D 5 s 9. ' 0 i O I 0 9L---” --.-.-°'_--.---_--_-.,_-.-------_-.- ..... ° : 5 i 9 I 9 -’ooe -o-'04 -o-'oz oroo 0102 604 0106 ousnnce non cmcx LINE (m) Figure 5.49 A Plot of Strain 5y on the Mid—plane. 172 9 6 x -. g 1 (D: 00026 III/In a o ®=O-0020In/In ,—_ 6' ®-o-oous III/In x ©=o—oouo III/In g ©=o-ooos III/In C o 3. \ 3 ° g ®® ® ® ® I I "' : u o l 5: ° 3 P : {’3 : O : I) I 9 I O . I . . '-c>|o -005 0-00 0-05 mo DISTANCE FROM CRACK LINE (IN) Figure 5.50 The Constant Strain Contours ey on the Mid-plane. 173 mcon mmcmamlpflz paw Hmpumso .mommusm map so ho mo comHHmmEOU < .maflq xomuo map 523 a: x0<¢0 Son: woznuu .. \235 56 :15 FSPL QPL 0 I. 3666 . 42m :- . $23 MPL BSPL ll J l1 44444 gamma mOSOm u. nun Figure 5.58 Positions of the Strain Gage Installed on NOTE: Each Plane of Specimen No. 2. FSPL = Front Surface Plane; QPL = Quarter Plane; MPL = Mid Plane; 3QPL = Three—Quarter Plane; BSPL = Back Surface Plane; T = Crack Tip; E = The End of the Specimen Width. 185 Table 5.2a 'Position of Strain Gages on Specimen No. 2 I Gage DISTANCE (IN) { Position FSPL QPL MPL BSPL ! 0-1 0.065 0.054 0.054 - T-l 0.067 0.058 0.050 0.038 1—2 0.124 0.102 0.114 0.120 , 2—3 0.118 0.098 0.126 1.164 g 3-4 0.140 - - - 5 4—5 0.906 — — - g 5-6 0.114 — - - f 6-E 0.061 - — - 3 3-E — 1.272 1.240 0.208 Table 5.2b Strain Results From Specimen No. 2 Gage STRAIN (IN/IN) position FSPL QPL MPL BSPL 0 0.00264 .... 0.00283 - 1 0.00079 0.00191 0.00315 0.00132 2 0.00022 0.00072 .... 0.00038 3 0.00003 . . 0.00052 0.00058* 4 -0.00010 - — - 5 -0.00100 - - - 6 _0.00138 - - — * The strain values of ex. 186 The differences between these two specimens are: 1) On specimen no. 1 all gages were installed to measure the strain along the crack line only, but on specimen no. 2 gage no. 0 was installed to measure the strain ey near the crack tip but not on the crack line (see Figure 5.58). ii) On specimen no. 1 all gages were installed to measure strain near the crack tip only, but on Specimen no. 2 two gages (gage no. 5 and no. 6) were installed to measure the strain ey near the end of the width of the specimen on one surface. (nIthe other surface, one gage (gage no. 3) was installed to measure the strain ex near the end of the width of the specimen (see Figure 5.58). The dimensions of the compact tension specimen withthe strain gages are the same as those of the test specimens with the gratings. The set-up of the specimen with the strain gage is the same as before except that no light source and no camera were used. Each gage was connected to a VISHAY 1011 (Vishay Instruments, Inc.) portable strain indicator A photograph of specimen set up is shown in Figure 5.59. Each specimen was loaded in tension to 143 ksi (9.10 MPa) the same as the specimen with the grating. The strain .results on each plane wereobtained from the strain indica- tors. The strain results from’all gages on specimen no. 1 «and no. 2 are shown in Table 5.1b and 5.2b, respectively. 'The plots of 8y on each plane are shown in Figure 5.60. 137 Figure 45.59 Overall View of the Specimen Set Up. 188 .mmmw camuum 0gp tom oocuoz oufioz map Eoum owcflmuno .mo .ucmcomfioocfimnum onu m0 COHmHummEOU d om.m musmfim “2: n2... xu I - I P O O m . O o 0 s m u n w N W m o m z 0 0 Jam 40 ‘ 4 * 4&2 no: ”02.0mmm _.02.0wmm w¢_02 20:..001 ”040 2215.0 9000 189 The results show that close to the crack tip, the strains 6y in the interior and on the surface are very large tensile strains when compared with those at a distance farther away from the tip. A high strain gradient occurred near the crack tip. The strain 6y is a maximum on the mid-plane and decreases to a minimum on the surface plane. On the quarter- plane, strain 5y changed to a compressive strain at about 0.55 in. (14 mm) from the crack tip. On the mid-plane, gage no. 4 did not work, but it seemed that the strain 8y changed to a compressive strain somewhere between 0.5 in. to 0.6 in. (12.7 nun to 15.2 mm) . On the surface plane, large tensile strain occurred near the crack tip then decreased and be- came a compressive strain at about 0.375 in. (9.5 mm) from the crack tip until the end of the width of Specimen. As pointed out by Irwin, Tada and Paris (43) applied stress on a compact tension specimen has both tensile and bending components. _ a h d From KI — 6N (I/w-a) “W W" W‘ where 0 = 0 + 0 N N(tension) N(bending) = ....P__ + 6P(a+(W-a)/2) W-a (W-a)2 if W = 2a _ E 2 ON — a + 9a vw‘ _—‘ ‘6‘ -v 190 Because the bending stress is much larger than the pure tensile stress, the bending stress causes the compressive strain from the end of the width of the specimen.' Tensile strain near the crack tip was caused by both applied ten- sile stress and bending stress. The strain plot of Ex is shown in Figure 5.61. From Table 5.1a, strain ex on gage no. 1 is slightly smaller than strain 8x on gage no. 2. Therefore, tensile strain ex in- creased with distance from the crack tip and the maximum strain occurred somewhere between gage no. 1 and gage no. 2. It then slightly decreased until the end of the width of specimen. The strain plots of Ex and CY on the surface plane from the crack tip to the end of the width of the specimen along the crack lineenxashown in Figure 5.62. The results show that the strain 8y is tensile strain in only a small area near the crack tip, but Ex is tensile strain from near the crack tip to the end of the width of specimen. Strain 8x close to the crack tip is much smaller than Ey, therefore, strain 8y is more important than EX in studying deformation in front of the crack tip. Fromlhble 5.2b the strain 6y of gage no. 0 on the surface plane is larger than that of gage no. 1 that lies on the crack line. Because the maximum strain 8y on the surface— plane did not lie along the crack line, there were two symmetrically located strain peaks above and below the crack line. This result is consistent with the result obtained from the moire method. On the mid—plane, the 191 .momw :amuum map can oozumz mufloz may Eoum owcflmuno umnu .xw .ucmcomfiou Cam-mum 93 m0 comwnmmfiou d $.m 0.30am 02: ...... ¥U<¢0 20¢... Nov-(Pm:— no so no «.ro .16 oo 0 mu m -I - - III-II------------------II----- .0 I -- 4m 6 m m .0 0 m u .m a II 0 0 4mm .M n I ....a .... .620qu m0¢0 2215.0 ”2.08 20.5.50...— 906-0 (NI/NI) NIVULS 192 2 8. o _A— 6x —0— 6, J o o o 3‘ \ 2 o . \o z o = 8 \ Z 0 ___________ ____________ __ ,__ _, __ _ z o \O " ” ” a: g. (I) on o 0 SI 9 8 o . - s (100 060 POO IQO DISTANCE FROM CRACK TIP (IN) Figure 5.62 The Plot of ex and a From the Crack Tip to the End of the Width on the Surface Plane of the Specimen with the Strain Gages. 193 resultant strain on gage no. 0 (at 0.054 in. (1.37 mm) from the crack tip, see Figure 5.58 and Table 5.2a) is about 7% smaller than that of gage no. 1 on the crack line. There- fore the peak of maximum strain in the interior lies on the crack line. This result is again consistent with the result obtained from the moire method, but the strain difference of these two positions obtained from the moire method is about 60%. The strain 8y obtained from gage no. 0 on the mid-plane is about 50% larger than that obtained from the moire method at the same position because the correction strain from the moire method is accUrate only on the area close to the crack line (within 0.02 in (0.51 mm) from the crack line). At a distance farther away from the crack line, the result is not quite as good because the curve of the strain error does not satisfy the equation (As = 0x2) that is used for correcting the interior strainIas shown in Figure 5.38). More data would be needed to get a good-fitting curve for each line. Many specimens would be required to get an equation for the curve of the strain error along the thickness for each point. The purpose of this research, however, is to study the strain close to the crack tip that causes severe deformation. The result then shows that in the interior of the specimen, the maximum strain occurred along the crack line. Because we are only concerned with the analysis of the strain along the crack line near the crack tip, it is not necessary to make more specimens to get better results for the area away from the crack line. 194 The measurements of strain 8y from the strain gages on the surface-plane, the quarter-plane and the mid-plane were compared with those values obtained from the moire method (see Figure 5.60). The moire method gives good results only from 0.025 in. (0.635 mm) on the surface plane and 0.05 in (1.27 mm) on the quarter and mid-planes to 0.125 in (3.175 mm) away from the crack tip. The strain values from the moire method are slightly higher (about 1%) than those from the strain gage, but the two methods agree quite well. The measurements of the strain ex obtained from the strain gages on the surface-plane and the quarter-planewere also compared with the moire method measurements (see Figure 5.61). The results show that, for both methods, the magnitude of the strain ex on the quarter-plane is slightly higher than on surface-plane. The results obtained from strain gage can not give a good strain plot near the crack tip along the crack line because the distance between the gages is too large. Only the results from the first gage from both the quarter-plane and the mid-plane were compared with the strain from the moire method. The magnitude of the strain gage value is in a good agreement with that of the moire method. 5.5 Discussions In this investigation, strain deformation was studied close to the crack tip along the thickness of a polycarbonate crack specimen. Experimental results are compared with 195 available theory. -From the theory of linear elastic frac- ture mechanics and the stress—strain relation as described in Section 5.112, the strain 8y on the surface is predicted to be larger than on the mid-plane. The strain measure- ments from the moire method are found to be in disagreement with the theory of linear elastic fracture mechanics be- cause, in almost every case, higher strain was measured near the crack tip in the interior, even though only a small load was applied to the specimen. Vincent (41) suggested that in some cases such as for the fracture of notched specimens of isotropic polyestyrene at room temperature, the zone of plastic deformation is small. It is possible, then, to calculate stress distribu— tions and to derive the stress intensity factor assuming linear material behavior. On the other hand, in many prac- tical cases, the size of the zone of plastic deformation is not negligible and theories based on the assumption of Hooke's law can be in significant disagreement with experi- mental values (42). Liu (33) pointed out that when the plastic zone size is small the elastic relationships may no longer be valid, but unique relationships between the stress intensity factor, stress and strain do exist. It follows, then, that the theory of linear elastic fracture mechanics cannot be applied in this case. Several research- ers (74, 75, 76, 77) have now determined the variation of the stress intensity factors along a straight crack in the 196 standard compact tension specimen defined in ASTM E 399-74 by using elastic three-dimensional finite element programs. All of their results show that the stress intensity factor is a maximum at the center of the specimen and a minimum at the surface. For the same condition, Schroedl and Smith (50) uSed photoelasticity techniques to study the stress intensity variation between the full thickness and a center slice of a compact tension specimen made from PLM-4B. In the center slice, K was found to be 5 to 10% higher than I average through the thickness. From these results and.the relation between local stress and the stress intensity fac- tor, the stress on the mid-plane is shown to be larger than on the surface plane. Experimental study of the defOrma- tion at the crack tip on the surface plane and the mid- plane of specimens made of steel were carried out by Lequear and Lubahn (48) and by Robinson and Tetleman (49). Lequear and Lubahn found that the radius of curvature of the notch in the bent specimen was greater at the specimen mid-section than at the outside. Robinson and Tetleman measured crack tip opening displacement at the tip of bent specimens by using the rubber infiltration technique. A low alloy pres- sure vessel steel, A533B, was used to make the specimen (0.394 in thick). The technique consisted of filling the crack or notch with a catalytically hardening silicone rubber. After the rubber had hardened, the casting was removed and examined in the scanning electron microsc0pe. They found the crack Opening displacement at the tip on the 197 mid-section of the specimen to be larger than on the sur- face. Considering all of these results, because the stress intensity factor (Kl)’ stress (0y) and crack tip opening displacement (CTOD) on the mid-plane are larger than on the surface plane, and because fracture initiates at the mid-plane, then the deformation on mid—plane near the crack tip should be larger than on the surface plane. Thus, strain on the mid-plane along the crack line near the crack tip must be larger than on the surface. The reason why strain 6y in the interior near the crack tip along the crack line is larger than on the surface was thought to be that, while the specimen was loaded, the contraction strain occurrs near the crack tip on the surface of the specimen. The contraction strain increases while increasing the load. This effect causes reduction of the crack tip opening dis- placement and ey near the crack tip on the surface plane. In the mid-plane, however, the transverse strain is zero. No contraction affects the mid-plane, and the crack tip opens easier than on the surface. At the same time, no strain reduction occurs on the mid-plane. Therefore strain deformation on the mid-plane is larger than on the surface, and it causes fracture initiation on the mid-plane. The elastic solution (Boyd (72)) predicted that, at A. the crack tip (r ~ 0), 0 = 0 and 0y on the mid-plane is X larger than 0y on the surface. This prediction agrees with the experimental result reported by Martoff, Leven, Ringler and Johnson (73). It is possible to have 0y on the 198 mid-plane larger than EY on the surface plane right at the crack tip. ‘For r < 0, that is, within a small but finite region near the crack tip, Irwin (78) and Cotterel and Rice (79) suggested that, K1 8 Cy (13,0) = m]?- + 0 (r ) K (5.7) -—l— + 0(r%) Ox (r,0) = /2FE ' Oox The added term, Oox’ corresponds to local stress acting parallel to the crack center at its tip. 0(rg) are higher order terms which customarily are assumed to be negligible If eq. (5.7) is substituted into eq. (5.3) (page 107) one obtains 1 K 5y (r,o) = §[I1'V) 21 + VOOX] for plane stress wr _ 1 2 K e (r,0) — — (l—v-Zv ) ( I ) + v(1+v) 0 for plane Y E ’2}; ox strain For the material used in this study,\)= 0.45. For the thick specimens used it is assumed that the surface-plane is in a state of plane stress, and the mid—plane is in plane strain. If KI on the mid-plane is larger than on the surface-plane, and if Cox is a large value, then it is possible to have 8y on the mid—plane larger than on the surface plane for r > 0. It follows that the strain 8y along the crack line on the mid—plane must be larger than 199 By on the surface-plane. This conclusion is supported by the results obtained in this study. The Cox seems to be an important factor for the conditions of this investigation. 5.6 Summary of Strain Field in Thick Compact Tension Specimens The embedded grid moire method was used to measurethe strain near the crack tip on the surface and in the interior of a compact tension specimen that was made of polycarbonate. Strain error was found in the interior resulting from the fact that the specimen thickness changed and its index of refraction changed near the crack tip. The results obtain- ed from the moire method in the interior of the compact tension specimens were not real strain. Other specimens were made and studied correct these interior results. Finally, compact tension speciemns with internal and exter- nal strain gages were made. The results of the strain plots from the crack tip along the crack line for both Ex and 8y on the surface and in the interior from the strain gage and the moire method were compared and found to be in good agreement. Near the crack tip the strain gradient of 8y is high. This strainnear the crack tip is tensile strain which de- creases and changes to a compressive strain at some distance from the crack tip. The maximum compressive strain occurs at the end of the specimen width. On the mid-plane the strain EX is compressive strain at the crack tip. It in- creases to a maximum tensile strain at about 0.07 in. 200 (1.8mm) from the crack tip, then decreases until the end of the specimen width. Strains 8x on the quarter-plane and on the surface—plane are tensile strain from the crack tip to the end of the width. ex on the quarter-plane and on the surface-plane increases from the crack tip to a maximum at about 0.09 and 0.07 in. (2.3 and 1.8 mm) from the crack tip, then slightly decreases until the end of the specimen width. Near the crack tip, tensile strain 8y is much larger than ex, and therefore, the strain value 6y is more impor- tant than ex. The severe deformation near the crack tip is due to strain 6y Near the crack tip maximum tensile strain 8y occurs on the mid-plane and decreases to a minimum on the surface- plane. Therefore, the crack front will grow on the mid- plane first, then extend to the surface-plane. In the in- terior, the maximum strain 8y starts out from the crack tip and lies on the crack-plane, but on the surface, maximum 8y splits and lies along two lines radiating from the crack tip. CHAPTER 6 CONCLUSIONS In this study, the embedded grid moire method was de- veloped. The specimen gratings in the interior and on the surface formed moire fringe patterns of comparable quality. This successful technique was successfully employed to measure the strain in the interior of coldworked specimens and compact tension specimens. Two types of transparent material, 60:40 of flexible-rigid polyester resins and poly- carbonate, were used to make the specimens. Strain gages were used to measure the strain in the compact tension specimen. 'Theresults were compared with the moire method and found to be in good agreement. First,the:strain around the coldworked specimen was studied. Radial strain, transverse strain and hoop strain were measured by using the embedded grid moire method. Good results were obtained by this method. The strain was different along the thickness depending on the thickness of specimen and.the shape of the mandrel while the specimen was being coldworked. The strain was not uniform along the thickness and the strain measurements on both sides were not quite the same. The strain in the radial direction 201 202 inside the specimen was smaller than on the surface after the specimen was coldworked. The maximum strain occurred near the edge of the hole and decreased with distance from the hole. The strain in the z-direction,cn:transversestrain, was tension near the top surface and changed to compression along the thickness; the maximum occured near the mid—plane and it decreased towards the bottom. After passing through a minimum, the transverse strain increased in compression again near the bottom (about 0.08 in (2 mm) from the bottom). The h00p strain was maximum on the top surface and decreased slightly to the minimum at mid-plane. The change in this component is only about four percent of maximum value. It then appeared to increase toward the bottom surface. Second, the compact tension specimens that were made of polycarbonate with the gratings on the surface and in the interior were studied. The moire method was used to measure the strain around the crack tip on the surface and in the interior. Good results were obtained directly on the surface plane. fNueinterior results were corrected by making one specimen with gratings on the quarter-plane only to provide data for a new correction procedure. IBM) polycarbonate compact tension specimens with strain gages were made to validate the results obtained from the moire method. The strain measurements near the crack tip along the crack line from both methods are in good agreement. The strain gradient of 8y is very high near the crack tip. Strain 8y was tensile in only a small area near the crack tip, then changed to a 203 compressive strain at a distance father away from the crack tip. The characteristics of the plot of 2 along the crack x line from the crack tip are the same for both types of test specimens (the specimen with the grating and the specimen with the strain gages) on the surface and in the interior. Close to the crack tip, strain 8y along the specimen thick- ness was maximum on the mid-plane and decreased to a mini— mum on the surface plane. ex on the quarter-plane and on the surface-plane was small tension at the crack tip. It increased to a maximum and then decreased at distances further away from the crack tip. On the mid-plane the strain ex was a compressive strain at the crack tip and increased to become the overall maximum tensile strain at about 0.07 in. (1.8 mm) from the crack tip. It decreased, like Ex’ on the quarter-plane, with distance from the crack tip. In this investigation, at the crack tip, 8y was much greater than 8x on both the surface and the interior planes. The severe deformation at the crack tip will, therefore, be caused primarily by the strain 8y. It is possible, then, to start fracture at the mid-plane while loading in tension, because a larger strain occurs on the mid-plane. It is not possible to present here a complete analysis of the three-dimensional strain around the crack tip. It is important to briefly mention, though, some basic obser- vations made in using the embedded grid moire method in the study of interior displacement and strain. The experimental 204 measurement by the embedded grid moire method of the in- ternal strain in the transparent compact tension Specimen of the area around the crack tip was found to be in very serious error due to the non-uniformity of the index of refraction caused by a craze zone (high strain gradient) and because the thickness changed around the crack tip. In such instances, the embedded grid appeared better suited to measure the development of the dilatation near the crack tip rather than to measure the occurring strains and dis- placements. For the study of coldworking, on the other hand, no correction was necessary. Most of the resultant strain from the coldworked specimen was found to be compressive strain (the craze zone does not form by compressive strain) and the specimen that was used to measure hoop strain is a lot thinner than compact tension specimen. No serious error was found in the coldworked specimen. It is shown that in some cases, however, static problems can be very seriously in error. In order to gain confidence in such techniques, further experimentation with known theoretical solutions is needed. One physical phenomenon observed during the experimen- tal measurement of the strain around the crack tip on the surface and in the interior of the compact tension specimen was a crack tip opening displacement in the interior of the specimen. It was thought that the crack tip opening displacement measurement in the interior could be obtained by using the moire technique as has been done on the 205 surface (44). The results illustrated in Figure 5.16.1, however, show that the grating record could not give good enough fringes for the interior immediately behind and ahead of the crack tip. Because of an optical problem, while the compact tension specimen was loaded diffusion occurred at the crack tip in the interior, the density was changed, the thickness was changed, and thus the index of refraction changed. Consequently, the moire fringe pattern in the immediate vicinity of the crack tip in the interior cannot be used for measuring crack tip opening displacement. 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W., "Plastic Strains and Energy Density in Crack Plates, Part 1 - Experimental Technique and Results", Exp. Mech., 335-344 (Nov. 1964). Kerber, R. C. and Whittier, J. S., "Moire Interferometry with Embedded Grids-Effect of Optical Refraction" Exp. Mech., (May 1969). Kambour, R. P., "Stress- Strain Behavior of the Craze", Polymer Engineering and Science, Vol.8, No. 4, 281- 289 (Oct. 1968). Hull, D., "The Microstructure and Pr0perties of Crazes", Deformation and Fracture of High Polymers, Edited by Kauch, Hassell, and Jeffee, Plenum Press, pp. 171-189. Kobayashi, A. S., Engstorn, E. L., and Simon, B. R., "Crack Opening Displacements and Normal Strains in , Centrally Notched Plates", Exp. Mech. Vol. 9, No. 4, ; 163-170 (1969). 1 Vincent, P. I., "Ductile Crack Growth inIkflQr(Ethy1ene Terephthalate) Film, Polymer, 12, 534-536 (1971). Vincent, P. I., "Load-Extension Curves and Fracture Toughness", Deformation and Fracture of High Polymers, Edited by Kausch, Hassell and Jaffee, Plenum Press, 287-300 (1972). Tada, H., Paris, P., and Irwin, G., "The Stress Analysis of Crack Handbook", Del Research Corporation, Hellertown, Pennsylvania. Luxmoore, A., and Wyatt, P. J., "Application of the Moire Technique to Fracture-Toughness Tests on Zirconium Alloys", J. of Strain Analysis, (Oct. 1970). Polymer Fracture, Edited by H. H. Kauseh, Springer— Verlag Berlin Heidelberg, Germany, (1978). Fatigue of Engineering Plastics, Edited by R. W. Hertzberg and J. A. Manson, Academic Press, (1980). Hahn, G. T., Sarate, M. and Rosenfield, A. R., "Plastic Zones in Fe-BSi Steel Double Cantilever-Beam Specimens", Int. J. of Fract. Mech., 7, 435-446 (1971). Lequear, H. A., and Lubahn, J. D., "Root Conditions in a V-Notch Charpy Impact Specimen", Weld. Res. Suppl., 33, pp. 585-8 to 588-8, (1954). Robinson, J. N. and Tetelman, A. S., "The Relationship Between Crack Tip Opening Displacement, Local Strain and Specimen Geometry", Int. J. of Fracture, 11, 453-468 (1975). 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 210 Schroedl, M. A. and Smith, C. W., "Influence of Three- Dimensional Effect on the Stress Intensity Factors for Compact Tension Specimen", Fract. Anal. ASTM STP 560, 64-80 (1974). Theory of Flow and Fracture of Solids", Edited by A. Nada1, McGraw-Hill Book Company, Inc. 1950. Plastici_y, Edited by A. Nadai, McGraw-Hill Book Company, Inc. 1931. Theory of Plasticit , Edited by O. Hoffman and G. Sachs, McGraw-HilI Book Company, Inc. 1953. Fracture Process in Polymeric Solid, Edited by B. Rosen, Interscience Publishers, John Wiley and Sons, Inc., 1964. Fracture and Fatigge Control in Structures, Edited by S. T. Rolfe and J. M. Barsom, Prentice-Hall, Inc., 1977. Fracture of Structural Materials, Edited by A. S. Tetelman and A. J. McEV1ly, Jr., John Wiley and Sons, Inc. 1967). Deformation and Fracture Mechanics of Engineering Materials, Edited by R. W. Hertzberg, John Wiley and Sons, 1976. The Physics of Glassy Polymers, Edited by R. N. Haward, AppIIed Science Publishers, Ltd. 1973. Elementary Engineering Fracture Mechanics, Edited by David Broek, Sijthoff and Noordhoff, International Publishers B. V., 1978. Experimental Techniques in Fracture Mechanics, 2, Edited by A. S.Kobayash1, The Iowa Stress Press, SESA, 1975. Numerical Methods in Fracture Mechanics, Edited by A. R. Luxmoore and D. R. J. Owen, Proceedings of the First International Conference held at the University of College Swansea, Swansea SA2 8PP, West Glamorgan, U.K. (Jan. 1978). Experimentaletress Analysis, Edited by J. W. Dally and W. F. Riley, McGraw-Hill Book Company, Inc., (1978). 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 211 Kambour, R. P., "A Review of Crazing and Fracture in Thermoplastics", Macromolecular RevieWs, Vol. 7, J. of Polymer Science, Edited by A. PeterlIn, John Wiley and Sons, pp. 1-154 (1973). Moire Fringes in Strain Analysis, edited by P. S. Theoearis, Pergamon Press (1969). Cloud, G. L., "Simple Optical Processing of Moire Grating Photographs", Experimental Mechanics, V. 20, No. 8 (August 1980). Irwin, G. R., "Plastic Zone Near a Crack and Fracture Mechanics", Seventh Sagamore Ordnance Material Research Conference, (1960). Fundamentals of Fracture Mechanics, edited by J. F. Knott, John Wiley and Sons, 1973. Liu, H. W., "Analysis of Fatigue Crack Propagation", NASA Contract Report, NASA CR—2032, National Aeronautics and Space Administration Washington, D. C., May (1972). G. T. Hahn and A. R. Rosenfield, Acta. Met., 13, p. 293 (1965). Sharpe, W. N., Jr., and Paleebut, S., "An Experimental Study of Dugdale Model", the Fifth International Conference on Fracture ICFS at Cannes (France), March 1981. Dally, J. W., and Mulc, A., "Polycarbonate as a Model Material for Three-Dimensional Photoplasticity", Transaction of the ASME, J. of Applied Mechanics, 600—605 (June 1973). Boyd, G. M., "From Griffith to COD and Beyond", Engineering Fracture Mechanics, 172, Vol. 4, pp. 459- 482. Marloff, R. H., Leven, M. M., Ringler, T. N., and Johnson, R. L., "Photoelastic Determination of Stress- intensity Factors", Experimental Mechanics, pp. 529- 539 (Dec. 1971). Tracey,1).lfl., "Three-Dimensional Elastic Crack Analysis," Nucl. Eng. and Design, Vol. 26, No. 2, pp. 282-290, 1974. ReyneanJ., "On the Use of Finite Elements in Fracture Analysis Cd'Pressure Vessel Components," Trans. ASME J. Press. vess. Tech., No-75-PVP-20, pp. 1—9, 1975. 76. 77. 78. 79. 212 Raju, I. S. and.Newman, J. C., "Three-Dimensional Finite Element Analysis of Finite Thickness Fracture Specimens. NASA Technical Note, No. NASA TN D-8414, 1977. Neale, B. K., "The Stress Intensity Factors Associated with Curve Crack Fronts in a Compact Tension Specimen." Numerical Methods in Fracture Mechanics, edited by A. R. Luxmoore and D. R. J. Owen, Proceedings of the First International Conference held at the University of College Swansea, Swansea SA2 8PP, West Glamorgan, U. K. (Jan. 1978). Irwin, G. R., "Analysis of Stresses and Strains Near the End of a Crack Traversing a Plate," J. of Appl. Mechs. (Sept. 1957), pp. 361-364. I Cotterell, B. and Rice, J. R., "Slightly Curved or U Kinked Cracks," Int. J. of Fracture, 16 (1980) 155- i 169. APPENDIX Computer Program and Subroutines 213 PROGRAM MOIRE (INPUT, OUTPUT=65) C C COMMON /INTP/ YINT(101.2) COMMON /DIFY/ YDIF(101)DY(100) COMMON /STNAM/ ISTNAM(9),IST COMMON /PLOTER/ XRAY(900),YRAY(900),INUM COMMON X(80,2),Y(80,2)NPTS(2),XPL(101),XL,XJ,YL,YH,XMIN, XMAX REAL M ' LOGICAL FIN C FIN=.FALSE. IC=0 INUM=0 C C ---ENTER RUN DATA C 100 IC=IC+1 CALL READIN(P,M,C,XO,IPR,FIN) IF (FIN) GO TO 1000 ISET=ISTNAM(IST) g ---DETERMINE X-RANGE FOR INTERPOLATION & DELTA VALUE C CALL RANGE(DEL) g ---COMPUT INTERPOLATED "SMOOTH" CURVES THRU DATA & BASE SETS C CALL INTERP(DEL) g ---PLOT INPUT & SMOOTHED DATA C CALL PLOTM(1,ISET) g ---COMPUTE CURVE DIFFERENCE & DERIVATIVES C PMC=P*M*C CALL DIFF(PMC,DEL) g ___CORRECT ABSCISSA ARRAY C CALL CORRECT (x0) 6 ---PLOT DIFFERENCE CURVE C CALL PLOTM(2,ISET)' E ---PLOT STRAIN CURVE CALL PLOTM(3,ISET) C 214 C ---PRINT OUTPUT IF DESIRED C 1000 IF(IPR.NE.1) GOTO 100 CALL WRITEOUT(IC,DEL) GO TO 100 CONTINUE STOP END 215 PROGRAM CLOUD(INPUT,OUTPUT=65) COMMON /INTP/ YINT(101,2) COMMON /DIFY/ YDIF(101),DY(100) COMMON /PLOTER/ XRAY(900),YRAY(900),INUM COMMON x(80,2),Y(80,2),NPTS(2),XPL(101),XL,XH,YL. YH, XMIN, XMAX REAL M LOGICAL FIN FIN=.FALSE. IC=0 INUM=0 ---ENTER RUN DATA READ 1, ISET FORMAT(A10) 00 CALL READIN(P,M,C,XO,IPR,PIN) IF(FIN)GO TO 500 ---DETERMINE X-RANGE FOR INTERPOLATION & DELTA VALUE CALL RANGE(DEL) ---COMPUTE INTERPOLATED 'SMOOTH' CURVES THRU DATA AND BASE SETS CALL INTERP(DEL) ---CORRECT ABSCISSA ARRAY CALL CORRECT(X0) —--COMPUTE CURVE DIFFERENCES & DERIVATIVES 000 000 O PMC=P*M*C CALL DIFF(PMC,DEL) IC=IC+1 IF(IPR.EQ.1) CALL WRITOUT(1C,DEL) GO TO 100 500 CALL PLOTR(IC,ISET) STOP END 216 PROGRAM HOOP (INPUT, OUTPUT = 65) COMMON /INTP/ TINT (101,2) COMMON /DIFY/ YDIF (101), DY (100) COMMON /PLOTER/ XRAY (900), YRAY (900), INUM COMMON X (80,2), Y (80,2), NPTS (2), XPL (101), XL,XH,YL, YH, XMIN, XMAx REAL M LOGICAL FIN FIN=. FALSE. IC = 0 INUM = 0 ---ENTER RUN DATA READ l, ISET FORMAT (A10) 00 CALL READIN (P,M,C,XO,IPR,FIN) IF (FIN) GO TO 500 XO=.2 C C C l l ---DETERMINE X-RANGE FOR INTERPOLATION & DELTA VALUE CALL RANGE (DEL) ---COMPUTE INTEROLATED SMOOTH CURVES THRU DATA AND BASE SETS CALL INTERP (DEL) ---CORRECT ABSCISSA ARRAY CALL CORRECT (X0) ---COMPUTE CURVE DIFFERENCES & DERIVATIVES 000 000 O CO 000 PMC=P*M*C CALL DIFF (PMC,DEL) O IC=IC+l GO TO 100 500 CALL PLOTH (IC,ISET) STOP END 217 (2),xpL(1o1),XL,xu,VL,vu,xnIN,anx 0 0 0 O p- 0 D 0 U’ 0 A A I: o I- 2 0-000 '13 H 000 < u. :00 --l s u: c: x 0’- GDP 0 In 0. C- I‘D-I- 0 I- Lu H In 0 c 0 _l D \ 000$ 0 D .J or—m D DUI: A < 3000- I- O A 0D I- 3 Ho. OAA I on- \ I u I: c o 0 - o H >- v‘ \ £9 PUMA O c v ‘0‘" 0 ' OI- DID X < Wm A I-I-I—ID A0- A I- m 3‘ \ 0 £900 0 F A Z ' < EU V20 0‘) "I! C ' '0 UI" \ m G Ct 2 Pk I- A UQILIUX I-ID ' P. 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ZXZXUZHGHJU O m c N m K w w HEIGHTS oESTDRE DATA HITH ZERO .- 0* O \ cow 0 0‘ ”C 0 co “1" .F0 AZ EH I 2 2m row N qu O 237 A MPO 0* '0 \O \ M+m AAO~AO~AAA M3~N~£2£ Avvm+m£££ MOGO‘IOVVV V>3 V CCC GHHA3AX>3 XAAEVI-I II II II M¢W 000 A P A4» 65 qvv o POJJC ,N,xo,vo,un,no,x~,FN,GN,DN,THETA,Q,IABS(IPRINT)) .91 93 fN3in MPHPNIhIAAAFZDAAAPMA m ZHAI£+ *A'I' +¥2¥¥¥¥ I I’D-xxx ' ' ‘54:) VV'ZfiXfififiVHVVVVXN A O G F v )- A JAG mm~ Q~A Irv UGO HFP VVV hhu 22H 0H0 h>> H «\x 3m» bu UZH 2H0 H\\ p... :22 ODD m2: VVVXQVAPGGSE mun"uuvvvuuuooouzcooounu1XSux>3xQHumwwuu If. 0 v— (' O~-O~ connon X(80,2),Y(80,2),NPTS(2),XPL(101),XL,XH,YL,YH,XMIN,XHAX UU XPL(K),VINT(K,1),YINI(K,2),YDIF(K),DY(K-1) , XPL(I),YINT(1,1),YINT(1,2),YDIF(1) _I w a \ x t I X \ 2 H U! ‘- HX O P \ S s \ omommm z I—I-I-I-OI-H zzzrmzr- o-w-n-u-I Hz KKKCDKD SRO-EGO“) U If C) PRINT 120 238 A \ \ \ \ X NX < HO I - X 8 \ § ‘0 2 0 H I—N I uu- \ X mm X s s In I I 8 I" )- I \ \ II 8 ..I W N )- H: >- \ VII-I 8 2 >1 \ X .JX X s <3 O ..I 2 \ P XA ‘3 ‘ V1 0A8 Ac ‘P\P PI- I-\\>- Ov (\O: P)- O s O \ VD : Nx -I.J In I PC no a: luv- XI HO: \ A \ \ DU 8 A OAA ImIIIU\ ONM D _I \ pvo I XC: VII)!- um 22 >I-V U< H AAagx P=Pl< I—N \20 3.4.401 U ‘A NJ lOM.IDA HOONZ 3 : S a: fire- s \ DD ND NA mVVOA : UXUX \ HI-u-QN N-BINO VZI-ovm X=\N mmCM- V0 I->-> \V 0V): 1- o Au. \OF-HI-U ..I\\N: \UPOZV 0.0.>- \m 5" Lu: unnA I'll-OH ‘- Q \ c ‘- UZHK‘ 3X: XLUXI 2HOV2 PU" #Usfr- H\\XD I—I— PHI-#2 32220) ( "112,8.6 635 eséS AA FF \ \ ZHI-I OO wv VV PX)- mean II II II HHPI-IAA XXV I-II-I ((MOV" PP!)- AAQPOO _I—IZ ..I-I ((IIC‘DC 0920:: O u UUU I. UU HUD!- 239 A A P N \ \ m m \ § F P \ \ 2 2 \ \ > > G D 4 4 D O z 3 Q \ m X AA m X OOOOG NN 00060 00004 §\ 0.004 OCOiO ZHH OOO¢D uuuu: QHV «nun: AAAAV PX» AAAAV PNFNmAnflfl PNPNUO U¢+++ZNHAAw++¢+ZL DZZZZHV HH22222Hm 2vvvvd¢°vvzvvvvd HXX>> PNX>HXX>> D hoooc4mwoohocc04p 2444442 44244444 CDCDG‘"DCODDOOO¢C UZIIIUZDIZUIZIZU¢ 110 120 A m 0 o s m G I I s .A OF ‘3 CO on \\ #P NO IO \- :I A\ O. 20 HO v\ I Nb 4\ AA CN NN 2P A¢€ \ 099 :3 ~00 O 0“ m2 INN km P¢¢ I ‘00 “I“ q... 09 O§§ 2‘ All (4 ‘FN pm >ss mm 900 HH 4GP DO 6“ =: P :00 \\ A m UP‘FU h 00 AH cmOOX\\ C OOPHV .0000!- 4 \JOVB 0.0.40? a OOw4H IGIOOVV OC\&G uuuuzzz m VVPX> AAAAV44 U mmnuu NMwaIIO 2 HHHAAwOOOtzmmU w XX HH=*PPPH<HXX>> O m 44Nochoocc444h K 44 4474444444 H <1:OOOOOCO<<HX 44MGOPD 44 4424 ((DDDDO 900219: 0 O u r m X,HOLDY,N,1,5,1) x,n0ch,~,1,5,2) m AA m cocoa NN 00060 0000-; \\ ....4 OCO¢O ZHH OOO¢O "NH“: IHV "NH“: AAAAV PX) AAAAV FNQ—AIIJJA II II II FRIFR'UO U++++2NHAAw++++ZL SZZZZHV HH22222Hm zvvvvdmovvzvvvvd HXX>> PNX>HXX>> O I—OCDC4QPCGI-DGOO4I- 2444442 44244444 COOOGCHOCDOODDO\\ WW Q00 HH 40? on C\\ :: P :00 00 AH CmOOX\\ OOPHV cocoa-o ssovm 000040? 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II 4444 H HP <UU%I>>>UUU?IUUUU I I I I I I I I I U UUU U MU U 241 LR5L0T(IZ.O,O.O,999) D 4I-DG 2222 II (LBZDOC'I-IHU UQUWUUOGZ "-EXIT PLOT ROUTINE C C --INIT PLOTTING PARAMETERS C ,25?,0) ELLA V‘. DOVN-N CD I0 I0 H0 00 o v I o I 0 VIII II II II I—AAAA Cc-filw-N 40000 QPPPF VS'VV 4>->->->- 4<<<< <0:me uxx>>l --PRINT AXES AND TITLES C ‘242 \ 0 O C‘ \ “I I \ 0 ‘0 \ I 4‘ 0 N 0 I \ \ 0 : o A h- I \ A A 2 no 0 \‘r H 1- ‘ A'- V \ I 00 3 o s 0 IL! 2 \ I \ _J H AA: A OQ' O < NN 0 ‘- 2 a: sfs‘r 0!- : 0 I— 000 \ 0 I Z W 'I I2 I 2 K C \ s 0 I4< C: I.“ NNLLI \ com is > ##4I— I4 I H DOOM-I 40 I.” (O ‘I 02m : \ U W \ 5" H w 2 In I O s s N o < a: \TNNN I0 I- LL \ s O 020 \ COAZA I IUL'L 00 «I Iv \ WOHNN (3).-o)- . o o o_; o I «x KOPrMzz_J_n—_Iz v \v \44ODHO4 In 04’00000 wHwAzzmmzm: BIN-B C‘I-C‘I- Hv-HPM'WEIHZWIAII II II II II II II XOXO<<>>~ >¢~7AAAAAAA (F4>4444UJ44 II 90050909 4<4<4444o44ommwwwmw - 022 < 0 C-++ 0:0 H 9‘5"? 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XXV HHSZZ a Z I <HXX 4 LU H H 44QJ-QGI—0C3 4 I- V U) 442 44244 < w u < <. >- 9 C _J 4 0 0 z, 2 ‘ \ x AA In X 000 NN 00000 c 0.4 s s 0 I I I4 osro ZHH COO¢O II II 2 WV II II II II 22 AAV 4x)- AAAAV q-NLUAII II II PNPR’UO ++zf\.I—IAALIJ++++Z'~~‘ zsz HHtZZZZHVfi >> h~x>Hxx>> o 00404-00I-00004I- 4442 44244444 oc- s \ my, G O 0 HH 40“- oc 9“ :: P 1" I s A In 0!- N-O I 0 AH omcox ‘ ‘ OOPHV IO '00 ’ . \ \cvu. C; 00 I4C'1- onng IOICOvv 00 “.0 II II II "222 va—x>- AAAAV44 WWII II II NMNMLUIIC‘ HHHAALUUUOC. ZWWC xx HHzo-v-I-I-H<HXX>> D 44NO0I-0000444I- 44 4424444444 ((COGDDDOO<<(C ugazzuzzzzuuuw 210 00 2 : A s s (- o o + 000HA \ (Ark-I I Iv-4V 00 IO.>- UVPXO AAAA Amy II II II PNPI'V HHHAAwOOCO xx HH:PF"""’ (HXX>>~ _JJMGOP-OGGG 44 4424444 ((00000000 uuozzuzzzz m m Cunt-N I0 no 0 'C‘ ' I o I C II II II II --STRAIN PLOT ice 0 r- M C C C HOLDY £.,9., CQ’1 C, ox 245 ”TII'ITJIMLTIL’IJIIMMEI’IILIIIJIII‘“