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'LLLL'L'LL'" '3 1.3.2222" 22232121215! 22233122323 I. llfllalllllll ljl Will llllfllllll 1932 0112 4794 "mg?“ 2 ,, km a. .,.u 1. i , P 1 a. .1. 1:- ..- . was. .51! 4‘ u‘ 5-3! “'1": ”noun-r I._._ , t, :nnuoo‘n-A-IO': 1"- VIII 'V—fl~"::l \ This is to certify that the dissertation entitled A STUDY OF MECHANICALLY FASTENED COMPOSITE USING HIGH SENSITIVITY INTERFEROMETRIC MOIRE TECHNIQUE presented by Pedro Jesus Herrera Franco has been accepted towards fulfillment of the requirements for Ph. D. . Mechanics degree in ajor professor Date April 2, 1986 MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 ‘ MSU RETURNING MATERIALS: 'Place in book drop to remove this checkout from LlBRARlE Azggzgga;:_ your record. FINES will be charged if book is returned after the date stamped below. NM"! 1 ff i951 , L315 ' Io 1_1:3?;2.,. A STUDY OF MECHANICALLY FASTENED COMPOSITE USING HIGH SENSITIVITY INTERFEROMETRIC MOIRE TECHNIQUE BY Pedro Jesus Herrera Franco A DISSERTATION 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 1985 ABSTRACT A STUDY OF MECHANICALLY FASTENED COMPOSITE USING HIGH SENSITIVITY INTERFEROMETRIC MOIRE TECHNIQUE BY Pedro Jesus Herrera Franco A High Sensitivity Interferometric Moire technique is presented and analyzed. A review of simple concepts and equations of diffraction by a grating serves to analyze how this technique can be applied to perform measurements of strain fields on the surface of specimens in order to study current problems in Mechanics and Structural design. A three-dimensional geometrical approach is used to describe this technique and to demonstrate the mechanics of Moire fringe formation. A review of simple concepts of the theory of elasticity for anisotropic bodies was utilized to characterize the material used in this research, and also to set the theoretical foundations of the experimental method used to measure its mechanical properties, which were needed to calculate stresses in the material. 'Ihe stress and strain distribution in the vicinity of single-pin lap Joints in an orthotropic laminate (glass-epoxy laminate) and in an isotropic plastic were studied using this experimental technique. The possibility of obtaining significant stress concentration relief through the use of isotropic inserts was investigated. Pedro Jesus Herrera Franco Surface strains in three directions were measured employing the high-sensitivity Moire technique. The strain results that were obtained corresponded to the insert ring and the adjacent composite laminate. Strain gage rosette equations were used to obtain shear strain contours. Measurements were performed for specimens without any insert, with a hard insert (Aluminum). and with a soft insert (epoxy) in an isotropic material and in the composite laminate. Stress reductions were observed for both types of inserts, with the hard insert giving the greatest advantage. A reduction of approximately 50 1 was obtained in the bearing region, owing to the presence of the plastic insert. In the case of the specimen which utilized the Aluminum insert, the stress concentration was reduced by approximately 75 i in the bearing region and 90 1 in the ligament regions. Shear strain ny was also calculated along the locus of shear-out failure; and, in the case of the specimen with the Aluminum insert, it was reduced by approximately 90 %. ACKNOWLEDGMENTS The author wishes to express his sincerest appreciations and gratitude to his advisor and major professor Dr. Gary Lee Cloud for his valuable guidance, encouragement and understanding during the course of this research. The author also wishes to thank his Committee, Drs.[L Yen, W. Bradley and especially R. Abeyaratne for their interest in his project. Finally, the author wishes to thank his wife Maria de Lourdes and sons Pedro Jesus, David Jose and Cristian Javier for their patience, understanding, help and encouragement. TABLE OF CONTENTS Page LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . v CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . .1 1.1 PROBLEM OUTLINE . . . . . . . . . . . . . . . .1 1.2 OBJECTIVES AND SCOPE . . . . . . . . . . . . . .2 1.3 RELEVANT LITERATURE . . . . . . . . . . . . . 3 1.” APPROACH TO THE PROBLEM . . . . . . . . . . 8 1.5 CHOICE OF THE EXPERIMENTAL TECHNIQUE . . . . . 10 CHAPTER 2 CHARACTERIZATION OF THE MATERIAL. . . . . . . 11 2.1 ELASTIC PROPERTIES OF COMPOSITES . . . . . . . 11 2.2 STRESS AND STRAIN IN THE LAMINATE . . . . . . 1H 2.3 EXPERIMENTAL EVALUATION OF MATERIAL PROPERTIES 18 2.3.1 Specimen Dimensions . . . . . . . . . 18 2.3.2 Experimental procedure . . . . . . . . 19 CHAPTER.3 OPTICAL PRINCIPLES OF THE MOIRE INTERFEROMETER . . . . . . . . . . . . . . . 25 3.1 MOIRE INTERFEROMETRY . . . . . . . . . . . . . 25 3.2 THE GRATING . . . . . . . . . . . . . . . . . 26 3.3 THE GENERAL GRATING EQUATION “OBLIQUE INCIDENCE“ . . . . . . . . . . . . . 30 3.3.1 Directional Relations . . . . . . . . 30 3.3.2 Derivation of the general grating equation . . . . . . . . . . . . . . . 33 3.” TWO BEAM INTERFERENCE . . . . . . . . . . . . 39 11 3.5 3.6 3.7 308 3.9 3.11 CHAPTER 9 “.1 GEOMETRY OF THE MOIRE INTERFEROMETER . . . . . DEFORMATION OF THE SPECIMEN GRATING . . . . . MOIRE FRINGES OF DISPLACEMENT Ux . . . . . . . 3.7.1 Fringes produced by normal strain ex . 3.7.2 Fringes due to normal strain ey. . . . 3.7.3 Fringes due to shear strain ny. . . . 3.7.” Fringes produced by a rigid body rotation . . . . . . . . . . . . . . . 3.7.5 Fringes produced by out-of~plane rotations . . . . . . . . . . . . . . MOIRE FRINGES OF DISPLACEMENT U y. . . . . . . MOIRE FRINGES OF DISPLACEMENT UAS' . . . . . . 3.9.1 Fringes due to normal strain ex, . . . 3.9.2 Fringes due to in-plane rotation w . . 3.9.3 Fringes due to shear strain Yx'y" . . 3.9.“ Fringes due to normal strain a , . . . 3.9.5 Fringes due to out~of~plane rotation . COMPLETE ANALYSIS OF TWO DIMENSIONAL STRAIN FIELDS . . . . . . . . . . . . . . . . USE OF PITCH MISMATCH IN MOIRE INTERFEROMETRY. 3.11.1 Analysis of the undeformed specimen . 3.11.2 Analysis of the deformed grating (Strain plus mismatch) . . . . . . . . EXPERIMENTAL APPARATUS . . . . . . . . . . . BASIC IDEA OF THE INTERFEROMETER AND OBJECTIVES O C C O O O O O O O O O O O O O O 0 111 R1 N3 as us 511 57 58 6O 60 62 6A 65 67 69 69 71 7H 75 80 86 86 “.2 “.3 CHAPTER 5 5.1 5.2 5.3 5.“ 5.5 5.6 5.7 5.8 CHAPTER 6 6.1 6.2 603 6.“ 6.5 6.6 6.7 CHAPTER 7 REFERENCES APPENDIX A APPENDIX B CONSTRUCTION OF THE MOIRE INTERFEROMETER . ADJUSTMENT OF THE MOIRE INTERFEROMETER . . EXPERIMENTAL METHODS . . . . . . . . . . FABRICATION OF MOIRE SPECIMENS . . . . . . SPECIMEN GRATINGS . . . . . . . . . . . . GRATINC REPLICATION PROCESS . . . . . . . SPECIMENS wITH ISOTROPIC MATERIAL INSERTS MOIRE FRINGE PATTERN PHOTOGRAPHY . . . . . DATA REDUCTION . . . . . . . . . . . . . . DIGITIZING MOIRE TEST DATA . . . . . . . . DATA REDUCTION AND PLOTTING STRAINS . . . EXPERIMENTAL RESULTS AND DISCUSSIONS . . COMPOSITE MATERIAL SPECIMEN wITHOUT ANY REINFORCEMENT STRESS AND STRAIN CONCENTRATION RELIEF BY THE USE OF ISOTROPIC MATERIAL INSERTS . ISOTROPIC MATERIAL NITH PLASTIC INSERT . . COMPOSITE MATERIAL SPECIMEN wITH PLASTIC INSERT . . . . . . . . . . . . . . . . . . COMPOSITE MATERIAL wITH ALUMINUM INSERT . COMPOSITE MATERIAL WITH ALUMINUM INSERT (AFTER GLUE FAILURE) . . . . . . . . . . . STRESS CONCENTRATION FACTORS . . . . . . . CONCLUSIONS AND RECOMMENDATIONS . . . . . iv .101 .105 .107 .110 .111 .112 .116 .120 .120 .138 .l“1 .152 .170 .180 .196 .20“ .208 .21“ .215 3.1 3.2 3.3 3.“ 3.5 3.6 3.7 3.8 3.13 LIST OF FIGURES Page Specimen Dimensions and Location of Strain Gages.............20 Photograph of Specimen Showing Strain Gages........... ..... ..21 Overall View of Specimen and Loading Rig.....................23 Typical Tensile StresseStrain Plots for Determination of Mechanical Properties of OrthotrOpic Composite Used in this Researcn.0......0..0.0.0.0...OOOOOOOOOOOOOOOOOOOOO0.0.0.2“ Phase Relation Between Rays From Two Grating Groves..........28 Three~Dimensional Geometry of the Incident and Diffracted Rays of a Grating.................................31 Oblique Incidence on a Plane Grating.........................3“ Constructive and Destructive Interference Produced by the Combination of Two Wavefronts.........................“0 Three-Beam Arrangement of The Moire Interferometer...........“2 Geometrical Representation of The Deformation of The Grating .00.000000000000000000000IOOOOOOOOOOOOOOOOOOOOOOOuu Geometry of The Incident and Diffracted Rays from a Grating Deformed by ex When Analyzed by Beams A0 and BO......51 Optical Arrangement Used to Create an Image of the Interference Pattern Produced by Two Diffracted orderSOOOOOO0......O0.0.0.0...O...OOOOIOOOOOOOOOOOOOOOOOO0...53 Geometry of the Incident and Diffracted Rays from a Grating Deformed by ey When Analyzed with Beams A0 and BO....56 Geometry of The Incident and Diffracted Rays From a Grating Deformed by ny when Analyzed with Beams A0 and BO...59 Geometry of the Incident and Diffracted Rays Affected by Rigid-Body Rotation of the Grating When Analyzed by Beams A0 and BOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO000.00.0061 Schematic of The Angles of Incidence of the Moire Interferometer When Analyzing the Specimen Grating "1th Beams A0 and CO 0.00.0.0.0000...0.0.0.000...00.000.000.063 Geometry of the Incident and Diffracted Rays From a Grating x' deformed by Ex' When Analyzing with Beams A0 andCO 0.0.00.00...O...OOIOOOOCOOOOOO0.00.00.00.000000000066 V 3.18 “.2 “.3 11.11 5.2 5.3 5.“ 5.5 5.6 Geometry of the Incident and Diffracted Rays Affected by a Rigid-Body Rotation of the Grating When Analyzing With Beams A0 and CO0.0.0.0...OOOOOOOOOOOOOOIOOOOOOOOO0.00.0068 Geometry of the Incident and Diffracted Rays From a Grating Deformed by Yx' , When Analyzing with Beams A0 and CO .00....OOOOOO¥OOOOOOOOCOOOOOOOOOOOOOOOOOOO0.0.0....70 Formation of Moire Fringes Caused by Pitch and Rotational MisthhOOI.0...OIOOOOOOOOOOOOOOOOOOOIOO0.00.00.00.76 Geometry of the Incident and Diffracted Rays Affected by a Fictitious Strain Produced by a Frequency and Rotational Mismatch When Analyzing With Beams A0 and BOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO..OOOOOOOOIOOOOOOO0......79 Geometry of the Incident and Diffracted Rays from a Grating Affected by Strain Plus Mismatch When Analyzing With Beams BO and CO I...00......OOOOOOOOOOOOOOOOOOOOOOOO0.0.81 Moire Fringes of Displacements for Two Different Amounts of Mismatch and Same Loading Conditions......................8“ Strain c Obtained from Two Sets of Moire Patterns from the SameySpecimen with Different Amounts of Mismatch.........85 Overall View of the Experimental Set Up Incorporating the ”Dire Interferometer.00.0.0.0...OOOOOOOOOOOOOOOOOO0.0.0.087 Schematic of the Basic Idea Used in the Construction Of theMO1r8 Interferometer OOOOOOOIOOOOOOOOOOOOOOO000.000.0088 Arrangement of the Optical Elements of the Moire Interferometer 0....OIOOOOOOOIOOO0.00.0.0...0.00000000000000090 Mechanical Elements Used in the Positioning of Optical Elements of the Moire Interferometer ................9“ Dimensions and Parameters Involved in the Design or the Test coupon O00.0.00...OOOOOOOIOOOOOOOOOOOOOOOO0.00.0100 Photomicrograph of Two-Way Phase Grating Using a Scanning Electron Microscope .............................10“ Production and Replication of Moire Gratings ...............106 Specimens Showing Reflective Gratings on their Surface......108 Steps in the Reduction of Moire Data Used to determine Strains in Pin-Loaded Holes ......................113 MiOFOdatatizer system OO0.00.0000...OOOOOOOOOOOOOOOOOOOOO0.0115 vi 6.2 6.3 6.& 6.5 6.6 6.7 6.8 6.8' 6.9 Moire Fringe Pattern of Displacements Perpendicular to Load Axis for Composite Specimen.........................121 Moire Fringe Pattern of Displacements Parallel to Load Axis for Composite Specimen.................................122 Moire Fringe Pattern of Displacements in the +u5 Degrees Direction for Composite Specimen............................123 Strain ex Along Lines Perpendicular to the Direction of Load and Located Above the Hole..........................126 Strain ex Along Parallel Lines Perpendicular to the Direction of the Applied Load and Encompassing the Bearing Region and Both Ligament Areas......................127 Strain 5 Along Parallel Lines Perpendicular to the Directioh of the Applied Load and Located in upper Portions Both Ligament Areas ..............................129 Strain 5 Along Parallel Lines Perpendicular to the Directiofi of the Applied Load and Located in Lower Portions of Both Ligament Areas ............................130 Strain 5 Along Lines Parallel to the Direction of the The Loadyand Located in the Bearing Region and Above the H018 Right Portion.O...O...OOOOOOOOOOOOOOOOOOOOOO0000000131 Strain 5 Along Lines Parallel to the Direction of the Loadyand Located in the Bearing Region and Above the H018 Left Portion.I...0.0.0000...OOOOOOOOOOOOOOOOOOOO0.0132 Strain e Along Lines Parallel to the Direction of the Load andyLocated in the Ligament Area on the Right Side Of the HOIe 0.0.00.0.0.....0OO...OOOOOOIOOOOIOIOOOOOCOOOO00.133 Strain 6 Along Lines Parallel to the Direction of the Load andyLocated in the Ligament Area on the Left Side Of the H018 OOI...OIOOOOOOIOOOOOOOOOOIOOOOOOOOOOIOOOOOOO000.13“ Strain e” Along Parallel Lines Oriented at “SrDegrees to Direct?on of Load and Located in Both Ligament Areas.....136 Strain en Along Parallel lines Oriented at NS-Degrees to Direct?on of Load and Encompassing The Bearing Region and Both Ligament Areas O0000......0....000.00.00.0000000000137 Strain 6 Along Parallel Lines Oriented at h5-Degrees to Direct on of Load and Located Above the Hole.............139 Moire Fringe Pattern of Displacements Parallel to the Direction or the LoadOOOOOOOOOOOOOOO.00.0.00...000.00.00.00011‘3 v11 6.15 6.16 6.20 6.21 6.22 6.23 6.2“ 6.25 6.26 6.27 6.28 6.29 6.30 Strain e for Two Lines Parallel to Direction of the Load andyLocated in the Bearing Region.............. ..... ...1uu Strain e for Lines Parallel to the Direction of the Load andyLocated in the Ligament Area to the Right of the ”Ole.0.0.0.0....OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOIOOOO..1u5 Moire Fringe Pattern of Displacements Parallel to the Direction of the Load...............C...’................C..1u6 Strain e for Lines Parallel to the Direction of the Load andyLocated in the Bearing Region......................1u7 Strain e for Lines Parallel to the Direction of the Load LocXted in the Ligament Region to the Right of the “018.000.0000...0....OOOOOOOOOOOOOOOOOOOO0.0...00.00.000.0001’48 Comparison of Strain 6 Between Plastic Specimens With and Without Insert foryLines Located in the Bearing Region..1u9 Comparison of Strain 5 Between Plastic Specimens with and Without insert foryLine x- 0.1250 Located in the Ligament Region to the Right or the ”Cleo.0.0000000000000000150 Comparison of Strain 5 Between Plastic Specimens with and Without Insert foryLine X-O.1875 Located in the Ligament Region to the Right of the Hole....................151 Moire Fringe Pattern of Displacements Perpendicular to Direction of the Load for Specimen With Plastic Insert......153 Moire Fringe Pattern of Displacements Parallel to the Direction of Load for Specimen With Plastic Insert..........15fl Moire Fringe Pattern of Displacements at ”SrDegrees to the Direction of Load for Specimen With Plastic Insert......156 Strain ex Along Lines Perpendicular to the Direction of Load and Located in Both Ligament Regions...................157 Strain ex Along Lines Perpendicular to the Direction of Load and Located in Both Ligament Regions...................158 Strain Ex Along Lines Perpendicular to the Direction of the Load and Located Above the Hole.........................1S9 Strain ex Along Lines Perpendicular to the Direction of Load and Located in the Bearing Region......................160 Strain 5 Along Lines Parallel to the Direction of Load and Located Above the Hole and in the Bearing Region........162 viii 6.31 6.32 6.33 6.3u 6.35 6.36 6.37 6.38 6.39 6.H0 6.u1 6.fl2 6.u3 6.uu 6.u5 Strain 5 Along Lines Parallel to the Direction of Load and Located in the Ligament Area to the Right or the Hole...163 Strain 5 Along Lines Parallel to the Direction of Load and Located in the Ligament Area to the Left of the Hole....16u Strain en Along Lines “SrDegrees to Direction of Load and Locatgd in Both Ligament Areas..........................165 Strain Eu Along Lines RS-Degrees to direction of Load and Locatgd in the Bearing Region and in the Right Hand Side Ligament O0.000.000...0..OOOOOOOOOOOOCOOOOO0.00.0...0.0166 Strain en Along Lines "S-Degrees to Direction of Load and Locatgd Above the H016...OIOOOOOOCOOOOO00.0.0000... ..... 167 Strain en Along Lines “SrDegrees to Direction of Load and Encomgassing Both Ligament Regions and the Bearing Region..0.00......O...OOOOOOOOIOOOOOOOOOOOOOOOOOOOOOOOOOOO..168 Strain Eu Along Lines h5-Degrees to Direction of Load and Locatgd in the Bearing Region...........................169 Moire Fringe Pattern of Displacements Perpendicular to the Direction of the Load for Composite Specimen with Aluminum Insert 0.0.0000000IOOOOOOOOOOOOOOOO0.0.0.0000000000171 Moire Fringe Pattern of Displacements Parallel to the Direction of the Load for Composite Specimen With Aluminum Insert 000.0.00...OOOOOOOOOOCOOOOOOOOOOOO..OOOOOOOOOOOOOI00.173 Moire Fringe Pattern of Displacements at “5“Degrees to the Direction of the Load for Composite Specimen With Aluminum Insert O0..0.000...OOOIOOIOOOOOOOOOOOOOOO00.0017“ Strain 8x Along Lines Perpendicular to the Direction of Load and Located in Both Ligament Regions...................175 Strain ex Along Lines Perpendicular to the Direction of the Load and Located in the Bearing Region..................177 Strain ex Along Lines Perpendicular to the Direction of the Load and Located Above the Hole.........................178 Strain 2 Along Lines Parallel to the Direction of Load and Located Above the Hole and in the Bearing Region........179 Strain e Along Lines Parallel to the Direction of Load and Located in the Ligament Area to the Right of the Hole...181 1x 6.u6 6.h7 6.u8 6.u9 6.50 6.51 6.52 6.53 6.5“ 6.55 6.56 6.57 6.58 6.59 Strain cu Along Lines Oriented at DS-Degrees to Direction of Load agd Located Both Ligament Areas.....................182 Strain en Along Lines Oriented at NS-Degrees to Direction of Load aid Encompassing Both Ligament Areas and the Bearing Region ..OOOOOOOOOOOOOOOOOOOOOOOOOOOOOIOOOOOOOO0.0.0.0000000183 Moire Fringe Pattern of Displacements Perpendicular to the Direction of Load for Composite Specimen with Aluminum Insert After Failure of the Glue...................185 Moire Fringe Pattern of Displacements Parallel to the Direction of Load for Composite Specimen with Aluminum Insert After Failure of the Glue............................186 Moire Fringe Pattern of Displacements At NS-Degrees to the Direction of Load for Composite Specimen with Aluminum Insert After Failure or the Glue.0..00.000.000.00...0.0.0.0187 Strain Ex Along Lines Perpendicular to the Direction of Load and Located in Both Ligament Areas.....................189 Strain ex Along Lines Perpendicular to the Direction of Load and Encompassing both Ligament Areas and the Bearing Region O0.0000000000000000000000000.00.0000...0.0.0.00000000190 Strain 6 Along Lines Parallel to the Direction of the Load and Located Above the Hole and in the Bearing Region........191 Strain 5 Along Lines Parallel to the Direction of the Load andyLocated in the Ligament Area to the Right of the H016 O00.0.0000...0.0.0.000...0.0.0.0000...00.0000000000192 Strain EH5 Along Lines uS-Degrees to the Direction of the Load and Encompassing Both Ligament Areas...............193 Strain Eu Along Lines “SrDegrees to the Direction of Load and Encompassing both Ligament Areas and the Bearing Region IO..0O...OOOOOOCOOOOOOOOOOOOOO0......00.00.000.00000019u Comparison of Strain e for Line XA- 0.0000 Located in the Bearing Region 0.00..000.0....OOOOOOOOIOOOOOOOOOOOO0.000195 Comparison of Strain e For Line X2- 0.1250 Located in the Right Hand Ligament Area..... ..........................197 Comparison of Strain e for Line X3- 0.1875 Located in the Right Hand L18amen¥ Area...OOOOOCOOO..OCOCOOUOOOOOOOOOOO198 6.60 Bearing Stress Concentration Factors .......................200 6.61 Stress Concentration Factor Along Line in the Direction of the Load Located in the Ligament Area ...................202 6.62 Shear Strain ny Along The Locus of Shear Out Failure ......203 xi CHAPTER 1 INTRODUCTION 1.1 PROBLEM OUTLINE. Advanced composites have been extensively used in aerospace applications during recent years. Because of their high-strength-to- weight ratdx>, the applications of modern composites have expanded into other areas of engineering, such as automotive and ground transportation vehicles, medical equipment and prostheses. sporting goods, and others. Mechanical fastening is one of the most frequently used Joining methods for composite laminates in structural applications. As indicated by Godwin and Matthews [1]. there is an almost unlimited number of possible combinations of composite materials, fiber patterns and lay up sequences in which bolted Joints of various sizes may be utilized. Fiber-reinforced composites can be considerably weakened when bolt or rivet holes pierce the laminate to form a mechanical Joint. The strength of some laminates can be further degraded, partly by the lack of plasticity and heterogeneity, and partly because of the high stress concentration that occurs near the hole by virtue of the material's high degree of anisotropy. In the fastening of most isotropic materials, normal plasticity acts as an "averaging" mechanism tending to relieve both high local stresses and uneven load distribution in the fasteners. If the degree of anisotropy in the region of a bolt hole could be reduced, and if the material could accomodate local stress concentration by ductility or some other mechanism of compliance, efficiency of the Joint would increase. 1.2 OBJECTIVES AND SCOPE. The main obJectives in this research are: a) To develop an experimental technique whhfliis suitable for measurement of surface strains in isotropic and anisotropic materials, which will provide high-sensitivity, and which also yields enough information for the determination of shear strains and principal strains. Effort is centered in a high-sensitivity interferometric Moire technique, where the specimen carries a phase-type grating and the information is obtained by overlapping beams of diffracted coherent light. b) To gain insight into the stress-strain field in the vicinity of pin-type fasteners of various kinds. 0) To explore a stress and strain concentration relief method. This method of stress modification uses an isotropic insert having modulus of elasticity different from those of the components being Jointed. This idea is applied to both isotropic and anisotropic materials. Important parameters included variation in the material used to construct the insert, which is in the form of a thin-bushing. The experimental results in this investigation cannot be directly compared with any existing study because there has been no research on this method of stress and strain concentration relief. Comparisons are made between fasteners with no insert and similar fasteners using different material inserts. 1.3 RELEVANT LITERATURE. Numerous previous studies of mechanical Joints and the associated problem of bolt or pin-loaded holes have been conducted. Some of the most relevant articles are mentioned here. Bickley [2], studied the problem of an infinite plate containing a loaded hole. For the case of a push fit he presumed contact over an arc of 180 degrees and a cosine distribution for the contact pressure. He obtained a solution in the form of polar trigonometric series by means of a Fourier analysis in a full 0-360 degrees range. He also considered a clearance fit, proposing a pressure distribution by analogy with the Hertzian solution, and proceeding as in the previous case to obtain a stress function. The problem of a smooth loose rigid pin pressed against the edge of a circular hole in an infinite plate was studied analytically by Tiffen and Sharfuddin [3]. They obtained an integro-differential equation for which there is no known solution except in the case when the region of contact can be taken to be equal to a semi-circle. The problem of a rough loosely fitted pin, pressed against one portion of the boundary of the hole was studied analytically by Sharfuddin [ll]. This solution considers a specified displacement on the contact area while the rest of the boundary is stress free, but it does not permit sliding in the rough contact zone. Theocaris [5] gives an exact analytical solution for the stress distribution resulting from loading a perforated strip in tension through a rigid pin filling the hole. He assumed an angle of contact between pin and hole of 180 degrees, no slip, and identical mechanical properties for pin and plate. It should be mentioned that these first four references used isotropic materials only. More recent studies have used numerical methods and computer techniques to solve pin fastener problems. Waszcak and Cruse [6] used finite elements in an attempt to ihmmher understand the failure characteristics of bolted Joints. A cosine distribution of normal stresses acting over half<fl’the hole surface was used to simulate the stress distribution caused by the bolt. De Jong [7] investigated the stress distribution around a pin-loaded hole in an elastically orthotropic or isotropic plate with a rigid pin and frictionless interface, using Lekhnitskii's method of complex functions. wong and Matthews [8], used a 2-dimensional finite element analysis for problems of bolted Joints in fiber reinforced plastic, and some correspondence is demonstrated between the calculated and experimental data for fiber glass reinforced epoxy resin. Matthews, et al [9]. employed a 3-dimensional finite element analysis and showed that the stress distribution around a loaded hole depends on whether the load is applied via.a pin or a bolt. Wilkinson and Rowlands [10] used a finite element analysis to study stresses and strains associated with a single-fastener mechanical Joint in wood; and they evaluated the effects of friction.and variation in Joint geometry as well as ratio of pin-to- hole diameter. They also presented experimental verification by means of Moire technique. Analytical approaches, including boundary collocation and finite element methods, were presented by Oplinger [11] to study the effects of edge distance and width in single-fastener lugs. Comparison of single-fastener and parallel fastener configurations was also made, and the analytical results were compared with Moire technique results. Chang et al [12] presented a method to first determine the stress distribution in the laminate by the use of a finite element method, and the failure mode and failure load are predicted. For this purpose they used Yamada's failure criterion [13], which is based on the assumption that, Just prior to failure of the laminate, every ply has failed as a result of cracks along the fibers. Pradhan and Ray [111] also investigated the stress distribution around pin-loaded holes for isotropic as well as fiber-reinforced plastic composite materials. The case of full contact angle, i.e. 180 degrees, and that of partial contact, i.e. the case for an angle of contact less than 180 degrees between hole and pin, was considered. They concluded that the maximum circumferential stress at the edge of the hole depends strongly on the material properties and on the ratio of hole diameter to width of plate, as well as on the angle of contact between pin and the hole. Chang and Scott [15, 16] presented a method based on finite elements to calculate the failure strength and the failure mode in the laminate. First they determined the stress distribution and then they predicted the failure load and the failure mode by means of a proposed failure hypothesis together with the Yamada-Sun failure Criterion. They also present a method to size laminates containing more than two pin-loaded holes which will result in the maximum failure load and the maximum failure load per unit of weight. Zhang and Ueng [17] obtained an analytical solution of stresses around a pin-loaded hole in orthotropic plates by the use of complex stress functions which satisfy the displacement boundary conditions along the hole boundary. Frictional effects are included. They also concluded that the distribution of stress around the pin-loaded hole is strongly affected by the presence of friction and the properties of the orthotropic material i.e. the ratio of moduli of elasticity along the two principal axes of the Joint. There have also been experimental studies of the pin-loaded hole problem. Frocht and Hill [18] used photoelasticity and strain gages on isotropic materials to study the influence of the material on the state of stress, and they also considered the effects of geometrical parameters of the Joint. Also using photoelasticity, Lambert and Brailey [19] and Jessop, Snell and Holister [20] studied the reduction in stress concentration factors resulting from the introduction of an interference-fit pin in a circular hole. Cox and Brown [21] used a photoelastic analysis to study the effects of variations in pin clearance and plate width. Nisida and Saito [22] combined photoelasticity and interferometry to separate the stresses in the neighborhood of pin-loaded holes in tensile plates. These studies were done using isotropic materials. More recent experimental studies have been applied to the problem of pin-loaded holes on anisotropic materials. Oplinger et a1 [23] utilized the Moire technique together with finite element techniques to study pin-loaded composite plates. Wilkinson, et al, [211] and Rowlands, et al [25], used the Moire technique and finite elements to analyze orthotropic materials for single-bolted and double-bolted mechanical fasteners in order to develop design information. Phabhakaran [26] used photo-orthotropic elasticity to study bolted Joints in composites. Quasi-orthotropic and unidirectionally reinforced specimens were also tested for different end-distance-to-bolt-hole ratios. Cloud, et al [27] in a combined experimental-theoretical research program which included classical Moire techniques, twoundary-element and finite elements studied the mechanics of fasteners in orthotropic composites to obtain stress-strain fields in the vicinity of pin-type fasteners. Serabian [28] also utilized classical Moire techniques in order to experimentally verify Oplinger's assumptions pertaining to bolted Joints on non-linear behavior observed in 0/90 and ins degrees laminates. The first laminate type is in shear primarily in the region in front of the pin, while the second laminate is in tension in the ligament area. Katz [29], described a semi- automatic system for data reduction of Moire fringe photographs from work on pin-loaded holes on composite materials at AMMRC. Various attempts at strengthening the Joint area with added material have been made. Dallas [30] proves that shim-reinforced laminates will show an increase in bearing strength of about 50 1. but this solution results in weight penalties and difficulties in the manufacturing process. Webb [31] concludes that the use of extra plies may be too expensive. Collings [32] suggests the inclusion of ins degrees plies as a means of reducing the stress concentration factor. Padawer [33] shows that addition of film plies in the highly directional fiber laminate increases the Joint strength and stiffness up to 200 %. Eisenmann and Leonhardt [3“] suggest that tailoring the laminate to uncouple bearing strength and axial strain gives an increase of about 60% to 75 1, but the production process is very complicated. 1.“ APPROACH TO THE PROBLEM. The strength and stiffness of load carrying Joints in composites is difficult to characterize because many parameters are involved in their design. These parameters affecting Joint strength can be arbitrarily divided into three groups [1]: 1. Material parameters a. Fiber type and form (unidirectional, woven fabric, etc.) b. Resin type c. Fiber orientation d. Laminate stacking sequence e. Fiber volume fraction f. Fiber surface treatment 2. Fastener parameters a. Fastener type (screw, bolt, rivet, etc.) b. Fastener material c. Fastener size d. Clamping force e. Washer size f. Hole size and tolerance 3. Design parameters a. Joint type (single lap, single cover butt, etc.) b. Laminate thickness and tolerance c. Geometry (pitch, edge distance, hole pattern, etc.) d. Load and rate of loading e. Failure criteria A maJ or problem in composite design is to satisfy the structural requirements i.e. to design a material which will perform satisfactorily under different loading and environmental conditions and at the same time satisfy the design requirements for an efficient Joint. Eisenmann and Leonhardt [311] pointed out the two characteristics highly desirable in the laminate in the bearing region: (1) it must have a low tensile modulus of elasticity in the primary load direction, because that is the key to isolating the bearing stresses from the high tensile loads carried in the ligament areas; and (2) the bearing laminate must have a high bearing capacity. They also pointed out that the ligament areas should have a high tensile modulus of elasticity in the primary direction of the load. The suggested approach to improve the Joint strength uses a concept simpler than changing the fiber orientation as suggested by Eisenmann and Leonhardt [311]. Suppose an insert in the form of a thin bushing of isotropic material is glued into the hole, then some improvements in the bearing capacity should result. Also, gained is the difference in modulus of elasticity which is required to uncouple the bearing strength and axial strain of the ligament areas. In other words, the difference of material properties between insert and the composite should create a more even distribution of stresses and strains around the hole area, producing a more efficient Joint. 10 1.5 CHOICE OF THE EXPERIMENTAL TECHNIQUE. To predict the strength of the mechanical Joint, detemmination of the deformation fields, which are probably nonlinear, is required. A full-field analysis seems to be the most suitable approach. Transmission.photoelasticity has been successfully utilized in studying isotropic-material mechanical Joints [18-22] but it cannot be used to study opaque composite materials. Photoelastic coatings have been applied to measurements in composites [35, 36], but they present problems on the free edges, producing distortions in the displacement field, especially in the boundary. The extension of transmission photoelasticity techniques to birefringent composites offers another method to determine stresses in composite Joints. This novel technique is called photo-orthotropic elasticity [26]. The major problem with this technique is to find a suitable combination of birefringent materials which will best serve to model the actual composite to be tested. Holography and Speckle interferometry have also been applied to perform measurements of strain in composites, but they lack the desired strain sensitivity [37, 38]. The Moire method yields full-field information of the in-plane surface displacements, and it appears to be the most suitable optical technique for this study [59-63,69]. It has a great potential for the macroscopic strain analysis of composites, and it does not suffer any limitations from anisotropy, inhomogeneity, or inelasticity of the composite material. CHAPTER 2 CHARACTERIZATION OF THE MATERIAL The material used in this investigation was a fiber glass-epoxy laminate with woven fibers (R1500/1S81, 13 plies, 0.1u in. thick) supplied by CIBA-GEIGY, Composite Materials Department, 10910 Talbert Avenue, Fountain Valley, California 92708. In order to calculate stresses in the material, it was necessary to know with precision the material properties of this composite laminate. A review of simple concepts of the theory of elasticity for anisotropic bodies was utilized to characterize the material used in this research, and also to set the theoretical foundations of the experimental method used to measure its mechanical properties. No information about the material properties was available from the manufacturer. 2.1 ELASTIC PROPERTIES OF COMPOSITES Generally composites are anisotropic materials, i.e. their strength and stiffness vary with the direction in which they are measured. If the matrix is a resin, the behavior in any direction is in fact visco-elastic. In most calculations, however, elastic constants are used. At low loads, for short duration and within a small 11 12 temperature range, it is appropriate to use Hooke's law for the description of the anisotropic behavior [A1]. 01- CIJEJ (2,1) or 61- SIJOJ (2.2) where i, J-1,2,...,6, and o are the stress components, 5 are the i .1 strain components, Cij is the stiffness matrix and SiJ is the compliance matrix. The double subscript indicates a summation. Referring to equation (2.2), it can be shown that the compliance matrix S has 36 1.1 constants. However, less than 36 of the constants can be shown to be actually independent for elastic materials when the strain energy is S iJ- Ji' Thus, considered, and it is shown that S is symmetric i.e. S 1.1 in the compliance matrix only 21 of the constants are independent. The reinforcing material of the composite used here was a cloth composed of two sets of interwoven fibers at right angles to each other. This means that each ply can be considered as an orthotropic material. For the assembly of plies, there are two orthogonal planes of material property symmetry, and symmetry will exist relative to a third mutually orthogonal plane. Also, if the principal material axes are aligned with the natural axes of the specimen used to characterize the material properties, each laminae can be called specially orthotropic. The stress-strain relations in coordinates aligned with the principal 13 material directions, i.e. parallel to the intersections of three orthogonal planes of material symmetry are: . _, (2.3) Cu 2823 Y23 Suu‘23 552 2831' Y31 855131 66' 2612 Y12 S66T12 where the Y (i a J) represents engineering shear strain, 5 (i a J) ij 11 represents tensor shear strain, and 91 is the contracted notation. It should be noticed that there is no interaction between normal stresses 0 (J '03 and shearing strains Y Similarly, there is no 1' 2' 23' Y31' Y12' interaction between shearing stresses and normal strains in different planes. There are now only nine independent constants in the compliance matrix SL1. For a lamina in the 1-2 plane, a plane stress state is defined by setting o3= O, 123- 0, 131a O and equation (2.3) expressed in matrix form reduces to e1 1 S11 S12 0 °1 e2 ' S12 S22 0 °2 (2.4) Y 0 S 12 O 66 T12 l4 and 3 13 1 23 2 (2.5) 2.2 STRESS AND STRAIN IN THE LAMINATE. Knowledge of the variation of stress and strain through the laminate is essential to the definition of the extensional and bending stiffnesses of a laminate. Inversion of equation 2.” yields °1 311 312 8 E1 °2 012 022 Q 52 (2.6) T12 66 12 where the QiJ’ the so-called reduced stiffnesses, are Q E1 Q . E2 11 1 -v12v21 22 1-v12v21 (2.7) v1232 v2151 Q12” 1 - v v °r Q12' 1 - v v 12 21 12 21 Q66” G12 and E1 E2= Young's Moduli in 1, 2 directions respectively. 15 v - Poisson's ratio for transverse strain in the 1J J-direction when stress is in the i-direction. G12- Shear Moduli in the 1-2 plane. In.this study the fibers in the material used to construct the specimens are aligned with the edges of the specimen; thus, there is no ‘need to use transformation of coordinates to find the stresses and strains along the fiber orientations. Now, the stress-strain relation of the nth. layer of a multilayered laminate is written as [0]“. [Q]n [EJn (2.8) From lamination theory, the resultant forces and moments acting on the laminate are given by, Nx A11 A12 A16 5; B11 B12 B16 Kx Ny ' A12 A22 A26 5y 1 B12 B22 B26 Ky (2.9) ny A16 A26 A56 Yx B15 B26 B66 Kx O Mx B11 B12 B16 8; D11 D12 D16 Kx My - B12 B22 826 2x + D12 D22 D26 Ky (2.10) ”xy B16 B26 B66 ny D16 D26 D66 K where 16 N AiJ ' Zn=1Qij)n(zn ' Zn-1) 1 N 2 2 BiJ "2 Zn-1QiJ)n(zn- zn-1) (2.11) 1 N 3 _ 3 DiJ "3 zn-1QiJ)n(zn zn The Aij are the extensional stiffnesses, the BiJ are called the coupling stiffnesses and the D are called the bending stiffnesses. 1J The laminate used in this study can be considered to be "balanced", i.e. one that contains an equal number of laminae of +6 degrees and -9 degrees fiber orientation. For this type of laminate, the shear coupling stiffnesses terms A16 and A 6 can be shown to vanish [111]. 2 Also since all the laminae within the laminate are positioned symmetrically with respect to the laminate mid-plane, the coupling terms BiJ vanish. It is important to note that the twisting-bending terms D16 and D26 vanish. This type of laminate could therefore be called a "specially orthotropic laminate". All of this means that extension forces will produce only extensions and bending moments will produce only curvature. The S or compliance constants can be expressed in terms of 1.1 engineering material properties as: .17 1 s -- 11 E1 5 .-2.21.-2_12_ 12 E2 E1 (2.12) 1 s -- 22 E2 1 Soo'E For the class of fibrous laminates where the tensile and compressive elastic properties are identical, the compliance matrix S 1.1 is symmetric. Hence S12— 821, and, V 2 —- = E— (2.13) —-b—l N The fundamental principle underlying test methods to characterize laminated composites states ['42]: The single lamina is the building block of the multidirectional laminate. Therefore, characterization of lamina material properties allows predictions of the properties of any laminate. Then a tensile test can be performed to determine uniaxial strength, effective Young's modulus and Poisson's ratio. For the case of uniaxial tension, 2 \ a . I bu.-j*\ d ’ \ I I A ”/L dsuux \\\\‘\sih\ / ousrnncreo LIGHT a on GRATING enooves Figure 3.1 Phase Relation Between Rays From Two Grating Grooves 29 grating surface and the diffraction equation, diffraction angles are in the range -90 S 8m 5 +90. Also by convention, the diffraction orders are numbered increasing in the counterclockwise direction, beginning with the zeroth order. For the zeroth order one has ma- 0 and Sin a= 2 Sin 8, which is Snell's law of reflection and defines the path of the direct or undiffracted light. In other words, the zeroth order always emerges at an angle which corresponds to the specular reflection angle and which is equal to the angle of incidence. The numbering of diffraction orders aids in following a diffraction sequence of a beam, particularly when two Moire gratings are involved. Equation 3.3 can be rearranged as: Sin sm= mlf + Sin a (3.4) where the :1; sign can be omitted according to the convention mentioned above and f- 1/d is the grating frequency. Interest will be centered in the special case of diffraction referred to as "symmetrical diffraction". This situation occurs when, for every order that emerges from the grating at an angle +8, there is another diffraction order that emerges at angle -8. If the diffraction order ms -1 is chosen to emerge symmetrically opposed to the zeroth order, then B_1= -a and the diffraction equation 3.A reduces to, or Sin a= -2— f (3.5) 04> Sin as 30 3.3 THE GENERAL GRATING EQUATION -OBLIQUE INCIDENCE- 3.3.1 Directional Relations. In the description of the behavior of the grating given before, it is assumed that the incident and emergent beams both lie in the same plane. More general relations between the direction of incidence and emergence have been given by Guild [A9]. Some of his fine explanations are adopted here. Referring to figure 3.2, the following definitions are important: A "principal plane" is defined as a plane normal to the surface of the grating which intersects it in a line perpendicular to the grating lines. A "secondary plane" is a plane normal to the surface of the grating which intersects it in a line parallel to the grating lines. An "incidence plane" is a plane formed by the incident ray and the grating normal; and a plane which is formed by an emergent rays of any spectral order and the grating normal is a "plane of emergence" for that order. To specify the direction of a ray of light in relatnnito the grating we use two parameters. First, the direction of the incident ray may be specified by the angle a between the ray and the normal to the grating surface, usually referred to as the angle of incidence, together with the angle wi between the plane of incidence and the principal plane. Similarly, the direction of emergence of rays of the mth. order, may be specified by the angle of emergence, 8m, between the mth order 31 SECONDARY PLANE —-1 PflflflflPlL PLANE Figure 3.2 Three Dimensional Geometry of the Incident and Diffracted Rays of a Grating 32 rays and the normal to the grating plus the angle 1pm between the plane of emergence and the principal plane. The direction of a ray may also be specified by its inclination to any two planes, an obvious choice being the principal and secondary planes of the grating. These two inclinations for the incident ray are a' , which is the angle between the incident ray and its proJection on the principal plane, and a", which is the angle between the incident ray and its proJection on the secondary plane. The corresponding specification for the emergent ray of the mth order, are 8' , which is the angle between the emergent ray and its proJection on the principal plane, and B", which is the angle between the emergent ray and its proJection on the secondary plane. The geometrical relationships between these parameters are very simple and can be obtained from figure 3.2. They are given here only for completeness. First, Sin a'- ad/ao, Sin a- ac/ao and Sin 1111- ad/ac which combined yield Sin a Sin *1 = Sin a' (3.6.a) also, Sin a"= ab/ao, Sin a= ac/ao and Cos wi= cd/ac= ab/ac, yield Sin a Cos mi - Sin a" (3.6.b) for the mth emergent order, Sin B'= eh/oe, Sin 8m: ce/oe, and Sin 1pm- eh/ce, 33 Sin 8m Sin Wm . Sin 8m (3.6.c) and Sin 8;: ef/oe, Sin Bm- ce/oe, Cos Wm. ch/ce- ef/ce, yield Sin 8m Cos Wm - Sin 8; (3.6.d) 3.3.2 Derivation of the general grating equation. The exposition below draws heavily from Stroke's [70] and James and Sternberg's [71] fine explanations. Consider a plane grating shown in Figure 3.3 whose surface lies on the plane z- 0 of a rectangular coordinate system. Also, consider a ray incident on the grating at the origin of the coordinate system, with direction cosines L1, L L . Let there be another ray parallel to the 2' 3 first, meeting the grating at point C(x, y, 0). Let them both be diffracted, with direction cosines L', Lé and Lé' As shown in figure 3.3, lines CA and CB are drawn from C to the first incident and cuffracted rays, meeting them perpendicularly. Then CA lies on the incident wavefront, and if the outgoing rays are in the direction of a principal maximum, CB lies on a diffracted wavefront. Therefore, for constructive interference, ACB must be an integral number of wavelengths long. Let the incident waves a O and at C be: L1x + L2y + L323 0 (3.6.e) 34 GRATING NE //—1 Y 4110051111 and “/ A r '1‘ , '1‘ . ‘ i \um I DIFFRACTED RAYS F18“re 3.3 Oblique Incidence on a Plane Grating 35 and L1x + L2y + L3z- CA respectively. Let the diffracted rays at 0 and at C be: L'x + Léy + L'z- 0 1 3 and 1 1 1. L1x + L2y + L3z CB Let the incident wave normal be: 112+ “+12 nx nyJ nz or Z I C" p... + 1 L2J + L3k and the diffracted wave normal be 1. 1- 1 1 N L l + L2J + L3k and on the first groove, at point C r- di + yJ then, for the incident beam (3.6.f) (3.6.g) (3.6.h) (3.6.1) (3.6.j) (3.6.k) (3.6.1) 36 + + 0A3 N P- L1G + L2y (3.6.m) and for the diffracted wave 03-13-11de 1 2y (3.6.n) then for constructive interference, 0A + 082 ml (3.6.0) or (L1 + L;)d + (L2 + Lé)y- ml ’ (3.6.p) Thls equation must hold for any value of y, and so, the first condition for OB to be an outgoing ray is L . - L' (3.6.q) and SO (3.6.r) Referring to figure 3.2, it is easy to see that L1- Cos (n/2 - a")= Sin a" L1: Cos (n/2 + B")= -Sin 8; (3.6.3) L2= Cos (n/2 - a')= Sin a' 37 Lé- Cos (n/2 + B')- -Sin Bé (3.6.s) Thus, equations 3.6.q and 3.6.r, can be expressed as: ml Sin 33' —d- + Sin a." (3.6.1:) Sin a'- Sin 6$ (3.6.n) Combining equation 3.6.a and 3.6.b, 3.6.5 and 3.6.t, mA ". :— Sin 8m d + Sin a C03 *1 Sin B$= Sin a Sin t1 (3.7) Note from figure 3.2 that for 1111- 0, Sin c"- ad/ao-cd/ad and Sin Bg- ef/oe= ch/oh, then, equation 3.7 reduces to Sin 3;- mAf + Sin d (3.4) which is the two-dimensional case. A very important idea is demonstrated by equation 3.7.b. The angle Br; i.e the angle between the emergent ray and its projection on the principal plane is a function only of the angle of incidence a and the angle 11 between the plane of incidence and the principal plane. It is not a function of the diffraction order. It is constant fxn~.any given arrangement of apparatus. This result means that the angle between the 38 projection of any diffracted ray and the principal plane is the same. For ma 0, equations 3.7.a and 3.7.b reduce to Snell's law, that is, o. - and 111i . 111 Also, the sine of the angle between any 81118 0 m- 0' diffraction order and its projection on the secondary plane will always be given by the summation of Sin 8111- 0 plus a quantity which depends on a multiple of the grating frequency. It is not the inclination of the incident and emergent beams to the normal, but their inclinations to the secondary plane which are relevant to the operation of the grating. The inclination of the incident beam to the principal plane is merely carried through to the emergent beams without modification. For the analysis of the formation of Moire fringes, a sign convention for the diffracted rays and the angles of incidence and emergence will be adopted here. Referring to figure 3.2, a diffracted order will be positive when it propagates with a component in the +2 direction and when it lies in a counterclockwise direction with respect to the zero order when viewed from the +y direction. For the incident and emergent rays, the following convention applies: the angle between the ray and the z-axis is positive when its projection on the xz-plane is rotated counterclockwise from the z-axis and it travels in the +2 directnnn when viewed from the +y direction. Its proJection on the yz-plane follows the same convention when viewed from the +x direction. In the case of a two-direction grating, the same convention applies if x and y are interchanged. 39 3.A. TWO BEAM INTERFERENCE When two coherent beams of collimated laser light intersect at an angle 2d, a volume of interference fringe planes is created [5A]. Figure 3.A shows in cross section how the two incident wavefronts combine to form a stationary system of parallel interference bands in space. The bands are not merely lines as shown, but they are planes lying perpendicular to the bisector of wavefronts and they always exhibit a sinusoidal intensity distribution. Such intensity distribution can be recorded if the two beams fall simultaneously on a screen. The distance p between the adjacent walls of interference is calculated from figure 3.3.a, to be 1 8" E's—1'5"; (3.8) where A is the wavelength of the laser light, a is half the angle between the propagation axes of the two beams of light. Equation 3.9 can be rewritten as Sin 1) . g P (3.9) / where P is the fringe density or number of fringes per unit of length. 40 WAVEFRONT l INTEN$TY SCREEN DBTMBUUON ORHlM umummmqmunwm .............. umnMMlmu IImNmmu "I" 1"?311111511'1'11H'IIHIII 1.11...111.11IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII \J 1111.111 11111IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII'II'IIIIIIIIIIIIIIIIII“ 2n 111111111111111IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII11IIIIIIIIIIIIIIIIIIIIIII 11111I|IIIII|W|Y 1111 ,,....1.£131111111:111111IIII 'III Wm»IMWMMWWWWWWWWWWWWWWMWMWMWWW 11111 ”IIIIII II 11111 I IIII1IIII l 11’1111111111111111|“m”’11I'1"11'1ll11III11111111'MMM“'HMVWMIHW PLANEOF ............ CONSTRUCTIVE - ‘~ INTERFERENCE ............. WAVEFRONT 2 PLANE 0F DESTRUCTIVE INTERFERENCE 1111111 '9 .111I1I1I1I1I11111111111 IIIIIIIIIIIIIIIIIIIIIIIIIIIIIII II» «- Figue 3.A Constructive and Destructive Interference Produced by the Combination of Two Wavefronts 41 3.5 GEOMETRY OF THE MOIRE INTERFEROMETER. Czarnek and Post [50] have suggested that specimen gratings with rulings oriented at 1115 degrees to the y-axis can be used to determine the Ux and Uy displacement fields, i.e. the displacement fields in the x and y directions. Figure 3.5 illustrates a three beam arrangement of a Moire interferometer with 1&5 degrees specimen gratnugs. By combining diffracted orders from beams A0 and BO, deformation fields in the horizontal direction x can be visualized; the combination of beam BO and CO provides the information for the vertical y-direction, and combining beams A0 and CO provides information in a third direction located at #5- degrees with respect to the x-axis. When the specimen grating is interrogated with beams A0 and 80, an interference pattern is formed. The frequency of this interference system is F, and it is calculated according to equation 3.9. This interference pattern will be referred to as "the reference grating". Now, the angle between these two beams of light is AOB= 2¢ (3.10) The first diffraction order (-1) of beam A0, incident on the specimen grating x' is desired to be perpendicular to the specimen grating surface, since for an oblique diffracted order the view is foreshortened and distorted. Also, a fixed distance from the observer or camera would make focusing easier. Then, the required angle of 42 Figure 3.5 Three- Beam Arrangement of The Moire Inte f r erometer 43 incidence is g, i.e. angle AOQ. The angle 25 is related to the angle 2¢ by Sin 5- /2 Sin 1b (3.11) Similarly, the angle of incidence for beam BO should be -g, in order for the first diffraction order (+1) to be perpendicular to the specimen grating surface. The frequency of the specimen grating can be determined using the grating equation by letting m- -1, a - -e; and B_1- 0, to get: f a Sin Ell. Recalling that F-(2/l) Sin ¢1 and using equation 3.11 one obtains, F f‘ 75 (3.12) The first diffraction orders (-1) and (+1) of beams A0 and 80, will be referred to as A0' and 80' respectively, and they will be perpendicular to the specimen grating surface if and only if the frequencies of both gratings x' and y' are related to f as in equation 3.12. 3.6 DEEURMATION OF THE SPECIMEN GRATING Consider one square formed by the intersection of two pairs of grating lines, originally parallel to the x' and y' axes as shown in figure 3.6.3. The specimen on which the grating is mounted is being 44 (b1Normal strain 6; 'a d11+em11 43411-161141 71" = 61/2 11V 1 7Xy‘: 9W2 (3 «E 1 ’3: .__‘___. 1‘2- 11 4- 6,1121 11’ 1 I? 11 + em (d)Shear strain 75w dad" ‘7 f’: F ..____‘ + "a 1’? 11 +7.,/21 d':dl1-1&y/21 f"=-E ‘ (0)". 'd b Y 115' 11 4.11/21 Jam? '1 IIY Flynn 3.6 Geometrical Representation of The Deformation of The Grating 45 deformed by the action of externally applied loads. Assume a state of plane strain parallel to the xy-plane exists at the point under consideration. In tensor form, the strain components can be written as [58]: (3.13) Ex and ey represent normal strains along the x and y axes respectively, and e . e . Y yx x xy /2 are the small changes in right angles whose sides Y were initially parallel to the x and y axes respectively. The strain along the sides of the square whose sides are parallel to the x' and y' axes can be calculated simply by applying the following transform to the strain in the specimen, since it is a second order tensor, +! or E = A E A0 (3.14) E' - A A E 13 ip Jq pq where A is the matrix of direction cosines and Ac is its transpose. Then Em A E Ac - (3.15) where m= Cos c and n8 Sin c degrees, F b For small Strains, write: P ex(m2) -ex(mn) b Each of the matrices represent the effect of Ex respectively, ex(m2)+ey(n2)+ny(mn) (ey-ex)mn ‘vxy1m2-n2) 1 2 2 2 2 ny(m -n ) ex(n )+ey(m )-ny(mn) -(ey-ex)mn and Y x on the deformations of the grating lines. this equation reduces to (3.16) the principle of superposition can be used to (3.17) Y For (a 45 47 (3.18) From these results, Ex and ey will produce both extension and rotation of the grating lines, and ny will produce extensions and contractions but not rotations. The geometrical representation of the changes in length and rotations of the sides of the square are shown in figure 3.6.a-3.6.e. The original position of the lines is indicated by dashed lines. The results shown in equation 3.18 apply to two sides of the square originally oriented along the positive direction of the x' and y' axes. From the first matrix, corresponding to the effect of ex, the two sides of the square are stretched by ex/Z and the final angle between them is (n/2 + ex). Similarly, the second matrix, which corresponds to the effect of ey, indicates that the two sides are stretched by ey/Z and that the final angle between them is (n/2 - ey). From the third matrix, it can be seen that ny will produce a tensile strain ny/2 to the line along the x'-axis and a compressive strain -ny/2 to the line along the y'-axis, but they will remain perpendicular to each other. If the frequency of the grating before deformation is fa 1/d, then the new frequency is given by f'- 1/d'. 48 3.7 MOIRE FRINGES OF DISPLACEMENT Ux' 3.7.1 Fringes produced by normal strain ex. As can be seen in figure 3.7. the specimen grating is interrogated with beams A0 and BO. Part of the incident beam A0 will be diffracted by the x' grating and its -1 diffraction order will be referred to as OA'. Similarly, 08' will be the +1 diffraction order of the incident beam BO, diffracted by the y' grating. A normal strain ex produces a change in frequency from 1/d to 1/d' (see Figure 3.6.b). and it also produce a rotation of the grating lines through an angle Yx,y,/2, (see figure 3.6.b). As a result the incident beams will not lie in the principal planes of their corresponding gratings. In order to determine the orientations of the emergent beams, it is necessary to use the three-dimensional grating equations. For beam A0, using equations 3.7, it can be written: Sin 8; (x'z)= mlf + Sin a cos 11 (3.19) Sin 8$ (y'z)= Sin a Sin 11 Similarly for beam 80 Sin 8; (y'z)- mlf + Sin a Cos 1: (3.20) , , g . . Sin 8m (x 2) Sin a Sin wi 49 Note that the direction of each diffracted ray is given by the direction of its projection in the x'z and y'z planes, as indicated in parentheses. For the case of infinitesimal normal strains, the 11 will be small and these equations reduce to the following: for beam A0 Sin B; (x'z)- mlf + Sin a (3.21) v v . Sin 8m (y z) *1 Sin a and for beam BO Sin 8; (y'z)- mAf + Sin a (3.22) Sin 8& (x'z)- 11 Sin a Figure 3.6 and the related equations show that it is appropriate to use the same 1111 for beams A0 and BO. Then, for beam A0, let ma -1, Sin o= Sin 1; =- /2 Sin 111- /2 (A/2)F= ”7/2, 1111- Yx,y,/2= ex/Z, and f'- (F//2)(1/(1 + ex/Z). which can be expressed as a continued firaction f'= (F//2)(1 - ex/Z + ei/H - ..... ). Equations 3.21 yield after neglecting the high order terms: AF: x Sin 821- (x'z)- 292?. (3.23.a) SO AFex Sin 811 (Y'Z)' 272— (3.23.b) Similarly for incident beam BO, letting m-+1, Sin a a Sin E; -- AF//2, wi- -Yx, /2 - -eX/2, and f' as given above, equation 3.22 yields y! AFe x 31“ 831W” ' 57'2— (3.24) APE: Sin 811(x'z)- - 57-23- It is not difficult to show that the diffracted rays A0' and B0' lie in the xz-plane (figure 3.7) and that the sine of the angle between each of them and the z-axis is APEX Sin s_1(z) = -Sln 8+1(z) = 2 (3.25) In an actual optical system, where a field lens decollimates beams A0' and 80', they converge to two bright points in the focal plane of the decollimating lens. Because of their angular separation, these two bright spots will be focused at a small distance apart. If they are close enough to overlap, then an interference pattern is produced. A more useful procedure is to use another lens and screen (that is, a camera) to construct images of the specimen gratings x' and y' fields with the light contained in the two diffracted wavefronts OA' and 08' 51 Figure 3.7 Geometry of the Incident and Diffracted Rays from a Grating Deformed by ex When Analyzed By Beams A0 and BO 52 [69]. Essentially, the camera forms two images which lie on top of one another (see Figure 3.8). The degree of interference in the image depends mainly upon the relative displacement of the two focal spots in the back focal plane of the decollimator which, it must be recalled, depends upon the relative inclinations of the diffracted beams. The fringe pattern is not affected by magnification of the lens system. The ratio B'/8 which governs the fringe spacing is exactly equal to M. That is, any magnification of the specimen is carried into the fringe pattern, so the ratio of fringe spacing to specimen dimension remains constant. The image in the camera displays, then, a pattern of interference fringes which are indicative of the local spatial frequency and Iorientation differences between the two gratings. The resulting interference system of vertical fringes, parallel to the y-axis, has a spatial frequency PE which can be calculated using equation 3.9 to be x 2 Fax - 7 Sin 8 (z) - ex F (3.26) Equation 3.23 is important because it relates the strain ex in the specimen to the known frequency F of the interference system produced by beams A0 and BO, and the observed Moire fringe density P'5 . Therefore, x the quantity Ex can be determined in terms of Pe and F. x The relative displacement Ux (perpendicular to the direction of the lines of the reference grating) between a pair of points a distance 53 1' I T I ' . I 1 I g 1 I I 1 I I | l I 1 1 I : I 3 I l l I 1 , : I I I I I I ' 1 L— ——l ' I I I I D“ I i 1 1 I 1““)? . I ' I I 1: f1 —1— 1' I1 — 1 f2 “'1 1 I 1 1 1 I l 44 L “5 1 #Dh ' IOI RAY TRACE IbI INTERFERENCE TRACE Figure 3.8 Optical Arrangement Used to Create an Image of the Interference Pattern Produced by Two Diffracted Orders 54 p apart (pa 1/Pe ) on two adjacent fringes of orders, say, N- O and N= 1 x will then be 1 U-EPIF (3-27) In general, the in-plane displacement in the direction normal to the grating lines is, in terms of the fringe order Nx u - ——F (3.28) 3.7.2 Fringes due to normal strain ey As shown in figure 3.6.0, a normal strain ey would also change the frequency of the specimen grating, and cause the grating lines to rotate through an angle Yx,y,/2. Again using equations 3.21 and 3.22, it is possible to calculate the orientations of the diffracted rays with respect to the x'z and y'z planes. For the incident beam A0, and letting m- -1, Sin a - Sin 5 - AF//2, wi= Yx,y,/2= -ey/2 and f'= (F/2)(1 - ey/2), equation 3.21 yields for small angles and rotations 55 AFey n v . Sin B_1 (X Z) 72— (3.29) and AFs Sin 811 (Y'Z)- - 2 (3.30) In a similar way for incident beam OB, letting m- +1, Sin a =Sin E . -AF//2, “Hf: ‘Yx,y,/2- ey/2, and f'- (F//2)(1 - ey/Z), equation 3.22 yields for small angles and deformations AFe Sin 8:1(Y'Z)- - 2 (3.31) APE v v . Sin 8+1(x z) 2 2 In this case, the two diffracted rays lie in a plane parallel to the yz-plane (figure 3.9) and the sine of angle between each of them and the z-axis is (for small wi) AFe Sin 3-1(2) - Sin 8+1(+1) - (3.32) Since the diffracted rays A0' and 80' in this case lie parallel one to the other, equation 3.9 yields 56 Figure 3.9 Geometry of the Incident and Diffracted Rays From A Grating Deformed by ey When Analyzed with Beams A0 and BO 57 Pg ' 0 (3.33) This means that when the specimen grating is interrogated by incident beams A0 and 80, the normal strain 6y will not contribute to the interference system. 3.7.3 Fringes due to shear strain ny As shown in figure 3.6.d, a shear strain ny would induce normal strains ex, and Ey' in the x' and y' directions respectively. Due to ex” the pitch of grating x' would increase, consequently decreasing its frequency. Similarly, due to ey, the pitch of grating y' would decrease and consequently increase its frequency. Since the shear strain does not produce any rotation, the two-dimensional grating equation can be used to determine the orientation of each diffracted ray. For incident ray A0, with m- -1, Sin a a Sin t; a ”7/2, 11:1- 0, f- (F//2)(1 - ny/2), and for incident beam OB, with m- +1, Sin a =- Sin E; - -AF//2, wia O, and f"- (F//2)(1 4- ny/Z), equation 3.“ yields AFY - I x Sln 331 (z) . Sin 811(2) - -§7§l- (3.34) 58 USing simple geometry (see figure 3.10), the sine of the angle between each of the diffracted orders and the xz-plane is found to be xrv Sin 3 - -7r—JL (3.35) The resulting interference fringe pattern is composed of horizontal fringes parallel to the xz-plane whose frequency can be calculated using equation 3.9 to be Y 2 (3-36) 3.7.“ Fringes produced by a rigid body rotation A rotation of the x' and y' gratings through an angle w, considered positive in the counterclockwise direction, causes the two diffracted rays OA' and 08' to diverge from their respective planes x'z and y'z, but their pitch and frequency will not be disturbed. Using equations 3.21.b and 3.22.b, the orientations of the two diffracted rays are found to be Sin 811(Y'Z)= $22 (3.37) Sin 8:1(x'z)= - $§$ I l. /'I'/" 0:) ‘u \‘. \ S. .i g... 4.1 Figure 3.10 Geometry of the Incident and Diffracted Rays From a Grating Deformed by ny When Analyzed with Beams A0 and BO 60 and the sine of the angle between each of the two diffracted orders auui the xz-plane is (see figure 3.11) . fl . Sin 8w 2 (3 38) Again, equation 3.9 gives the frequency of the resulting interference system P - w? (3.39) 3..7.5 Fringes produced by out-of-plane rotations Any small out-of-plane rotation about an in-plane axis, will introduce equal deviations of beams A0' and 80' and it will not make a contribution to the interference pattern, thus it can be neglected [50]. This analysis has shown that by interrogating the specimen grating with beams A0 and BO, Moire patterns of displacement, Ux’ can be obtained due to the effect of normal strain ex, shear strain ny, and in-plane rigid body rotation w. 3.8 MOIRE FRINGES OF DISPLACEMENTS UY' To obtain the Moire fringes of displacements Uy, the specnmui grating is interrogated with beams BO and CO. Due to the symmetry, tflua 61 \ _._--._.—-...-.- u-..—._—._.—.. 1 I 1 I I I I Figure 3.11 Geometry of the Incident and Diffracted Rays Affected by Rigi-Body Rotation of the Grating When Analyzed by Beams A0 and B0 62 deformation.of the grating lines will be the same as those obtained for ex, and there is no need to repeat the analysis. 3.9 MOIRE FRINGES OF DISPLACEMENT UHS' The displacements UHS’ are obtained by interrogating the specimen grating with coherent beams A0 and CO. The basic geometry of the Moire interferometer is reproduced again in figure 3.12 ‘The angle ADC is equal to 2:, thus the frequency of the reference grating produced by the interference of beams A0 and CO is given by equation 3.9, as: F115. 3.3—13.3 (3,40) Using equation 3.10, it can be rewritten as v =3? (/2 sm (1 ) - 2r (3,41) HS i.e. exactly twice the frequency f of the specimen grating. 63 Figure 3.12 Schematic of the Angles of Incidence of the Moire Interferometer When Analyzing the Specimen Grating with Beams A0 and CO 64 3.9.1 Fringes due to normal strain €x" Due to the action of normal strain ex, the pitch of the x' grating lines will change by a factor of (1 +ex,), so its frequency will decrease to f'- (F /2)(1 - ex,). MS Since both incident beams A0 and CO lie in the x'z plane, the angle of emergence of the diffracted rays can be determined using the two-dimensional grating equation 3.“ or Sin 8m (x'z)- mAf + Sin a (3.4) Then for beam AO, let m- -1, Sin a - Sin E - AF /2, and f' given HS above, the grating equation yields AF 6 , Sin 821(x'z) - —112——1‘— (3.42) Similarly, for beam CO, let m- +1, Sin a a Sin 5 - AF x,/2, and 1458 f as above, to get AFHSEX' 2 (3.43) Sin 821(x'z) - - 65 Bonicuffracted rays A0' and CO' lie in the x'z-plane (figure 3.13), and they are located at the same distance on opposite sides of the z-axis. The resulting interference pattern has frequency F e 344) NS x' ( ° and the fringes are perpendicular to the x'z-plane. Again, it is seen that the frequency of the interference pattern is a functnnicfi'the strain ex, and the frequency Fu of the reference grating, and the 5 displacement along the x'-direction is given by 1 UUS- (Nx,)'f (3.45) 3.9.2 Fringes due to in-plane rotation w. When the specimen grating is unstrained but undergoes a rigid body rotation in its own plane, by a small counterclockwise angle 11), the three-dimensional grating equations 3.21.b and 3.22.b define the direction of the diffracted rays. For A0 and CO then, Sin 821(x'z)= O (3.46) *F45w Sin 8:1(y'2)= Y Y \ A Y' "";Elllll';r c \\ W. x \\.. V. m IVK‘ Figure 3.13 Geometry of the Incident and Diffracted Rays From a Grating x' Deformed by e , When Analyzing with Beams A0 and co x 67 and Sin 8:1(x'z)= O. (3.47) *Fusw 2 Sin 811(y'z)- - Since both rays AO'and CO' lie in the y'z-plane, at the same distance on opposite sides of the z-axis, (figure 3.14), the fringes of’ resulting interference pattern are perpendicular to the y'z-plane, and their frequency is w 45 (3.48) 3.9.3 Fringes due to shear strain Yx'y" .A positive shear strain Yx'y" will cause the x' grating lines to rotate clockwise through an angle -Yx,y,/2. Then, equation 3.18.b yields . . - M 3 49) and 68 Figure 3.14 Geometry of the Incident and Diffracted Rays Affected by a Rigid—Body Rotation of the Grating when Analyzing with Beams A0 and BO 69 Sin 8+1(y z)- 4 (3.50) Again, both diffracted rays are located at the same distance on opposite sides of the z-axis (see figure 3.15), and the frequency of the resulting interference pattern is FuSY , , x'y' 3.9.4 Fringes due to normal strain ey,. When the specimen grating is deformed in the y'-direction by the action of a normal strain ey,, the lines of the x' grating are stretched, but the frequency does not change. Then bouicnffracted orders leave the grating surface along its normal, and since they are parallel, the frequency of the interference pattern is equal to zero. 3.9.5 Fringes due to out-of-plane rotation. It has been shown by Basehore and Post [52] that a rotation about the x'-axis causes identical angular deviations in the diffracted rays A0' and CO', but the angle between them remains unchanged. Also Walker and McKelvie [53]. showed that a rotation about an axis perpendicular to 70 Figure 3.15 Geometry of the Incident and Diffracted Rays from a Grating Deformed by y , , When Analyzing with Beams A0 and CO x y 71 the plane under examination will produce some fringes, but, for the case of small rotations, they are negligible. 3.10 COMPLETE ANALYSIS OF TWO DIMENSIONAL STRAIN FIELDS. The theory presented in previous sections demonstrates that the displacement component in a given direction can be measured by interrogation of the specimen grating with different pairs of beams. That is, in order to measure Ux’ the specimen grating is interrogated by beams A0 and BO. Similarly, Uy is obtained by combining beams BO and CO, and U by combining beams A0 and CO. 45 The state of strain throughout a general two-dimensional strain- field can now be determined. Recognize that there are three unknown strain components ex, ey, and ny, at every point in "plane" elasticity problems, authout distinguishing between plane stress, plane strain and generalized plane stress at this stage. Differentiation of the displacement components with respect to the appropriate space variables yields ex, ey and 54 i.e. the three strain 5! components which define the complete state of surface strain [58] throughout the field of view. Then 1 GNX BN 1 EN Ex"F 3x EY BY 845' F ax' (3.52) nan-4 It was also demonstrated that when interrogating the specimen grating with beams A0 and BO, the resulting interference Moire pattern 72 is formedlnrfringes of extension produced by normal strain ex; and fringes of rotations are produced by shear strain ny and the rigid body rotation w. It was also shown that the fringes of extension are perpendicular to the xz-plane, while those caused by the small rotations are nearly parallel to it. Also, the interference fringes created by the intersection of beams A0 and BO were perpendicular to the xz-plane. Since the quantity to be measured is a normal strain, e.g. ex, the needed final result is the gradient or spacing of the Moire fringes in the x-direction. The rotation-induced fringes run roughly parallel to the x-axis and, therefore, have no gradient in that direction. D1 algebraic form, the total fringe order is expressed as the sum of a part resulting from extension and another part caused by rotation: Nx- (Nx)€ + (Nx)R (3.53) 3x '1: 3x 3x (3'54) .1. aux 1 [30198 + 3("x)n] F 3(N ) 3(N ) x e x R but 3x >>> 3x The conclusion is that rotation does not affect the measurement of normal strain [54]. In practice, the Moire fringe order is not separated.into two parts. The rotation element merely causes the Moire strain fringes to 73 deviate from the usual orientations parallel to the grill. This effect is clearly seen in figure 3.16. The shear strain ny could be evaluated by cross derivatives of displacements as ny' 3';— + 3X (3.55) The partial derivatives in the shear equation are gradients in the direction parallel to the analyzer grating lines. The gradient of the rotation-induced fringes in that direction may be at least as large as the gradient of extension-induced fringes [54]. In algebraic notation ny' 2exy- F 3y + 3x (3'56) 01" 3(N ) 8(N ) 3(N ) 3(N ) y -l___X_§.._.L§..__L_€.__¥_B (3.57) X)’ F 3y 3y 3x 3x 3(N ) 3(N ) 6(N ) 3(N ) but x. __"_.*3. 2 _’_<_8 and ___J_§ 2 __L_€_ (3.58) 3y 3)! 3x 3x so the error may be very great. This situation is worsened by the lack of any direct way of estimating or eliminating the error. Thus, the 74 rectangular strain gage rosette method involving three measurements of normal strain is superior for evaluating shear strain ny, as xy HS - (ex + 6y) (3.59) This approach also circumvents the experimental difficulties associated with rigid-body motion and accidental misalignment of the reference gratings [55]. In order to differentiate the displacement components with respect to their appropriate space variables, graphical differentiation can be utilized. Observe that: (a) Moire fringe order can be plotted as a function of position coordinate: (b) the slope of this curve can be evaluated point by point and the result plotted; (c) when divided by the frequency of the reference grating, this last plot becomes a plot of strain along a given axis. 3.11 USE OF PITCH MISMATCH IN MOIRE INTERFEROMETRY. The accuracy of the in-plane Moire method depends largely on how accurately the displacement curve is plotted; and the accuracy of the displacement curve in turn, depends on the number of available points. For a given displacement field, the number of fringes depends on the grating used. 'nuafdner the grating, the greater the number of resulting fringes. Using the interferometric technique, the frequency of the reference grating and the specimen gratings can be adjusted 75. easily to suit the problem. However, the finer the grating, the greater are the requirements in terms of handling, optical bench stability, etc. For practical purposes, it is convenient to use frequencies in the range of 15000 to 60000 lines per inch, (600 to 2400 lines per mm). So far, it has been assumed that the reference grating and the specimen grating are related by F- /2 f, and that the angle of incidence of the laser light onto the specimen grating is exactly 5. If these conditions are not satisfied by a small difference, pitch and rotational mismatches are present. These initial differences are called "linear" and "rotational" mismatch respectively, and of course they are measured against the reference grating. The resulting fringes are called "linear mismatch fringes" and "rotational mismatch fringes". An illustration of the fringes produced by the mismatch is shown in figure 3.16 3.11.1 Analysis of the undeformed specimen. In order to illustrate the use of the mismatch, an undeformed specimen grating being interrogated with beams BO and CO, i.e. to obtain the Uy displacement field will be considered. The relationship between the specimen grating and the reference grating frequencies is 5F- F - /2 r (3.60) 11 1111111111111111111111 111111111111 111111111111111111111111111111111WW 1111111111111111111111111111111111111111111111111 WWWWWWWWWWW1 1111111111111111111111111 111 1111111 11111 111 11111 11' 111 11 111 1W1WWWWM W11111111 11:11 11111111 11“” 111 11 11111111111 111111 11111 IIIIIIIIIII'III 11 1111111” 111111111 IIIIIIIIIIIIIIIIIII IIIIIIIIII 11111111111111111111111111111111111111111 W "'1 W 01 rotatlon and 77 where 6F is the frequency mismatch, F ,is the reference grating frequency, and f is the specimen grating frequency. This equation can be rewritten as F 65‘ r- 72(1 - E—J (3.61) It should be noted that the second term in parentheses is a component of the frequency of the reference grating with its lines parallel to the xz-plane. To obtain the components along the ins-degrees orientation, it should be considered the result of a rotation through HS-degrees. This will yield a component of extension equal to 6F/2F and a component of rotation of the same magnitude. The first component will produce the same effect as a fictitious tensile strain, while the second is equivalent to a fictitious rotation of the specimen grating lines. Using the three-dimensional grating equation, it is possible to determine the frequency of the resulting interference pattern. For beam 80, let ms +1, Sin a - Sing - -AF//2, f'- (F//2)(1 - GF/ZF). 11- 6F/2F, then Sin 8:1(y'z)- - T111152) (3.62) A(6F) Sin 8:1(x'z)= 2 Similarly, for beam co, let ma +1, Sin (1 = Sin 2; . -11F//2, f'= (F//2)(1 - (SF/2F) and 11113 - (SF/2F, then 78 Sin 8:1(x'z) - - Aéggl (3.63) S ,, , ms) “Be”z)"yr It can be shown that the diffracted rays 08' and OC' lie in the yz-plane (figure 3.17), and the sine of the angle between them and the z-axis is Sin 86F(z)- liggl (3.64) The frequency of the resulting initial fringe pattern with its fringes parallel to the xz-plane is Par' 6F (3.65) and the fictitious displacement is given by y 5F"? (Ny16F (3.66) Thus, the fictitious strain produced by the mismatch is 3(U ) 8(N ) . ___1_§§...l.__JL§§L (5y)6F 3y F 8y (3.67) 79 Figure 3.17 Geometry of the Incident and Diffracted Rays Affected by a Fictitious Strain Produced by a Frequency and Rotational Mismatch When Analyzing with Beams B0 and CO 80 3.11.2 Analysis of the deformed grating (strain plus mismatch). Now, Unacombined effect of strain plus mismatch will be considered. In this case, f'- (F//2)(1 + (8Y15 + (5F)/2F), Then for beam BO, Sin (1 - sm r, - -AF//2, 1111- [(ey)e/2 + (6F/2F)], m- 1, the three-dimensional grating equation yields AF (Ey)e 6F n I ,. .. __ 51“ 3+1111 21 75 2 ‘1 2F (3.68) (e ) 5111 3116.121 - 17“;- --%’—6+ 353 and for beam CO, with m- +1, Sin 6 - Sin E - - AF//2, and f' and 11 as above, - AF y e g5 Si“ 3311x121 73 2 + 2F (3.69) AF y e ‘QE 31“ 311(3'121‘ 75 2 + 21“ Again, the diffracted rays 80' and CO', lie in the yz-plane (figure 3.18), and the sine of the angle between each of them and the z- axis is 81 Figure 3.18 Geometry of the Incident and Diffracted Rays from a Grating Affected by Strain Plus Mismatch When Analyzing with beams BO and C0 82 (e ) 6F . Y8 ...... Sin 3(5 + 61,) AF 2 + 2F (3.70) and the frequency of the resulting interference pattern whose lines are parallel to the xz-plane. is P15 + 5F)- F [(ey)e + 6F] (3.71) and the displacement is 1(N) I — 0 2 (Uy)(e + 6F) F y (e + 6F) (3 7 ) and the strain is a(Ny)(e + 6F) 1 1 (e + or)” F 3y (3°73) 5y) Computation of the true displacement and strain at any point follows directly from prior derivations. (U ) (”1'1Uy)(e + 6F) 7 y 65‘ yt (3.74) 1 (”11112. F (We + 6F) - (”3115? where: (Uy)t is the true strain. The true strain a is thus computed as t follows, 83 8t. (Ey)(e + 6F) - (6:311SF (3.75) BUN ) - (N ) J _ l y (e + 61“) j 6F 6t I" 3x (3.76) Equation 3.66 demonstrates that the spacing between fringes is smaller than that given in equation 3.611, which means that there is an increase in the number of fringes in a Moire measurement. Notice that the sign of the fictitious strain should be of the same sign as that of the true strain, otherwise a reduction in the number of fringes will result. Mismatch of the opposite sign can be used efficiently if its magnitude»is at least twice the magnitude of the strain to be measured. Nevertheless, caution should be exercised, because the use of more mismatch would overshadow the information. Note also that it is not absolutely necessary to know the exact size of the mismatch in order to calculate and eliminate the pitch difference and initial rotation effect. "Before strain" and "after strain" fringe photographs may be used directly. A detailed procedure for the elimination of the fictitious strain is given in Chapter 5. Figure 3.19 shows photographs of two different amounts of mismatch for the same loading condition in a pin-loaded hole. Figure 3.20 shows the strain plots obtained for these photographs, and it should be noticed that the strain plot obtained for the case of more mismatch gives a more detailed strain contour than the other case. 84 /: ’_W ’11-” NO-LOAD AT-LOAD AT-LOAD .Figure 3.19 Moire Fringes of Displacements for Two Different Amounts of Mismatch and Same Loading Conditions 85 LEGEND H—9 0 LOT O—H LITTLE '11 1111111111‘1'11111111‘ J 240 .00 160.00 J n 5 . X D ‘5’ x2 F‘ 0'1 H CO I ... .. 1 . "‘ 1 1 .. , | '_I l.‘ 1‘ :28 : a = ” H ‘..11. _______________ / _______ I ________________________ (IO " 1 m -_~ H : D— I (Do '-‘ E 80. : NOTE: The curves were drawn 0% : using Ioo cannot" H I '1 : generated points. The Z : whole an tor ldontltlcauon O I only. I O I 5 1 (D 1 wt. 1 ' 1 : Cy for o I X2=O.125 s : g 1 COMPRRISUN I N 1 I I 1—0 0.60 -h. 30 000 0.30 0.60 DISTRNCE FROM CENTERLINE (IN 1 Figure 3.20 Strain e Obtained from Two Sets Of Moire Patterns from the SameySpecimen with Different Amounts of Mismatch CHAPTER U EXPERIMENTAL APPARATUS “.1 BASIC IDEA OF THE INTERFEROMETER AND OBJECTIVES. The main objectives pursued in the construction of the Moire interferometer were: (1) to avoid a rigid connection between the optical elements and the specimen being tested, (2) to be able to perform measurements in three different directions, (3) to obtain a good efficiency of light Ludlization, (A) to develop an experimental set-up suitable to be adapted to perform measurements in environments not so ideal as an optics laboratory. Figure 11.1, shows a photograph of the overall view of the experimental apparatus. Figure “.2 shows a two-dimensional schematic of the basic idea of the experimental apparatus as follows: a 20 mw Helium- Neon laser 1 produced a beam of coherent light having a wavelength of 632.8 nm (24.913 x 10-6in.). Using two front surface mirrors 2 and 3, the beam was directed to the spatial filter where it was filtered and expanded with a hOx microscope objective and a pinhole H. A collimating lens 5 (a 13 in. diameter plano-convex lens, focal length- 1.0 m) changed the expanding beam into a parallel beam. This collimated beam 86 _ ,_fi»_’_—.——r—..L. 87 Figure 4.1 Overall View of the Experimental Set Up Incorporating the Mbire Interferometer 2:3 6 -9 11' 88 HELIUM NEON LASER MIRROR SPATIAL FILTER COLLIMATING LENS FRONT SURFACE FLAT MIRRORS SPECIMEN GRATING DECOLLI MATING LENS Figure 4.2 Schematic of the Basic Idea Used in the Construction of the Moire Interferometer 89 reached two first surface flat mirrors 6 and 7 oriented at 145 degrees with respect to the optical axis of the lens. Each of these two mirrors directed portions of the parallel rays to another two mirrors 8 and 9 which could be oriented easily to direct the light towards the specimen grating 10, and to obtain the desired angle of interference between the two beams. These two beams incident onto the specimen grating were diffracted. The diffracted rays passed through a converging lens 11 (focal length- 75 cm.), and were directed by a front surface mirror, oriented at #5 degrees with respect to the normal of the grating, to the observing system or photographic camera. The beam narrowed and passed through an aperture before reaching the camera. “.2 CONSTRUCTION OF THE MOIRE INTERFEROMETER Figure “.3 shows the arrangement, viewed along the z-axis, of the elements forming the interferometer. In order to describe the orientation of the flat mirrors, consider a set of axes with the positive z-axis going into the page, and the x and y axis located on the plane of the paper as a reference system. The three flat mirrors upon which the plane wavefront from the collimator is incident are denoted by the numbers 1, 2 and 3. The other three mirrors used to direct the beams of light towards the specimen 'grating are denoted by the letters A, B, and C. To direct a portion of the light to mirror A, mirror 1 is rotated IVS-degrees from the xy-plane in the clockwise direction as viewed from the positive y-axis and also rotated 1&5 degrees counterclockwise from the xy-plane as viewed from the positive z-axis. In order to direct Figure 4.3 Arrangement of the Optical Elements of the Moire Interferometer 91 light to mirror B, mirror 2 is rotated 145 degrees from the xy-plane counterclockwise as viewed from the positive y-axis and 115 degrees from the xy-plane in the clockwise direction as viewed from the positive 2- axis. Similarly, mirror 3 is rotated 145 degrees counterclockwise from the xy-plane as viewed from the positive y axis and 115 degrees from the xy-plane counterclockwise as viewed from the positive z-axis to direct a portion of the light to mirror C. Mirrors A, B, and C are located at the three corners of a square. The distance between their centers can be adjusted easily. For this research, it was set at eleven inches. These three mirrors provide the three beams of light which converge on the surface of the specimen forming three sides of a square-based pyramid. The angle of interference between them can be adjusted by changing the distance between the interferometer and the specimen grating according to equation 3.9, which relates the angle of incidence and the desired interference frequency. All the mechanical components used to position and mount the mirrors were purchased from the Newport Corporation, 18235 Mt. Baldy Circle, Fountain Valley, CA 92708. The mirrors were purchased from Melles Griot, Optical Components, 1770 Kettering Street, Irving, California 927111. They were front surface mirrors. Their flatness was one quarter of a wavelength, and their dimensions were 100 x 100 mm. All the mirrors were glued on mirror mounts type Newport MM-2. The glue used was Pliobond Adhesive, manufactured by the Goodyear Tire 8: Rubber Company, Akron, Ohio, “11301. The tilting element of the mirror mounts provided a kinematic orthogonal adjustment using balls that are spring- loaded between a conical recess and a flat. Orientation is smooth to 92 adjust with low friction, fine pitch (80 tpi), lapped stainless steel drive screws, which push the alignment mechanism via hardened steel balls. The angular range is H.145 degrees about each of the orthogonal axes. The RMS sensitivity is 2/3.5 arc-seconds. Mirrors mounts 1, 2 and 3 are fixed permanently to the supporting aluminum frame of the interferometer. Mirrors A, B, C are connected to the aluminum frame by support posts which are held by post holders (Newport type VPH). The posts are held together using universal clamps (Newport model CA-2) . These universal clamps allow free positioning and orientation of mirrors A, B, and C. The post holders of mirrors A, and B are mounted on optical carriers (Newport type MTF) which allow horizontal adjustment up to 0.25 inch by an 80-pitch screw. These carriers are mounted on an optical rail (Newport model MEL-12). The post holder for mirror C is fixed directly to the aluminum frame using a screw which can slide horizontally. The aluminum frame holding all the optical components is mounted on a tilt platform (Newport model 37) which offers three axis adjustments: two tilts, and one in-plane rotation. The tilt range in each direction is -6 to +8 degrees. The body of the platform is hard anodized aluminum and it has hardened steel inserts to interface with the micrometer spherical tips. The platform is oriented to provide tilt about the x and z axes and in-plane rotation (xz-plane) about the y- axis. I-I.3 ADJUSTMENT OF THE MOIRE INTERFEROMETER. In order to adjust the positions of the mirrors A, B and C, the following procedure was used: 93 1. Using the formula given in equation 3.9, calculate the distance from the interferometer to the specimen grating that will render the desired angle of interference between the two incident beams. 2. Using this result, move the specimen together with the loading frame to the calculated distance. Loosen the universal clamps and orient manually the mirrors to direct the light towards the specimen grating. Care should be taken to avoid touching the surface of the mirrors, since any hand grease cannot be removed without damage to the surfaces. This positioning will give a rough adjustment of the orientation of the mirrors. To describe the process of fine adjustment of the orientation of the mirrors, let the knobs of the mirror mount adjustment screws be c and c for mirrors A, B and C denoted by a and a b and b 1 2, I 2’ 1 2' respectively, (see figure u.u.a). 3. On the focal plane of the decollimating lens, notice three bright spots. They correspond to the first diffraction orders of the three incident beams. Adjust mirrors 8 and C first. Block the light from mirror A, then only two bright spots will remain on the focal plane of the decollimator. These bright spots should be in a vertical position, that is, one above the other. In order to get the correct sign of the pitch mismatch, that is, the mismatch that will result in an increase in the number of fringes per unit of space due to a tensile strain, and a decrease in the number of fringes per unit of space due to a compressive strain, the bright spot corresponding to the diffracted order 08' should be located above the one corresponding to the E ‘Figure 4.4 94 Mechanical Elements Used in the Positioning of Optical Elements of the Moire Interferometer 95 diffracted order OC' Then, both spots should be identified before continuing. The position and orientation of the bright spot 08' can be adjusted using knobs b1and b2. Knob b1 will control the separation between the two bright spots. A clockwise rotation of this knob will move the bright spot 08' upwards. The angular orientation of the bright spot can be adjusted with knob b A clockwise rotation of this knob 2. will rotate the‘interference fringes clockwise as viewed from the 0 camera. The function of the first knob is to produce extensional pitch mismatch and rotational pitch mismatch is produced by the second, as in traditional Moire. Once the two bright spots have been positioned correctly, the resulting interference pattern should be in the form of horizontal fringes. 14. Now, to adjust mirrors A and B, block the light from mirror C and uncover mirror A. In this step, mirror 8 should not be touched at all, since that would produce a misalignment of the horizontal fringes. Next, block any of the two mirrors to identify their corresponding diffraction orders. Again, to get the correct sign of the pitch mismatch, spot OA' should be to the right of spot 08', that is, in a horizontal position. Knob a2 can be used to produce extensional pitch mismatch, and a clockwise rotation will move spot OA' farther to the right of spot 08'. Knob a1 will produce a rotational pitch mismatch and a clockwise rotation of it will produce a counterclockwise rotation of the interference pattern as viewed from the camera. When these two bright spots have been positioned correctly, the resulting interference pattern should be formed of vertical fringes. 96 5. After adjusting mirrors A and B, and B and C, for vertical and horizontal fringes respectively, a third set of interference fringes can be obtained at forty-five degrees. To obtain this set of fringes, simply block the light reaching mirror B and uncover mirror C. While doing this, extreme care should be taken to avoid disturbing the adjustments for the other two orientations. 6. Also, for further alignment of the fringe pattern, the micrometers of the tilt table on which the frame is mounted can be used. These: micrometers add three degrees of freedom and are denoted by D, E, and F. hflcrometer D provides tilting about the z-axis and will rotate the interference pattern in the xy-plane, that is in the surface of the specimen, as viewed from the camera. Micrometer E tilts the interferometer about the x-axis and it will produce both extensional and rotational pitch mismatch. Micrometer E will rotate the interferometer in the xz-plane. This rotation is used to position the surface of the specimen grating perpendicular to the optical axis of the collimator. For the no-load stage, the diffracted orders should retrace their path to the pinhole of the spatial filter. 7. A 35 mm Canon A-1 Camera and a 70-210 mm zoom lens with a 2x focal length converter were used to record the fringe patterns. A cable shutter release was used to avoid any vibration of the camera. The specimen was loaded using a modified loading frame manufactured by Scott-Engineering Sciences, 11400 S. W. 8th. Street, Pompano Beach, Florida, 33060. Tension was applied using a hydraulic 97 (double-acting) cylinder ram with the hydraulic pressure supplied by a hand pump. To measure the applied load, a force transducer was made by the investigator. This transducer was connected to a strain indicator model 3270, made by Daytronic Corporation, Miamisburg, Ohio. The span of the conditioner was adjusted to display the actual magnitude of the applied load. The sensitivity of this transducer was :5 pounds, and the range was 12200 lbs. The virtues of this system can be summarized as follows: 1. It allows a good efficiency of light utilization. 2. No rigid connection is required between the specimen and the system. 3. With small improvements, it can be utilized to perform measurements in environments not so ideal as an optics laboratory. H. Grating frequencies are easily adapted to suit the problem. 5. Measurements can be performed in three different directions giving a map of strains in the same number of directions and allowing calculations of maximum strains. 6. Pitch mismatch can be easily adjusted. 98 7. Amenable to various methods of data recording, such as film, television camera, etc. CHAPTER 5 EXPERIMENTAL METHODS 5.1 FABRICATION OF MOIRE SPECIMENS The dimensions of a test coupon should be selected very carefully in order to assure that the planned test will yield results which truly represent the expected material behavior. When working with composites, care should be taken when dimensioning the test coupon. Horgan [£15] . showed that end effects are more severe for fiber reinforced composites than for isotropic materials. Also, it is the geometric factors that usually render one of the failure modes predominant. These factors are the width of the specimen and the distance between hole center and the free end which is perpendicular to the loading axis. Figure 5.1 shows these geometric parameters considered in the design of the specimen. The effect of end distance was investigated by Collins [32]. He suggested that a minimum ratio e/d > 3 was needed to develop full bearing strength. He also studied the effect of specimen width in single-hole specimens. A minimum w/d of 8 was required if full bearing strength was to be developed. The dimensions of the specimen adopted in this study are also shown in figure 5.1. In the spirit of avoiding end effects and to provide full bearing strength, the specimen used to obtain the stress and strain distribution around the pin-loaded hole was: 50.8 mm (2.0 99 lOO $ L” d = 0.25 In. O*=:1.0 In. L = 8.0 In. w: 2.0 In, L 14: DRILL THRU d / >59 —T e i I Figure 5.1 Dimensions and Parameters Involved in the Design of the Test Coupon 101 in.) wide, 203.2 mm (8.0 in.) long; a 6.35 m'm (0.25 in.) diameter hole was drilled through this composite laminate, and the distance from the center of the hole to the free end was 25.“ mm (1.0 in.). The specimens were cut slightly oversized, with fibers perpendicular and parallel to the axis of loading, from a large panel using a band saw. The cutting process was carried out slowly, and the composite was frequently wetted with water to prevent both the saw and the laminate from overheating. The sides were finished by hand to the specified dimensions, using first a wet/dry, 200 grit paper, and, for final finish, a wet/dry, “00 grit paper. Again, water was used to prevent overheating. Since the finishing procedure was performed by hand, in order to obtain square sides, the composite laminate was sandwiched between two steel templates, whose edges had been aligned, one with respect to the other. During this cutting and finishing procedure, extra caution was required to avoid delamination of the material. Also, the drilling process had to be carried out with extreme care. The finished blank was sandwiched between one of the steel templates and a piece of particle board, and clamped together. This pressure helped to avoid delaminations during the drilling operation. The steel template was used as a reference to position the holes in the specimen. 5.2 SPECIMEN GRATINGS Application of the Moire effect to any problem depends on the successful deposition of line grids (or dots) on the surface of the specimen material. 102 All specimens in this study used reflective diffraction gratings. Following a method developed by Basehore and Post [51], the master gratings or molds were produced as follows: A high resolution photographic plate (Agfa type 8E75), was exposed to two beams of collimated coherent laser light. The two beams intersected at the surface of the emulsion. These interfering beams generated a three-dimensional pattern of constructive and destructive interference. The frequency of this interference pattern could be calculated using equatdrni 3.9. When the exposed plate was developed, a silver compound remained in the areas exposed to constructive interference, while the silver was washed out elsewhere. Upon drying, the gelatin of the emulsion shrunk, but a reinforcing effect of the silver crystals (”1 the exposed areas caused a non-uniform shrinkage which is termed the lenticulation effect. The effect was a textured surface which had the same frequency as the interference system. Since a cross-line grating was desired, the plate was rotated 90-degrees after the first exposure and a second exposure was made. In order to produce these photoplates, the Moire interferometer was used. It was adjusted to produced two horizontal beams using mirrors A and B, following the procedure described in chapter “. Since no reference grating of the desired known frequency was available, the angles were adjusted, and using a traveling microscope with a “0x microscope objective, with a scale in millimeters, the number of fringes per millimeter in the interference system was counted. Then, the mirrors were fine-adjusted until the desired frequency was obtained. A second checking was done on the previous counting results using a 75x microscope objective lens. As a final verification step, a small piece 103 of the overcoated plate was placed in a scanning electron microscope, and a photomicrograph (figure 5.2) with a precisely known magnification was taken. Based on this photomicrograph, calculation of the frequency of the grating was verified. The frequency used in this study for the specimen gratings was 6“0 lines per mm (16,256 lines per inch). The diffraction efficiency of a grating depends partly--£>--4D xs=0.107s a——~——a X8=0.2600 I: a: 4: x7=0.3128 o x——-x———x x0=0.3760 O . Y 33* : 'P 1 ' 111111111111111111111 8. O {9: c u e a L T N ”TX 8. ' " "08‘ g 3111 p... I ’ I 28 H 0 (1° ' ' I: p. l 88 :1 831 1 hdl ' z. 1 9 1 8 1 '7‘ . 1 8 1 SE 1 N I / 1 OO/ey y ' -060 -'0. 30 0'. 60 DISTRNCE FROM CENU'JTERLII‘iE0 (IN 1 lfigune 6.9 Strain ey Along Lines Parallel to the Direction of the Load and Located in the Ligament Area on the Right Side of the Hole 134 LEGEND B—H X4=-0 . I 250 H—O X5=-0 . I 075 45—4—15 X8=-0 . 2500 vL F #1 X7=-0 o 3125 X—--)(—-X X0=-0 .3750 11111111111111 240.00 160.00 J _ ------—--- 00.00 J xIO' 0.00 MICROSTRHIN -60.00 I -160 000 NI/lOO/ey -0.60 -0. 30 0'60 DISTRNCE FROM CENOOTERLINEo [IN 0) o..------------------- .-240 000 Figure 6.10 Strain 6 Along Lines Parallel to the Direction of Load and Locazed in the Ligament Area on the Left Side of the Hole 135 noticed that the highest value does not occur at the ends of the horizontal diameter of the hole but slightly below as was reported elsewhere [5]. This characteristic can be observed on both sides of the hole. This behavior is an indication that, due to the deformation of the hole, some clearance developed in between the pun and the hole. Frocht and Hill [18] noticed the same behavior. De Jong [7] reported that the maximum tangential stress concentration did not always occur in the minimum net area of the plate, i.e. the ends of the horizontal diameter of the hole, and that the place where the maximum occurs depended on the material properties and probably on the width of the plate. It should be noted that the strain concentration is more noticeable along these lines x- t 0.125; and the farther away the line is from the edge of the hole, the more the strain decreases to an average value. In these two figures, the strain is shown only for lines located up to x- i 0.375 inches away from the center of the hole where the strains represent the average strain. Inclusion of more lines in the same figure would have decreased the clarity of the drawings. A careful comparison between these last two figures shows a slight asymmetry in the strain distributions for the two ligament areas. This difference can be attributed to variation in thetmechanical properties of the material. Figure 6.11 shows strains 6“ for parallel lines in the upper 5 right portion of the hole, i.e. in the right ligament area and also to the left of the bearing region. Figure 6.12 shows strain 845 for parallel lines which go from one ligament area to the other passing through the bearing region. This figure shows interesting features such 136 I‘M u—H v'0.-.- 0.0000 O—H Y' n: 0. 0000 b——-b-—fi Y1=-0.0292 P T Y'B=-O. 0292 x———-x-—-I< v’a=-0. 0664 o—o——o Y'3=-0.0584 'Y 111111011 111111111 CENWEHUNE ”$2 .2 l -00.00 MICROSTRR -100.00 J 1 III/1001s,5 .‘240 o 00 0.60 -0. 30 0'. 60 DISTHNCE FROM CENOOTERLINE0 (IN. ) Figure 6.11 Strain €05 Along Parallel Lines Oriented at 45-Degreees to Direction of Load and Located in Both Ligament Areas 137 LEGEND B——B-—-fl Y' =-0. 1325 O—H Y'4=-0. 1707 ‘—.'——‘ fiY’ 5: -0.2200 fi 1.1-0.2051 1i'3 ’” 3- P ~ 1 mm 11111111111 8 8“ : MINE '3 1 r. 6 ' F" “qu VII 1'0 3' - 28 '\\ EM / 55.1.” --------- 0: 1 I— I (Do 1 0c; : g z 1 8 E a? 1 '1“ E a; 1 N I ‘-0.60 -0. 30 0' 0' 30 0' 60 DISTHNCE FROM CENOTERLINE [IN- 0] Figure 6.12 Strain 64 Along Parallel Lines Oriented at 45-Degrees to Direction of Load and Encompassing the Bearing Region and Both Ligament Areas 138 as a very high tensile strain on the lower right portion below the hole, which rapidly changes to a high compressive strain. Figure 6.13 also shows strain 511 for parallel lines located above the hole. The 5 relative magnitudes of the strains in this figure with those shown in figure 6.12 make more evident the highly localized nature of the stress and strain distributions around a pin-loaded hole. One important feature which was observed in figures 6.9 and 6.12 was that the points along the vertical lines in figure 6.9 where the strain 8y changed sign coincided with the location of the maximum strain values cu thus 5’ providing the locus of maximum shear strain ny by simply applying the strain gage rosette equation for maximum shear. 6.2 STRESS AND STRAIN CONCENTRATION RELIEF BY THE USE OF ISOTROPIC MATERIAL INSERTS . From the last section, it was evident that the stress and strain concentration is highly localized around the region of the hole. Based on these findings, one possible solution to the stress and strain concentration problem would be to produce a redistribution of the stresses and strains in the vicinity of the hole, and at the same time try to keep or increase the strength of the Joint. The dependance of the stresses on the elastic constants of the material has always interested the analyst and the experimentalist [2, 18, 65-69]. The theorem of Michell [68] asserts that when a homogeneous and isotropic elastic body is in a state of either plane strain or 139 LEGEND B—-—H Y' 5:0.2209 o—e—o y' 7:0.2945 h—h—d Y' 930.3838 1? #:~ ##iY'10=0.3977 V p 11111111111111111111111 1 J 240.00 100.00 I éyb ' X 00.00 I CENTERLINE ulO‘ I 0.00 “’80 000 I MICROSTRRIN “160 .00 I NI/100/645 |"2‘0 600 0.60 -b.30 0.00 0330 0360 DISTRNCE FROM CENTERLINE (IN.) Figure 6.13 Strain 6 Along Parallel Lines Oriented at 45-Degrees to DirecEIon of Load and Located Above the Hole 140 generalized plane stress induced by prescribed surface tractions, the in-plane stresses do not depend on the elastic constants of the material, provided there are no net forces transmitted by closed contours lying within the body. As indicated by Theocaris [5]. a pin-connected joint is a multiply connected body, and, since the resultant of the forces applied to its individual boundaries do not vanish, the stress distribution depends on the elastic constants of the material and especially on Poisson's ratio. The concept presented by Eisenmann and Leonhardt [3“] on tailoring the laminate to uncouple bearing strength and axial strain in order to increase the strength of the Joint made use of this concept of redistribution of the stresses around the hole. Also, they introduced a change in the effective mechanical properties of the material via a rearrangement of the fibers within the lamina forming the laminate. Several other authors, in their work on composite material Joints, have noticed that the stress and strain distribution around the hole depends on the material properties. Pradhan and Ray [1“] concluded that the maximum circumferential stresses at the edge of the hole depend strongly on the material properties. The maximum stresses are seen to be high for graphite/epoxy and boron/epoxy unidirectional laminae. De Jong [7], utilized three different types of laminates to obtain numerical results: unidirectional, a (90u°/ 1H5°)s and a quasi-isotropic laminate of carbon-reinforced plastic. He reported stress concentration factors for pin-loaded unreinforced holes in these laminates based on the maximum value of the tangential stresses at the edge of the hole. The values given for the unidirectional laminate were almost twice as 141 large as for the (903 / i 115°)s laminate, thus demonstrating that the normal stress distributionstrongly depends on the properties of the plate material. Collings [32] measured the bearing strength of 0° :A5°, 0° 90° and 90° 1 115° carbon fiber-reinforced plastic laminates and concluded that 0 it was dependent on four material properties, namely: (1) O longitudinal compression strength, (2) 0° constrained transverse compression strength. (3) 0° constrained bearing strength (laterally restrained from splitting) and (A) 1 45° constrained bearing strength. In this investigation, a simpler approach is used to produce the redistribution of stresses and strains around the hole. Instead of trying to improve Joint strength by changing fiber orientation, an insert in the form of a thin bushing made of an isotropic material is glued into the hole. 6.3 ISOTROPIC MATERIAL WITH PLASTIC INSERT. Before applying this concept to a composite material, one experiment was performed using an isotropic material, (photoelastic plastic PSM-S, E- 500 K61). One specimen without any insert was tested, and the strains at two points in the boundary were analyzed. These points were: (a) two vertical lines located below the pin hole in the bearing region, and (b) two vertical lines in the ligament area, one of them being tangent to the hole. 142 Figure 6.111 shows typical Moire patterns for displacements parallel to the direction of the load axis. Figures 6.15 and 6.16 show strain 6y for two lines located in the bearing region, below the vertical diameter of the hole, and for lines located in the ligament region respectively. A second specimen also made of photoelastic material PSM-S but containing an insert made of photoelastic material PS-3A (E- 30 Ksi) , was tested. This insert provided a difference in material properties around the hole. For the same loading conditions, the displacement Moire pattern is shown in figure 6.17. The corresponding plots of strain 8y along the same lines as for the specimen without any reinforcement are shown in figures 6.18 and 6.19. In order to appreciate the effect of the material properties on the strain distribution, the strain Ey along the two lines xA- 0.0000 and x8- 0.0625, located in the bearing region were combined in one single figure 6.20. Notice the considerable reduction of the strain level in the plastic PSM-S, owing to the addition of the insert of different material properties. Also, in figure 6.21, the strain along' the line x2- 0.125, which is located at the edge of the hole, shows an excellent agreement with the strain values for the portion of the line closer to the center of the specimen. But, the portion of this line located in the bearing area shows a considerable decrease in the strain level. This means that the insert is indeed producing a redistribution of the strains around the hole. Similarly in the line located at x3= 0.1875. that is, at the interface of the insert and the plastic PSM-S. as shown in figure 6.22, the same behavior is apparent. 111111 11111111 0 1,... 143 I _ sl.\ e E I \\ : $53: ,\ l: _ ex&§xehw\\e ll .s§ .rfig Figure 6.14 Moire Fringe Pattern of Displacements Parallel to the Direction of the Load 144 LEGEND B—H XR:0 .0000 0—0—0 X0=0 . 0025 ’11 P’ '101111111111111111' 240.00 J 160.00 _L f'\ ____Jfi#r )¢ 09.00 #10' ----------------------- 0.00 J '80 000 MICROSTRRIN “160000 J PSMS/c, /6o -p----------------------------------1.----------------------------------- i '-240 000 .00 0130 0160 0.60 -0. 0 M CENTERLINE (IN.] DISTHNCE Figure 6.15 Strain e for Two Lines Parallel to Direction of the Load and LocaXed in the Bearing Region 145 LEGEND 0+—-—£}-—-40 X2=0.125 G%--4}--4D X3=0.1075 O—5—fi X4=0 . 2500 YB P 1111111111111111111 240.00 4_I CENTERLINE 160.00 _I seen S X 0.00 * I 0 J \_ f ; 0.00 1 I I I I I I I I I I I I I I I I I I I ..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 HICROSTRRIN —160.00 -po.00 PSMS/ey /00 0.60 -0. 30 0'60 DISTRNCE FROM CENOOTERLINE0 (IN-1 Dd .--------------------- ,-240.00 Figure 6.16 Strain e for Lines Parallel to Direction of Load and Located 1n the Ligament Area to the Right of the Hole fig 1111111111111111 Figure 6.17 Moire Fringe Pattern Of Displacements Parallel to the Direction 'of the Load 147 LEEENQ. G-—B-——El “II-10.0000 O-—-Q——0 XB=0.0825 fl> 1111111111111111111111 240.00 I CENTERLINE me 160.00 80.00 1 uli)‘ -80 000 010 00 i 1 l 1 1 1 1 1 MICROSTRHIN 1 -160.00 PSMS/INS/ey/so -10.60 -0. 30 o. oo 0.30 0. so DISTRNCE FROM CENTERLINE (IN. 1 Figure 6.18 Strain ey for Lines Parallel to the Direction 1 _-2‘0 000 of the Load and Located in the Bearing Region 148 LEGEND B—B——B X230 . 1250 O——O—0 X3=0.1875 H—‘ X4:0.2500 v FF""" 11111111111111111111111 1 240.00 CENTERLINE IF 160.00 I x“ x4 09.00 !1(3‘ 0.00 1P---- 1 HICROSTRRIN -160 000 l -0. 60 -0. 30 0'. 60 DISTRNCE FROM CENOOTERLINE0 (IN. 1 Figure 6.19 Strain e for Lines Parallel to the Direction of the Load Located in the Ligament Region to the Right of the Hole 240.00 149 LEGEND B—H PSH-B NONE +—+—-+ PSfl-E INS Y P 1 1111111111111111 240.00 J 160.00 1 CENTER! IMF 80.00 1 X3 xlll‘ 0.00 ] MICROSTRRIN -00.00 Cy- XA : 0.0000 X3 = 0.0625 —160-00 CDMPRRISDN -240.00 -0.50 -b. Sqé 0.00 0130 0350 DISTRNCE M CENTERLINE (IN.) Figure 6.20 Comparison of Strain 6 Between Plastic Spcimens With and Without Insert for Lings Located in the Bearing Region 150 LEGEND. H—B PSH-E NONE 0—-G-—-—0 PSN-S INS Y *nnnnmfinnm ” CENTERL::1J\ 1 vi x 240.00 160 . 00 1 - 1| . ------------—-------- M ----- / alt? 09.00 I r I I I I I I I I I I I I I I I I I I I I 0.00 1 MICRDSTRRIN -80.00 -160 000 C y for X 230 .1 25 CDNPHRISDN 0.60 -0. 30 0.30 0'.60 DISTRNCE FROM CENOTERLINE (IN. 1 Figure 6.21 Comparison of Strains Between Plastic Specimens with and without Insert foryLine x- 0.1250 Located in the Ligament Region to the Right of the Hole o4P------------------------. '-2‘0 000 151 LEGEND H—El PSH-B NONE 0—-0-—0 PSH-B INS ‘Y P 11111111111 11111111111 240.00 J 80.00 J 160.00 J '1 a: E, 3:57;. 1 xICP 0.00 i I I I I I I I I -80 000 NICROSTRRIN Ehy $00 -160000 1 COMPHRISON dP--------------------- |-240 000 0.60 -h.30 0.00 0730 0160 DISTRNCE FROM CENTERLINE (IN.) Figure 6.22 Comparison of Strain s Between Plastic Specimens with and Without insert for Ling X33 0.1875 Located in the Ligament Region to the Right of the Hole 152 One common characteristic is observed in these last three figures. Especially in the region below the horizontal diameter of the hole, the slope of the strain curves is steeper for the case of the specimen without insert than for the specimen with it. This can be interpreted as a more uniform strain distribution. 6." COMPOSITE MATERIAL SPECIMEN WITH PLASTIC INSERT. One of the first characteristics desirable in the laminate, in the bearing region, cited by Eisenmann and Leonhardt [314], is that it must have a low tensile modulus of elasticity in the primary load direction. Fortunately, such "softening" can be readily achieved by the incorporation of a thin, molded in-place isotropic plastic bushing. The low modulus of elasticity (Ep- 500 K31) of the plastic would introduce some plastic or pseudOplastic behavior in the joint, thus tending to act as the averaging mechanism to relieve both local stress concentration and uneven load distribution. A typical Moire fringe pattern for displacements in the transverse direction of the load is shown in figure 6.23. Especially in the region of contact between pin and plastic, the fringes tend to rotate due to the high shear strain, and they also tend to increase in number due to the tensile strain ex in that direction. Figure 6.2” shows a Moire fringe pattern for displacements parallel to the direction of the loading axis. This photograph shows a very interesting aspect of the behavior of the plastic insert. Because of the large deformations in the insert, the fringes reversed sign; that ‘III. 111 .“A 3 \ . \. 2. \I \ 6 \\ \I P \ ‘\ \k‘ I\\‘ \\\ I‘ \‘\ u u \\\\ \\ “NI I \“\ \‘I g i \H \|\ “ .\ \ m 154 1 \1 \ \1 \\m\\\ 111 =flé/2fivg¥%’% 4 _ / \\.\\\\\\ III/7 111' \ \ § \ . -\. _ S S ':\‘ 1“ Q .1 Figure 6.24 Moire Fringe Pattern of Displacements Parallel to the Direction of Load for Specimen with Plastic Insert 11111 1' 1111111 X1 155 is, the initial mismatch was cancelled and the deformation gave fringes of opposite sign. Then, instead of decreasing, the fringes increased in number per unit of length, despite having a compressive strain. This sign reversal does not imply that the measurement is wrong, but that, during the data reduction, care should be taken to obtain the correct sign of strain. The fringe number reduction in the laminate is not as noticeable in the composite laminate as for the case of no reinforcement; but it is difficult to tell by simple inspection of the photographs the degree of deformation due to the difference in initial mismatch. On both sides of the hole, the number of fringes per unit of length increases considerably, especially at the rim of the hole. Figure 6.25 shows Moire fringes of displacements at H‘s-degrees from the vertical. In this case, the observed behavior of the insert is similar to that observed in the previous case. Figures 6.26 and 6.27 show strain ex along parallel lines located in the ligament area, above and below the horizontal diameter of the hole. Figure 6.28 shows strain Ex for parallel lines located above the hole, and figure 6.29 shows strain ex for parallel lines located in the bearing region. In this figure, the magnitude of the compressive strain for line y11- -O.125 seems to be excessively large. One possible source of error could be that the Moire fringes in that area (see figure 6.211) are almost horizontal because of the large shear strain ny. During the digitization process, the points corresponding to those horizontal fringes were taken at the middle point of the horizontal portion. The magnitude of the strain decreases rapidly as the lines are located farther away from the hole. 156 \ . 3 asks... _ is s \ MKQW. \\\ \V \ \kvwx .\., R“ : . . 1 c .s.\\\\. \\ ass as NsNNMWNM \ Ass s... xx. \ l \\.\N\\ .\ J a .. \ M» \ xx s as \.\ as as. . v \\ \ . \M\\\\\*k _ x a. §., . ®§¥x$m .MWN“\\\&m§sw “WWV \\ s\_\\. a sin \. \. \\\\.s\ \\.\. saw \\.§\\\\\\\\\\u .\ \\ x \ is g as a _ s§s§ \\ \\\\\ V. 3. . .x . we . .\\\.\\\\\\. .. \s rs \x. \ \\ \ . E. \\\ \\\ \ : . \ \ ,,\\\\\. the Direction of Load for Specimen with Plastic Insert Figure 6.25 Moire Fringe Pattern of Displacements at 45-Degrees to 157 LEGEND B--B'—El Y1 :0 .0441 9 0—0—0 YB=O .0441 9 b-—-‘-—‘ Y2=O .0825 A? % ‘ir YC=0.0825 X—H Y3=O .00839 O—-O-—-O YO=O .08030 p 1111111111 240.00 1 180.00 {g1%? ,4:- l 80.00 ultl' 0.00 HICROSTRHIN -po.00 1 ”160000 Ex/100/F’L1113TIC -0.60 -0. 30 0'. 30 0.80 DISTRNCE FROH CENOTERLINE (IN. 1 Figure 6. 26 Strain 6 Along Lines Perpendicular to the Direction of Load andeocated in Both Ligament Region g..----------------------------------11----------------------------------- "240 o 00 158 LEGEND m—a—m Y0: 0.0000 O—O—O YR: 0.0000 Y2=-0.0825 # .r 4 YC-‘P’OOOBZS '1. 111111111111111111111 240.00 A 1 180.00 ENTERLINE 09.00 £10‘ I 0.00 '80 000 l 11 ICROSTRHIN ‘160000 L 6x/100/F’LF13T I C -0. 60 —b .30 0. 00 0'. ' 00 DISTRNCE FROM CENTERLIDNE0 (IN. 0) Figure 6. 27 Strain 6 Along Lines Perpendicular to the Direction of Load andeocated in Both Ligament Regions _.--__---__._____--_--_--_--..__._..__---+-_--_--------_--_--_----------_--_- "2‘0 000 159 L_E_C;_E_N_D_ B—G—El Y4=0.1280 O—H Y8=0.1878 b—‘——-A Y8=0.2800 e e: :: 77:0.3126 O X——)(-—-X Y8=0.3780 o O—F—O Y9=0.4375 o. 4-—-4——+ 710:0.500 §‘ 1 W} 5 1111111111 1111111111 8 1 8?. 1 V" I YIO 1 ”v5 z: . W 3 1 J " i a 1 J "001 E lean-m H I *1 s 5.8 1 co-w-m-u- . ---- tr 1 '56 1 2°- s 5:?- 5 z 1 s a a? 1 7- : 1 c. 1 9 1 3 1 ExflOO/PLHSTIC N . ' -0.60 -'0. 30 0100 DISTRNCE FROM CENOOTERLINE0 (IN. 1 Figure 6. 28 Strain ex Along Lines Perpendicular to the Direction of the Load and xlocated Above the Hole 160 LEGEND H—fl Y4=-0 .1250 0—9—0 Y5=-0 - 1075 ‘——‘—‘ Y8=-0 .2500 +—I——+ Y7=-O .3125 X—¥——X Y8=-O .3750 8 O—k—O Y9=-0.4375 . 4——¢——+ Y10=-0.600 9 Y N1 p ' 1111111111111111111111 8 3?- 2:315:1qu 8 5:" X _ 6‘ ::::m 20 _Y8° I 2:8 aa----- CK I.— 03c: DO 5;; Z O O s? g . g? Gx/‘OO/PLRSTIC T 160 -0.60 -b.30 0.00 0130 0 DISTRNCE FROM CENTERLINE (IN.) Figure 6.29 Strain 5 Along Lines Perpendicular to the Direction of Load andeocated in the Bearing Region 161 Figure 6.30 shows strains 6y for parallel lines located to the right of the vertical diameter above and below the hole. The lines xA- 0.0000 and xB- 0.01116 represent strains for the composite laminate in the bearing region only, whereas the line xD- 0.0833 represents strains for both the insert and the composite laminate. Figures 6.31 and 6.32 show strain 8y for parallel lines in both ligament areas. It can be noticed that the strain attains its maximum value slightly below the horizontal diameter of the hole; and, going towards the bearing region, the strain suddenly rises again. This behavior can be noticed in lines x11:- 1 0.125 at approximately y- -0.113. These points are located very close to the locus of maximum shear strain and the interface of the insert and the laminate. This sudden change in magnitude is an indication of the highly nonlinear behavior of the soft plastic in that area, produced by the shear strain. Figures 6.33 and 6.311 show strain 511 for parallel lines oriented 5 at US-degrees with respect to the vertical axis and encompassing part of the ligament and bearing regions. It should be noticed that the magnitude of the strain 8115 in the bearing region is large, especially along lines y'A= 0.000. Figure 6.35 shows strain a“ for parallel lines located above the 5 hole and oriented at 115-degrees. It should be noticed that the strain along line y'9= 0.3535 is the average strain in that direction. Other lines parallel to this one and located farther away from the hole showed the same value of strain, and they are not included for clarity. Figures 6.36 and 6.37, show strain Ell along parallel lines located in 5 162 LEGEND B-——-B——El XO=0 .0000 O—O——0 XH=O .0000 b—fi—fi XI=O .0418 «#vr #, i‘ XB=Oo0418 X—H X3=O .0833 O—+—O X0=0 .0033 ’P HMHUH1HHHH 4, —T——_— 'vv—Frv— T—v-v— '— CENTERLINE O l O 1 o 1 cu O "160 000 I PI/iooks -0060 “003 0.00 0230 0160 DISTHNCE FROM CENTERLIONE (IN 1 Figure 6.30 Strain 8 Along Lines Parallel to the Direction of Load and Located Xbove the Hole and in the Bearing Region -?40.00 163 LEGEND s——a——m “=04st o—e——o xs=0.1s75 4—H x0=0.2500 # . :1 x7=0.3125 H—x xe=0.3750 8 1 e-—+—-o xs=0.4376 ‘3 . N1 : Yfi 1 1111111111 111111111. ‘3 1 C3 .. O m. CENTERLIN O ‘3’ -cm 0“ v-l ! 28 hd a. 05:) m: p. 03c: :23 . 084 ' hfl I 1 Z 1 8. 11 o 1 £2 1 '1 1 1 g 1 . 1 g i 6woe/01.0mm: ea 1-0.00 -0. 30 0.00 0'. 00 DISTHNCE FROM CENTERLIONaEo (IN 1 Figure 6.31 Strain 6 Along Lines Parallel to the Direction of Load and Loca¥ed in the Ligament Area to the Right of the Hole 164 LE GE N D ur—-£}--4n X4=-0.1250 o—o—o nae-0.1075 .__...__.. xa=-0.2500 e...” 3 1 0:23.312: 3 o——o—o xs=-0:437s _ c; +——+——+ nos-0.500 '11 g y P N . ’ g 111111111111111111111 O I 9 “ O 23‘ x9 : T1 cemyud I I 3 1 x 158' 1 #1 fl ‘ H F 1 . A ' ” 28 ”#57:? 25"} =_ 1‘: . H o ______ ==_ J ' “‘1‘ _-_ (:61 ’ 1 "' tr 4 1 l— “ U 1 (Do 1 ‘99 1 83 1 a—c IT 1 ’3 1 s 1 O CO 1 T‘ 1 1 O 1 9 1 3 1 6y/‘W/F’LRSTIC 91 ' -0.60 -'0.30 0.00 0'. 30 0' 80 DISTRNCE FROM CENTERLINE (IN. 1 Figure 6.32 Strain 6 Along Lines Parallel to the Direction of Load and Located In the Ligament Area to the Left of the Hole 165 1111111111111111111111 JCENTERUNE Y'O 320100 590000 1. '7 W 109000 ulEP STRHIN e'1'5/100/F’LFIST I C .1 b-----—--—---c---Inc---—--—---+L-a-u-n-un-u-u-un-c-c-n-u——-------- ‘ 0 00 -0. 30 0. 00 0' 30 0'.60 DISTHNCE FROM CENTERLINE (IN 1 Figure 6.33 Strain 6 Along Lines 45-Degrees to Direction of Load and Located in Both Ligament Areas 166 LEGEND H——El Y'0= 0.0000 G—O——Q Y'fl: 0.0000 t—+—‘ Y1=0o04419 ‘ ‘ YB=O. 04419 X—-—X—X Y'3=0. 08839 C--~.--O11fl0=0o 08839 W 1111111111 1111111111 240 .00 160.00 1 1 80.00 11(3' 01. 00 i 1‘ 1 % ”80 o 00 J HICRUSTRRIN "160-00 4 645/100/PLFISTIC -D.60 -0. 30 0.30 0'00 DISTRNCE FROM CENOTERLINE (IN 1 Figure 6.34 Strain 6 Along Lines 45-Degrees to Direction of Load and Located in the Bearing Region and in the Right Hand Side Ligament 0‘ .------------------- 162 LEGEND B—H Y'4=001325 H—-O y'5=0.1707 fi—h——‘ y'0=0. 2209 i i ‘fi'Y' 7:00 2851 X—*—X y'.0=0.3093 H—O v'9=0.3535 Y1 p 1 WHMHHIIHHHH 240.00 I 160.00 J 80.00 L ulO‘ qb---- ' , ‘ .I- _".__- - - "l ---- 0.00 ’80 000 l HICRDSTRHIN ‘160 .00 l ‘45/1W/PLHST [C 0.00 -0. 30 0'.60 DISTRNCE FROM CENHOTERLINE‘J (IN 1 Figure 6.35 Strain 6 Along Lines 45-Degrees to Direction of Load and Located ove the Hole 0‘ .------------------- '0-240 000 168 L_E_Ci_E_1\1_D- B—H Y' 4=-0 .1325 O—H Y'5=-0 . 1707 5—H Y' 8=-0 .2209 t i 9% Y7:—0.2851 H——-X Y'8=-0 .3093 Y P 111111111111111111111 240 .00 CENTERLINE 160.00 v'e .00 ‘ o" r/ '1 '- 28 cr0'1 I TR 030 00 m C 08 o—cl I ”160000 l 645/100/PLF18T I C d B-QD-D-u-------¢-------+--- ‘»_ ('2‘0 .00 0.60 -'0.30 0.00 0'.30 0'.00 DISTRNCE FROM CENTERLINE (IN.) Figure 6.36 Strain £4 Along Lines 45-Degrees to Direction of Load and Encompassing Both Ligament Regions and the Bearing Region 169 LEGEND m—e—m w 9=-0.3636 o—o—-o y' 10=-0.390 b—t—«s yin-0.442 '19 111111111111111111111 105N750 E X 0 0 -160.00 ‘45 /|00/PLnsnc -.0 60 -'0. 30 0. 00 30 0'. 60 DISTHNCE FROM CENTERLIONE (IN 1 C ...-..........--....-----..--....---..--------..-+-.. -240.00 Figure 6.37 Strain 3 Along Lines 45-Degrees to Direction of Load and Located in the Bearing Region 170 the bearing region. The strains in this region are shown in two figures to avoid overcrowding. 6.5 COMPOSITE MATERIAL WITH ALUMINUM INSERT. The second property the laminate should have in the bearing region is high bearing capacity. Also, the ligament areas must have a high tensile modulus of elasticity in the primary load direction. This combination of material properties around the hole can be accomplished with the addition of an isotropic material having a higher modulus of elasticity, in the form of a thin bushing. This concept will provide both high bearing capacity and high modulus of elasticity in the direction of the load. The material chosen for this insert was Aluminum (6061-alloy). with a modulus of elasticity (average of tension and compression moduli) equal to 107 psi. Some of the softening required in the bearing area can still be provided by the plastic glue used to mount the Aluminum ring in place. The glue used to mount the Aluminum bushing in place was the same photoelastic cement used to produce the plastic insert. Figure 6.38 shows Moire fringe patterns of displacements in the x-direction, that is, perpendicular to the loading axis. It can be seen that there is not much change in the number of fringes because of the induced deformatdxni. Only a slight curvature of the fringes is noticeable in the bearing region. 172 Also, in figure 6.39, Moire fringes of displacement in the y-direction do not show much change in number, but only a slight change in orientation, especially in the bearing region. Similarly, in figure 6.110, The Moire fringes of displacement in the HIS-degrees orientation show only a small change in orientation in the bearing region. Plots of strain ex for parallel lines located in the ligament area are shown in figure 6.111. Owing to the symmetry of the loading, a fair symmetry is expected in the distribution of the strain in this direction. In this figure, the strain plots reveal some asymmetry, especially for the lines located at the left of the hole. This behavior can be explained if one takes into consideration that the load is being transferred to the ligament areas through the bearing region via the insert, and that the insert transmits the forces via shear stresses. The bushing is glued to the laminate, and it is constrained in a circular boundary on its outer circumference. But, at the same time, it is being acted upon by a non-uniformly distributed loading on the lower half of its inner circumference. Because the insert is much stiffer than the composite material or the glue, the deformations in it will be much smaller than those in the composite. But then, the forces have to be transmitted to the laminate via shear stresses and strains in the glue. If the insert was not glued to the laminate, it would act just like another rigid pin; but, in this case, the glue acts as a restraint, and any motion of the ring produces shear in the glue. Because of the stress concentration in the direction of the loading, the tensile stresses are larger at the edgeof the hole; but since the insert is made with a high modulus of elasticity, it does not 173 4 111111 1__... —. —. Figure 6.39 Moire Fringe Pattern of Displacements Parallel to the Direction of the Load for Composite Specimen with Aluminum Insert ' 174 11111 L—. \ \\\~_\.. ‘\ "““ \ .~ \ "‘ ‘ ‘\ ' \ \. \ C \ \{z .Figure 6.40 Moire Fringe Pattern of Displacements at 45-Degrees to the Direction of the Load for Composite Specimen with Aluminum Insert 175 LEGEND B—H Y0: 0.0000 0+0 YR: 0. 0000 O—O—i Y1: 0. 0825 i— Y8: 0. 0825 )9--96--%( Y1=-0. 0825 O—-O——O Y8=-0. 0825 o o C o ‘2. Y1 1 N 1111111111'1111111111111 N ”160000 1 TRH I N DJ. 00 . :l ' ‘ HL/lDOAfi '-2‘0 000 -.060 -'0. 30 0'.30 0' 80 DISTRNCE FROM CENOTERLINE (IN 0) Figure 6.41 Strain 6 Along Lines Perpendicular to the Direction of Load and focated in Both Ligament Regions 176 undergo large deformations. For some value of the applied force, the glue starts1yielding; and, once it deforms plastically, its restraining role isruuzperformed as efficiently on the upper half of the outer circumference where tensile stresses are acting. When this restraint failure occurs, the direction of shear stresses changes and the stress and strain distributions around the hole are affected. This behavior is evidenced by the change from negative strain to positive strain ex along lines located in the ligament region, to the right of the hole, and close to the interface of insert and laminate. Figures 6.112 and 6.113, show plots of strain ex along lines perpendicular to the direction of the load in the bearing region and above the hole, respectively. It should be pointed out that line y- - 0.125 shows variations, especially at the right side of the hole, at x- O.28116. This variation is produced by the imperfection in the grating which produced some discontinuity in the fringes, and it is not ruelated to the behavior of the Joint. Figure 6.NU shows strain ey along lines above and below the hole, and parallel to the direction of the load. Again, some discrepancy is observed in the sign of the strain, especially in the bearing region, where it changes from compressive strain to tensile strain. This change in sign occurs in the region of the interface of the insert and the laminate, and apparently it is caused by the shear stresses in the glue. Also, for the lines located above the hole, the strain shows a change in sign in the vicinity of the interface. It should be noticed that, unlike the Joint without an insert, the bearing strains in this case are almost of the same magnitude for 177 LEGEND DISTHNCE FROM CENOOTERLINE0 ( IN] m—-e—m vz:-0.1250 o—e—e Y3=-0.1875 b—d—a Y4s-0.2600 % : 1 vs=-0.3125 o x—x——x 1ro=-0.3750 o O——+——O Y7=-0.4375 5 11111111.. 11111111 I o 1 C? I o 1 21 1 1 emails: 1 \\ I o 1 112 i c. : 1: . 0 I Y. 238" 1 '7 ,.. . at 28 H o 0:07 x F- 03:: DO 0: ‘ ' 08 ha 11 Z 11 O c.’ O to 1 'r- I 1 s 1 ~ . I S E HL/lDD/EX N 1 ' -D.60 -'0. 30 0' 0' 30 Figure 6.42 Strain 6 Along Lines Perpendicular to the Direction of the Load andeocated in the Bearing Region 178 LEGEND m—H 12:0.1250 o—o——e Yaw-1876 1 .——.——.. 14:0.2500 : a; 11: 15:0.3125 ‘ x——-x——-x Y8=O.3760 Y1 1111111111 11111111111 240.00 180.00 80.00 1 alt“ 0.00 J 1 1 I 1 1 1 I 1 L I flzf :~ \E . A >0 ~80.00 l MICROSTRRIN 4 “160000 I I RL/lOO/EX D|-240 000 0.60 -0. 30 0' 00 0'30 DISTRNCE FROM CENTERLIONE0 (IN I Figure 6. 43 Strain 3 Along Lines Perpendicular to the Direction of Load andeocated Above the Hole 179 LEGEND m—a—m x0:0.0000 o—o—o xa=0.0000 X1=0.0825 H—X X1=-0.0825 8 o——o—o x32-0.0826 91 1 P 1 .1 1 11111111111111111111 E o 1 9 1 o I <0. 1 "‘ 1 5 :fl: 1 n 7 “08‘ 1 ’° H I at 1 1 :28 1 hfl 0. ---- """ (IF: 1- a: 1 1— 1 03:3 ' «so 1 tr' ' (2?. 1 "£- 1 c, 1 9 1 o 1 2 1 1‘ 1 1 o 1 C: 1 3 : RL/lOO/EY 9. 1 '-10.00 -'0. 30 0' 0' 3° DISTRNCE FROM CENOTERLI‘NE0 (IN-1 Figure 6.44 Strain 6 Along Lines Parallel to the Direction of Load and Locazed Above the Hole and in the Bearing Region 180 different lines. This evidence is an indication of a better distribution.of the strains, and that the distribution depends strongly on the material properties. Figure 6.115 shows strain ey for lines in the ligament region to the right side of the hole. In this case, the only discrepancy is observed in line x2- 0.125, which shows some sign variatnnifrcm y- +0.08 to y- -0.06, and it attains a maximum compressive strain at x- - 0.1025. This could be caused by the deformation of the Aluminum ring. Figure 6.116 shows plots of strain €115 for lines located in both ligament regions. Lines y'1- 0.011112 and y'2- -0.08811 show also a high tensile strain at approximately x'- 0.2, i.e. close to the interface of the insert and the laminate. Figure 6.“? shows plots of strain ‘4 along lines in the bearing 5 area. Line y'3= -O.1325 shows peak values at x'- i 0.13, i.e. at the interface of the insert and the laminate. Similarly, fkn~iline y'us - 0.1767, the peak values are obtained at the same place. From these results, it is evident that because of the incorporation of the high modulus of elasticity isotropic material, there is a redistribution of strains around the hole. 6.6 COMPOSITE MATERIAL WITH ALUMINUM INSERT (AFTER GLUE FAILURE). Since the strain levels were so low in the specimen with the Aluminum insert, this specimen was loaded until the glue failed. The detached area was limited to the upper portion of the outer circumference of the ring. Also, since the glue did not separate in the 181 LEGEND D—H X2=0 . 1 250 0—0—0 X3=0 o 1875 L—fi—fi X4=0 . 2500 #e 4 1» X6:0.3125 X——X——X X8=0 o 3750 O—-O—O X7=0 . 4375 H—O “M 5000 11111111111111111111’ a fi\ “zflé u x1 . -----------—-------------- -160.00 1 RL/100/6, 0‘ h---------“------------'- u‘ l-240'00 -n.so -o. 30 o'.eo DISTHNCE FROM CENOOTERLINE0 (IN 1 Figure 6.45 Strain 6 Along Lines Parallel to the Direction of Load and Locazed in the Ligament Area to the Right of the Hole . -. 0' ‘ V . w u, ‘ i. I‘v , , . - 11 . - - n-v - . . " \- 4. c '«i‘ r ’u - .- 1- v . 'rr- » ' f ‘ " - . v. A) . . .“., i ' a ‘ A ‘ I 182 LEGEND X-—-)(-—x y’2=-0.0884 F—O—o 11' «0.0884 Y1 1 P 111111111111111111111' .00 2_40 190.00 so .00 1 l x101 0.00 TRRIN 03 00 CR 80. .1. 11 .00 -160 eufloo/RLUHINUM -0.so -b.3 0'.00 0.6 60 DISTHNCE FROM CEN'I'ERLIONE0 (IN I Figure 6.46 Strain 6 Along Lines Oriented at 45-Degrees to Direction of Load and Located in Both Ligament Areas -.----------------—‘_— '-240 000 183 1 L__E_§_E_ND_ B—B—fl Y' 3=-0 . 1325 O—Q——O Y 4=-0 . 1707 H——¢ Y' 5=-0 .2209 1 e : _Y'8=-0.2881 11111111111111.1111 240 .00 j 100.00 1 80.00 L 11101 0.00 1 -80 .00 L MICROSTRHIN -160 .00 1 :65/100 /RLUM I NUM 0.60 ~11 .30 01.00 0'.30 0160 DISTHNCE FROM CENTERLINE (IN.) {-240 .00 Figure 6.47 Strain e Along Lines Oriented at 45-Degrees to Direction of Load 53d Encompassing Both Ligament Areas and the Bearing Region 184 lower portion, i.e. in the bearing region, another test sequence was started to check the performance of the insert. This specimen was loaded and tested again with the same conditions as before the failure of the glue. As will be seen later, the glue left intact in the bearing region did work as efficiently in the transfer of the load as before. There seemed to be some permanent deformation in the bearing region as can be seen in the Moire fringes in figure 6.fl6.a. This permanent deformation did not affect the subsequent measurements, because it was subtracted together with the initial pitch mismatch. Figure 6.u8 shows Moire fringes of displacements perpendicular to the load direction. It is very noticeable from the curvature of the fringes in the Aluminum ring that it is undergoing some deformation. Also, the curvature of the fringes on both sides of the bearing region indicates a fair amount of shear deformation. Figure 6.1-19 shows Moire fringes of displacement parallel to the loading axis. Again, the ring shows some deformation, especially in the upper and lower portions. The fringes in the bearing region show sharp corners in the Aluminum/laminate interface, but they are still continuous. Figure 6.50 shows moire fringes of displacement in the +u5-degrees orientation. The characteristics observed in these photograph are the same as those observed in figure 6.fl9. One common characteristic which can be observed in these photographs is that, in the detached region, continuity of the fringes is lost. This means that the deformations in the composite material are different from those in the Aluminum insert. 185 \\\\\\\\\\\\\\m\1m11 \1 1 11111 111/11110111100004 / 11v 11111111111 Figure 6.48 Moire Fringe Pattern of Displacements Perpendicular to the Direction of Load for Composite Specimen with Aluminum Insert After Failure of the Glue. 186 1111111 11v ’1111111 C: 1 111 1 11 Insert After Failure of the Glue Figure 6.49 Moire Fringe Pattern of Displacements Parallel to the Direction of Load For Composite Specimen with Aluminum 187 45 1 Y I 0'. l 1. .l. I .I I I .II .1. 0| 1.. I I .lll I \\§§\\\9 \\.\_\\\\\\\\\s 0.1.x “\kuxVQ1 . .0. \z \\\u\ ‘ \\\ _ \ \ . . _ \\\;W\\\ . 1 \\ 1H.:\,\:.: \\\\\H \ wmwfiyyllfl \\ . l ,w .aWMV .mNflNNWHNwQ“MWNV\ “mm - \ \~\ \ \. A. \ . \ ‘ \\\.\\\\ . .Ss A v \ x .. \ \ C 1.1 . 1.}1 d .. . v \ . \s\ \ ~ . . \\ . c \W&\\ \w. \. . . *Figure 6.50 Moire Fringe Pattern of Displacements at 45-Degrees to the Direction of Load for Composite Specimen with Aluminum Insert After Failure of the Glue 188 Figure 6.51 shows plots of strain ex on both sides of the hole in the ligament regions. Again, some discrepancy is observed in the symmetry of these lines, especially at the edge of the hole. Plots of strain Ex for parallel lines located in the bearing region are shown in figure 6.52. A small asymmetry is observed for lines y3 and ya, but this can be caused by the imperfection of the grating. Figures 6.53 and 6.514 show strain ey for lines located above and below the hole and for lines in the ligament area to the right of the hole and in the bearing region respectively. It is seen that the deformation in the ligament is slightly larger at the insert/laminate interface than at the edge of the hole. Figures 6.55 and 6.56, show strain 8115 for lines located in the bearing region and both ligament areas. In comparison with the case of the specimen with no insert, the strain magnitude does not change noticeably in this case, especially for lines y'A, y'B, and y'C. In order to appreciate better the effect of the inserts in the behavior of the Joint, some of the more relevant deformations along some of the lines are plotted together. Figure 6.57 shows strain 6y in the bearing region for line xA= 0.0000 for all three specimens. A drastic reduction in the deformation of the bearing area can be noticed for both specimens with plastic and Aluminum inserts, this last one even after failure of the glue. A conclusion is that the inserts do improve the bearing capacity of the joint. 189 LEQENQ. B-—H Y0: 0.0000 O——O——0 YR: 0. 0000 H—15 Y1=-0~ 0825 # % # YB=-0- 0825 g; 1v . P S! tfliftlnllflit efl c: 9 c: E? ‘ 15::ig: o 1 9 30°01 -160000 4 RHIN 0:00 €vfiLUM/RF/50 -240.00 -n.so -o. 30 o'.so DISTHNCE FROM CENOOTERLINE0 (IN I Figure 6.51 Strain ex Along Lines Perpendicular to the Direction of The Loadx and Located in both Ligament Areas 190 LEGEND B——B‘—B Y2=-0 . 1250 GF--4D--4D Y3=-0.1875 h—‘——‘ Y4=-0 .2500 #f 4; %* Y5=-0o3125 X——*——-X YB=-0 .3750 O—O——O Y7=-0 . 4375 O O O O N 11111111111711111 Contomm Rx _ 3:35 1133 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 U “-160.00 A fidfiLUH/HF/SU 0.60 -E.3o 0.00 0130 0160 DISTRNCE FROM CENTERLINE (IN.] . ---------------------. .-24o.oo Figure 6.52 Strain e Along Lines Perpendicular to the Direction of Load andencompassing Both Ligament Areas and the Bearing Region 191 LEGEND B——H X0=0 . 0000 O——O—0 Xfl=0o 0000 ‘-—-‘—¢ X1=0o 0825 r i 4 X0=0 0025 111111 11111111 x1 [Content". I l 7X MKS 240.00 160.00 J ------------F---- 80.00 ulJJ‘ 0.00 TRRIN ICROS -po.oo 11 1 '160-00 €v/HLU11/F1F/100 0.60 -o. 30 O'.so DISTRNCE FROM CENDOTERLINE0 (IN- 1 Figure 6.53 Strain 6 Along Lines Parallel to the Direction of the Load andyLocated Above the Hole and in the Bearing Region 0‘ b----------------------- ’2‘0 000 192 LE ND W X230.1250 P—O—O X3:0.1575 b—H X4=o.2500 X——*—-K X880. 3750 8 O—O—-§ X7: 0. 4375 é . P - g 1 I 11 ~ 1 11111111 11111111 1 o I c: I o 1 231 : an: E «mm». 1 e o 1 =2 : .0081 ". “#117 H 3' 28 H 0..---- 4- < xlo‘ V", p 0.00 J b--- --- W1 " “'80 .00 J MICROSTRRIN -160.00 6y -X2:O.125 COMPHRISON 0. 60 -0. 30 0. 00 0' 30 0. 00 DISTRNCE FROM CENTERLIONE (IN- 1 Figure 6. 58 Comparison of Strain e for Line XZ- 0.1250 Located in the Right-Hand Ligament Area .Jt---..-..----..-------..--..-------..---- (240 o 00 198 LEGEND B—H NONE H—O PLROTIC h—O—i RLUII I NUM # #9L 90L 240.00 J P ‘JHHHIH IHHHIII‘ 160.00 J 80.00 I a10‘ 0-00 1 l MICROSTRHIN -00.00 -160000 I e, — x3: 0.1875 COMPHRISON d p-------—--—----------- {-240 000 -0. 00 -0. 30 0. 00 0.30 0'. 00 DISTRNCE FROM CENTERLINE (IN 1 Figure 6.59 Comparison of Strain e for Line X3: 0.1875 Located in the Right Hand Ligament Area 199 Figure 6.60 shows the bearing stresses along the line parallel tc> the loading axis. These stresses are non-dimensionalized by the bearing stress 0b, defined as where P is the applied force, d is the diameter of the hole and t is the thickness of the laminate. The position is non-dimensionalized by the radius a of the hole. The stresses were calculated using equations 2.55 and 2.6, and the strain values obtained from the Moire experiment. As can be seen, the stress concentration factor (SCF) at the edge of the hole is slightly larger for the specimen with the plastic insert, but farther away from the hole, it is lower than the SCF for the specimen without insert. The SCF for the specimen with aluminum insert changes sign in the region of the insert/laminate interface. This same specimen, after failure of the glue, still shows a considerable decrease of SCF in the composite material laminate. In the case of the horizontal line located in the ligament region, the tensile stresses were non-dimensionalized by the far-field stress 0 0 defined as where P is the applied load, w is the width of the specimen and t is the thickness of the laminate. Position was non-dimensionalized by the 200 e None . Plastlc Insert 0 Aluminum Insert belore glue fallure A Aluminum Insert after glue faIlure q/a; 3.0 20 1.0 0.0 a L Figure 6.60 Bearing Stress Concentration Factors 2 31/3 201 radius of the hole. Figure 6.61 shows that the specimen with the plastic insert yielded a higher SCF at the edge of the hole and also in the laminate. The SCF for the specimen with the Aluminum insert again shows a decrease with respect to the specimen without reinforcement, and also a sign reversal at the insert/laminate interface. The fourth curve shows that, even when the glue has failed, there is still a decrease in SCF. Figure 6.62 shows the shear strain ny versus position for the four specimen possibilities studied here. Again, the specimen with the plastic insert shows a tendency to fail under a shear-out mode at points located close to the hole. But, at points located farther away from the hole, the shear strain becomes smaller. The Aluminum insert yields the best results, because of the considerable reduction of the shear strain. 202 16.0 Q I None c.) lil|astllc Insert rt 14.0L beggengme [Fagin re fl ‘ 233332.30? .0 fl 1mum I'mily II © 6.0 4.0 0.0 I Figure 6.61 Stress Concentration Factor Along Line in the Direction of the Load Located in the Ligament Area 203 e None e Plastic Insert 0 Aluminun Insert before glue failure A Aluminum insert aiter glue failure Ky .03. _ -500 A H (I 00.1254,- -0.250« A 1 “Y p HHlll iii!!! 7 O 41375" 13 O 0 Figure 6.62 Shear Strain ny Along the Locus of Shear-Out Failure CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS A high sensitivity interferometric Moire technique was developed and analyzed. A review of simple concepts and equations of diffraction by a grating demonstrated how this technique could be applied to perform measurements of strain fields on the surface of specimens in order to study current problems in Mechanics and Structural Design. A three-dimensional geometrical approach was used to present this technique and to demonstrate the mechanics of Moire fringe formation. Moire patterns of displacements could be obtained for three different orientations, allowing calculation of strain in the same number of directions, and also allowing the calculation of shear strains by the use of the strain gage rosette equations. A three beam Moire Interferometer was constructed to perform the strain measurements while, at the same time, retaining important characteristics such as: good efficiency of light utilization, simple to adjust, suitable to be adapted to hostile environments, etc. The Moire fringe patterns obtained by this technique exhibited high contrast and quality, making the data reduction process very easy. Since the sensitivity of the interferometric Moire Technique depends on the frequency of the interference pattern produced by the mutual interference of pairs of beams, the construction of the Moire Interferometer was designed to allow the sensitivity to be easily 204 205 adjusted to fit the problem. Also, by utilizing an initial pitch mismatch, the measurement was shown to provide more information for a more accurate construction of strain plots. Using this experimental technique, a method for stress and strain concentration relief in a pinsloaded hole was analyzed. The proposed method suggested the use of thin bushings made of isotropic materials ‘ and glued to the composite laminate. The insert was used as a means to produce a redistribution of the stresses and strains around the hole. In order to verify the effect of the insert on the stress and strain distribution around the hole, an insert was glued to an isotropic material. The modulus of elasticity of the insert was lower than that of the plastic used for the specimen. A reduction of approximately 50 1 was obtained in the bearing strains owing to the presence of the insert. Then, based on these results, the idea of the insert was also applied to an orthotropic material. For this type of material two possibilities were explored: first, an in-‘place molded plastic bushing with a modulus of elasticity lower than that of the composite laminate was analyzed. A reduction of the stress concentration factor of about 50 S is obtained at a distance of approximately two radii from the hole, but a slight increase of the stress concentration factor is obtained in the ligament regions. Also, the shear strains which would cause a shear-out type failure are reduced to approximately 50 1 in the bearing region, at approximately two radii from the hole. A second possibility explored was the use of a thin Aluminum bushing glued into the laminate. In this case the insert has a higher modulus of elasticity than the laminate. The stress concentration 206 factor was reduced by approximately 75 1 in the bearing region and approximately 90 1 at the edge of the hole in the ligament regions. Also the shear strain value was reduced considerably (9O 1 approximately) in the bearing region, thus, reducing the possibility of a shear-out failure. In view of the promising results obtained with the Aluminum insert, the specimen was loaded until the glue failed, and it was retested. Happily, the stresses and strains around the hole were still considerably lower than those obtained in the specimen without any reinforcement around the hole. In the bearing region the stress concentration factor was reduced by 75 1 and, in the ligament region, by approximately 90 1. Also the shear strain which would cause a shear-out failure was reduced by approximately 50 1. Thus, it has been shown that it is possible to reduce by approximately 90 1 the stress and strain concentration factors by producing a redistribution of the stresses and strains around the hole through huflusion of different material properties in the laminate. This idea was proven to be feasible for both isotropic and orthotropic materials. This method of stress concentration relief around the pin-loaded hole bears further study. Amongst the several points that need further study are: 1. Optimization of the size of the insert. 2. Better selection of the material used to build the insert. 3. Study of the effect of the insert in the failure strength of the joint. 207 A. A theoretical study to develop a suitable failure criteria. 5. The glue used to hold the insert in place, in this study, was not the most ideal one. The adhesive industry has developed strong cements; and, depending on the particular application, a better cement can probably be found. 6. Variation of the laminate parameters should be analyzed to assess their influence in the stress and strain fields when using an insert as a means of stress and strain concentration relief. 7. Manufacturing parameters, such as load eccentricity and thermal stability should also be studied. Recommendations for further improvement of the technique: 1. The digitization process can be improved considerably by the use of a computerized system of data aquisition. 2. Construction of a portable optical table tormnunzthe Moire Interferometer, in order to allow its use in hostile environments. 3. The high-resolution plates used as a mold to replicate the grating on the surface of the specimen should be produced using a reference grating of known frequency in order to obtain more accurate results in the strain measurements. REFERENCES REFERENCES 1. Godwin, E. W. and Matthews, F. L., "A Review of the Strerugth of'.JoirH:s ir1 Fibrwe-Reiwiforwzed I’last:ics", Composites, Volume 11, July 1980, pp. 155-160. 2. Bickley,ih G», " Distribution of Stress Around a Circular Hole in a Plate", Phil. Trans. Roy. Soc. London, Ser. A, 227, 1928, pp 383-"15. 3. Tiffen. R. and Sharfuddin, S. M., "A mixed Boundary Value Problem of Two Dimensional Elasticity", Mathematika, Vol. 11, 196A, pp. 151-15A. A. Sharfuddin, S. M., "Loaded Loose-Fitted Rough Circular Rigid Pin 111a: Circular Hole", Acta Mechanica, Vol. 3, No. A,1967. pp. 376-38". 5. Theocaris, P. S., "The Stress Distribution in a Strip Loaded in Tension by Means of a Central Pin", J. Applied Mechanics, Vol 23, 1956, pp. 85-90. 6, Waszcak, J. P. and Cruse, T. A. "Failure Mode and Strength Predictions of Anisotropic Bolted Bearing Specimens", J. Composite Materials, Vol. 5, (July 1971), p. A21. 7. De Jong, Theo, "Stresses Around Pin-Loaded Holes in Elastically Orthotropic or Isotropic Plates", J. Composite Materials, Vol. 11 (July 1977). pp. 313-331- 8. Wong, C. M. S. and Matthews, F. L. "A Finite Element Analysis of Single and Two-hole Bolted Joints in Fibre Reinforced Plastic", J. Composite Materials, Vol. 15 (September 1981), p.u81. 9. Matthews, F. L., Wong, C. M. S. and Chrysafitys, C. "Stress Distribution Around a Single Bolt in Fibre-Reinforced Plastic", Composites, Vol. 13, pp 316-322, (1982). 10. Wilkinson, T. L. and Rowlands, R. E. "Analysis of Mechanical Joints in Wood", Experimental Mechanics, Vol. 21, pp. HOB-AIR, (November, 1981). 11. Oplinger, D. W. "0n the Structural Behavior of Mechanically Fastened Joints in Composite Structures" Fibrous Composites in Structural Design, Edited by Edward M. Lenoe, Donal W. Oplinger and John J. Burke, Plenum Press, New York, 1980. 208 209 12. Chang Fu-Kuo, Scott, Richard A. and Springer, George 8., "Strength of Mechanically Fastened Composite Joints", J. Composite Materials, Vol. 16 (November 1982). p.u70. 13. Yamada,EL.E., "Analysis of Laminate Strength and its Distribution", J. Composite Materials, Vol. 12, 1978, pp. 1”. Pradhan, B. and Ray, Kumar,‘"Stresses Around Partial Contact Pin-loaded Holes in FRP Composite Plates", Journal of Reinforced Plastics and Composites, Vol. 3, (January 1980): PP. 69’8“. 15. Chang, Fu-Kuo and Scott, Richard A., "Failure of Composite Laminates Containing Pin Loaded Holes - Method of Solution", J. Composite Materials, Vol. 18, May 198A, pp. 255-278. 16. Chang, Fu-Kuo and Scott, Richard, A., "Design of Composite Laminates Containing Pin Loaded Holes", J. Composite Materials, Vol. 18, May 198“, pp. 279-289. 17. Zhang, Kai-Da and Ueng, Charles E., "Stresses Around a Pin-loaded Hole in Orthotropic Plates", J. Composite Materials, Vol. 18, September 198A, pp. A32-hu6. 18. Frocht, M. M. and Hill, H. N., "Stress Concentration Factors Around a Circular Role in a Plate Loaded Through Pin in the Hole", Journal of Applied Mechanics, Vol 7, March 19A0. pp. 5-9. 19. Lambert, T. H. and Brailey, R. J. "The influence of Coefficient of Friction on the Elastic Stress Concentration Factor for a Pin-jointed Connection", Aeronautical Quart., Vol. 13, 1962, pp. 17-29. 20. Jessop, R. T., Snell, C. and Holister, G. S., "Photoelastic Investigation of Plates with Single Interference-fit Pins with Load Applied (a) to pin only and (b) to Pin and Plate Simultaneously", Aeronautical Quarterly, Vol. 9, 1958, pp. 147-163. 21. Cox, H. L. and Brown, A. F., "Stresses Round Pins in Holes", Aeronautical Quarterly, Vol. 15, Nov. 196A, pp. 357-372. 22. Nisida, M. and Saito, H., "Stress Distribution in a Semi-infinite Plate Due to a Pin Determined by Interferometric Method", Experimental Mechanics, Vol. 23, (May 1966). pp. 273-779. 23. Oplinger, D. W., Parker, B. S., Katz, A. H., "Moire Measurements of Strain and Deformations in Pin-loaded 210 Composite Plates", Presented at the SESA 1979 Spring Meeting. San Francisco California (May 1979). 2A. Wilkinson, T.L., Fuchs, E. A. and Rowlands, R. E. "Photomechanical Determination of Stress in the Neighborhood of Loaded Holes in Anisotropic Media", Proc. 6th. Int. Conf. on Stress Analysis, Munich, 121~126, (1978). 25. Rowlands, R. E., Rahaman, M. J., Wilkinson,’T.Ln and Chiang, Y. I., "Single-and-Multiple-Bolted Joints in Orthotropic Material", Composites, Vol. 13, Number 3, July 1982, pp. 273-279. 26. Prabhakaran, R., "Photoelastic Investigation of Bolted Joints in Composites", Composites, Vol. 13, Num. 3, July 1982, pp. 253-256. 27. Cloud, G. L., Sikarskie, D., Mahajerin, E., Herrera, P., "Theoretical and Experimental Investigation of Mechanically Fastened Composites", Technical Report No. 1300”, U. 8. Army Tank-Automotive Command Research Center, Warren, Michigan, u8890, 198A. 28. Serabian, S. M., "Experimental Verification of Analytical Bolted Joint Methodologies", Proceedings of the Sixth Conference on Fibrous Composites in Structural Design, Army Materials and Mechanics Research Center, Watertown, Mass., AMMRC MS 83-2, November 198“. 29. Katz, Alan R., "A Semi-Automated System for Moire Strain Analysis", Proceedings of the Sixth ConferenceéC\O O \ OmOOOOO- O o. O O O 0 C 00 10 I-I OI XPL(I)= -XM1N+DEL*(I-1) CONTINUE PTSII).KNOTS.XD.YD.WD.RD,XN.FN.GN.DN,THETA.U. IN2(XPL(K). KNOTS. XN. FN. GN, 1) RETURN SUBROUTINE CORRECTSXO) COMMON X1150. 2) Y( DO 10 121.101 XPL II=xo- XPL(I) CON RETURN END FUNCTION SPLIN2(X.N.XN.FN.GN,ITYPE) t t t 0 t I t t t t t t t t I I t t t t - A FUNCTION UTILITY TO 8E1USED IN CONJUNCTION WITH THE SPLINE ROUTINES SPLIN4 OR SPLI c so 2).NPTS(2).XPL(101L XL. XH VL YH XMIN. XMAX E ----THIS ROUTINE COMPUTES CORRECTION FOR DISTANCE FROM CENTERLINE t 0 0000000000000 201 100 110 120 301 000 000 O + GN(150). ON 150). 5(7 218 AT WHICH THE SPLINE X MUST BE BETWEEN X ' THE VALUE OF THE INDEPENDENT VARI OR ITS DERIVATIVE ARE TO BE EVALE A A N 1) AND XN N) FOR CORRECT RESU ; D -< DUI-Ila) LE ED. LI BY S N - NUMBER 0 KNOTS (RETURNED BY NING RDUTINESI XN - ARRAV OF KNOTS POSITION? (RETURNE SPLINING RDUTINESI FN - ARRAV OF SPLINE VALUES RETURNED B PLINE RDUTINES I G . ARRAV OF DERIVATIVE VALUESx (RETURNED BY SPLINE RDUTINESI ITYPE . I RETURNS SPLINE VALUE AT 2 RETURNS THE DERIVATIVE VALUE AT x. t t t t t t t . t I a t t t A: DIMENSIONN XN 150). .FNSTSO). .GN(150I IF‘X LT ”3 gm IF x GT 1.XNN GOTO 110 Do 8 K- IF(x. NE .XN(K)) GOTO 8 KK-K GO TO CONTINUE DD IO K-2.N IF(X.GT.XN(K)) GOTO IO UKaK GO TO N2O CDNT UE HHIXN(JK3 XN(dK-1) HH3=HH .. T1=HH*GN dK- I +FN(JK 1)FN(JK) T2=HH*GN UKI+ N(dK-1)-—FN UK) T3= XN(U -x T4=x-XN(UK- I IF(ITYPE. ED. 160 201 ggLég§a(T3tFN dK-I)+T4*FN(JK))/HH+T3¢T4I(T3¢T1-T4tT2I/HH3 SPLIN2= (FNSUK)-FN(UK- 1))/HH-T3tT4'(HH‘(GN§dK- 1)+GN(JK))+2. ‘(FN(JK- ég;5N(u dK)) /HH3+(XN(dK-1)+XN(JK) 2 th (T vTI- T4:T2I/HH3 UK: 2 GO TO 20 UK-N GO TD 20 IF(ITVPEN ED. 2) GOTO 301 SPLIN2= N(K KI RETURN SPLIN2'GN(KK) EESURN SUBROUTINE SPLIN3(M. N. XD. YO. WO. RD. XN. FN. ON. ON. THETA. W. IPRINT) DIMENSION W 2500). XDI'$8gTX?(1SO)' .WD(150). RD(1SO). XN(150). FN(150). TRANSFER KNOTS TO N. AND INSERT EXTRA KNOTS AT ENDS 0F RANGE SN(N);AMA§1(XN(N)..XD(MI) :§I+3)=XN31) §(523W(K+1)+W(K+1)-W(K+2) w(K+3I-w(K+2)+w(K+2I-M(K+II SET UP THE LEAST SQUARES EQUATIONS IN w 121 u-2 IigI'g/ (W(3) I-w 4 (w (s I (W(3)- W(6 I: (w(3 I-w( 7 ) g-AAD/(IH(SI- N(IIIh (éI- )w(2)I (WISI- W(3) h( (5) -W(4 III AA-I O/((w(u+2I-V(u+3IIt(w(u+2I-V(U+4II*(w(u+2I- w(u+5II* ( {0+2)'W(d+6) I SEAS: W(d+2)-W(d+6 )/(W(d+2)'W(d+1)) D-I. /(SW(J+4)-ng))‘(W(d+4)-W(d+1))'(W(d+4)-W(d+2))* (W + I-V(u+3I I C=OD*KK+3;‘W(IPR+2)'W(KK+6)'V(IPR+3)*W(KK+9) IPR+4)‘U(K K+12)) N(L 1IPR'IPR! GO TO 35 KRES=K CALCULATE SPLINE VALUES AT THE DATA POINTS. AND SCALAR PRODUCTS ¥SOATA=KRES‘3*M D. 43 L'1. 5 KKI)§% AA=1. 0/((U(3) W(4)) (W(3)- W(5))* (W(3) W(6)) (W(3)- -W(7))) 000 000 000 221 o-1.o/((w(5)-w(1))~(w(5)-w(2))~(w(5)-w(3))t(w(5)-w(4))) 39 u=u+1 K=K+3 é¥3d+3))*(W(d+2)-w(d+4))*(w(u+2)-W(u+5))* 4+5 )/(w(u+2)-w(u+1)) y;g))r(w(u+4)-w(u+1))t(w(d+4)-w(u+2))1 a-1 41 ))/(H(d+3)-H(d+4)) .41.39 sI)-wfu+1 )sw3+(81w(LK+3g+AAtw(LK+6)1: **3+ C*H§LK)+DO*V(LK-3) 1(w(u+3)-xo I))**3 +4)-XD(I) -*3 HXHUIMUIMU‘OW)” flXufl-~vanm0K3 a~u~thush.uuul~ 40.40.44 CALCULATE THE RESIDUALs ANo PARAMETERS OF THE REQUIRED SPLINE 44 DET- (1): 3)-S(2)‘ 2 A: s 3 -s -s 2 ts /0ET 8- s 1 t5 5 -s 2 cs 4 /DET KsKSOA A K0 :5 1'1." 45 RD(I)8RD(I)-A*W(K-1)'B'H(K) KL=KRE +1 00 46 K-KL KUMAX.3 46 w(K)=w(K)+A:w(K+1)+a:w(K+2) CDMPUTE ELEMENTS OF FN.GN AND ON KSKRES 6a;:.96((w(2)-V(3))‘(W(2)-H(4))‘(W(2)-W(5))'(w(2)-W(6))) oo 4; u-1 N A=AA:w(K+42t(XN(u -w(u+1I):t2 AA=€QOQ££¥ 3:325: u+3))v H(d+2)-U(d+4))‘(w(d+2)'w(d+5))* BaAAaéggxa75Igg§tgét(w(0+2)-w(d+6))/(w(u+2)-w(u+1)))- ‘DD=Y&?312;$:}3:313§N(J))/((H(d+4)-w(d))‘(U(d+4)-W(d+1))* FN ugsAtéXN(d)-w {+1))+Bt(XN(d)-H(d+2))+DD’(W(J+4)-XN(J)) 1': 3.313213% 49 DN ?)'6.0*TGNgg-1)+GN(d)+2.0‘(FN(d-1)-FN(d))/(XN(J)'XN(d-1)))/ 1(XN d)-XN(U-1 -*2 0N(u—1)=0N(u)-DN(d-1) K 47 CONTINUE PROVIDE PRINT IF REQUESTED IF$IPRINT) 50.50.51 50 RE URN 51 PRINT 52 52 FORMAT(1H1.35X.*SPL NE APPROXIMATION OBTAINED BY SPLIN3tl/4X,THI. 19x SHXN21;.18X.SHFN 1).18x.5HGN(x).13x.16H3Ro DERIv CHANGE.11X. 28HTHETA I //) 1 1 PRINT 63.1.XN(I).FN(I).GN(I) 54 1-I+1 IF I-N) 55.56 56. 55 PR NT 63.1.XNlI).FN(I).GN(I).oN(1).THETA(I) GO To 54 222 56 PRIuT 63 I.XN(I).FN(I).GN(I) //4X,1HI 9X.5HXD(I).18X.SHYD(I).18X.5HUD(I).19X.3HFIT, SIDUAL//) 1 M\~J .M x~(u)) 60.60.61 X) GO TO 59 60 YDRD - 3v0(I - ROII) PRIN a“ ). YD (I). wO(I) YDRD .RD(I) 63 FORMAT1IS. SE2 4) 58 CONTINUE GO TO 50 END SUBROUTIN E F COMMON /IE I DI F PMC.DEL TP/ v NT(1 OTER/ XRAY YRAY S900). INUM FY/ v IF(10153W3 C 50.2) v(150 ). N Ts(2). XPL(101). DUM6(6) g ----COMPUTE DIFFERENCES AND OIVIOE 3v PMc DO 20 K-1 HO YDIF(K)= (Y NT (K.1)'VINT(K.2))/PMC O CONTINUE --°-COMPUTE DERIVATIVES (USING FINITE DIFFERENCES) ----COMPUTE DERIVATIVE OF FIRST AND SECOND POINTS USING FORWARD DIFF. DO Ks 2 +”gm; (S2'YDIF(K+3))' -(9tvoIF(K+2))+(18cYDIFIK+1))- (11~vDIF(K))) INUM=INUM+1 XRAY INUM -XPL (K) YRAY INUM -DY( K) o CONTINUE ----COMPUTE DERIVATIVE USING CENTRAL DIFFERENCES. DO 40K DY(K)- -(vDiF(K- -2)- (84VDIF(K 1))+(8-YDIF(K+1))- YDIF(K+2))/(12*DEL) INUMzINUM M+1 XRAY INUM)-XPL(K) YRAY INUM)=DY(K) o CONTINUE ----COMPUTE DERIVATIVES OF THE LAST TWO POINTS USING BACKUARD DIFF. DD 50 K=99.1OO /( K6;($11*YDIF(K))- (189YDIF(K-1))+(9*YDIF(K- 2))- (2*YDIF(K- 3))) + t INUM=INUM+1 XRAY INUM -XP VRAvI INUM sDY so CONT RETURNU END SUBROUTINE PLO COMMON /STNAM/ COMMON /PLOTER UF EG I 000008) 0000.) 0005 ‘Ar .11 DIMENSION IB ;.YRAY(102) DIMENSION XL NC= IC~1oo DO I=1 NC CONTINUE --—-INIT PLOTTING PARAMETERS CALL PLOTS(IBUF.257.0) XRAY 101 05 101 YRAY 102 :0 O1 ----PRINT AXES AND TITLES CALL AXIS(O..O..'DISTANCE FROM HOLE (IN.)'.-24.9.0.0.0. 000d 000 223 ) MPRESSIVE STRAIN'.18.7.0.90.0. . .042. .042 . .042. .04 2 .' HDL E No. ".0..9) .ISET. O. 10) .2. ”LEGEND“. ..6) 6. .6. 56. .014. .014) <>m>>> mmmmmmmOrrmrrr omooooourrmrrr mbmaund- ----PRINT PLOT LINES DO 30 I'1.IC K'I-1 N'100*K DO 20 '1 100 XRAY d 'X d+N YRAY d V d+N 20 CONTINUE CALL LINE(XRAY.YRAY.100.1.5.K) C ----:RINT LEgE N3 HTD HTD NE XLEG. YLEG. 3. EB L(8. .HTD- .042. ”b 084 ISTNAM(I). o. 10) E LEGEND 8. 7. 000 O ermmmo OAPPQDQN WI 4MZmrun‘4 04C