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Measurement of surface displacements and strains by the double aperture speckle shearing camer Brdicko, Jan 1977

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MEASUREMENT OF SURFACE DISPLACEMENTS AND STRAINS BY THE DOUBLE APERTURE SPECKLE SHEARING CAMERA by JAN BRDICKO B.Sc, University of I l l i n o i s , 1970 M.A.Sc., University of British Columbia, 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of C i v i l Engineering) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1977 © Jan Brdicko, 1977 In present ing th is thes is in p a r t i a l fu l f i lment o f the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make i t f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho la r ly purposes may be granted by the Head of my Department or by h is representa t ives . It is understood that copying or pub l ica t ion of th is thes is fo r f inanc ia l gain sha l l not be allowed without my wr i t ten permission. Department of C i v i l Engineering The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date October 11 197? i i MEASUREMENT OF SURFACE DISPLACEMENTS AND STRAINS BY THE DOUBLE APERTURE SPECKLE SHEARING CAMERA ABSTRACT In the testing of materials, structures and structural components i t is often desired to determine the surface displacement and strain fields due to some external loading. Numerous optical techniques have been developed for this purpose and successfully used in particular applications. Unfortunate-ly, when the surface deformation is quite large, as is usually the case in practical testing, most of these methods f a i l and only a few suitable opti-cal interferometric techniques w i l l work. Two of the recently developed techniques that seem to work are based on laser speckle interferometry. The f i r s t technique was described in 1972 by Duffy [ l ] who showed that a Double Aperture Speckle Camera (DASC) is suitable for measurement of a reasonably large in-plane displacement having i t s direction parallel to the line connecting the two apertures of the camera. A second technique was described in 1973 by Hung [3] . He showed that a Double Aperture Speckle Shearing Camera (DASSC) may be used to measure both the in and out-of-plane strains of planar surfaces. Duffy has not considered the fringe formation by DASC due to the displa-cement normal to the surface and the displacement normal to the line connecting the two apertures of DASC. Hung, in turn, has not considered the effect on fringe formation of either the in and out-of-plane displacements, or the in-plane strain, which is the partial derivative w,v (see Fig. 3.11 for the definition of w,y). Because of the great potential of DASC and DASSC stemming from their a b i l i t y to measure displacements and strains over many orders of magnitude, a considerable effort was made to determine the fringe formation of the two i i i cameras due to a l l displacements and strains occuring in a general deformation of a specimen surface. The theoretical analysis of models of DASC and DASSC was performed and resulted in two "new" equations describing the fringe formation by these cameras. The equations take into account the effect of a l l displacements , and strains on the fringe formation; in addition, the equations are "symme-t r i c " and the equation governing DASSC reduces to the one governing DASC for the lateral shear set equal to zero. The accuracy of these equations was then verified by a number of simple experiments. Various ways of using the two cameras were proposed so that the unknown displacements and strains in the specimen surface may be calculated from the least number of fringe patterns. Computer programs based on these proposed methods were written and used in several experiments. In a l l instances the actual and the calculated displacements and strains agreed quite well. iv TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS iv LIST OF TABLES v i i i LIST OF FIGURES ix NOTATION xiv ACKNOWLEDGEMENTS xv CHAPTER 1. INTRODUCTION 1 1.1 Background 1 1.2 Holographic Interferometry 3 1.3 Holographic Contouring Techniques 4 1.4 Measurement of Displacements by Speckle Interferometry 5 1.5 Measurement of Strains by Speckle Interferometry 9 1.6 Limits of Investigation 9 2. THEORETICAL PRELIMINARIES 12 2.1 Introduction 12 2.2 Light 12 2.3 Geometrical Optics 13 2.4 The Ray 16 2.5 Fermat's Principle 16 2.6 Point Source of Light 16 2.7 The Principle of Linear Superposition 18 2.8 Diffraction i 8 2.9 Huygen's Principle 20 V CHAPTER Page 2.10 Coherence 20 2.11 Imaging Properties of a Thin Lens 22 2.12 Aberrations in Optical Systems 24 2.13 Generalized Coherent Interferometer 26 3. ANALYSIS OF THE FRINGE FORMATION BY DASC AND DASSC 30 3.1 Preliminary Remarks 30 3.2 Image Formation by a Single Aperture Camera ... 36 3.3 Image Formation by a Double Aperture Speckle Camera (DASC) 36 3.4 Formation of Moire Fringes by DASC 40 3.5 Image Formation by DASSC 46 3.6 Formation of Moire Fringes by DASSC 53 3.7 Imaging of Real Surfaces by DASC and DASSC 62 4. CALCULATION OF DISPLACEMENTS AND STRAINS 65 4.1 Preliminary Remarks 65 4.2 Use of DASC to Measure General Deformation .... 67 4.3 Use of DASC to Measure Plane Strain and Plane Stress Deformation 69 4.4 Use of DASC to Measure Out-of-Plane Bending ... 71 4.5 Use of DASSC to Measure Specimen Deformation -Theoretical Considerations 73 4.6 Use of DASSC to Measure General Deformation (Algebraic Solution) 76 4.7 Use of DASSC to Measure Plane Strain and Plane Stress Deformation (Algebraic Solution) 77 v i CHAPTER Page 4.8 Use of DASSC to Measure Out-of-Plane Bending ( A l g e b r a i c S o l u t i o n ) 78 4.9 Use of DASSC to Measure General Deformation ... 78 4.10 Use of DASSC to Measure u,u, y ,v,,and v , y from Two Photographs 81 4.11 Use of DASSC to Measure Out-of-Plane Bending . 85 4.12 Use of DASSC to Measure Plane Stress and Plane S t r a i n Deformation 87 5. EXPERIMENTAL APPARATUS AND PROCEDURE 89 5.1 The Camera 89 5.2 The Recording System 95 5.3 The F i l t e r i n g System 95 5.4 The Specimen Loading Systems 97 5.5 Specimens 104 5.6 Experimental Procedure 104 6. EXPERIMENTAL WORK 107 6.1 P r e l i m i n a r y Remarks 107 6.2 R i g i d Body, Out-of-Plane T r a n s l a t i o n of a P l a t e Specimen 107 6.3 R i g i d Body, In-Plane R o t a t i o n of a P l a t e Specimen H I 6.4 Out-of-Plane Bending of a Thin Beam w i t h a Rectangular Cross-Section 115 6.5 In-Plane S t r e t c h i n g of a Thin Beam w i t h a Rectangular Cross-Section 119 6.6 In-Plane S t r e t c h i n g of a Beam w i t h a V a r i a b l e Cross-Section 123 v i i CHAPTER ' Page 6.7 In-Plane Stretching of a Wooden Beam 126 6.8 Error Analysis 129 7. CONCLUSIONS 167 7.1 Summary and Conclusions 167 7.2 Suggestions for Future Research 169 BIBLIOGRAPHY 171 APPENDICES A DERIVATION OF EQUATION (2.9) 175 B DERIVATION OF EQUATIONS (2.18) AND (2.19) ...... 176 C DERIVATION OF EQUATION (3.5) 178 D DERIVATION OF EQUATION (3.7) .v. 181 E DERIVATION OF EQUATIONS (3.10) AND (3.11) 182 F ...... . ..... 183 G DERIVATION OF r g l AND r £ 2 FOR DASC DURING THE FIRST EXPOSURE 184 H DERIVATION OF EQUATION (3.12) - THE FIRST EXPOSURE I r 187 I DERIVATION OF EQUATIONS (3.14) AND (3.15) 188 J DERIVATION OF r e l AND r e 2 FOR DASC DURING THE SECOND EXPOSURE ••••• 1 8 9 K DERIVATION OF EQUATION (3.17) - THE SECOND EXPOSURE I 192 L DERIVATION OF EQUATIONS (3.19) AND (3.20) 193 M DERIVATION OF EQUATION (3.21) 195 N DERIVATION OF-r e ij FOR DASSC DURING THE FIRST EXPOSURE 197 v i i i APPENDIX Page 0 DERIVATION OF EQUATIONS (3.23) AND (3.24) - THE FIRST EXPOSURE I r •••• 203 P DERIVATION OF r e i j FOR DASSC DURING THE SECOND EXPOSURE 206 Q DERIVATION OF EQUATIONS (3.25) AND (3.28) -THE SECOND EXPOSURE I r . . 212 R DERIVATION OF EQUATIONS (3.26) AND (3.29) 214 S DERIVATION OF EQUATIONS (3.27a) AND (3.27b) 217 T DERIVATION OF EQUATION (4.13) ... 219 U 221 V DERIVATION OF THE DISPLACEMENTS AND STRAINS CAUSED BY THE OUT-OF-PLANE BENDING OF BEAMS 222 LIST OF TABLES Table 6.1 Fringe Data of Exp. 19 . 110 ix LIST OF FIGURES Figure Page 2.1 Refraction of light 14 2.2 Optical path length of a ray 14 2.3 Point source 15 2.4 Fraunhofer diffraction by an aperture 15 2.5 Fresnel diffraction by an aperture 15 2.6 Illustration of the Huygen's principle 19 2.7 Coherence of light 19 2.8 Longitudinal and lateral coherence of electric fields 19 2.9 Focal length of a thin lens 23 2.10 Image formation by a thin lens 23 2.11 Lateral magnification by a thin lens 23 2.12 Curvature of f i e l d aberration of a thin lens 25 2.13 Generalized coherent interferometer 25 3.1 Single aperture camera 31 3.2 Diffraction in a single aperture camera 31 3.3 Coordinate system of the circular aperture 33 3.4 Diffraction pattern of a single circular aperture . 33 3.5 Double aperture camera 36 3.6 Diffraction in a double aperture camera 38 3.7 Diffraction pattern of two circular apertures 39 3.8 Elevation of the f i r s t exposure speckle 41 3.9 General deformation of the specimen surface 41 3.10 Elevation of the f i r s t (1) and the second (2^,2g) exposure speckles 45 3.11 The schematic of DASSC 47 X Figure Page 3.12 Diffraction in DASSC 48 3.13 Intensity distribution I r for DASSC with Ay s>D s s .. 50 3.14 General deformation of the specimen surface 56 4.1 Normal view of the aperture plane 66 4.2 Normal view of the specimen showing coordinate systems y,z and y±,z^ 66 4.3 Geometry of specimen illumination 70 4.4 Rotated coordinate system 70 5.1 Double aperture speckle shearing camera (DASSC) .... 90 5.2 Schematic of DASSC 90 5.3 Photographic plate holder assembly 91 5.4 Shutter assembly 91 5.5 Schematic of the recording system 92 5.6 Recording system 93 5.7 Filtering system 93 5.8 Schematic of the f i l t e r i n g system 94 5.9 Plate specimen positioned on translation and rotary tables 96 5.10 The arangement in bending of the beam experiments .. 96 5.11 Schematic of the tensile loading apparatus 98 5.12 Right side view of the loading apparatus 99 5.13 Left side view of the loading apparatus 99 5.14 Measurement of displacements u(y,0) by di a l gages .. 100 5.15 Variable cross-section specimen 100 5.16 Central part of the beam used in the beam bending experiments 102 5.17 Tensile specimen with the uniform cross-section .... 102 x i Figure Page 5.18 Wooden beam specimen 103 6.1 Measurement of the out-of-plane displacement u by DASC 134 6.2 Fringe pattern of Exp. 19 134 6.3 Microdensitometer trace of Exp. 19 135 6.4 Fringe pattern of Exp. 22 135 6.5 Fringe pattern of Exp. 17 136 6.6 Fringe pattern of Exp. 18 136 6.7 Predicted n vs. experimental n - Exp. 19 137 6.8 Predicted u vs. experimental u - Exp. 19 137 6.9 Rotation of a plate about x-axis 138 6.10 Measurement of the in-plane displacements v and w by DASC 138 6.11 Fringe pattern of Exp. 24 139 6.12 Fringe pattern of Exp. 25 139 6.13 Fringe pattern of Exp. 26 140 6.14 Fringe pattern of Exp. 2 140 6.15 Predicted v vs. experimental v - Exp. 26 141 6.16 Measurement of the out-of-plane displacement by DASSC by DASSC 141 6.17 Fringe pattern of Exp. 16 142 6.18 Predicted u vs. experimental u - Exp. 16 142 6.19 Fringe pattern of Exp. 10IC 143 6.20 Predicted u vs. experimental u - Exp. 10IC 143 6.21 Predicted u, v vs. experimental u, v - Exp. 101C .... 144 6.22 Fringe pattern of Exp. 10IA 144 6.23 Fringe pattern of Exp. 101B 144 x i i F i g u r e Page 6.24 P r e d i c t e d u vs. experimental u - Exp. 10 1 A @ 101B ... 145 6.25 P r e d i c t e d u, y vs. experimental u, y - Exp. 1 0 1 A @ 1 0 1 B . 145 6.26 Measurement of the in-plane deformation by DASSC .... 146 6.27 Fringe p a t t e r n of Exp. 114C 147 6.28 Fringe p a t t e r n of Exp. 114D 147 6.29 P r e d i c t e d u vs. experimental u - Exp. 1 1 4 C @ H 4 D . . . . 148 6.30 P r e d i c t e d u , y vs. experimental u , y - Exp. 114C@ 114D. 148 6.31 P r e d i c t e d v vs. experimental v - Exp. 1 1 4 C @ 1 1 4 D .... 149 6.32 P r e d i c t e d v , y vs. experimental v , y - Exp. 1 1 4 C @ 1 1 4 D . 149 6.33 Fringe p a t t e r n of Exp. 114B 150 6.34 P r e d i c t e d u vs. e x p t l . u - Exp. 114B@ 114C@ 114D 151 6.35 P r e d i c t e d u , y vs. e x p t l . u , y - Exp. 114B@ 114C@ 114D. 151 6.36 P r e d i c t e d v vs. e x p t l . v - E x p . 114B@ 114C@ 114D 152 6.37 P r e d i c t e d v , y vs. e x p t l . v , y - Exp. 114B@ 114C@ 114D. 152 6.38 P r e d i c t e d u vs. experimental u - Exp. 1 1 4 C @ 1 1 4 D 153 6.39 P r e d i c t e d u , y vs. experimental u , y - Exp. 1 1 4 C @ 1 1 4 D . 153 6.40 P r e d i c t e d v vs. experimental v - Exp. 114C@ 114D .... 154 6.41 P r e d i c t e d v , y vs. experimental v , y - Exp. 114C@ 114D. 154 6.42 T e n s i l e specimen of Exp. 122 1 5 5 6.43 Normal view of the aperture screen 1 5 6 6.44 Fringe p a t t e r n of Exp. 122S1 157 6.45 Fringe p a t t e r n of Exp. 122S2 157 6.46 P r e d i c t e d u vs. e x p t l . u - Exp. 122S1@ 122S2 158 6.47 P r e d i c t e d u , y vs. e x p t l . u , y - Exp. 122S1@ 122 S 2 1 58 6.48 P r e d i c t e d v vs. e x p t l . v - Exp. 122S1 @ 122S2 159 6.49 P r e d i c t e d v , y vs. e x p t l . v , y - Exp. 122S1@ 122 S 2 1 5 9 6.50 Fringe p a t t e r n of Exp. 12 2 S 3 1 6 0 x i i i F i gure Page 6.51 Fringe p a t t e r n of Exp. 122S3 160 6.52 The par t of the specimen surface where the displacements and s t r a i n s were c a l c u l a t e d 161 6.53 Contours of constant displacement v(y,z) i n the v a r i a b l e c r o s s - s e c t i o n specimen 162 6.54 S t r a i n v , v ( y , l ) i n the v a r i a b l e c r o s s - s e c t i o n specimen 163 6.55 S t r a i n v, v(y,0) i n the v a r i a b l e c r o s s - s e c t i o n specimen 164 6.56 Fringe p a t t e r n of Exp. 132D1 165 6.57 Fringe p a t t e r n of Exp. 132D2 165 6.58 Fringe p a t t e r n of Exp. 132S2 165 6.59 Fringe p a t t e r n of Exp. 132S1 166 C.l D i f f r a c t i o n i n a s i n g l e aperture camera 178 F . l U n i t v e c t o r s of e l e c t r i c f i e l d s 183 V . l Out-of-plane bending of a p r i s m a t i c beam 223 NOTATION The meaning of symbols is defined in the text where they are introduc The summation convention applies to subscripted variables with lower case indices with the range of the subscripts usually indicated. XV ACKNOWLEDGEMENTS The author would like to sincerely thank to his advisors, Dr. M. D. Olson and Dr. C. R. Hazell, for their helpful advice and guidance given during the course of the research and preparation of this thesis. He also wishes to thank to Dr. D. L. Anderson for his assistance during the research work. The author also wishes to thank to Mr. L. E. Dery, Mr. Phil Hurren, Mr. John Hoar and Mr. Dick Postgate for their valuable technical assistance. This study was made possible through a research grant provided by the National Research Council of Canada. 1 1. INTRODUCTION 1.1 Background There are many experimental techniques [ 7 , . . . , l l ] for determining the displacements and strains in materials or structural components subjected to various loads. Certain tests, for various reasons, must be. noncon-tacting, i.e. i t is not possible to use stress coatings, photo-elastic coatings, strain gages, displacement gages or other contact probes. In such instances optical interferometry may often be used successfully. Its pre-vious use in practical testing had been hindered by i t s excessive sensitivity, the need for complicated and expensive instrumentation and i t s susceptibility to disturbing effects of environment. While some shortcomings remain, the use of optical interferometry has spread dramatically since the invention of the laser in the 1960's. Numerous optical interferometric techniques using laser generated coherent light were developed and proved extremely valuable for specific applications. Most of the techniques that have been developed for the measurement of displacements to date are, unfortunately, only sui-table for the measurement of very small displacements. Alternatively some techniques have been developed to measure large displacements along the line of sight. Therefore, there has existed a need to develop a technique for the measurement of the displacements and strains of the magnitudes encountered in practical testing. Duffy [l,2] described the double aperture camera and showed that i t is suitable for the measurement of reasonably large in-plane displacements. Hung [3,4,5,6] has shown that a somewhat modified double aperture camera, DASSC, may be used to measure in and out-of-plane strains. Other researchers have proven the f e a s i b i l i t y of different methods for measuring displacements and strains, but these methods seemed less powerful and promising than Hung's. This thesis objective, in 1974, was to examine the mechanism of failure 2 of wooden beams subjected to various loads and, at that time, Hung's method appeared to be the most suitable noncontacting experimental technique avai-lable. Subsequently, DASSC was built and i t s performance tested on a c a l i -bration specimen. The surface strains were then calculated from the fringe patterns produced by the camera in the way suggested by Hung. Unfortunately, the calculated strains and the actual surface strains (determined by strain gages and d i a l gages) were in a considerable disagreement. A decision was made to find the cause of this disagreement, and the subsequent theoretical analysis of DASSC resulted in a new equation governing the fringe formation. The accuracy of the new equation was then thoroughly tested both experiment-all y and by computer simulation. However, the above analysis had been very time consuming and l e f t l i t t l e time for the actual investigation of the failure mechanism of wooden beams. The research which was done, though, has given some insight into the behaviour of wood and provides a starting point for other researchers considering this area of study with the use of DASSC. Literature Survey Because both optical interferometry and even coherent light interfero-metry encompass such a great number of various techniques i t is not possible to review a l l of them. Instead, only the best known techniques suitable for measurement of displacements and strains in planar surfaces are investiga-ted- with special emphasis in the review attached to speckle interferometry. Hopefully, the review of a l l techniques relevant to the research pre-sented in this thesis i s complete. If an omission has been made, i t is quite unintentional. The techniques are reviewed in chronological order, star-ting with holographic interferometry and ending with DASSC. 3 1.2 Holographic Interferometry One of the f i r s t demonstrations of holographic interferometry was done by Powell and Stetson [12] in the early 1960's. The technique is similar to conventional holography, except that two exposures (and, hence, two holo-grams) of the object are recorded on the same holographic plate; the surface of the object is deformed or displaced between the two exposures. Upon recon-struction of the hologram, two three-dimensional images of the object are formed, interfere with each other, and produce a set of fringes. The fringes represent areas of the same change in total optical path length and, with knowledge of the parameters of the experimental setup, a component of the surface displacement along a line of sight may be calculated. Since three displacement components must be calculated to determine the surface defor-mation, in general, three holograms are needed. The surface strains may then be obtained by differentiation of the displacements. Since 1965 holographic interferometry has been applied successfully to a study of transient and steady vibration [l2,...19] and in a wide variety of materials testing [20,23,24,29,30,31,32] . Special techniques were deve-loped for the measurement of the wave propagation using pulsed lasers [25,26] . By illuminating the specimen surface by two beams inclined at equal angles to the surface normal, or by other means, experimenters were able to develop several holographic techniques for the measurement of i n -plane displacements [27,28,33] . A number of techniques for the measurement of in-plane strain were developed as well [34,35,36] . The main advantage of holographic interferometry is i t s a b i l i t y to work with an arbitrary three dimensional surface. Another advantage i s , in many cases, i t s high sensitivity as the fringes usually represent displacements - 5 of the order of 1 x 10 in. Unfortunately, such a high sensitivity makes this method unsuitable for ordinary engineering testing where displacements 4 — 3 — 2 of the order of 1 x 10 in. and 1 x 10 in. are commonly encountered. A l l holographic techniques are also quite sensitive to the disturbing effects of the environment. Finally, a major problem associated with using holography to obtain numerical measurements stems from the fact that the fringes may not be localized [21,22] on the specimen surface and, consequently, the fringe positions cannot be clearly established. 1.3 Holographic Contouring Techniques There are several holographic interferometric techniques suitable for measurement of relatively large static or dynamic changes in the shape of an object. In the absence of large in-plane motion, the fringes produced by these techniques are related to the out-of-plane displacements of the object. The methods known as contouring techniques, [37,40,41,42] are based on pro-ducing an illuminated volume of space in which the apparent illumination of any point in that space is some function of position alone. If the function is known, the shape of the illuminated area of the object can be determined. Most optical f i e l d contouring techniques result in sinusoidal functions of position. Although in theory the sensitivity of such a continuous function is unlimited, in practice a sensitivity of one-half of the period i s used. There are numerous ways in which contours can be formed [38,39] . Two holograms can be recorded on the same plate. The object can be illuminated with two wavelengths simultaneously, with only one wavelength but from two directions, or with one wavelength but with a medium of different retractive index surrounding the object. Alternatively, the hologram can be recorded with one wavelength, developed and replaced, and both i t and the object illuminated with a second wavelength. The interference between the wave appearing to originate from the image (magnified because of the shift in wavelength) and the wave actually coming from the object i t s e l f causes 5 contours to appear on the image. Similar results are obtained i f the holo-gram is recorded while the object i s in a medium of one refractive index,and the hologram and the object then illuminated while the object is in another medium. A l l of these techniques generate Moire-type of fringes which are rela-ted to the change in shape of the object; the techniques are usually insensi-tive to the in-plane displacements. Out-of-plane displacements cannot be calculated from these fringes i f deformation of the object involves large i n -plane displacements. 1.4 Measurement of Displacements by Speckle Interferometry Anyone working with lasers is familiar with the speckle phenomenon which causes a grainy appearence of the laser illuminated surface. Laser speckle (or speckle pattern) is formed when coherent light i s either scattered from a diffusely reflecting surface or propagates through a medium with random refractive index fluctuations. The speckle size is usually defined as the s t a t i s t i c a l average distance between adjacent regions of maximum and minimum brightness. If a diffusely reflecting surface is imaged by a lens on a -screen, the speckle size D s is related to the effective numerical aperture NA of the lens by X Ds - .6 — b NA The speckle pattern depends on the properties of the scattering surface and this fact i s u t i l i z e d by speckle interferometers which relate the fringes created by changes in the speckle pattern to the surface deformation. Speckle interferometers may be classified as those suitable for measurement of displacements smaller than speckle size and those suitable for measurement of displacements larger than speckle size. Alternatively, the speckle interferometers could be classified according to the type of light fields that the interferometers employ, i.e., either Las interferometers combining speckle and uniform fields or as interferometers combining two speckle ' fields. The main advantage of speckle interferometers is their a b i l i t y to vary their sensitivity by changing speckle size and other optical parameters i t is also claimed that these interferometers measure the in-plane displa-cements of the tested surface independently of any displacement taking place in the direction normal to the object surface. There are a great number of interferometric techniques u t i l i z i n g laser speckle .[1,2,3,4,5,6,44,45,46,47] . These techniques are used for displa-cement measurements, steady state vibration analysis and qualitative testing An excellent up-to-date review of existing techniques using speckle inter-ferometry for measurement of displacements and strains has been done by A.E.Ennos [43] . Several of the numerous techniques w i l l now be briefly described. The speckle interferometer described by Leendertz [49] uses two illumi-nating beams incident at equal angles on either side of the normal to the object surface; the illuminated surface is then imaged by a lens on a photo-graphic plate. The two speckle patterns, one due to each illuminating beam, interfere coherently, producing fringes according to the rule 2u sinS^ = nX where u, 0i 1 > n and X are defined as u ... in plane displacement e ± ... angle of illumination n . .. fringe number wavelength of illuminating light 7 The s e n s i t i v i t y of the i n t e r f e r o m e t e r may be changed p r i m a r i l y by changing the angle 6 i . The maximum all o w a b l e in-plane displacement u must be smaller than the apparent speckle diameter. Fringes produced by t h i s i n t e r f e r o m e t e r are of low v i s i b i l i t y . Duffy [ l , 2 ] has proven the f e a s i b i l i t y of a double aperture speckle i n t e r f e r o m e t e r . One beam i s needed to i l l u m i n a t e the object s u r f a c e , and the surface i s imaged on a photographic p l a t e by a lens having two r a d i a l l y opposed c i r c u l a r apertures i n i t s entrance p u p i l . To each aperture c o r r e s -ponds a p a r t i c u l a r speckle p a t t e r n , and the two patterns i n t e r f e r e cohe-r e n t l y i n the photographic p l a t e emulsion to produce a f i n e g r a t i n g - l i k e g r i d over an area of each speckle. In a double exposure method two g r i d s are formed over the area of speckle, one g r i d corresponding to the unde-formed surface and the other to the deformed surface. The two g r i d s add e i t h e r c o n s t r u c t i v e l y or d e s t r u c t i v e l y . The c o n s t r u c t i v e a d d i t i o n preserves the g r i d - l i k e s t r u c t u r e of the speckle whereas t h i s s t r u c t u r e vanishes i n d e s t r u c t i v e a d d i t i o n . The a d d i t i o n takes place over a l l speckles comprising the "speckled" image of the o b j e c t , and generates f r i n g e s r e l a t e d to the sur-face in-plane displacement i n the d i r e c t i o n of the l i n e connecting the two apertures. However, o p t i c a l f i l t e r i n g i s necessary to view these f r i n g e s . This i s done by i l l u m i n a t i n g the photographic p l a t e w i t h p a r a l l e l l i g h t and viewing i n the d i r e c t i o n of the f i r s t order d i f f r a c t e d beams. Using the n o t a t i o n shown i n F i g . 3.11 and employed throughout t h i s t h e s i s Duffy showed that a small displacement v along the y-axis causes f r i n g e s according to Ax s v(y,z) = — n(y,z) (1.1) Duffy d i d not consider the e f f e c t on f r i n g e formation by displacements u and w which may occur i n a d d i t i o n to the displacement v. This method i s l e s s s e n s i t i v e than the preceeding one and the f r i n g e s are of high 8 v i s i b i l i t y . It, too, is limited to the measurement of displacements smaller than the object speckle size and the system is further handicapped by i t s failure to work when slope changes exceed a certain magnitude; in addition, long exposures are necessary i f small apertures are used. Duffy has also described an alternative technique which uses one illumi-nating beam and a lens with a single aperture to image the surface on a photographic plate. A double exposure is used to record surface deformation. This recording contains a l l spatial frequencies from zero to the highest frequency which the aperture limited lens is capable of passing. Through the use of a double aperture screen optical f i l t e r i n g of the recording i s used to produce fringes related to the displacements along the line con-necting the two apertures; the sensitivity is determined by the separation of the apertures. The advantage of this method i s that the sensitivity can be chosen after the deformation of the object was recorded to obtain the most desirable fringe spacing. The disadvantage is that only a small part of the surface can be viewed at a time. Ennos [48] has demonstrated a speckle interferometric technique which produces fringes when the lateral component of surface displacement is grea-ter than the object speckle size. The object is illuminated by one beam and a double exposure photograph, one each before and after straining, is recorded on high resolution film. The optical transform of this recording is a pattern of parallel fringes of angular distribution a given as v — sina = nX m where m is the demagnification factor. The examination of the recording on a point by point basis thus yields the magnitude and direction of the lateral movement of the object surface. Alternatively, the recorded image may be spatially f i l t e r e d to yield a 9 contour map showing the surface displacement along a chosen direction. The technique is insensitive to displacements in the direction of the line of sight; however, i t too f a i l s to work when slope changes are large. The number of fringes that the method generates is limited and the fringes are usually of low v i s i b i l i t y . 1.5 Measurement of Strains by Speckle Interferometry There seems to be, at present, only one speckle interferometer capable of direct measurements of surface strains. It is called the Double Aperture Speckle Shearing Camera (DASSC) and is described by Hung in several papers [3,4,5,6]. The camera i s the same as DASC used by Duffy except for the lateral shear which is produced either by placing inclined glass blocks in front of apertures or by defocussing, i.e. , positioning the photographic plate a small distance away from the focal plane of the lens. Regardless of how the lateral shear i s produced, Hung showed that the fringes are formed according to Ay s ( l + cos9 x)u, v + Ay scos9yV,y = - An(y,z) (1.2) with the notation being that of Fig. 3.11. Hung has not considered the effect on fringe formation by the displacements u,v and w and the strain w,y a l l of which, in general, are present in a specimen deformation. He then solved equation (1.2) for u,y and v,y "algebraically" from two fringe patterns. 1.6 Limits of Investigation The work presented in this thesis i s devoted solely to the theoretical and experimental investigation of the use of DASC and DASSC for the measure-ment of displacements and strains in planar surfaces. The theoretical investigation of the fringe formation by DASC and DASSC is restricted to the 1 0 analysis of the simplest possible models of the two cameras. In the analysis the following assumptions (restrictions) are made: a) A small area of the specimen surface may be represented by a point source of light in the analysis of DASC and by two point sources (reflectors) when DASSC is considered. b) The imaging lens(es) i s negligibly thin, free of aberrations and coincident with the aperture plane. c) The only significant diffraction occurring in the camera takes place in the aperture(s). d) The specimen illumination is collimated, monochromatic and perfectly coherent. e) The recording medium (photographic plate emulsion) is negligibly thin and records the intensity of the incident light in a linear fashion. f) The two cameras are used only in the double exposure method. The results of the theoretical investigation are two equations descri-bing the fringe formation by the models of DASC and DASSC. Then, various ways in which the unknown displacements and strains may be determined from the smallest number of "photographs" made by the two cameras are considered. In general, the recorded fringe patterns represent partial differential equations which are solved here only by the f i n i t e difference method. Lastly some special cases are considered leading either to a set of algeb-raic equations or to an ordinary f i r s t order differential equation with a variable coefficient. The experimental work is limited to seven experiments involving various specimens, a l l with planar surfaces coated with a f l a t white paint to approximate the diffusely reflecting surface. Light of wavelength 5145 A provided by an argon gas laser was the only illumination used. The f i e l d of view (defined as /y 2 + z 2 /xg) is smaller than 1 : 5 and in most cases less than 1 : 10 . A l l the fringe patterns are obtained by the Fourier f i l t e r i n g of the photographic plate(s) made by the two cameras. 12 2. THEORETICAL PRELIMINARIES 2.1 Introduction The basic concepts of geometrical and wave optics are reviewed in the f i r s t part of this chapter to familiarize the reader having l i t t l e or no background in optics with those aspects of optics which are used later in the analysis of DASC and DASSC. Those readers acquainted with optics and, in particular, interferometry may wish to proceed directly to Section 2.13 where the basic concepts of a general coherent interferometer are developed, although a brief review of the whole chapter might be helpful, as the terminology and notation introduced here is used in the subsequent chapters. The chapter starts with a discussion of light waves and their representation and behaviour, such as reflection,diffraction and interference. The coherence of light is then defined and developed, an examination of some of the imaging properties of a thin lens follows and, in addition, various types of specimen surfaces are defined. In the last part of this chapter the properties of a general coherent interferometer are derived. Some of the material presented there i s original and is essential for the analysis of DASC and DASSC. So that this chapter may be kept to a reasonable length, many topics are discussed only briefly and, hence, the presentation may be at times overly simplistic. However, most topics presented here are well known and are discussed in depth in numerous textbooks and source books on optics [50, 56,60,61]. 2.2 Light Visible light is a form of electromagnetic energy usually described 13 as electromagnetic waves. The behaviour of light i s governed by Maxwell's electromagnetic theory and quantum theory; Maxwell's theory describes the wave-like aspects of light, while quantum theory describes the particle-like nature of.light. Even though light i s an electromagnetic nature i t w i l l be represented here, without loss of generality, by i t s electric component only. This is done both to simplify the notation and because the photographic plate emulsion, ..used as a recording medium, is sensitive only to the intensity of the electric f i e l d component of incident light. 2.3 Geometrical Optics There i s a class of optical phenomena which may be described without taking into account any hypotheses concerning the wave nature of light or it s interaction with material bodies. This division of optics concerned with the image formation by optical systems i s called geometrical optics since i t s description is founded almost entirely on geometrical relations. The laws of geometrical optics may be stated as follows: 1. Light i s propagated in straight lines in homogeneous medium. 2. Two independent beams of light may intersect each other and thereafter be propagated as independent beams. 3. The angle of incidence of light upon a reflecting surface i s equal to the angle of reflection. 4. On refraction, as is shown in Fig. 2.1, the ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant, depending only on the nature of the media. This relationship is known as Snell's law and is given by F i g . 2.1 Refraction of l i g h t . Fig. 2.3 Point source. Fig. 2.4 Fraunhofer diffraction by an aperture. Fig. 2.5 Fresnel diffraction by an aperture. r^sincj)! = n2sinc}>2 (2.1) where and n 2 are the indices of refraction of the media. 2.4 The Ray The ray may be defined as the path along which light travels or, alternatively, i t may be said that the ray is the direction in which the wave motion propagates. The optical length of a ray of length 1 in a medium of index n is defined as the product n l . For example, the optical length SR shown in Fig. 2.2, is given as SR = n j l j + n 2 l 2 + n 3 l 3 2.5 Fermat's Principle This principle, sometimes called the law of extreme path, states that the path taken by light in passing between two points is that which i t w i l l traverse in the least time. 2.6 Point Source of Light It can be shown that, in free space, the electric f i e l d component of light, E(r,t), emitted by a point source S radiating uniformly in a l l directions may be expressed as E(r,t) = ^ f (r - ct) (2.2) where r denotes the distance from the point source , t denotes time, and c is the speed of wave propagation. The point source S is shown in Fig. 2.3, with R being the receiving point where E(r,t) is measured. Solution (2.2) satisfies Maxwell's equations everywhere except at r = 0. This singularity is unimportant since any real source cannot have a zero radius. The form of f depends on the nature of the source; i f the source radiates a monochromatic wave then f is of the form f(r-.--ct). =a;:cos [k(r - ct) + ty] (2.3) where a .. . amplitude of radiation ty ... phase angle determined from the value of E(r,t) at r = r 0 and t = t 0 k ... wave number X ... wavelength of radiation CO. . . . angular frequency 2TT and k,c,X, and cu are related as k = — (2.4) A co = kc (2.5) Using equations (2.3),(2.4), and (2.5) we may write the electric f i e l d at R(r) in the form E(r,t) = ^  cos(kr - cot + ty) (2.6) Equation (2.6) describes a circularly polarized electric f i e l d . If the f i e l d is polarized in any other way i t is necessary to introduce vector notation: n ... unit vector normal to wavefront k ... propagation vector defined as: k = kn r ... position vector e ... unit vector normal to propagation vector and oriented so that i t l i e s in the plane of polarization. With this notation a polarized electric f i e l d due to a point source may be written as 18 E(r,t) = ^ e cos(k.r - cot + ty) (2.7a) and, in general, the polarized electric f i e l d may be described by E(r,t) = E(r) ecos(k«r - oJt + ip) (2.7b) where E(r) is the amplitude of electric f i e l d at r. 2.7 The Principle of Linear Superposition The theory of optical interference is based essentially on the principle of linear superposition of electromagnetic fi e l d s . According to this principle, the electric f i e l d E produced at a point in empty space due to n different sources is equal to the vector sum. The same principle holds for the magnetic f i e l d . In the presence of matter, however, the principle of linear superposition is only approximate-ly true. 2.8 Diffraction When waves pass through an aperture or past the edge of an obstacle they always spread to some extent into the region which is not directly exposed to the oncoming waves. This phenomenon is called diffraction. In the study of diffraction i t i s customary to distinguish between two general cases known as Fraunhofer diffraction and Fresnel diffraction. Fraunhofer diffraction, shown in Fig. 2.4, occurs when the source of light and the screen on which the diffraction pattern is observed are effectively at i n f i n i t e distances from the aperture causing the diffraction. If either the source or the screen, or both, are at f i n i t e distances from the aperture then Fresnel diffraction occurs. An example of Fresnel E = Ej + E 2 + + E. 'n (2.8) 19 diffraction is shown in Fig. 2.5. There is no sharp line of distinction between the two cases of diffraction and, i f i t is at a l l possible, the Fresnel diffraction is approximated by the Fraunhofer diffraction as the Fraunhofer case is much simpler to treat theoretically. The approximation is appropriate only i f the actual optical path from the source to the screen and the optical path given by the Fraunhofer approximation differ by much less than the wavelength of light. 2.9 Huygen's Principle This principle says that each point on a wavefront may be considered as being instantaneously and continuously the origin of a new spherical wavefront moving outward from that point. The secondary wavelets from a l l points along the wavefront overlap and the superposition of a l l of them accounts for the forward motion of the original wavefront. This principle is useful in the explanation of diffraction and the determination of diffraction patterns of various apertures. The i l l u s t r a t i o n of the Huygen's principle is shown in Fig. 2.6. 2.10 Coherence In discussing the idea of coherence of light i t is convenient to consider two identical point sources Sj and S 2 at different locations, each radiating harmonic travelling waves of the same frequency co, as is shown in Fig. 2.7, and generating an electric f i e l d at point R Ej = — ej cos (krj - cot + ^ j ) 1 E 2 = — e 2 cos (kr 2 -r 2 cot + \p2) 21 The resultant electric f i e l d at R is given by the principle of superposition as E = Ej + E 2 The instantaneous intensity at R is given by I(t) = |E| 2= ( E : + E 2).( Ei + E 2) = |E:|?+ I E2|?+ 2E 1«E 2 and the intensity recorded over "exposure" time T>>T, (T being the period of light wave), i s derived in Appendix A as I r 2 f -1 + 7 a 2 T ai a 2 t r l j 2 l r 2 j 2 l r i j [ r 2 J ( e i.e 2) cos (kr 1 - kr 2 + - ty2) = Ij + I 2 + 2 / I ^ a '( e 2) cos (krj - k r 2 + tyx - ip2) (2.9) The term 2 / I I ' ( e^ e 2) cos (kr 2 - kr 2 + x - ty2) is called the interference term, and i t s presence causes the resultant intensity to be greater than or less than the sum of I : + I 2 . In the derivation of equation (2.9) i t was assumed that the phase difference ty1 - i>z did not change during the "exposure" time. If the two sources behave in such a way, they are said to be mutually coherent. If the phase difference ty1 - ty2 does change in a random fashion with time during the "exposure", then the mean value of the cosine term would be zero and the two sources would be called mutually incoherent. The product of the unit vectors ej»e 2 depends on the relative polarization of the two electric f i e l d s . If the polarization of these two fields are mutually orthogonal, then ex • e 2 = 0. In many instances the two propagation vectors, k : and k 2, are nearly parallel (G i s very small) and, i f the two fields are circularly polarized or are polarized in the same way, then e^e^ = 1 and equation (2.9) is reduced to the form I r = Ij + I 2 + 2 /ijl/cosCkr-j - kr 2 + ^  - ij>2) (2.10) Since the argument of the cosine depends on r1 and r 2 , periodic spatial variations in intensity occur; these variations are the familiar fringes that are seen when two mutually coherent beams of light interfere. If the two independent sources are not purely monochromatic but have, instead, a dominant frequency and the same f i n i t e frequency bandwidth Av, then the relative phase difference ty1 - ty2 w i l l remain constant over a time of the order of (Av) This time i s usually referred to as the coherence time, and the distance that the radiation traverses in the coherence time is called the coherence length. Often the two sources may be "locked" in phase with one another i f they are "driven" by a common driving force. In this case, even though the phase constant of each source may change in a random manner in time (Av) , where Av is now the bandwidth of the common driving force, the relative phase difference w i l l remain constant. There are several more aspects to the coherence of light that need to be mentioned here. In Fig. 2.8, S is a point source of monochromatic radiation. The two points, Pr and P 3, l i e in the same direction from the source; they differ only in their distance from S . The electric f i e l d at P 1 is Ej and the f i e l d at P 3 is E 3. The coherence between the fields E1 and E measures the longitudinal spatial coherence at the f i e l d . Point P„ is at the same distance from S as Pj , but i t li e s in a different direction. In this case the coherence between fields Ej at Pj and E 2 at P 2 measures the lateral spatial coherence of the f i e l d . 2.11 Imaging Properties of a Thin Lens A lens is a most common element occuring in optical systems. It i s made of a transparent, optically dense material, usually glass, having an 23 Fig. 2.11 Lateral magnification by a thin lens. index of refraction greater than one. Usually the two surfaces of a lens are spherical. In a simple thin lens the line through the center of the lens joining the centers of curvature of the lens surface is called the optical axis. The imaging "power" of a lens is defined by i t s focal length f, which is the distance from the lens at which a l l incident rays parallel to the optical axis w i l l meet after passing through the lens as is shown in Fig. 2.9 . If a thin lens is used to image a source point at distance x s from the lens, then the image w i l l be formed at distance x-^  behind the lens, as is shown in Fig. 2.10 . The two distances x s and x^ are related to the focal length f by the equation An important imaging property of a thin lens is that a l l rays emitted by S and passing through the lens to the receiving point R are of equal (or nearly equal) optical path length. The image size of an object is usually different from the actual size of the object. This imaging property of a thin lens is called the lateral magnification m . By considering the geometry of Fig. 2.11 , equation (2.12) relating the object size and the image size to the lens parameters is obtained. The minus sign in the equation (2.12) means that the image of an object is inverted. 2.12 Aberrations in Optical Systems Optical systems in which thin spherical lenses are used have a number of aberrations or faults which impair or limit the imaging quality of the (2.11) (2.12) Fig. 2.12 Curvature of f i e l d aberration of a thin lens. Fig. 2.13 Generalized coherent interferometer. 26 system. The most common aberrations are spherical aberration, astigmatism, comma, curvature of the image f i e l d , distortion of the image, and chromatic aberration. A l l aberrations are analyzed in great detail in a number of optics textbooks and other. The only aberration that needs to be mentioned here is curvature of the f i e l d . It arises i f the object is an extended plane; in that case the astigmatic images w i l l not be planes but curved surfaces. For object points on or near the optical axis, there w i l l be sharp point-to-point representa-tion in the image plane, but, as the distance from the axis is increased, the sharpness of the image w i l l decrease. Each point of the object w i l l be represented by a blurred patch, the size of which w i l l be greater for greater distances from the axis. Even i f the defects of spherical aberration, astigmatism, and comma are corrected, this patch w i l l be the clsest approach to a sharp point focus. The surface containing the best possible focus for a l l parts of the image w i l l not be a plane but a surface of revolution of a curved line about the axis. An example of this aberration in a system using a lens with two relatively small apertures in it s entrance pupil is shown in Fig. 2.12 . The curvature of f i e l d aberration may be corrected i f more than one thin lens is used. 2.13 Generalized Coherent Interferometer We w i l l consider here a coherent interferometer with one point source of polarized monochromatic coherent light. The schematic of the interfero-meter is shown in Fig. 2.16 . S is the point source of light, and r x , r 2 , ... r n are the light rays passing through the interferometer and reaching the receiving point R. In Section 2.6 , i t was shown that these rays may be described by equation (2.7a). Our task is to determine the expression for the intensity of light which would be recorded at the receiving point over some "exposure" time T much greater than the period of light radiated by the source. The analysis w i l l eventually be restricted to the case where a l l rays reaching the receiving point are nearly parallel so that the scalar description of light may be used. The resultant electric f i e l d at the receiving point is given by the principle of superposition as n E r(t) = I Y~ e i c o s ^ . ^ i - bit + ty) (2.13) i =1 1 The phase angle ty and the amplitude a are the same for a l l rays since they originate from one source. For a continuous f i e l d , the number of rays is i n f i n i t e and, hence, the summation sign in equation (2.13) must be replaced by the integral sign. Equation (2.14) results. E r(t) = Y7y~y e(y,z) cos [k(y,z).r(yyz) - tot. + ty] dA (2.14) The integration'extends over the area A(y,z) of the aperture(s), with the understanding that the integration i s to include only those parts of the aperture area which are traversed by those rays eventually reaching the receiving point. The optical path length of a ray, and the propagation vector are expressed as functions of the coordinates (y,z) of the aperture in the entrance pupil of the interferometer. The intensity recorded at R over "exposure" time T is given by 28 E r ( t ) | dt (2.15) 0 Let us now consider an interferometer where the light rays reaching the receiving point R are nearly parallel and polarized in the same way. In addition the optical path lengths of the rays are almost equal so that the "mean" optical path length r Q may be defined as the optical path length of a typical ray. With these assumptions, and using r 0 , we may closely approximate equation (2.14) by E r(t) cos[kr(y,z) - cot + ty] dA (2,16) It is now convenient to introduce the path length variation r„(y,z) defined as r e(y,z) = r(y,z) - r. (2.17) In Appendix B equations (2 .15),(2 .16), and (2.17) were used to derive equations (2.18) and (2.19) giving I r as r ^ a l r o J T 2 j cos kr g(y,z) dA "A + j sin kr £(y, z) dA ^ A (2.18) With the use of complex notation, equation (2.18) may be written as Ir = a T l r o J 2 ikr e(y,z) dA v A ikr e(y,z) dA (2.19) It should be noted that the intensity I r depends, in general, on the positions of the source point S and the receiving point R, since any change in their positions w i l l cause a change in optical path length variation r e ( y , z ) . The recorded intensity I r often varies in some reasonable fashion 29 with y±>z± thus producing a pattern of dark and bright fringes usually referred to as an "interference pattern". If the image formation or characteristics of the interferometer are known, i t may be possible to obtain some information about the source of light from the interference pattern, and indeed, most interferometers are used in this way. Various techniques are used to produce interference patterns which provide information about the source (with respect to the object's size, position, displacement, etc.). Reseachers usually strive to design an "ideal" interferometer which would be sensitive to only one particular aspect of the source or object behaviour which i s of interest, while being completely insensitive to the other aspects. Unfortunately, in most cases this cannot be achieved, and the object behaviour which is of interest must be "extracted" from the interference pattern by an additional data processing, f i l t e r i n g , etc. 3. ANALYSIS OF THE FRINGE FORMATION BY DASC AND DASSC 3.1 Preliminary Remarks In Chapter 2 the principles of physical optics were reviewed, and the equations describing the image formation by the generalized coherent interferometer were derived. Using these results, we w i l l now examine the image formation of three particular interferometers. In the study, the actual interferometers are represented by their mathematical models, and the object surface is represented by one or two points. The equations describing the image formation by these mathematical models are approximate, but should approach the image formation of real interferometers with sufficient accuracy for practical testing. The chapter starts with an analysis of image formation by a camera having one small circular aperture in i t s entrance pupil. The basic principles, notation, and approximations which are made here are then used throughout this chapter. The analysis of the Double Aperture Speckle Camera (DASC) and Double Aperture Speckle Shearing Camera (DASSC) follows, and the equations governing the fringe formation by these two cameras in the double exposure process are derived. 3.2 Image Formation by a Single Aperture Camera The model of the camera is shown in Fig. 3.1 . The cartesian coordinate system x,y,z is set up with x-axis coincident with the optical axis of the system and the y and z-axes in the source plane. The point source S ( y g , Z g ) radiates a monochromatic, coherent light of wavelength X and amplitude a at the unit distance from S. The aberration free lens has a focal length f, diameter D^ , and is negligibly thin as compared to the distances x g and x i . Placing the lens . immediately to the right of the aperture plane permits one to say that both the aperture plane and the lens V d -s ( V z s } y a A 1 1 1 0,z z a source aperture plane plane x s lens V x i z ± x , R ( y i c z i c ) image plane F i g . 3.1 S i n g l e aperture camera. < dw S W N \ 6 ^ y \\7 s (v z s } undiJ d i f f i A P : f r a c t e d ray (tc •acted ray (to C 1 > R) )) ' Q(y ±, ' R ( y i c °> z x z a s x i 1 X F i g . 3.2 D i f f r a c t i o n i n a s i n g l e aperture camera. 32 are at approximately the same distance x s from the source plane. The aperture plane is opaque, i n f i n i t e in extent, and contains a circular aperture of a diameter d with i t s center at (y^,0). The geometric image of S(yg,Zg) is in the image plane at R(y- c^> Z-j_c) • It is assumed that the aperture diameter d is very much smaller that the lens diameter D-^  and, hence, the only significant diffraction in the camera occurs in the aperture i t s e l f . This diffraction causes the image, R, of S to be "blurred" rather than being a point. We wish to determine the recorded intensity distribution I r around the geometric image R. One of the imaging properties of a thin lens is that the optical path lengths of a l l undiffracted rays from a point source to i t s geometric image are equal. Thus in Fig. 3.2, which shows the diffraction in the camera, a l l rays from S to R are of equal optical length, that is S-l-R = S-2-R = ... . Since the source wavefront sw reaching the aperture is spherical, S-l = S-2 = ... and, correspondingly, 1-R = 2-R = ... . If there were another point source at D(y^,Zp), similar arguments would hold; but of course S-R f D-Q, where Q(y^,z^) would be the image of D. Let us consider a diffracted spherical wavefront dw'of such curvature and orientation that i t would appear to originate from the point D. The optical path lengths from the wavefront dw to the virtual image of D at Q are a l l equal, that is 6-Q = 7-Q = ... . However, the diffracted wavefront dw is derived from the spherical, constant phase , source wavefront sw and, therefore, in general is not a wavefront of constant phase. At point 6 dw leads sw by the distance 1-6, at point 7 i t leads sw by the distance 2-7 and so on. The distances 1-6, 2-7, ...may be expressed in terms of the system geometry, and the intensity at point Q due to the diffracted wavefront dw can be calculated according to the equation (2.18). The analysis is restricted to those systems where xs» xi > y yi>ys»yD' zi' zs> zD» d» D»yA ( 3- 1) 33 Fig. 3.4 Diffraction pattern of a single circular aperture. 34 Let us now determine an approximate expression for the optical path length variation r e(y,z) which was defined in Chapter 2 as r e(y,z) = r(y,z) - r. (2.17) Here r(y,z) i s the optical path length from the point source S(y^,z^) in the source plane, through a point (y,z) in the aperture plane, to the image point Q(y^,z^) in the image plane; i t i s given by r(y,z) = S-A + dw-sw + A-Q (3.2) where dw-sw is the distance by which the diffracted wavefront dw leads the source wavefront sw and A is the aperture centre. For example, r(y,z) for the ray 1 i s r^y.z) = S-l + 1-6 + 6-Q = S-A + dWj-sw + A-Q If we choose r. as r n = S-A + A-Q (3.3) then the substitution of equations (3.1) and (3.2) in equation (3.3) yields r e(y,z) = dw-sw (3.4) It is shown in Appendix C that because of equation (3.1) the optical path length variation r e is very closely approximated by r e = r 1 -3ys^ 2x"j A ys zs Ay — Az 3 z s cose + 1 2 2SYS A \z 2~~ Ay s i n t (3.5) The radius r and angle 8 of equation (3.5) are shown in Fig. 3.3, and Ay and Az are defined as A y = yp_JLZS A z = ^ ^ (3.6) In Appendix F i t is shown that a l l rays reaching a point in the image plane 35 are nearly parallel, thereby making i t possible to disregard the vector nature of light in the calculation of the light intensity at that point and consequently, equation (2.19) may be used to calculate I r . The intensity is calculated in Appendix D from equation (2.19) with r •given by equation (3.3) and r e(y,z) given by equation (3.5). The integration i s done over the circular area A of the aperture with the result MP) = i c f2J 1 ( p)l (3.7) [ a l 2 T frrd2! 2 I Q and p are given by l r oJ 2 { * J kd P = 2x ± /(y± -- y i c )2 + - H,? (3.8) (3.9) Equation (3.7) describes the recorded intensity distribution I r as a function of the image plane coordinates y^ and z^ . Jj is the f i r s t order Bessel function of the f i r s t kind. The amplitude of the distribution is proportional to the square of the aperture area Tfd /4 and to the amplitude 'a' of the radiation. It decreases with the square of the source-image distance r Q .. I r is linearly proportional to the exposure time T. The distribution is of the shape shown in Fig. 3.4 and is symmetric about the geometric image R ( y i c > Z i c ) of the source point; here i t also attains i t s maximum value. The f i r s t minimum in the distribution occurs when p = 3.83 , and the area within this perimeter minimum is known as the Airy disk. Its diameter Dg is given in terms of the system geometry by equation (3.10) derived in Appendix E . D s = 2.44A^ (3.10) The apparent diameter of the Airy disk in the source plane is given by (3.11) As i s shown in Fig. 3.4 , the values of the maxima of the intensity distribution decrease rapidly with increasing distance from the centre o the pattern and, hence, the diffraction pattern of the circular aperture may be approximated by the Airy disk alone. The results of this section may be summarized by the following: When the camera with a single circular aperture in i t s entrance pupil is used to image a point source of monochromatic coherent light, the image the point source i s essentially a cir c l e , sometimes referred to as a speckle, of diameter D s . Fig. 3.5 Double aperture camera. The model of DASC is shown in Fig. 3.5 . With the exception of the two apertures, i t has the same geometry as the single aperture camera. The two apertures are circular, of diameter d, and their centers are at (xs,-D/2,0) and (xs,+D/2,0) where D is the separation of aperture centers. Again we wish to determine the recorded intensity distribution around R(y^ c,z^ c), the geometric image of S . The diffraction process in the two apertures is shown in Fig. 3.6 . The optical path length variations, r for aperture 1 and r g 2 for aperture 2, are derived in Appendix G as r e i = re r e 2 = r e - D6 r e is given by equation (3.5) and 6 is given by equation (3.13). Because of the assumptions stated in equation (3.1), the electric f i e l d unit vectors are nearly parallel and equation (2.19) may again be used to calculate the recorded intensity distribution I . The calculations are done in Appendix H with the result I r = 4I n 2J1(p)^|2 kD6 cos —T- (3.12) In equation (3.12) I is given by equation (3.8), p by equation (3.9), and 6 L 0 by y± - yic 6 = - ~ - ^ (3.13) Equation (3.12) describes the recorded intensity distribution in the image plane as a function of the image coordinates (y-j^z-^) . The amplitude of the distribution i s proportional to the square of the area of the two apertures 2 ^ and to the amplitude a' of the radiation. It diminishes with the square of the distance r Q from the source to i t s image and is linearly proportional to the exposure time T . An example of the typical shape of Fig. 3.6 Diffraction in a double aperture camera. 39 the distribution i s shown in Fig. 3.7 for the case of D/d = 4 . The envelope of the intensity distribution is of the same shape as that for the single aperture case shown i n Fig. 3.4 . However, because the intensity distribution i s modulated by the cosine term of higher frequency, i t causes the grid-like appearance of the speckle. This grid i s normal to the y£ coordinate and is centered on and symmetric about the geometric image Fig. 3.7 Diffraction pattern of two circular apertures. 40 of the source point. The speckle diameter D s is the same as that in case of the single aperture, i.e. D s = 2.44 A^p (3.10) The "grid pitch" G s in the image plane i s given by two successive zeros of the modulating cosine term and is calculated in Appendix I as G s = A ^  (3.14) The apparent "grid pitch" G s s in the source plane is given by G S S = A ^ - (3.15) The result of this section may be summarized as follows: When a camera with two radially opposed circular apertures in i t s entrance pupil is used to image a point source of monochromatic coherent light, the image of the point source is essentially a speckle of diameter D g modulated by a grid of pitch G g which i s perpendicular to the line connecting the centers of the two apertures. 3.4 Formation of Moire Fringes by DASC The equation governing the formation of Moire-type fringes by DASC in the double exposure process are derived here. The equations relate the magnitude of the in-plane and out-of-plane motion of a point source and the parameters of the system to the Moire fringe number n . During the f i r s t exposure the coordinates of a point source S are (0,y ,z„). We know from the preceeding section that the recorded image of S 41 FigT 3.9 General deformation of the specimen surface. 42 formed by the DASC is centered about the geometric image of S in the image plane at the point R(y l c> zi c) where the coordinates y^ and z^ c are given by equation (2.12) as v i c ys jxc The image i s a speckle of diameter D s and is modulated by a grid of pitch G s perpendicular to the y-axis. The elevation of this speckle i s shown in Fig. 3.8 . Between the f i r s t and second exposure of the double exposure recording method the specimen is deformed in a general manner. Point S , which repre-sents the specimen surface, is therefore displaced both in and out-of-plane and i t s new coordinates are (u,yg+v,Zg+w) as shown in Fig. 3.9 . The three components comprising the displacement vector of S are u,v and w oriented along x,y and z-axes respectively. The optical path length variations, r and r g 2 , occuring during the second exposure are derived to within an accuracy of A/30 in Appendix J as r e i = E [ ( e I + F)cos6 + E 2sin0] r e 2 = l [ ( E i " F)cos9 + E 2sin9] - D(6 - £) where E1,E2,F and E, are defined in Appendix J with £ given as 5 = u X so ys X so u 1 X + so X so D X so ys X + w so X so + V X so y| xsoJ _w_ y s z s  xso Xso I r is calculated in Appendix K with the result (3.16) x r = 4 Io JAP,) J ^ P P I2 , r 2 J 1 ( p 1 h + ,kD(6 -O cos (3.17) speckle grid term 43 where p x and p 2 are given as kd / F) 2 + E 2 kd P2 = — /(Ej - F) 2 + E 2 The term ^ 2 is of much lower "frequency" than the speckle Pi P2 J grid term and i t s magnitude is usually smaller than that of the term f2J 1(p 1)'| f2J 1 ( p 2 ) ] Pi Ps Since we are interested in the recorded intensity pattern, I , acting as a diffraction grating, the former term may be thought of as a low frequency "background noise" which does not appreciably affect the diffractive efficiency of the speckle grid described by the term 2J 1(P 2)i kD(6 - O cos Comparison of the speckle grid terms recorded during the f i r s t and second exposures reveals that the speckle grid recorded during the second exposure has been "shifted" by the amount £ with respect to i t s f i r s t exposure position. This " s h i f t " and the simplified plan view of the speckles recorded in the two exposures is shown in Fig. 3.10 . In the double exposure method the images of the object in the positions which i t occupied during the two exposures are both recorded on the same photographic plate. Usually the two exposures are of equal duration and, hence, the resultant image intensity distribution i s the sum of the two distributions, as is shown in Fig. 3.10 . The two speckles overlap, and in the overlaping area the two grids, each belonging to one speckle, add. This addition is defined here as constructive when the high intensity regions of one speckle overlap the high intensity regions of the other speckle. This occurs when speckle grid shift = nGc n = 0, ± 1, ± 2, 44 In a similar manner, the addition i s defined as destructive when the high intensity regions of one speckle overlap the low intensity regions of the other speckle; this destructive addition occurs when speckle grid shift = nGc 1 3 n = + — + — Thus, "n" indicates the type of addition which takes place in a double expo-sure method. In general, n is continuous and the product nGg is equal to the shift between the two grids caused by the deformation of the specimen surface. In particular we may write speckle grid shift = y^an - yn-CT, = nG„ n continuous (3.18) In equation (3.18) Y-^ gj i s the speckle grid "center" position during the j-th exposure. Equation (3.18) i s solved for n in Appendix L with the result D n = -A.xc 1 -2 . - 1 ys XS0J u r y s x s + u 2xs vy s + wzs ,-^XS0 X S0 X so ys zs vso + V where X g Q is defined as x s o = 4 + y| + 4 + yl (3.19) Equation (3.19) may be closely approximated by n -D A.xc ys ^ ys zs — u + v - — — w x s x s (3.20) The accuracy of equation (3.20) should be sufficient for any laboratory testing; both equations (3.19) and (3.20) are applicable as long as the speckles recorded during the two exposures overlap and, to ensure this, the displacements must be sufficiently small. Equation (3.21), which restricts the size of the surface displacements, was derived in Appendix M as The addition of the speckles recorded during the two exposures is shown in Fig. 3.10 . The out-of-plane component u of the displacement vector causes the second exposure speckle envelope to "divide" into two circles 2^ and 2g . The Moire fringes are formed by the addition of the speckle grids created during the f i r s t and second exposures. This addition takes place in the area which is common to speckles (circles) 1,2 and 2 B . x i z i c - w 1- u x s x s z i c z i 5w M") i i mum 1C x i . yic y . - V + U •'IC Xe Xq - u x± D x s 2xs 2 second exposure speckle V f i r s t exposure speckle x i . A yic x i D y i c - v — + u + u — - — x s x s x s zx s Fig. 3.10 Elevation of the f i r s t (1) and the second (2 A,2 g) exposure speckles. 46 The result of this section may be summarized by the following: When DASC is used to record the displacement f i e l d of a specimen surface illuminated by a laser light, Moire fringes w i l l be produced according to equation (3.20) or, more accurately, according to equation (3.19). The surface displacements must be sufficiently small to satisfy equation (3.21). 3.5 Image Formation by DASSC The double aperture speckle shearing camera (DASSC) is similar to DASC except for the lateral shear of the images. This lateral shear may be achieved either by placing inclined glass plates in front of the apertures or by "defocusing". Both methods of producing the lateral shear were tried and the defocusing method was found to be more convenient for practical material testing, mainly because of the ease with which the size and the sign of the lateral shear may be adjusted to suit a specific test. The schematic of DASSC employing defocusing to produce the lateral shear is shown in Fig. 3.11 . With the exception of the position of the photographic plate, the geometry of DASSC is the same as that of DASC shown in Fig. 3.5 . To determine the equation governing the formation of Moire fringes by this camera we w i l l represent the surface of a specimen by two point sources. The diffraction process that actually occurs in DASSC is closely represented by the model shown in Fig. 3.12 . The schematic shows the speci-men plane y,z and the image plane y±,z^ which are familiar from the prior analysis of DASC. There are, however, two more planes in the system. One is called the "photo" plane Y^,Z^ and i t coincides with the emulsion of the photographic plate located the distance y^ from the image (focal) plane; the other plane Y,Z is called the object plane, which is an imaginary plane located at x = - y , where y is such that the object plane would be focused by the system lens on the photo plane. The system lens(es) is represented Fig. 3.11 The schematic of DASSC. Fig. 3.12 Diffraction in DASSC. 49 by a thin lens so that the axial distances are related according to equation (2.11) as 1 1 1 1 1 ,„ , ,x __ + _ = + (2.11) x s x ± f X s Xi where X g and Xi are defined as X g = x s + y X ± = x ± - Y i Let us consider an imaginary point source S(-y,Yg,Zg) in the object plane. The geometric image of S is in the photo plane at R ( Y i c , Z i c ) . If S were a real point source, then the intensity distribution around R at Q(Yi»Zi) would be found by considering an apparent source at D(-y,Y D,Z D), the geometric image of which would be at Q . This is what was done in the analysis of DASC previously. In the case of DASSC, the specimen surface is represented by two real point sources SjXC^y ^ j Z g ) and S. 2(0,yg 2 ,Zg ) . S 2 l i e s where the line from S to the center of aperture 1 (at y = y^ = D/2) intersects the specimen plane, and S 1 l i e s where the line from S to the center of aperture 2 (at y = - y^ = - D/2) intersects the specimen plane. The two real point sources Sx and S 2 are separated by the distance Ayc = y c ~ Yc • The choice of positions of S and S is based on the S o 2. o 1 1 2 experience gained in the analysis of DASC. It is anticipated (and con-firmed later) that the somewhat blurred image of S 2 made by light passing through aperture 1 is :centered•about R; as is.the image of Sx -made by.light passing through aperture 2 . It is reasonable to expect that the intensity distribution around R w i l l depend greatly on the magnitude of the apparent speckle diameter D g gand the separation Ayg of the two real sources and S 2; therefore, two cases w i l l be considered. The f i r s t case occurs when Ay g > D g g . From the analysis of DASC i t is known that the intensity distribution around the geometric image of a point source is of negligible magnitude at distances greater than Dg/2 from the 50 geometric image. The intensity distribution around R w i l l therefore be produced essentially by the interference of the light radiated by S 2 and passing through aperture 1 with the light radiated by Sj and passing through aperture 2 . The interference pattern that i s produced i s shown in Fig. 3.13 and is similar to the one produced by DASC shown in Fig.3.7 . unmodulated speckle modulated speckle unmodulated speckle 4 \ L 1 C x i A s Fig. 3.13 Intensity distribution I r for DASSC with Ay s > D s s Because of defocusing, the envelope of the pattern has changed; however, the speckle grid of pitch G s is s t i l l present. Light radiated by S 2 and passing through aperture 2 and light radiated by Si and passing through aperture 1 are both imaged as "blurred" unmodulated speckles similar to the speckle shown in Fig. 3.4 produced by a single aperture camera. These two speckles are not of interest in as much as they do not diffract light once the re-cording i s developed. In the calculation of the intensity distribution I r , computations similar to those done previously were made. The derivation of I r is,how-ever, more tedious as two point sources must be considered and some non-linear terms representing "blurring" of images may not be neglected. The calculation of the optical path length variation i s performed in Appendix N, and the intensity distributions are determined in Appendix 0 . The f o l l o -wing notation was used: sw^j . .. source wavefront'originating from and reaching aperture i dw^  ... wavefront appearing to originate from D ( - Y , Y Q , Z Q ) and caused by diffraction of a source wavefront in aperture i r e£j ... the optical path-length variation of rays radiated by source j and passing through aperture i The mean optical path length r was chosen arbitrarily as r Q = S 2-sw 1 2 + dWj-Q (3.22) r e i 2 and r g 2 j are derived in Appendix N to an accuracy of A/30 or better as r e i 2 = IE 2 + E ^ c o s e + K 2sin0) = r £ r e 2 1 = qr 2 + r(K l Cos9 + K 2sin6) - Dp where q,K1,K2, and p are defined in Appendix N and I • is calculated in Appendix 0 as X r = f " l 2 rji 2 rd/2 2TT . , fd/2 2TT | / e l k re.rdrd9 . / /e" i k l e r dr dO L 0 0 0 0 4 c o s 2 ^ (3.23) speckle envelope term speckle grid term 52 The integrals comprising the speckle envelope term cannot be evaluated directly. While some approximate solutions are certainly possible, just a one step integration by parts done in Appendix 0 yields results from which i t appears that the speckle envelope term is made up of terms having a much lower frequency than the speckle grid term. This is especially true for a small defocussing. In as much as the speckle grid i s our main interest, equation (3.23) describes the recorded I sufficiently well. Fig. 3.13 shows 1^ for this case; the drawing is only an approximation of I , and i t s sole purpose is to show graphically what I might be like and to aid in the discussion of the image formation by DASSC. Let us now consider the situation where Ay s < D s s • ^ n this case light radiated by Sj and passing through apertures 1 and 2 and light radia-ted by S 2 and passing through apertures 1 and 2 a l l interfere with one another and contribute to I r . Since Ay s < D s s , the parameter q is sufficiently small so that i t may be neglected in expressions for r and r £ 2 i . In Appendix N r e i l and r e 2 2 are derived and the four optical path length differences are given as r = r ei l K lsJ cos0 + K2sin£ - a A + a s r e i 2 = £ ( K i c o s 0 + K 2sin0) r e 2 i = rtKjCOsB + K 2sin9) - Dp D 1 r = r e 2 2 i VSJ cos0 + K 2sin0 Dp + a A + a c Also define p as kd / 2 2 p = — /K + K 2 l 2 53 The parameters and a g are defined in Appendix N, and the parameters f5 , Pj, and P 2 are defined in Appendix 0. I is derived in Appendix 0 as 1^ = T 2 d/2 2rr ei k remn r d r d 0 161, 2 0 0 ? p i ( p ) l 2 , i P J All 2rr I . / / e~ i k r e m n r dr d9 m = 1,2 o o n= 1,2 Pj + P 2cos 2 - (Dp - a A) (3.24) speckle grid term From equation (3.24) i t i s evident that the speckle grid with the same frequency as in the previous case i s again present. The speckle envelope and the actual intensity variation are rather complicated in shape; this fact i s not of great interest to us, however, since we are mainly interested in the diffractive properties of the recorded I r . Since the speckle grid term i s present, the intensity pattern I r may again be closely approximated by a diffraction grating of pitch G s over the speckle area. 3.6 Formation of Moire Fringes by DASSC In this section an equation i s derived governing the formation of Moire fringes by the DASSC in a double exposure process. This equation relates the Moire fringe number n to the in and out-of-plane displacements and strains of a surface represented by two points. In the analysis of image formation by DASC the surface was represented by a point source of light. Even though this point source was in fact i l l u -minated by a laser, the change in the distance between the point and the 54 laser had no effect on the recorded I r . This conclusion i s arrived at as follows: Define the terms: r- ... optical path length from the laser to the point S on the speci-men surface Ar ... increase in r T due to the displacement of S ... optical path length from S to point Q in the image plane through aperture 1 r 2 ... optical path length from S to point Q in the image plane through aperture 2 r ... increase in r, due to the displacement of S e i l c r^ ... increase i n r ? due to the displacement of S The difference in the lengths of the two optical paths (one' through each aperture) from the laser to the point Q in the image plane is given by Aj = (r^ + r ) - ( r L + r 2 ) ... difference during the f i r s t exposure difference during A 2 = ( r L + Ar L + r : + -r e i) - ( r L + Ar^ + r 2 + r e 2 ) ... the second exposure The relative change between the two optical paths due to the displacement of S during the time between the two exposures is given by A 1 2 = A 2 - Ax = (r1 + r e i ) - ( r 2 + r £ 2 ) - (^ - r 2 ) = r e i - r £ 2 Since i t i s A 1 2 that determines the relative shift between the speckle grids recorded during the two exposures, i t may be concluded that the distance from the point S on the specimen to the laser and the change in this dis-tance due to the displacement of S has no effect on the Moire Fringe formation of DASC. The situation is somewhat different in the case of DASSC. The specimen 55 surface i s represented by two point sources S{ and S 2 illuminated by one driving force - the laser. This time, the changes in distances between each of the point sources S and S 2 and the laser, arising from the surface dis-placements and strains, w i l l have to be considered in determining the fringe formation. Define the terms: r^ ... optical path length from the laser to Sl on the specimen surface r L 2 ' " ' ° P t l c a l P ath length from the laser to S 2 on the specimen surface Ar ... increase in r due to the displacement of Sn Li Li 1 Ar^ 2 ... increase in r ^ due to the displacement of S 2 r^j ... optical path length from Sj to Q in the image plane through aperture i r increase in r.. due to the displacement of S. e i j i j J It i s sufficient, for now, to look at the case where Ay g>D g s and, there-c -fore, we need to consider only two optical paths. One path is from the laser to S i and from S 2 through aperture 2 to Q in the image plane. The other path is from the laser to S 2 and from S 2 through aperture 1 to Q in the image plane. As before, let us define A and A 2 as'the differences between these two paths in the two exposures and A J 2 as the relative change between the two optical paths. A ! = ( r L l + r*i> " ( r L 2 + ri*> A 2 = < r L l + A r L i + r 2 i + r e2i> - ( r L 2 + A r L 2 + r 1 2 + r e 1 2 ) A 1 2 = A 2 - Ax = A r L i - A r L 2 + r £ 2 i - r ^ 56 A similar A 1 2 could be formulated for the case Ay g < D g g . It is not necessary to do so, however, as the only purpose of the above exercise was to show the need to consider the changes in distances between points on the specimen surface and the laser in the analysis of fringe formation by the DASSC. It should be noted that the actual distances between the laser and points S2 and S2 are of no importance and, hence, w i l l be arbitrarily defined as zero; therefore, equation (3.24) need not be altered. When a specimen i s deformed, i t s surface i s , in general, displaced and strained both in and out-of-plane. The deformation changes the coordinates of the two points Sj and S 2 , representing the surface, as follows S^O.yg^Zg) S*(u,y S i+v,z s + w) S 2(0,y S 2,z s) + s£(u + Su,yg +v + 6v,zg+w+Sw) Fig. 3.14 General deformation of the specimen surface. 57 where 6u , 6v , and Sw are defined as du 8v 8w 97 Ay s 6v = ^ Ay s 6w = ^ F ig . 3.14 shows the specimen surface with the two points Sx and S2 during the two exposures. The surface is illuminated by a collimated laser l ight with i t s orientation defined by the " i l lumination" vector . The i l lumi-nation vector 1 i s a unit vector perpendicular to the illuminating plane wavefront and oriented from the specimen surface toward the laser. It makes angles 9 X , 9 y and 9 Z with the x, y and z axes respectively; con-sequently 1 may be written as /\ S\ /N 1 = cos9 x i + cos9yj + cos9zk Due to the displacements and strains, the increments Ar^ and Ar-^2 are given by /\ /\ /\ / \ A r L i = (ui + vj + wk) •(-. !) = -ucos0 x - vcos8 y - wcos9z A r L 2 = [(u .+ 6u)i + (v + 6v)j + (w + 6w)kJ • ( - 1 ) - (u + <5u)cos9x - (v + 6v)cos9y - (w + 6w)cos( ArT - 6u c o s 9 „ - 6v cos9,7 - 6wcos9„ Li l A y We are now in a position to calculate ^e±j and I r ; f i r s t , the case Ay s > D g s i s considered. r e i 2 a n d Tezi ' a c c u r a t e to A/10 , are derived in Appendix P as 58 " e i 2 2XC + r X g X g + fZD - Z s Z s u + 6u I X c x s x s (1+3,) w + 6wl X sin8 s > + + Ar-^ - 6u cos0 x - 6v cos0y - <5wcos6z " e 2 i hi 2XC + r -J> i _ _3 — — - (1 + 3x) -I X g X g X g X COS0 + fzD - Zg I X c Zc u - — — - ( l + 3 i ) x s x s w X g J sin9 + A r L i - D(p - £*) where E,*' is defined in Appendix P . Note that now when we are considering DASSC the optical path length variations ^e±2 a r e functions of both the displacements and their increments related to the surface strains, whereas when DASC was considered r • .= were functions of displacements only. e J-j I is calculated in Appendix Q as 1^  = Rx + R2cos y D(p - £ * ) + A* + + R,sin k D(p - C*) + A * (3.25) I r consists of three terms containing Rj ,R2 and R3 defined in Appendix Q and A* is defined in Appendix P. The f i r s t term Rx may be thought of as a low frequency background noise; the second term R2 is likely to have the largest magnitude of the three terms and is modulated by the cosine term producing the speckle grid; f i n a l l y , the third term R3 is lik e l y to be of lower magnitude than the second term and i t is also modulated by a grid 59 producing term (but) of twice the frequency than the second term. Therefore , , i n the sense of I r a c t i n g as a d i f f r a c t i o n g r a t i n g of p i t c h G g , i t i s described by the term 2 k cos — D(p - E*) + A* ,kD = cos l * \ P - e + In the above speckle g r i d term, the presence of E* and A* i s due to the specimen deformation that took place between the two exposures. Through a comparison of the speckle g r i d terms i n the two exposures i t i s obvious that the second exposure speckle g r i d has been " s h i f t e d " by an amount E, D wi t h respect to the f i r s t exposure g r i d . From t h i s ' ' s h i f t " of the speckle g r i d the Moire number i s c a l c u l a t e d i n Appendix R as n(y,z) = -x c u + v - yz Ay £ (1 + c o s 0 x ) u , v -— — cos9,T Xo. y - cost w 'y (3.26) Equation (3.26) i s a c l o s e approximation of a more complicated "exact" equation derived i n Appendix R . The "exact" equation i s of l i t t l e i n t e r e s t to us, though, as i t contains a number of higher order terms which c o n t r i -bute n e g l i g i b l y to the equation; the accuracy of equation (3.26) should be s u f f i c i e n t f o r o r d i n a r y l a b o r a t o r y t e s t i n g . As i n the case of DASC, equation (3.26) i s v a l i d as long as the speckles recorded during the two exposures overlap. The equations c o n s t r a i n i n g the s i z e of specimen de-formation are derived i n Appendix S as y - D/2 Xc (u + 6u) + v + Sv + (Xs (u + Su) + w + 6w 'ss (3.27a) 60 y + D/2 u + v + f \ z —— u + w < D s s [ X s J . 2 . (3.27b) The addition of the speckles recorded in the double exposure process i s of the same nature as in DASC and therefore need not be examined again. Let us now consider the case Ay g < D g s . In this case the parameters q and 3X are so small that r £ 1 2 and r e 2 1 , accurate to A/10 and derived for the preceeding case (Ay s > D g s ) , may be approximated as r = r Y D - Y S Y g - y ^ u + 6 u v + Svl X c X c X c COS0 + s^ J Z D ~ Z S Xc Zg u + fiu w + 6w"| X s X s X s sin0 + Ar-^ - 6u cos0 x - 6v cos0y - <5wcos8z r e 2 1 -fYp ~ Y S _ YS + YA _ Z . X o X o X o X o COS0 + Z D - Z S Z S u Xc x s x s w Xe sin0 + + A r L i - D(p - T ) In addition to r e i 2 and r e 2 1 , r e and r £ 2 2 were derived in Appendix P and the results (accurate to A/10) are presented here. r = r ei i fYD ~ Y S Y S ~ YA u X c v X S Xg Xg COS0 + fz D - z s X c Zg _u_ x s x s w X s i n e s> + + A r L i - a s + 3 n r e 2 2 = £ rYD - Y s Y s + y A u + 6u v + Svl X c X a Xc COS0 + ••s J Z D ~ Z S Xc Zg u + 6u w + 6w' X s X s Xo J s m t + A r L i + a* + 3 2 2 - D( P - E*) 61 In the r e i j above, the parameters 3 1 X and 3 2 2 are defined in Appendix P I r is derived in Appendix Q as 16Ir Qt + Q2cos2-2- D(p - £*) - a* + 3n- 3 2 21 (3.28) I r again contains the low frequency background noise term called here Qx and another low frequency term Q2 which is modulated so that the speckle grid i s produced again. Q1 and Q2 are defined in Appendix Q . The Moire number is obtained in Appendix R by relating the shift between the two exposure .speckle grid terms as n(y,z) = - yz u + v Y W + Ayc 2x„ u, + T V , y yz —2 w > i (3.29) Equations similar to equation (3.27) could be derived again. However, as four speckles would have to be considered, four equations of constraint would have to be derived. With the case Ay s < D s s of l i t t l e practical interest and because the system is so much more sensitive to displacements than to strains, i t is sufficient to use equation (3.17), derived for DASC, as a guide. The results of this section may be summarized by the following: When DASSC is used to record the displacement and strain fields of a laser illuminated specimen surface the Moire fringes w i l l be produced according to equation (3.26) which is applicable as long as the lateral shear Ay„ > D„„ . If Ay„ < D o a , the Moire fringes w i l l be produced according to equation (3.29). The surface displacements and strains must be sufficiently small to satisfy equation (3.27) for the case Ays > D s s , and equation (3.17) is used as a guide for the Ay s < D g s case. 62 3.7 Imaging of Real Surfaces by DASC and DASSC Equations (2.18) and (2.19) were derived for a coherent interferometer with one point source. When a real, diffusely reflecting surface i s considered, these, and other equations must be modified to take into account a l l the light reaching a point (Q) in the image plane. This i s done by integrating over the illuminated area of the specimen, and consequently equation (2.14) becomes E r(t) =/ / 5 e ( y a ' Z a ) cos [ k ( y a , Z a ) - r ( y , Z , y a , z a ) -cot+^(y, Z)]dA sdA a (3.30) A s A a r ( y ' Z ' y a ' Z a } where A g is the part of the area of the specimen surface illuminated by a laser light, A a is the aperture(s) area, y a , z a are the aperture coordinates and y,z are the specimen coordinates. Note that f and ty are functions of the specimen surface coordinates. As in Section 2.13 we consider such an interferometer where a l l the rays reaching point Q are nearly parallel, and r Q denotes some typical object-image distance. Equation (3.30) i s then written as — cl E r(t) = / / — cos[kr(y,z,y a,z a) - cot + ty(y,z)] dAs dA£ (3.31) •^s A a Assuming that a l l the points on the specimen surface reflect (radiate) light with nearly the same amplitude a - a , the recorded intensity is approximately given by I ^ / /e i k redA sdA £ A-s A a / Je~ ± k r e dAs dAa A s A a (3.32) where r p(y,z,y a,z a) = r(y,z,y a,z a) - r a' "a' 63 We know, from the analysis of the image formation by DASC, that the recorded intensity distribution (due to a point source) is of a significant amplitude only within the speckle perimeter Dg . It is therefore sufficient to confine the integration over the specimen surface to an area of approxi-mately diameter D g s about the particular point under consideration. r„(y,z,y o,z) and iKy,z) may then be related to the (microscopic) surface c a. a. geometry, and specimen illumination, and I r is calculated from equation (3.32). Unfortunately, in most instances the surface geometry is not known, and since any real surfaces are extremely rough on the scale of an optical wavelength, no smooth-surface approximations are possible and i t is necessary to rely on statistics to derive an amplitude, intensity and phase of a speckle. The s t a t i s t i c a l properties of laser speckle patterns were studied by J. W. Goodman [43] and others, with results agreeing with the experimental observation that the "speckle pattern moves as the source is moved" [43] . Alternatively, i t could be said that when the surface is moved, the speckle pattern moves with i t , providing the illumination remains unchanged. Depending on the type of the motion of the speckle, the phase of the speckle observed at a point w i l l change accordingly. The phase variation of the whole speckle then may be approximated by a phase variation of a single "typical" point - a point source(s) used in this chapter. We may therefore assume that equations (3.20) , (3.26) and (3.29) derived for a point source, should also describe the fringe formation by the two cameras when they are used to image real diffusely reflecting surfaces. The results of a number of experiments described in Chapter 6. were found to support the validity of this assumption. The approximation of the speckle by a point source f a i l s , however, to explain the sensitivity of DASC or DASSC to the out-of-plane t i l t s of the specimen surface. It was noted and discussed by A. E. Ennos [43] who determined the maximum allowable surface t i l t angle as M 6<r < (1 + M)F where F is the aperture ratio of the lens, (f/d), and M is the system magnification. Using the notation of this thesis the maximum permissible surface t i l t angle i s given by d 6y < — (3.33) 65 4. CALCULATION OF DISPLACEMENTS AND STRAINS. 4.1 Preliminary Remarks The objective of Chapter 3 was to derive an expression n(y,z) describing the fringe pattern due to a general deformation of a specimen surface. The objective of this chapter is just the opposite, namely, the derivation of various methods of calculating the unknown surface deformation from a given pattern. When DASC or DASSC is used to record the deformation of the specimen surface, the end product is a photographic recording or recordings, showing a fringe pattern superimposed on the specimen surface. This fringe pattern is in fact "our" function n(y,z). When testing materials, a reseacher usually has some knowledge about the size of displacements and strains that are li k e l y to occur during the testing. This knowledge may come from the theoretical considerations of the test, from direct measurements, or from similar testing done previously by the reseacher. Knowing that the two cameras form fringes according to equations (3.20) and (3.26) and having some knowledge about the specimen deformation, the reseacher is able to assign numerical values to the fringes that make up the fringe pattern. Often this i s a t r i a l and error procedure and, usually, only the fringe centers are numbered and located; i f a continuous n(y,z) is desirable, a suitable curve f i t t i n g technique may be used. Alternatively, n(y,z) may be determined with reasonable accuracy at any point on the surface with the 2 knowledge that the fringe density variation is close to the cos type of variation. In any case, n(y,z) must be uniquely "numbered" by the reseacher before any calculations of the unknown surface displacements and strains are attempted. It i s desirable to be able to determine the specimen deformation using the smallest possible number of photographs. It w i l l be shown that usually i g . 4.1 Normal view of the aperture plane. Fig. 4.2 Normal view of the specimen showing coordinate systems y,z and y±,z± . 67 no more than three photographs obtained by DASC or DASSC are needed, and in certain cases of specimen deformation only one or two photographs are sufficient. The calculation of the surface displacements from the photographs obtained by DASC w i l l be discussed f i r s t and then calculations involving the use of DASSC w i l l be considered. 4.2 Use of DASC to Measure General Deformation The fringes are formed by DASC according to equation (3.20) which relates the in and out-of-plane surface displacements u(y,z),y(y,z) and w(y,z) to the fringe pattern function n(y,z). n(y,z) = Axs ;7 U + V - W (3.20) Treating n(y,z) as a known function, we wish to determine the unknown displacements u,v ,and w. Since u,v and w are independent, three independent equations like equation (3.20) are required to provide a unique solution. By varying the parameters D,X and x g we can obtain different patterns n(y,z); however, these would a l l be linearly dependent. One possible way of producing three independent patterns n(y,z) is to "photograph" the specimen deformation with a camera having three aperture sets with different rotations in the y,z plane-as is shown in Fig. 4.1 . The surface displacements and the three fringe patterns obtained by recording the specimen deformation through the three sets of apertures are related by equation (4.1). y * y * z • ^ u i(y i,z i) + v 1 ( y 1 , z 1 ) - w i(y i,z 1) = - -^f~ n i ( y i > z i ) ( 4- 1) i = 1,2,3 68 In equation (4.1) Uj ,vx , and wx are the three components of the displacement vector along the x,yj , and zl axes, and similarly u 2 ,v2 , and w2 are the displacement components along the x,y 2 , and z 2 axes and so on. is the aperture separation D for the case where (j) = (j)^  . From Fig. 4.2 i t i s apparent that the coordinate systems y,z and y-^  , z^ are related as y^ = y costj)^ + z sincf)^ (4.2a) z^ = - y sincj)^ + z coscj)^ (4.2b) At the same point on the specimen surface the displacement components u^ ,v^ , and w^  are related to the components u,v and w by u i(y i,z i) = u(y,z) (4.3a) v i ( y 1 , z 1 ) = v(y,z)coscj)i + w(y, z)sin<j>± (4.3b) w i ( v i > z i ) = ~ v(y,z)sin<j)i + w(y, z)cos§± (4.3c) where the (y^z^) are related to (y,z) by equations (4.2a) and (4.2b). Substitution of equations (4.2) and (4.3) in equation (4.1) yields a ±u(y,z) + b ±v(y,z) + ciw(y,z) = N ±(y,z) i = 1,2,3 (4.4) where a^ ,b^ ,c^ , and N-^  are defined as Yi a i = — (4.5a) A s y i z i b ± = c o s ^ +—72- s i n ^ (4.5b) s y i z i c i = s l n <J>i - -T72— cosa)± (4.5c) x s Xx s N i ( y ' z ) = - ~tfT n i ( y i > z i ) (4.5d) i where, again, the (.y±>z±) are related to (y,z) by equations (4.2). Provided the determinant of the coefficients of the set of equations (4.4) is not equal to zero, the set may be solved for the unknown displacements u,v, and w for any value of (y,z). Thus, a minimum of three photographs must be taken and processed to obtain three fringe pattern functions n^(y,z) to determine the general three dimensional surface deformation. When the deformation of the specimen i s of a special nature, such as plain strain deformation where the out-of-plane displacement vanishes, i t may be possible to determine some displacement components from only one or two photographs. The examination of some of these special cases follows. 4.3 Use of DASC to Measure Plane Strain and Plane Stress Deformation Under these circumstances the out-of-plane displacement u of the specimen surface is zero, or so small compared to the in-plane displacements that i t does not contribute significantly to the formation of fringes, and the term a^u in equation (4.4) may be neglected. Then, only two photographs of the deformation are required through two sets of apertures rotated by $l and (j> ; v(y,z) and w(y,z) may be solved from equation (4.6) b iv(y,z) + c i W(y,z) = N ±(y,z) i = 1,2 (4.6) If the magnitudes of the in-plane displacements v^ and w^  are of the same 1 , 1 W-j over most of the specimen surface being y-j ZA photographed, the term j , 2 A w-^  may be neglected, and v. can be determined x s 1 from a single photograph taken by DASC using one set of apertures rotated by <J>^  . With terms involving u^ and w^  neglected, equation (4.1) becomes vi ( y i . z i ) = - n i ( y i > z i ) (4.7) It is obvious that v^ is given by equation (4.7) with the absolute error y± y±z± equal to the sum of the — un- and ~— w,- terms. If this absolute error is A S A S Fig. 4.4 Rotated coordinate system. 71 acceptable equation (4.7) should be used, with the understanding that i t may result in large relative errors at those points of the specimen surface where v^(y£,z^) i s small or zero. While equation (4.7) relates v^y^>z^) t o n^(y^z^) nothing i s determined about the other two displacement components u^(y^,z^) and w^(y^,z^). For the important case of ty^ = 0° equation (4.7) becomes v(y,z) - - - — n(y,z) (4.8a) If w(y,z) i s to be determined, rather than v(y,z), an aperture set rotated by ty = 90° is used and w(y,z) is determined from w(y,z) - - n(y,z) (4.8b) This time the displacement components u(y,z) and v(y,z) remain undetermined. Similar considerations pertaining to accuracy that were made about equation (4.7) apply to equations (4.8a) and (4.8b). 4.4 Use of DASC to Measure Out-of-Plane Bending The out-of-plane bending of the specimen surface is usually accompanied by small in-plane displacements. If these displacements contribute negligibly y±z± — 2 - w i(y i,z i) x - u i(y i,z i) » |v i(y i,z i) to the formation of fringe patterns, that is i f over most of the specimen surface being photographed, the two terms involving v^ and w^  may be neglected in equation (4.1). Under these circumstances one can determine the out-of-plane displacement f i e l d quite accurately from one photograph of the specimen surface, by using DASC having a set of apertures rotated by ty = ty^_ . With the terms involving v^ and w^  72 neglected, equation (4.1) reduces to y i _ N „ * xs u i(y i,z i) = - — r n i(y i,z i) (4.9) y • The term — u±(y±,Zj) is given by equation (4.9) with the absolute error x s Y i z - i equal to the sum of the two neglected terms v^(y^,z^) and 2 w^y^z^) , y * This may again cause large relative errors in the calculated — u-^(y^,z^) x s term in those regions of the specimen surface where this term i s small or equal to zero. From equation (4.9) u^y^, z^) i s calculated as Ax2, n,- (y-:, z-j ) u i(y i,z i) - - lKyx' ± J y + 0 (4.10a) D i Yi Equation (4.10a) relates the out-of-plane displacement f i e l d u^(y^,z^) to the fringe pattern function n ^ y ^ j Z ^ ) . The in-plane displacements v^ and w^  remain undetermined. Due to the peculiar form of equation (4.10a), a small error in n^(y^,z^) may cause a large error in the calculated u^ when y^ is small. For the case of <j> = 0° equation (4.10a) becomes Ax 2 n(y,z ) u(y.z) = - — - — y^o (4.10b) Equation (4.10b) relates the out-of-plane displacement f i e l d u(y,z) to the fringe pattern function n(y,z). The in plane displacements v and w remain undetermined. Similar considerations pertaining to accuracy that were made about equation (4.10a) apply to equation (4.10b). There are certainly other "special" cases where one or two photographs taken by DASC may be used to calculate some components of the specimen deformation. However, the general case and the two special cases ought to demonstrate sufficiently the use of 73 DASC as an experimental testing device. To test the validity of equation (3.20) and get the "feeling" for DASC several experiments were performed with a known specimen deformation. These experiments and the numerical results obtained from the photographs taken by DASC are presented in Chapter 6. 4.5 Use of DASSC to Measure Specimen Deformation - Theoretical Considerations The fringes are formed by DASSC according to equations (3.26) or (3.29), depending on the size of the lateral shear used. We shall restrict our attention to the more practical case of Ay g > D s g , for which equation (3.26) is applicable. n(y,z) = - D y yz U + V 2~ w A X S [xs X S — — - COS0 V xc y Ay£ z (1 + COS9 X)u,y -COS0, w 'y (3.26) Equation (3.26) relates the surface displacements and strains to the fringe pattern function n(y,z) and for Ay s = 0 , i t reduces to equation (3.20) . Equation (3.26) may be written in the form similar to equation (4 .1 ) as y yz Ay sX s — u + v 5- w -I x s x| D (1 + cos0 x)u,y -Ay sX £ y - YA x s - cost v, Ay sx s - cost X X c w *y n(y,z) (4 .11 ) A set of apertures rotated by cj)^  with respect to the y,z coordinate system produces a fringe pattern according to x s y i z i u i + v i " — X" w± + AysjXsi n , n n e f l , „ Ay s lX s i 7 * (1 + c o s 0 x i ) u i > y i -COS0yi A y s ix s i ' u y i - cost Z l WH-I J y i ^ X s i , . — — ni(yi,zi) D? y i - yAi x s This may be written in a more compact form as A i u i + B i v i + c i w i + D i u i > y i + Fi vi»yi + H i w i > y i = N i ( y i > z i ) (4.12) where the coefficients H-^  are defined as Ai = Yi B • = 1 C-; = - y-iz 1 1 D i = M 1 + cos0 x i) y i - y A i l F i = S i Hi = S ± COS0 y i cosfc)zi - Xc N-; = -and S^ is defined as Si = A y S ixS i D? In Appendix T i t is shown that, when a l l quantities in equation (4.12) are transformed in the y,z system, equation (4.12) becomes a ±u + b ±v + c-jw + d ±u, y + e.jU, z + fj^v, + + 8 i V , z + h±w,y + k±w,z = N ±(y,z) (4.13) In equation (4.13) the coefficients a i,b i,c i are given by equation (4.5) and the coefficients d^,....,^ are given by d-^  = S^(l + cos0 xi) cos(j)£ = S^(l + cos0 xi) sincfi^ 75 = S^ [ ( r - c o s 0 z i ) sintjj^ - (s - c o s 0 y i ) cosdpj coscj^ S i = S i [ ( r " c o s 0 z ± ) s i n d ) : L - (s - c o s 0 y ± ) coscjjj sln<$>± hj_ = - S-^[(r - cos0 z^) coscj)-}. + (s - cosGy-^) sin<J>jJ cosc^ kj_ = - S±\_(* - cos0zi)coscj)i + (s - cos6 y^) sind)-jj sind)^  N ± = - — 5 5 - niCyi.Zi) Di z i w i t h r and s defined as r = — x s x s Due to a l a c k of equipment or for other reasons a s i t u a t i o n may a r i s e where the specimen i l l u m i n a t i o n i s i n the x,y plane, i . e . 9 Z=90° and c o s 0 z = 0 . For t h i s s p e c i a l , but important, case of specimen i l l u m i n a t i o n the c o e f f i c i e n t s f ^ g ^ j h - j ^ k ^ reduce to the f o l l o w i n g simpler forms as i s shown i n Appendix U. f x = S-£(rsin<J>-£ - scoscj)^ - sin0 x^)costj)^ g^ = S^ Crsincj)-^  - scoscf)^  - sin0x£)sincj^ h± = - S£(rcos(j>i + ssincj)i)cosc})^ k^ = - S-^ (rcos(J)£ + s.sind)^ )sincj)^  Since Yi Z i < -jr , . i t i s obvious that the c o e f f i c i e n t s h^ and k^ w i l l 5 probably be very small ( e s p e c i a l l y near o r i g i n ) when compared to the other c o e f f i c i e n t s , consequently, the DASSC using the in-plane i l l u m i n a t i o n w i l l be r e l a t i v e l y i n s e n s i t i v e to the s t r a i n s w,y and w,z . There are three displacements and s i x s t r a i n components i n equation (4.13) and, i f the specimen deformation i s general i n nature, a l l nine terms contribute significantly. Equation (4.13) may be solved in two possible ways. One way is to regard strains and displacements as mutually independent and, consequently, nine equations (photographs) are needed to solve for the unknowns. The other way is to take advantage of the fact that equation (4.13) is really a set of partial differential equations and, i f the required boundary conditions are readily available, a set of three equations (photographs) is sufficient to solve for a l l nine unknowns. The most advantageous approach to solving equation (4.13) depends greatly on a particular test situation but, in general, the f i r s t "algebraic" method of solving equation (4.13) should be undertaken only for special cases, and when there is no point or line of known displacements (boundary condition) on the specimen surface. When there is a point or a line of known displacements on the specimen surface the second method should be used. 4.6 Use of DASSC to Measure General Deformation (Algebraic Solution) Nine independent equations (photographs) could be obtained by using a number of aperture sets of different inclination to the y-axis and by using different illuminating beams associated with each aperture set, etc. In principle, the resultant set of nine independent equations (4.13) could be solved for the nine displacements and strains; however, because of the unavoidable errors in the determination of the camera parameters and in the location of the fringes, this method of solution of equation (4.13) is so inaccurate as to be of l i t t l e practical use. The method is reasonably accurate only when the specimen deformation is of a special kind, or only i f the strains along a particular direction are to be calculated, since under such circumstances fewer photographs need to be "taken" and processed. Using three aperture sets inclined by the same angle cj) to the y-axis, three illuminating beams, each assoc iated with one aperture set, and two photographic plates in series, one may calculate the three strain components from four photographs. Three of these photographs are recorded on one photo-graphic plate and one photograph i s recorded on the other plate. Fringe patterns in these photographs satisfy equation (4.12) as R(y i,z 1) + D - jU j.yj + F - j V j . y j + H i W l, y i = N i(y 1,z 1) i = 1,2,3,4 (4.14a) Equation (4.14a) may be solved for the three strains and the "displacement sum" R(y ,z 1) defined as R(y1,z1) = A j U l + B J V J + C L W J When the illumination is in the x,y plane and the aperture sets are in the d) = 0° position, the camera i s insensitive to w and w,y for small values of z; i f the term H^ w^ ,^  i s negligible, equation (4.14a) reduces to R(y,z) + D ±u, y + F ±v, y = N ±(y,z) i = 1,2,3 (4.14b) Equation (4.14b) may be solved for the two strains u, y , v,y and the "displacement sum" R(y,z) from three independent photographs, with the absolute error being most lik e l y of the same order as the terms that were neglected. 4.7 Use of DASSC to Measure Plane Strain and Plane Stress Deformation  (Algebraic Solution) In the case of plane stress or strain where |ux| << |vx|,|wx|, and V j and Wj are such that |v x| » y iz i w. over most of the specimen surface x s being photographed, the displacement sum R(y 1,z 1) to v r Therefore,'in addition to the three strains, a reasonably accurate v J ( y 1 , z 1 ) is also calculated by solving the set of four equations (4.14). If the magnitudes of the in-plane strains are of similar order such that |v,y| > |w,y|, and i f the illuminating beams are in the x,y plane and the aperture set rotations (J) are equal to zero, the term H^w,y i s so small that i t may be neglected and equation (4.14b) i s then c l o s e l y approximated by v + D i U, y + F ± v , y = N ±(y,z) 1=1,2,3 (4.15) In t h i s case, three independent photographs are s u f f i c i e n t f o r the so l u t i o n of v,u, y , and v , y . At those points of the specimen surface where v,v, y or u,y are small or zero, these v a r i a b l e s w i l l be calculated from equation (4.15) with a large r e l a t i v e error, but are l i k e l y to be determined every-where with an acceptable absolute error. 4.8 Use of DASSC to Measure Out-of-Plane Bending (Algebraic Solution) In t h i s case of specimen deformation the out-of-plane displacements and st r a i n s are usually much larger than the in-plane displacements and s t r a i n s , so that I v . I . I v . z . I (except f o r y small) K'yil » K'yil ' K'yJ and equation (4.12) may be c l o s e l y approximated by A i u i + D i u i > y i = N i C y ^ Z j ) 1 = 1» 2 » 1 v I , y i i — 3 - U - w x 1 x s i i i » x s (4.16) :om By using two independent photographs, one may calculate• u 1 and U j j y j frc equation (4.16) with the absolute error being probably of the same magnitude as the terms that were neglected. Due to the pecu l i a r form of the c o e f f i c i e n t A^ , a large error i n u t should be expected where yx i s small. 4.9 Use of DASSC to Measure General Deformation The main purpose of deriving equation (4.13) was to show that a photo-graph taken through a set of rotated apertures displays a fri n g e pattern which is due to a combination of the three displacements u,v, .and w and the partial derivatives of these displacements with respect to both y and z. By using three aperture sets, each with different rotation cf) , and by using three illuminating beams (at least one of them not in the x,y plane), three independent photographs may be obtained. If there is a line of known displacements u,v, and w (boundary condition) on the specimen surface i t i s possible, in principle, to solve the set of equations (4.13). For this the fi n i t e difference equations w i l l be used. The forward f i n i t e difference analogs for f i r s t order derivatives are u, y(y,z) = u(y + A»z) ~ u(y,z) A u, z(y,z) = u(y,z + A) - u(y,z) where A-is the grid spacing in both y and z directions. By replacing the partial derivatives with these f i n i t e difference analogs equation (4.13) becomes A ^ i A J u(y,z) + u(y + A,z) + u(y,z + A) + A A = N ±(y,z) i = 1,2,3 (4.17) If the displacements u,v, and w are known, for example, along the line z = z 0 then the displacements may be found row by row from e ±u(y,z + A) + g ±v(y,z + A) + k±w(y,z + A) = N i " ( a i A _ d i " e ±)u(y,z) - i = 1,2,3 (4.18) : i §i k ] g, k 2 e 3 § 3 k-3 * o Similar equations could be written for the case of the boundary condition being a line y = y 0 or an inclined straight line. A set of three photographs obtained by a computer simulation of the DASSC was solved by using equation (4.18) and i t was found that solution tended to diverge rather quickly with increasing distance from the boundary condition line. In contrast to this, a f i n i t e difference scheme based on equation (4.12) was found to be much more accurate and only slowly diverging. As equation (4.12) involves partial derivatives with respect to y only, the integration starts at a point on the boundary condition line and progresses away along a line z^ = constant as i s shown in Fig. 4.4 . The partial derivatives in equation (4.12) are replaced by their backward difference analogs and, with these, equation (4.12) may be written as (A±A + D i ) u i ( y 1 , z 1 ) + (B ±A + F i ) v 1 ( y 1 , z 1 ) + (C-jA + H i)w 1(y 1,z 1) = R± i = 1,2,3 (4.19) where R^  i s defined as R i = N i ( y 1 , z 1 ) A + D-jU^yj - A, Z l) + Fiy1 (y x - A,z x) + H ± W l (y : - A, Z l) (AjA + Dj) (BjA + F,) (C,A + Hj) (A2A + D2) (B 2A + F 2) (C 2A + H2) (A 3A + D3) (B 3A + F 3) (C 3A + H3) * 0 (4.20) To satisfy equation (4.20) three illuminating beams are needed and the DASSC shutter must have three sets of apertures inclined at the angle cf> with respect to the y axis. The advantage of this approach is that the boundary condition line C need not be straight and, since the f i n i t e differences are along a line, i t is possible to vary line spacing as well. A further advantage of this approach i s the smaller size of the equations and the fact that a l l three coefficients D^ , F^, and may be adjusted to be of the same magnitudes by a suitable choice of the aperture 81 rotation angle <$>1 . The programming of this scheme is also simple. If some of the displacement fields u,v, and w are known or determined by other means, the terms containing the known displacement and i t s derivative are put on the right side of the equations and the reduced set is then solved; for example, i f u i s known, equation (4.19) is reduced to (B.jA + F i ) v 1 ( y 1 , z 1 ) + (C-jA + H i)w 1(y 1,z 1) = R± 1=1,2 (4.21) where R^  is now defined as Ri = N i(y 1,z 1)A - A-iAujtyj .Z j ) - D iAu 1 , y i (y 1 , zj + Fiy1(y1,z1) + Riy1(y1,z1) (B:A + F J (CjA + Hx) (B2A + F 2) (C2A + H2) / 0 Once u ,v and w1 are found over the specimen area these displacements may be transformed into the y,z coordinate system to u,v and w , and through numerical differentiation (using central f i n i t e differences for example) the strains u,y ,u, z w,z are obtained. There may be other methods or choices of DASSC parameters which would permit solution of either equation (4.12) or equation (4.13) for a general case of deformation but, before any of these methods are used, their sensitivity to the inaccuracies in measurement of DASSC parameters and fringe locations should be established. In practice, the fringe locations and the size of DASSC parameters may be determined with only a limited accuracy, a fact which may make some numerical schemes rather useless. 4.10 Use of DASSC to Measure u,u,y ,v, and v,y from Two Photographs When a specimen illumination is in the x,y plane and the aperture sets are at zero inclination to the y-axis ((J) = 0°), the terms in equation (4.12) involving w and w,y are often so small that they may be neglected (especially 82 for z small) and i t is then possible to solve for u and v from two independent photographs as follows: ^- u + v - 1^ w + d j U , y + f-jv, y + hjV y = Nj j = 1,2 (4.22) \ - 0 • \ - 0 Replace equations (4.22) by a set of f i n i t e difference equations, using the forward f i n i t e difference analogs of the f i r s t derivatives which may be written, using f i n i t e difference notation as _ u i + l ~ u i _ v i + l ~ v i u ' y i A V ' y ± A With these analogs equation (4.22) becomes y i I . , u i + l ~ u i , f v i + l ~ v i „ , _ i n ( , 0 ^ — u ± + v ± + d-j + f j = Nj 3 = 1,2 (4.23) Equation (4.23) is solved for the unknowns U i + x and v^ + x " i + 1 = U 1 ± + U 2 ± U i + U 3 . V i (4.24) v i + i = v i i + V 2 i U i + V 3 i v i where the coefficients U ji V 3 i are given by U 1 ± = - r ( N 1 f 2 - N 2f x) V 1 ± = r(N xd 2 - N 2d x) U 2 i = - r ( f x - f 2) Z i + 1 v 2 ± = r ( d : - d 2) | i s s U 3 i = - - f 2 ) V 3 i = r ( d X - d 2 ) + 1 with r defined as r = A / ( f x d 2 - f 2 d x ) In equation (4.24) the coefficients V 3 i are evaluated at y = y i . If, at some point y = yj , both displacements u and v are known, then u and v may be found elsewhere by starting at this point and evaluating the displacements at the point y = yj + 1 by using equation (4.24). This pro-cedure, is repeated until'.the displacements at a l l points to the right of are found. To find the displacements to the l e f t of yj one must solve equation (4.23) for u^ and v^ u i = H i i + H 2 i u i + i + U 3 i v i + i v i = I i i + li±u± + l + I s i v i + l where the coefficients IJ ^  are given by (4.25) U 1 ± = s|N2(A - fj) - N,(A - f 2 ) U 2 i = s d, - d, + l 3 i = s ( f x - f 2 ) - i i = s - N, A x0' - d, v 2 i " S ? " ( d l " d2> X S n V 3 i = s with s defined as s = 1/ + d i " d 2 " ( f i - f 2 ) In equation (4.25) the coefficients V_3± a r e evaluated at y = y^ By setting i + 1 = j the displacements to the l e f t of y.. are found by successive applications of equation (4.25). In the actual experimental work a situation may arise when the two displacements u and v are known at two different locations; for example, u is known at y = yj and v is known at y = . This case is handled as follows: At point i + 2 equation (4.24) becomes U i + 2 = U l i + 1 + U 2 i + l U i + 1 + U 3 i + l V i + 1 = ( U n + i + U 2 i + i U 1 ± + u 3 i + 1 v l i ) + ( u 2 i + l U z i + u 3 i + l V 2 i ) u + ( U 2 i + l U 3 i + U 3 i + l V 3 i > V 84 and a similar expression could be written for + 2 . This process i s repeated n times until i + n = k whereupon we get u i + n = uk = U i k - i + U 2 k - i u i + U 3 k - i v i (4.26) v i + n = vk = V i k _ ! + V 2 k _ l U ± + V 3 k _ l V i The coefficients U l k _ x . V 3 k _ j are obtained by repeated use of the recursion relationship between coefficients + x and U ^ > V \ ± t ^ z ± + l » V 2£ + j and so on. From equation (4.26) i t i s possible to solve for any of u^ ,v^ ,u k , or v^ i f any two of these are known and, once two displacements at the same point are determined, displacements u and v may be calculated elsewhere by using equations (4.24) and (4.25). The slopes and strains may then be calculated numerically by using the central f i n i t e differences = u i + i ~ u i - i = v i + i ~ v i - i U ' y i 2A 'y 1 2A If the out-of-plane displacement u i s small, i t is possible to solve equation (4.22) for v and v, y as follows: y — u + v + d l U , + f v, = N 7 7 (4.22) y — u + v + d 2u, y + f 2 v , y = N 2 By multiplying the f i r s t equation by d 2 and the second equation by (-dx) and adding the two equations "we get (d 2 - d l ) u + (d 2 - d l ) v + ( f x d 2 - f 2 d l ) v , = N j d 2 - V 2 s By choosing the parameters of DASSC such that the term |d2 - d J| is as small as possible and the term | f 1 d 2 - f 2& l \ i s as large as possible, we may closely approximate the above equation by (d 2 - d x)v + ( f j d 2 - f 2 d j ) v , = N xd 2 - N 2d t which may be written as with p(y) and q(y) defined as v, y + p(y)v = q(y) p(y) = q(y) = d 2 - d t f x d 2 - £ z d l N i d 2 - N2d,  f l d 2 - f 2 d , (4.27) f i d 2 " f 2 d i * 0 Equation (4.27) is an ordinary, f i r s t order differential equation with a variable coefficient; i t s solution [63] i s v(y) = 1 y(y) /y(t) q(t) dt + y(y 0)v(y 0) v y 0 (4.28) where y(y) is the integrating factor y(y) = exp /p(t) dt In equation (4.28) v(y Q) i s a known displacement v at one point on the specimen surface on the line (z = z Q = constant) upon which v(y) is sought. Once v(y) isfound, the strain v, y(y) may be obtained by the numerical differentiation of v(y) or calculated from equation (4.27). After v and v,y are determined, the two remaining quantities u and u,y are found from equations (4.22). 4.11 Use of DASSC to Measure Out-of-Plane Bending In the cases where the out-of-plane displacements and strains are much greater than the in-plane displacements and strains, equation (4.16) is again applicable. Equation (4.16) may be written in the standard form given by equation (4.29) and, since equation (4.29) is of the same type as equation (4.27), i t i s also solved in the same way, with the solution given by equation (4.30). 86 u i »yi + Pi ( y i ^ u i = ^ 1 <yi> (4.29) where P ^ y ^ and q^Yj) are defined as P I ( Y I ) = T>1 + 0 u^y,) = U i ( y x ) y i / y 1 ( t ) q 1 ( t ) dt + y x ( y 1 0 ) u 1 (y 1 0) v y i o (4.30) where the integrating factor P^Yj) is given by ^ 1 ^ 1 ) = e xP / Pi( f c) dt Yi In equation (4.30) u 1 ( y 1 0 ) i s the known out-of-plane displacement u x at y 1 0 on a line (z x = z = constant) in the rotated coordinate system y 1 , z 1 where u 1(y 1) is being sought. If the out-of-plane displacement at one point of the beam surface is not available then two photographs are necessary for an approximate solution of U j and u 1 > y l . If we use two sets of apertures at <j> = 0°, the two fringe patterns are given by equation (4.13) as y y z — u + v - — w + dj_u,y + f;jV, v + h £ W , v = N-£ i = 1,2 (4.31) If the second of these equations i s subtracted from the f i r s t we get (d x - d 2)u, y + (f j - f 2 ) v , y + (h x - h 2)w, y = Nx - N 2 By an appropriate choice of the DASSC parameters the term | (d1 - d 2)u, y| may be made much larger than the other two terms, and an approximate value of u,^ is then given by equation (4.33): i f |(dj - d 2)u, y| » \(f1 - f 2 ) v , y | , | ( h 1 - h2)w,; (4.32) 87 u »y ~ d 1 - d 2 ± 0 (4.33) Once u, y is calculated by using equation (4.33) i t i s substituted back in equation (4.31) and an approximate solution for u (except near the origin) In some tests, the out-of-plane bending of the specimen may be accompanied by quite large in-plane displacements and strains of i t s illuminated surface. By the use of specimen illumination in the x,y, plane and the aperture sets rotated by a) = 0° the camera is made insensitive to w and w,y and, consequently, the terms involving w and w,y in equation (4.12) are often negligible. When this i s the case, equation (4.12) reduces to equation (4.22), the solution of which was derived and discussed in Section 4.10 . 4.12 Use of DASSC to Measure Plane Stress and Plane Strain Deformation When the specimen deformation i s of the plane stress or plane strain type, the out-of-plane displacement component u^ is usually very small or zero. If the terms involving u^ and i t s derivative(s) are so small that they do not contribute significantly to the fringe formation, these terms may be neglected in equations (4.12) and (4.13), and the displacement components v^ and w^  can be determined from only two photographs. These photographs could be obtained, for example, by DASSC with two illuminating beams, each beam being used with one of the two aperture sets which are rotated by the same angle (J)^  . Use of the f i n i t e difference scheme discussed in Section 4.9 allows v x and wx to be calculated from equation (4.21) with U j and U j j y j set equal to zero. As mentioned in Section 4.10, when the specimen illumination is in the. x,y plane and the aperture sets are in cj)^  = 0° position, the terms in is found by neglecting small terms involving v,v »y >w> , and w equation (4.12) involving w and are often negligible; and, when i t i s possible, equation (4.12) with the u, u, y ,w, and w,y terms neglected reduces to V + f f V j y = Ni which may be written as v, y + p(y)v = q(y) (4.34) where p(y) and q(y) are defined as p(y) = f~ i ,w - f E i Equation (4.34) is of the same type as equation (4.27) and, hence, i t is solved in the same way, with the solution given by equation (4.28) and the integrating factor given by y(y) = exp / P(t) dt Unfortunately in most practical testing, while i t is often possible to neglect the terms involving w and w,y , the term d i U , y is not sufficiently small and may not be neglected. If this i s the case, equation (4.12) reduces to equation (4.22), the solution of which was discussed in Section 4.10. 5. EXPERIMENTAL APPARATUS AND PROCEDURE 89 In the f i r s t part of this chapter the various apparatus used in the experimental work is described and discussed in the following-order: the camera; the recording system; the f i l t e r i n g system; the specimen loading systems; and f i n a l l y the various specimens themselves. The latter part of the chapter contains a description of the experimental procedure which was used. 5.1 The Camera The photograph of the camera is shown in Fig. 5.1 and i t s schematic in Fig. 5.2 . The camera may function as either DASC or DASSC, depending on the position of the photographic plate upon which the specimen is imaged. The position of the photographic plate may be varied along the x-axis with the use of the adjustable slide. The sub-assembly of the photographic plate holder and the slide is shown in Fig. 5.3 . The slide, lens assembly, and shutter assembly are each mounted on in. diameter steel rods f i t t i n g into holes in the camera frame. These holes, spaced 1 in. apart, are along the f u l l length of the frame, and are located precisely in the centre of the frame to ensure the axial alignment of the camera components while allowing their mounting at desired heights and spacings. The lens assembly, uncorrected for spherical aberration, consists of three high quality lenses mounted in a tabular frame. The shutter assembly shown in Fig. 5.4 is positioned in front of the lens assembly and u t i l i z e s a rotating shutter to open or close various aperture sets as required in a particular experi-ment. The stationary part of the shutter assembly accomodates a number of interchangeable aperture plates each having apertures of various diameters. The camera enclosure is used to prevent any unwanted light from reaching the photographic plate. F i g . 5.1 Double aperture speckle shearing camera (DASSC). enclosure ape r t u r e te ^ ^ h o t o g lens ass 'y raphic p l a t e s p l a t e holder IT ad j ustable s l i d e iii i frame i • 11 i M l 1 | - M 1 W/////////A F i g . 5.2 Schematic of DASSC. Photographic plate holder assembly Fig. 5 . 6 Recording system. Fig. 5 . 7 Filtering system. beam expander mirror mirror photographic recording made by DASSC diffracted light of f i r s t order frosted glass f i l t e r i n g system camera laser Fig. 5.8, Schematic of the f i l t e r i n g system. -p-95 5.2 The Recording System A typical recording system is pictured in Fig. 5.6 and i t s schematic in Fig. 5.5 . The system consists of the camera, the optical components, the laser providing the specimen illumination, and the loading apparatus with the specimen. The camera was described in the previous section and the loading system w i l l be described in Section 5.4 . The laser used i s an o argon laser providing a source of coherent light of wavelength 5140 A delivered at a maximum continuous power of approximately .6 watts. The laser beam of approximately .1 in. diameter i s directed by the adjustable mirrors and divided by the beam splitters. Once the laser beam has been properly oriented, i t i s expanded by the beam expander and then collimated by a large diameter lens. Depending on the experiment, one or two specimen illuminating beams were obtained in this way. The purpose of the recording system is to produce one or two photographic plates storing information ' about the specimen deformation that took place between the two recorded exposures. The emulsion of the photographic plate, once i t is developed, contains two speckle grids, each corresponding to one exposure; as described in Chapter 3, the two speckle grids add and thus form a resultant grid of variable diffractive efficiency. 5.3 The Filtering System Once the photographic, double exposure recording of the specimen deformation is made and subsequently developed, i t must be "processed" in the f i l t e r i n g system so that the various fringe patterns may be separated, displayed and recorded. The f i l t e r i n g system, shown in Figs. 5.7 and 5.8, uses again the laser as a source of light and mirrors to direct the laser beam. After properly oriented, the laser beam is expanded by the beam expander and, upon reaching a sufficient diameter, is made to converge 97 by use of a l a r g e l e n s . The photographic p l a t e c o n t a i n i n g the rec o r d i n g made by DASSC i s placed i n the converging l i g h t and, because i t contains a speckle g r i d , the re c o r d i n g d i f f r a c t s l i g h t i n a d i r e c t i o n depending on the speckle g r i d d e n s i t y and o r i e n t a t i o n . An opaque screen upon which the d i f f r a c t e d l i g h t orders are focused i s placed i n the f o c a l plane of the l e n s . A c i r c u l a r aperture i n the screen permits one of these d i f f r a c t e d l i g h t orders to pass through and be focused by the lens of the f i l t e r i n g system camera onto the f r o s t e d g l a s s where the specimen image may be viewed and recorded. As was mentioned i n the previous s e c t i o n , the speckle g r i d has v a r i a b l e d i f f r a c t i v e e f f i c i e n c y over the area of the recorded image of the specimen, and t h i s causes the i n t e n s i t y of the specimen image (formed by the d i f f r a c t e d l i g h t ) to vary i n some systematic manner. This l a s t image of the specimen (appearing on the f r o s t e d g l a s s ) i s , of course, the f r i n g e p a t t e r n n ( y , z ) . 5.4 The Specimen Loading Systems In the experiments to be described i n s e c t i o n s 6.2 and 6.3, i n v o l v i n g the out-of-plane r i g i d body t r a n s l a t i o n and the in-plane r o t a t i o n of a p l a t e specimen r e s p e c t i v e l y , a Kinematic micro t r a n s l a t i o n t a b l e (model TT-102) coupled w i t h a r o t a r y t a b l e (model RT-200) was used to impose d i s p l a -cements to w i t h i n an accuracy of ± .00025 i n . and r o t a t i o n s to w i t h i n an accuracy of ± .1 min. . The assembly w i t h the p l a t e specimen i s shown i n F i g . 5.9 . The schematic of the experimental apparatus used i n the out-of-plane beam bending experiments i s given i n F i g . 6.16 and i t s photograph i s shown i n F i g . 5.10 . Two C-clamps at each end of the beam were used to clamp i t to an aluminum channel of much greater bending s t i f f n e s s than the beam. A small c i r c u l a r hole was machined at the center of the channel so that the -thrust bearing axial displacement ^fotonic /-dial gage dial gage ^ s e n s o r h o l d e r load load-\ load spring-^ nut \ shaft-; -load c e l l lateral displacement dial gage I T 1 ~ ' L lateral adjustment ^ p i n screws Fig. 5.11 Schematic of the tensile loading apparatus. Fig. 5.12 Right side view of the loading apparatus. Fig. 5.13 Left side view of the loading apparatus. Fig. 5.15 Variable cross-section specimen. 101 micrometer t i p could contact and d i s p l a c e the center of the beam out of the plane by a known amount to w i t h i n an accuracy of approximately ± .00025 i n . I n a l l the remaining experiments (to be described i n s e c t i o n s 6.5 through 6.7) the l o a d i n g apparatus shown i n F i g s . 5.11 , 5.12 and 5.13 was used. I t was designed so that adjustments of the specimen shape and p o s i t i o n could be made w h i l e the specimen was being subjected to as much as 6000 l b s . of s t a t i c t e n s i l e a x i a l l o a d . The load was imposed by t u r n i n g the l o a d i n g nut which i n turn compressed a c o i l s p r i n g . The load was t r a n s -m i t t e d to the specimen through a load s h a f t , p i n , and clamps which were constrained to move a x i a l l y by s t e e l b a l l s f i t t i n g i n t o V grooves made i n the frame and i n the s i d e of the clamps. Such an arrangement permitted an a x i a l movement w i t h a minimum amount of " p l a y " i n other d i r e c t i o n s . The l o a d i n g , however, o f t e n r e s u l t e d i n an unwanted a x i a l t r a n s l a t i o n of the specimen. This a x i a l t r a n s l a t i o n was monitored a t one p o i n t by a d i s p l a c e -ment transducer (Fotonic Sensor) mounted on a kinematic t a b l e so that the Sensor probe could be kept a t a constant d i s t a n c e from the specimen s u r f a c e . By making adjustments w i t h the t r a n s l a t i o n c o n t r o l screw and observing the Fo t o n i c Sensor reading, we could zero out the a x i a l displacement of t h i s p a r t i c u l a r p o i n t . The i n i t i a l crookedness of the specimen or a small mis-alignment of the clamps a l s o caused a s m a l l l a t e r a l displacement of the specimen. This undesirable displacement was measured by a number of d i a l gages, shown i n F i g . 5.14, and minimized by adjustments made w i t h the l a t e r a l adjustment screws. In a d d i t i o n to the a x i a l t e n s i l e l o a d , the specimen was a l s o subjected to some bending caused by the asymmetry,of the l o a d i n g apparatus. To minimize t h i s bending deformation, a d i a l gage was used to measure the v e r t i c a l displacement of the centre of the supporting frame and t h i s displacement was then zeroed w i t h the bending c o n t r o l screw. Once a l l these adjustments were completed, the load increment and the instrument F i g . 5.17 T e n s i l e specimen w i t h the uniform c r o s s - s e c t i o n . Fig. 5.18 Wooden beam specimen. 104 readings were recorded and the second exposure of the specimen surface was made. 5.5 Specimens A l l specimens, i n c l u d i n g the wooden beam were coated w i t h a f l a t white enamel p a i n t to provide a d i f f u s e l y r e f l e c t i n g s u r f a ce. D e t a i l e d d e s c r i p t -ions of the specimens are i n Chapter 6; the p l a t e specimen used i n the experiments of s e c t i o n s 6.2 and 6.3 i s shown i n F i g . 5.9 and the remaining specimens are shown i n F i g s . 5.15 through 5.18 . 5.6 Experimental Procedure The work performed i n a l l the experiments described i n Chapter 6 fo l l o w e d , i n gene r a l , the same p a t t e r n whether DASC or DASSC was used. The experimental procedure can be o u t l i n e d i n the f o l l o w i n g steps: a) Using knowledge that the camera forms f r i n g e s according to equation (3.20) or (3.26) and having some idea about the s i z e of the s p e c i -men deformation i n the intended t e s t i n g , the parameters of the camera were chosen so that a d e s i r e d number of independent recordings having d e s i r a b l e f r i n g e d e n s i t i e s could be made. b) The specimen was coated w i t h white enamel p a i n t to r e f l e c t the l a s l a s e r l i g h t d i f f u s e l y over the area of i n t e r e s t w i t h approximately the same i n t e n s i t y . I t was then clamped or attached i n the l o a d i n g apparatus which had been p r e v i o u s l y set up to accommodate the specimen. c) Various sensors and d i a l gages were p o s i t i o n e d and t h e i r i n i t i a l ( f i r s t exposure) readings were recorded. With the use of a telescope w i t h dual c r o s s - h a i r s or some l a s e r alignment technique the center of the frame of DASSC was made to l i e i n the x,z plane of the specimen coordinate system. The o p t i c a l components that make up the camera were then mounted at the 105 required height and orientation so that they were aligned along the x-axis. An aluminum ruler coated with f l a t black paint and having scratch marks made at 1 in. increments was positioned next to the specimen (usually just below) so that i t s plane was coincident with the y,z plane. A frosted glass plate was inserted in the plate holder of the camera and the specimen with the ruler was illuminated by one of the collimated laser beams. The image of the specimen and the ruler made by light passing through a particular aperture set was viewed on the frosted glass, and the size of the lateral shear Ayi was adjusted by positioning the plate holder with the adjustable slide and observing the amount by which the bright images of the ruler scratch marks doubled. d) With both the room lights and the laser turned off and in the near or total darkness one or, i f desired, two photographic plates were inserted in the plate holder and the camera was then covered with the enclosure. The specimen was then illuminated by one illuminating beam at a time for the required exposure period at an appropriate shutter setting. Once the desired number of f i r s t exposures was made the laser was turned off and the specimen was loaded. After a l l the specimen adjustments described in Section 5.4 were made, the readings of the dial gages and other sensors were recorded. The specimen was again illuminated and the second exposures were made,using the same illuminating beams and shutter settings as in the f i r s t exposures. A l l exposures were usually of the same duration. e) The laser was turned off and the photographic plates were removed from the holder and developed according to manufacturers specifications. f) After development, the photographic plates were inserted in the f i l t e r i n g system (described in Section 5.3) where the fringe patterns cor-responding to the specimen deformation could be viewed and permanently recorded on the film. 106 g) The lateral shear Ay^ was measured on a traversing microscope either from the photographic plate or preferably from the film recording; using the film recording was easier because each recording had only one lateral shear corresponding to a particular aperture set. In contrast to this, the photographic plate contained a l l the images of the ruler scratch marks making i t d i f f i c u l t to identify and measure the various lateral shears. h) With the aid of a" microdensitometer the film negatives with the fringe patterns were scanned and, by scanning the image of the ruler too, i t was possible to establish the location of the fringes on the specimen surface. i) With the known fringe locations and DASSC parameters as the input data,an appropriate computer program was used to calculate the unknown specimen deformation. 107 6. EXPERIMENTAL WORK 6.1 Preliminary Remarks This chapter consists essentially of two parts. The f i r s t part is devoted to the description and discussion of the results of a number of relatively simple experiments, the purpose of which was to verify experimentally the validity and accuracy of the equations governing the . fringe formation of DASC and DASSC. The second part of this chapter deals with several f a i r l y complicated experiments which served to test the f e a s i b i l i t y and accuracy of DASSC for measurement of displacements and strains in applications similar to those encountered in the practical testing of materials. In contrast to the other chapters, in this chapter a l l figures are placed at i t s end so that the continuity of reading may not be interrupted by their excessive number. 6.2 Rigid Body, Out-of-Plane Translation of a Plate Specimen The purpose of this experiment was to verify the formation of fringes by DASC and DASSC due to the out-of-plane displacement of the specimen surface. For the r i g i d body, out-of-plane translation of the specimen surface, that i s for u = u Q , and for other displacements and strains being identically zero, both equations (4.4) and (4.13) governing the fringe formation by DASC and DASSC respectively reduce to a i u 0 = Ni(y> z) (6.1) By using the definitions (4.5a) for a^ and (4.5d) for N^ , we may write equation (6.1) as equation (6.2) for DASC and as equation (6.3) for DASSC. ycoso)^ + zsina)i Xxs : — u Q = - •—n^ycoscj)^ + zsintj)^ , - ysincj)^ + zcoscj)^) (6.2) ycoscj)^ + zsinct^ XXg - u Q •= - — — n^ycoscj)^ + zsincj)^ , - ysinc})^ + zcostj)^) (6.3) 108 Two aperture sets, one rotated by 0° and the other by 90°, were used to photograph the out-of-plane displacements u Q of the plate. The plan view of the experimental setup for DASC is shown in Fig. 6.1 . One way to verify the fringe formation by the two cameras is to compare the predicted and actual fringe positions and spacings. From equations (6.2) and (6.3) the predicted fringe patterns for this experiment are given by ,o D DASC: cj> = 0 n(y,z) = - u 0y (6.4a) * = 90° n(y,z) = - ^ | u 0z (6.4b) DASSC cf) = 0° n(y,z) = - J ^ J - u Qy (6.5) The DASC parameters for Experiment (Exp.) 19 (<$> = 0°) and for Exp. 22 (<j> = 90°) were x & = 38 in. D = 2.5 in. u Q = .025 in. For these parameters the fringe patterns are calculated from equations (6.4a) and (6.4b) as Exp. 19: n(y,z) = - 2.164 y Exp. 22: n(y,z) = - 2.164 z Hence, for Exp. 19 the fringes are predicted to be parallel to the z-axis and spaced .462 in. apart, and for Exp. 22 the fringes are predicted to be parallel to the y-axis and also spaced .462 in. apart. The actual fringe patterns for these two experiments are shown in Fig. 6.2 and Fig. 6.4 . From the microdensitometer traces of Fig. 6.2 and Fig. 6.4 the actual 109 fringe spacing was found to be .45 in. in both cases, which is close to the predicted spacing of .462 in. To show a typical microdensitometer trace, we have presented in Fig. 6.3 the trace of the fringe pattern of Exp. 19 shown in Fig. 6.2 . The DASSC parameters for Exp. 17 and Exp. 18 (ty = 0°) were x g = 38 in. u Q = .025 in. Exp. 17: D = 1.25 in , Ay s = - .0257 in. , X s = 37.22 in. Exp. 18: D = 2.50 in. , Ay g = - .0453 in. , X g = 37.31 in. The fringe patterns are determined by equation (6.5) as Exp. 17: n(y,z) = - 1.105 y Exp. 18 n(y,z) = - 2.204 y Hence, for Exp. 17 the fringes are expected to be parallel to the z-axis and spaced .905 in.apart; for Exp. 18 the fringes are also expected to be parallel to the z-axis but spaced .454 in. apart. The fringe patterns for these two cases, obtained with DASSC, are shown in Fig. 6.5 and Fig. 6.6 Note the "doubling" or "shearing" of lines and numbers in these two figures caused by the defocussing of the system. From the microdensitometer traces and after an appropriate scaling, the actual fringe spacing in Exp. 17 was found to be approximately .89 in., and in Exp. 18 the actual fringe spacing was found to be approximately .46 in. . Again the agreement between the predicted fringe spacing and the actual fringe spacing is acceptable. A computer program, WONLY.S, to calculate u from equations (6.4) and (6.5) was written; i t calculates u from data consisting of the parameters of DASC or DASSC and fringe center numbers and positions obtained from the microdensitometer trace. For example, the numerical data for Exp. 19 read 110 off Fig. 6.3 are given in Table 6.1 Fringe Location (in.) Fringe Number - 2.56 5.50 - 2.32 5.00 - 2.06 4.5o - 1.88 4.00 - 1.60 3.50 - 1.40 3.00 - 1.14 2.50 - .94 2.00 - .68 1.50 - .50 1.00 - .24 .50 .04 .00 .24 - .50 .42 - 1.00 .68 - 1.50 .85 - 2.00 1.14 - 2.50 1.32 - 3.00 1.60 - 3.50 1.80 - 4.00 2.02 - 4.50 2.24 - 5.00 2.48 - 5.50 2.68 - 6.00 2.96 - 6.50 Table 6.1 Fringe data of Exp. 19 The fringe function n i s approximated by a piecewise continuous cubic based on the above numerical data. For Exp. 19 this fringe function i s compared to I l l the f r i n g e f u n c t i o n p r e d i c t e d by equation (6.4a) i n F i g . 6.7 . The two fu n c t i o n s are i n good agreement and the a c t u a l d i f f e r e n c e between them i s shown by the t h i r d curve. This t h i r d curve has i t s o r d i n a t e on the r i g h t s i d e of the graph and the s c a l e of the " d i f f e r e n c e " curve i s u s u a l l y h i g h l y exaggerated. The reason f o r the exaggerated s c a l e i s that o f t e n the p l o t s of p r e d i c t e d and experimental ( i . e . obtained w i t h the use of DASC or DASSC) curves are so clo s e together that i t would be d i f f i c u l t to e s t a b l i s h the a c t u a l numerical d i f f e r e n c e between them. The p l o t of the displacement, u = u 0 v s . u, c a l c u l a t e d by WONLY.S i s shown i n F i g . 6.8 . Note that the two displacements are very c l o s e except near the o r i g i n where even a very small e r r o r i n the f r i n g e p o s i t i o n produces a l a r g e e r r o r i n the c a l c u l a t e d displacement. The reason f o r t h i s i s obvious from the form of equation Ax? n(y,z) u ( y , z ) = - - ^ — ^ y ^ O (4.10b) Therefore, i f DASC or DASSC i s used to measure the out-of-plane displacement i t i s e s s e n t i a l to a l i g n the camera p e r f e c t l y and to e x t r a p o l a t e the displacement at the o r i g i n from the displacements c a l c u l a t e d nearby. The r e s u l t of t h i s experiment agree very w e l l w i t h the t h e o r e t i c a l p r e d i c t i o n s and confirm the v a l i d i t y of the c o e f f i c i e n t a i n equations (4.4) and (4.13). 6.3 R i g i d Body, In-Plane R o t a t i o n of a P l a t e Specimen The purpose of t h i s experiment was to v e r i f y the formation of f r i n g e s by the two cameras due to the in-plane displacements of the specimen surface. The in-plane displacements v and w are produced by the r i g i d body r o t a t i o n , a, of a p l a t e about the x-axis as i s shown i n F i g . 6.9 . A poi n t which was i n i t i a l l y i n the y,z plane at (0,y,z) was d i s p l a c e d between the two exposures 112 to the (0,y*,z*) position. The displacements v and w are related to the coordinates (y,z) of an arbitrary point by u = u, y = u, z = 0 v = y * - y = y ( C O S a - 1) - Z S i n C l w = z* - z = z(cosa - 1) +ysina (6.6a) (6.6b) (6.6c) The partial derivatives of these displacements are v, y = w,z = cosa -1 v,, - w,v = - sma (6.6d (6.6e) A l l partial derivatives are constant and, for a small angle of rotation a as in the case, the terms in equation (4.13) involving these derivatives are negligible. For the angle of rotation, a =2minutes, the numerical values of the displacements and strains are v(y,z) = - 1.7 x 10 7y - 5.82 x 10 *z w(y,z) = - 5.82 x lO'^y - 1.7 x 10 _ 7z .-7 V , y = W, z 1.7 x 10 v, z = - w,v = - 5.82 x 10 -4 (6.7a) (6.7b) (6.7c) (6.7d) A schematic diagram for the experiments done in this section is shown in Fig. 6.10 . DASC with three aperture sets rotated by 0°,45°, and 90° was used to photograph the rotation of the plate which caused the displacements u,v and w given by equations (6.6). For these displacements equation (4.1); which governs the fringe formation by DASC, becomes CD = 0 v - yz Xxc w x c n(y,z) (6.8a) <j> = 45 + z2 - y 2 /2 2/2 x' V + 1 z 2 - y 2 ^  2/2 Xx, w n(y,z) "s ' (6.8b) 113 o v z A x s * = 90 : - — v + w = - — - n(y,z) (6.8c) The actual values of displacements v and w given by equations (6.7a) and (6.7b) are substituted in equations (6.8) and, using the fact that the viewing angle in these experiments was such that 1 < To ' w e m a y l x S neglect some small terms. Once this was done, the equations were solved for the fringe functions n(y,z) as Exp. 24 (({> = 0° ) : n(y,z) - (5.82 x 10 * ) Z Ax c Exp. 25 (ct = 45°) : n(y,z) = (5.82 x 10 " ) D Z Y Ax., /2 Exp. 26 (<j) = 90°) : n(y,z) = (5.82 x 10 4 ) -r^- y A X S Thus for Exp. 24 the fringes are predicted to be parallel to the y-axis, for Exp. 25 the fringes are predicted to be straight and inclined at 45° to the y-axis, and for Exp. 26 the fringes are predicted to be parallel to the z-axis. The DASC parameters used for the three experiments were x s = 39 i n . D = 2.5 in. With these parameters the predicted fringe spacing is .536 in. - the same for a l l three experiments. The actual fringe patterns obtained by DASC are shown in Figs. 6.11, 6.12 and 6.13 . The fringes are oriented as predicted and, from the microdensitometer traces of these figures, the fringe spacing was found to be .55 in., close to the predicted spacing of .536 in. Let us now predict the positions and spacing of fringes formed by DASSC, 114 which for Exp. 2 considered here had the following parameters: x s = 32 in. D = 2.5 in. ex = 20° , ey = no° , ez = 9 0 0 Ay s = .0608 in. 4> = o° The displacements and strains given by equations (6.7) are substituted in equation (4.11); we can make use of the fact that the viewing angle was again such that < Y Q to neglect many small terms and solve the equation for the fringe function n(y,z) as Exp. 2 : n(y,z) - 2.328 z . For this experiment the fringes are predicted to be parallel to the y-axis and spaced .429 in. apart. The actual fringe pattern i s shown in Fig. 6.14 . The fringes are parallel to the y-axis and the fringe spacing was found from the microdensitometer trace to be .42 in. which compares well with the predicted spacing of .429 in. A computer program, U0NLY.S, was written to calculate v from equation (6.8a) with the w term neglected, or to calculate w from equation (6.8c) with the v term neglected. It computes the displacements from the given parameters of DASC or DASSC and from the fringe center numbers and positions obtained from the microdensitometer trace. The plot of the exact displacement v(z) and the plot of v calculated by UONLY.S, using data obtained by DASC, is shown in Fig. 6.15 . The results of these experiments confirm that the response of both DASC and DASSC to the in-plane displacements is accurately described by equations (4.4) and (4.13). In particular, the accuracy of the coefficients b^ and c^ 115 of equations (4.4) and (4.13) was verified. 6.4 Out-of-Plane Bending of a Thin Beam with a Rectangular Cross-Section The purpose of this experiment was to verify the formation of fringes by DASSC due to the variable out-of-plane displacement of a specimen surface. A diagram of the experiment is shown in Fig. 6.16 . A thin alluminum beam with a rectangular cross-section was clamped at the ends and i t s center was displaced, by use of a micrometer, a known distance 6 out of the plane. In Appendix V strength of materials theory was used to derive expressions closely approximating the actual displacements and strains that occur in the vis i b l e surface of the beam. The strains in the z direction were not derived as the beam deformation w i l l be viewed through an aperture set(s) at zero inclination to the y-axis. The parameters of DASSC used in the beam bending experiments considered in this section were similar to those of Exp. 101C given here as beam: % x 2 x 50 in. V = .25 x s =39.75 in. , Ay s = - .12 in. , X s = 37.84 in. D = 2.5 in. 0 X = 76.8° , 0 y = 166.8° , 0 Z = 90° With these parameters equation (4.11) becomes 39^75 U + V " iffo W " 2 ' 2 3 U>y + U 8 2 ^ + 1.82 3 9 Z 7 5 w,y = - .00032 n(y,z) With the displacements and strains derived in Appendix T the above equation becomes y - 1.25 39.75 - .919 v, y + 116 2.5 x 10 2 y 1 - 3 25 + 2 25 + 1.2 x 10 3 y 1 + 25 + 1.9 x 10 7 yz 2 1 - 2 25 •+ 2.1 x 10 1 - 25 w 2.2 x 10 - 3 fy - 1.25 39.75 v, y - .919 1 - 2 25 + 1.1 x 10 6 z 2 w, .00032 n It i s obvious that, in the case of out-of-plane bending of beams, the fringe equation may be approximated by only two terms involving u and u, v . A more accurate approximation would involve u,u,v,v and v, v terms. Note that for a beam these four quantities are closely approximated by functions of y only. Let us f i r s t consider the use of DASC to determine the beam deformation. In the beam bending experiments considered here, the term involving u i s so much larger than the terms involving v and w that equation (4.10b) determines u with sufficient accuracy. u(y,z) Axf. n(y,z) y * 0 (4.10b) In Exp. 16 considered here the DASC parameters were beam: g- x 2 x 55 in. x s = 29.5 in. D = 2.5 in. 6 = .50 mm = .019685 in. The fringe pattern of this experiment is shown in Fig. 6.17 . Once the fringe centers are correctly numbered and their coordinates are read off the microdensitometer trace, the displacement u is obtained by the use of 117 equation (4.10). A computer program, BEAM1.S, was written to calculate • u(y,z) this way, and u determined by BEAM1.S and u obtained from the strength of materials theory are both plotted in Fig. 6.18. The two displacements correlate quite well. When DASSC is used to "photograph" the out-of-plane bending of the beam, a number of ways to determine the out-of-plane displacement and slope i s available. We shall consider several of these and point out their advantages and disadvantages. It was shown in Section 4.8 that, for the beam bending experiments considered here, equation (4.11) may be accurately approximated by keeping only the terms involving u and u, ; hence, equation (4.11) reduces to equation (4.16) , which for (j) = 0° becomes au + du, y = N(y,z) (6.9) In the case of beam bending, u and u, y are essentially functions of y only and, therefore, the partial derivative in equation (6.9) may be replaced by an ordinary derivative. After some rearranging we obtain du a N(y) d 7 + d u = ^ <6-10> In Section 4.11 is shown that, i f the out-of-plane displacement at one point, y = y 0 is known, the solution to equation (6.10) is given by equation (4.32) with p = a/d and q = N(y)/d . This approach was used to solve for u and u, y in Exp. 101C. The DASSC parameters used in Exp. 101C are given on page 118 and the fringe pattern is shown in Fig. 6.19 . A program, OUTlM.S, which was written solves,equation (6.10) according tor, equation (4.30) with <j> = 0° . The program accepts the system parameters, displacement at one point, and fringe centers as data and calculates u and u, y . The graphs of 118 predicted u and u, y vs. the actual u and u, y obtained by DASSC and OUTIM.S are shown in Figs. 6.20 and 6.21 . Note that in both cases the agreement between predicted and actual values is quite good. In the out-of-plane displacement u is not known at any point of the visible part of the beam, the two photographs must be taken and use made of the approach of Section 4.8 . Quantities u and u, y are determined from equation (4.16) which for the aperture set rotation (j) = 0° becomes A ±u + D i U, y = N ±(y,z) i = 1,2 (6.11) A program, BEAM2.S, which solves equation (6.11) was written to determine u and u,y for Exp. 101; the system parameters were as follows: beam: \ x 2 x 50 in. x g = 39.75 in. (illumination in x,y plane) D = 2.5 in. ex = 76.8° Ay s = - .012 i n . Exp. 101A D = 1.75 i n . 9 X = 26° Ay s = - .097 in. Exp. 101B D = 2.5 in 0 X = 76.8° Ays•= - .12 in Exp. 101C The two fringe patterns for Exp. 101A and Exp. 101B are shown in Figs. 6.22 and 6.23 . The plots of the predicted u and u, y and the experimentally found u and u,y (calculated by BEAM2.S) are shown in Fig.6.24 and 6.25 . Again the agreement between the predicted and the experimental results is good. It is important to note that i t is not enough to know that |u|,:|'u,y| >> | v"| , | v, y | , | w | ,.|w',y.|. and to ensure that dj- d 2 ^ 0 ; equation (4.32) must be satisfied as well to obtain accurate solutions for u and u,y . For example, the combi-nation of Exp. 101A and Exp. 101B satisfies this requirement while the combi-nation of Exp. 101B and Exp. 101C does not. The experiments described in this section have confirmed that DASSC forms fringes according to equation (4.13) for the case of out-of-plane 119 bending and have thus verified the accuracy of coefficients a^ and d^ of equation (4.13)• 6.5 In-Plane Stretching of a Thin Beam with a Rectangular Cross-Section The purpose of this experiment was to verify the formation of fringes by DASSC due to the in-plane displacement and straining of the specimen surface. The schematic diagram of the experiment i s shown in Fig. 6.26 . A thin, f a i r l y wide, acrylic beam with a rectangular cross-section was clamped at i t s ends, and a tensile load was applied between the two exposures in such a way that the center of the beam remained stationary while i t s ends were displaced a known amount. Due to the i n i t i a l crookedness of the beam, the poisson effect, and some misalignment of the system, the beam was also displaced out-of-plane by a small amount. This out-of-plane displacement was monitored at the z = 0 line by a number of dial gages. At the point (-t/2,0,h/2) there was also a di a l gage which measured the displacement of this point in the z direction; this last displacement was caused mainly by the poisson effect. A strain gage was cemented to the illuminated surface of the beam and i t s reading served as a standard against which the strain determined experimen-tally by DASSC was compared. The beam deformation i s essentially a special case of the plane stress, and the strains of the neutral surface are related to the imposed increase V in the length L of the beam by _ V £yo ~ L £ xo = " v e y o = " vl Since the in-plane stretching i s accompanied by a small unknown out-of-plane displacement u , the displacements and strains of the illuminated surface of the specimen are given as 120 t _ V t v, y - £ y - £ y 0 " u>yy ~ ^  ~ 2 U'yy v = y V £ L " 2 u'y v(0,z) = 0 w»z ~ e z ~ ~ V£y = ~ V W = - V rv L 2 u'yy 2 u'yy w(y,0) = 0 w,v = Vz — u, 2 u'yyy The deformation of the beam was photographed by DASSC with the apertures at < -^ jr . For these y z x s 9 ' x s 0 = 0 , and the f i e l d of view was such that parameters and the assumed form of the displacements and strains, and the specimen illumination in x,y plane equation (4.11) reduces to u + v yz V t I + 2 u'yy z + Ay s iX s i (1 + cos0 x i)u, y -w •y - Dj/2 - cost y i V t L 2 u'yy z t — Vz — u, yyy sx Di w, The out-of-plane displacement u caused by the i n i t i a l crookedness of the beam or by a misalignment of the loading mechanism was quite small, and i t s shape was found to be smooth and sinusoidal-like with one half period over the beam length. Because of these characteristics, and since the beam was thin (t = .25 in.), the terms %tu,y ,%tu,yy and ^ tu,yyy are very small. Thus the terms involving w and w,y may be neglected from the last equation and for this case equation (4.11) reduces to the form u + v + d-ju,y + f j V . y = N ± (4.22) 121 If u and u,y are very small, so that they are also negligible, equation (4.22) reduces to the simple form v + f i V , y = N ± (6.13) Equation (6.13) is analogous to equation (4.34) and can be solved for v in the same manner. Unfortunately, in the actual testing the term d^u,y was found to be large in magnitude and equation (6.13) could not be used. How-ever, by a careful alignment of the loading mechanism i t was possible to limit the out-of-plane displacement u to magnitudes less than . 0015 in. and y for such a small value of u, the term — u is very small, and i t is them x s possible to solve for u,u,y,v, and v, y from equation (4.22) in the manner discussed in Section 4.10 . A computer program PLATE2.S was written to solve equation (4.22) this way and was used to calculate the displacements and strains from the data of Exp. 114C and Exp. 114D which had the following parameters: beam: h x 6 x 32 in. x s = 45 in. D 2.5 in. Ay s = .07 in. X s = 46.26 in. Exp. 114B ex = - 21.2° ey = 68.8° ez = 90° ' > D 1.75 in. Ay s = .14 in. x s = 48.6 in. Exp. 114C 6 X = - 21.2° ey = 68.8° ez = 90° D 2.5 in. Ay s = .23 in. x s = 49.1 in. Exp. 114D ex = 23.7° ' ey = 113.7° ez = 90° v( - 3,0) = 0 u( - 1,0) = - . 00102 in. 122 v,y(-3.625,0) = 413 x 10~ 6 The fringe patterns obtained by DASSC are shown in Fig. 6.27 and Fig.6.28 . The computer plots of predicted and experimental displacements and strains, done by PLATE2.S are shown in Figs. 6.29 through Fig. 6.32 . The value of v,y(-3.625,0) calculated by the program from the two fringe patterns is - 6 - 6 419 x 10 , which compares favorably with the strain gage reading of 413 x 10 there. If we do not wish to neglect the out-of-plane displacement and strain, then three independent and properly chosen photographs are needed to solve for the displacements and strains. A computer program, PLATE3.S, which solves equation (4.14b) for u,y and v,y in the way discussed in Section 4.6 (and for u and v by numerical intergration), was used to calculate these unknowns for Exp. 114B, Exp. 114C and Exp. 114D. The fringe pattern of Exp. 114B is shown in Fig. 6.33 and the displacements and strains calculated by PLATE3.S are shown in Figs. 6.34 through Fig. 6.37 . This time the experiment-- 6 a l l y determined strain at y = -3.625 in. is 414 x 10 , which again compares well with the strain gage reading of 413 x 10 6 there. It should be realized, however, that i t is a coincidence that the two strains are this close at y = -3.625 in., as usually the two strains differ elsewhere by as much as five percent or more. Alternatively, the two displacements u and v and the strains u,y and v,y may be calculated by the f i n i t e difference method discussed in Section 4.10 . A computer program, FD2, based on equations (4.23), (4.24), and (4.25) was written and used to calculate u,v,u,y , and v,y from the fringe patterns of Exp. 114C and Exp. 114D. Since the program required one boundary condition on u and another on v, u(-:l,0) (measured by one of the dial gages) and v(-3,0) were used. Figures 6.38 through 6.41 show the comparison of the actual and calculated displacements and strains. The experimentally determined strain 123 at y = -3.625 in. i s 420 x 10 6 vs. 413 x 10 6 measured by the strain gage there. A l l three approaches used to calculate the displacements and strains for this particular example yield reasonably accurate solutions. If the displacements u(y Q,0) and v(y Q,0) are available, the program FD2 should be used as i t i s , in general, more accurate, easier to write and more efficient than the program PLATE2.S. If the displacements u(y Q,0) and v(y Q,0) are not available, then one must, of necessity, use the program PLATE3.S. to determine the two strains and the "displacement sum". These experiments have demonstrated the f e a s i b i l i t y of using DASSC in testing involving primarily the plane stress deformation and in particular the accuracy of the coefficient f was verified. 6.6 In-Plane Stretching of a Beam with a Variable Cross-Section The purpose of this experiment was to determine the performance of DASSC in a more "practical" type of investigation and to test the accuracy of the two-dimensional solution scheme(s) for displacements and strains. Unfortunate-ly, at the time this experiment was done, the theory of the fringe formation by the DASSC had not yet been f u l l y developed and, hence, the experiment was not set up in the way which would allow accurate determination of a l l displa-cements and strains. In particular, the effect of the displacement w and i t s two derivatives, w,v and w,z , on the fringe formation by DASSC was not known. The dimensions of the acrylic specimen used in this experiment are shown in Fig. 6.42. Two strain gages were cemented to the illuminated surface of the beam at the (-3,0) and (5,0) locations to measure the surface strains there for later comparison with the strains obtained with DASSC. An axial tensile load was imposed on the specimen by the use of the same loading apparatus as the one described in the preceeding section and shown in Fig.6.26. 124 The out-of-plane displacements of the beam were measured by a set of dial gages. The beam center, coincident with the coordinate origin, was kept in the same position during the two exposures with the aid of the Fotonic Sensor. DASSC used in this experiment was equipped with the shutter shown in Fig. 6.43 so that i t was possible to use two illuminating beams in the x,y plane in such a way that one illuminating beam was used with the inner aperture sets and the other illuminating beam was used with the outer aperture sets. We know now that the use of illumination in the x,y plane was unfortunate, as i t makes DASSC insensitive to the strains w,y and w,z . In this experiment DASSC was used to take four photographs with the following parameters: 45 in. D = 1.75 in A y s = .0914 in. Exp. 122S1 ex = -20.5° 6 Y = 69.5° 6 Z = 90° <t> = 0° D = 2.50 in A y s = .162 in. Exp. 122S2 = 24° 9 Y = 114° ez = 90° * = 0° D = 1.75 in. A y s = .104 in Exp. 122S3 = -20.5° 6 Y = 69.5° ez = 90° = 135° D = 2.5 in. A y s = .171 in Exp. 122S6 6x = 24° ey = ii4° 0 Z = 90° = 135° 125 The computer program FD2, based on equations (4.24), and (4.26), was used to calculate u(y,0) and v(y,0) from the photographs of Exp. 122S1 and Exp. 122S2 shown in Figs. 6.44 and 6.45. The two displacements, u(y,0) and v(y,0), and their derivatives calculated by FD2 are compared to those displacements and their derivatives determined from the dial gages and also to the f i n i t e element solution of this problem. These comparisons are shown in Figs. 6.46 through 6.49 and as can be seen from the graphs, the displacements and strains agree quite well. We would now like to calculate u(y,z), v(y,z) and w(y,z). To do this, we need three "independent" photographs taken by DASSC having three aperture sets and three illuminating beams with at least one of them not being in the x,y plane. The three displacements may then be calculated with the use of the scheme discussed in Section 4.9 and based on equation (4.19). However, as was already mentioned, when this experiment was done DASSC was equipped with two aperture sets and two illuminating beams in the x,y plane were used. Thus, only two independent photographs are available, and therefore the scheme based on equation (4.23) must be used to calculate v(y,z) and w(y,z), with u(y,z) assumed to be known. The displacement f i e l d u(y,z) was set equal to u(y,0) as calculated by FD2 and this has introduced a small but negligible error as, obviously, u(y,z) ^ u(y,0) for z 4- 0 . The error is expected to be small because u(y,z) is caused mainly by the out-of-plane bending due to a system misalignment and by the crookedness of the specimen and hence the out-of-plane displacement was most lik e l y the same for a l l points on lines y = constant. The variation in u(y>z), for y = constant, is caused by the reduction of the specimen thickness due to the poisson effect, but this variation is small enough so that i t may be neglected. Since the boundary condition w(y,0) could not be found, i t was therefore set equal to the dial gage reading at the point (0,1) and corrected 126 for the poisson reduction of the beam half-width there. Using the fringe patterns of Exp. 122S3 and Exp. 122S6, shown in Fig. 6.50 and Fig. 6.51, and the solution scheme, based on equation (4.21), we may calculate the displacements v(y,z) and w(y,z) over a part of one quarter of the beam surface shown in Fig. 6.52 . Since the specimen illumination was in the x,y plane, w(y,z) was not calculated accurately enough and is not shown. The displacement v(y,z) and i t s partial derivative v,y(y,z) are compared to the f i n i t e element solution in Figs. 6.53 trough 6.55 . So that we might see the effect of the accuracy of the boundary condition on v(y,z), two solutions for v(y,z) and v,y(y,z) were found. We obtained one solution by using u(y,0) and v(y,0) as calculated by FD2, and the other solution by using u(y,0) and v(y,0) as given by the f i n i t e element solution and dial gage readings. From the two plots i t is obvious that either of the two boundary conditions yields reasonably accurate v(y,z) and v,y(y,z). The result of this particular experiment confirm that the two dimensional computing schemes can produce accurate solutions for v(y,z) and v, y(y,z). Had the experiment been set up properly - and the computer simulation of the experiment confirms this - a l l displacements and strains could be calculated with acceptable accuracy. 6.7 In-Plane Stretching of a Wooden Beam The aim of this experiment was to test the possibility of using DASC or DASSC to measure the surface deformation of specimens made of materials such as wood. The wooden beam used in this experiment shown in Fig.5. ;was 4 in. wide, % in. thick, and 48 in. long, with a knot of approximately 1 in. diameter located at the center of the beam. The beam was subjected to an axial tensile load using the same experimental setup as i s shown in Fig. 6.26. The beam center, which was coincident with the coordinate origin, was 127 maintained stationary during the two exposures with the aid of the Fotonic Sensor. The out-of-plane displacement u(-.5,y,0) of the shadow side of the beam was measured by a number of dial gages. Due to the lack of time, and also because of inadequate control over boundary conditions, no numerical calculations of displacements and strains were performed. S t i l l , the photo-graphs which were taken provide useful information about the application of DASC and DASSC in the testing of highly inhomogeneous materials like wood. DASC used to produce the photographs of Exp. 132D1 and Exp. 132D2 (shown in Fig. 6.56 and Fig. 6.57) had the following parameters: x s = 45 in. D = 1.75 in. 9 X = - 22.5° ( 3 y = 67.5° e z = 90° Exp. 132D1 ej> = 0 ° D = 2.5 in. 0 X = 24.3° ( }y = 114.3 0 e z = 9 0 ° Exp. 132D2 4> = 0 ° axial load increment between the two exposures was approximately from 1600 lb. to 1800 lb.; this corresponds to the increase of 100 l b / i n 2 in tensile stress. The axial load caused some out-of-plane bending which was monitored by the dial gages with the results v(-.5,0,0) = .0 i n . v(0,16,0) = .00226 in. v(0,-16,0) = -.00307 in. u(-.5,-7,0) = .0 in. u(-.5,-5,0) = -.0000787 in. u(-.5,-2,0) = -.0000787 in. . u(-.5';l,0) = -.000315 in. 128 u(-.5,4,0) = -.000394 in. u(-.5,7,0) = -.000866 in. w(-.25,0,2) = .0 in. The fringe patterns in photographs taken by DASC are f a i r l y simple and show that DASC works quite well in testing specimens having highly nonuniform material properties. DASSC was used to produce the photographs shown in Fig. 6.58 and Fig.6.59, corresponding to Exp. 132S2 and Exp. 132S1, for which i t had the parameters: x s = 45 in. D = 1.75 in. = .0488 in. Exp. 132S1 9x = -22.5° 6 y = 67.5° ez = 90° = 0° D = 2.5 in. = .0878 in. Exp. 132S2 9x = 24.3° ey = 114.3° 0 Z = 90° = 0° The two fringe patterns obtained by DASSC are extremely complex, which suggest that the tensile loading of the wooden beam causes a complicated strain f i e l d in the illuminated surface of the beam. This is most likely due to the "aligning" process of the wood fibers involving large changes in surface slopes to which DASSC is most sensitive. In this particular experi-ment, the lateral shear Ay s was obviously set too large, thus making DASSC too sensitive; the resultant fringe patterns were of such complexity as to be of no use in the quantitive analysis, since there was no hope of successfully numbering the fringes in these patterns. In future experiments of this type, the sensitivity of DASSC would have 129 to be decreased or the load increment would have to be smaller to produce fringe patterns which could be interpreted. An inclusion of some simple boundary condition, such as a clamped end, would also be helpful. 6.8 Error Analysis The results of the experiments discussed in this chapter have ascer-tained that the fringe formation of DASC and DASSC is described with sufficient accuracy for an ordinary laboratory testing by equations (3.20) and (3.26), or their equivalent forms given by equations (4.4) and (4.13). a-^ u + b-jV + c-jW = N-L (4.4) a^u + b^v + ....+' k.jw,z = (4.13) By using either of the two cameras in a particular experiment we obtain a number of fringe patterns from which the displacements and strains of the specimen surface may then be determined by making use of the various solution schemes derived and discussed in Chapter 4. However, i t must be realized that these displacements and strains can be calculated with only a limited accuracy because of the following errors: 1. Errors caused by the approximations made in the derivation of equations (4.4) and (4.13). In the f i r s t approximation the real cameras were replaced by the physical models shown in Fig. 3.5 and Fig. 3.11. The second approximation involved deletion of the high order terms throughout the derivation of equations (4.4) and (4.13). To assign a numerical bound on the errors due to these two approxi-mations is very d i f f i c u l t . In principle, i t could be done by comparing the results obtained from equations (4.4) and (4.13) with those determined from some "exact" equations derived for more accurate 130 physical models of the two cameras. It was, however, more convenient to do this experimentally, and by a computer simulation (not discussed in this thesis) of the two cameras, with the results indicating that these errors are less significant than those discussed in the sub-sequent paragraphs. 2. Errors in coefficients a-^,..., k-^  caused by the inaccuracies in the measurement of the parameters of the two cameras. These parameters were usually determined with the following accuracies: D ... ±.005 in. x s ... ± .5, in. ex,6y,ez ... ±i? Ay s ... ±.001 in. 3. Errors in (related to the Moire fringe numbers n^) caused by a limited accuracy with which the location of the fringe centers may be determined. In the work presented in this thesis the fringe centers were usually located within ±.02 in. from the microdensito-meter traces. 4. Errors in the calculated displacements and strains caused by the approximate nature of some of the solution schemes for the displace-ments and strains. The size of these errors depends on the particular solution scheme, and also on the actual location (y,z) on the specimen surface where the displacements and strains are being calculated. From the experiments and calculations that were done, and from the computer analysis of the two cameras, i t appears that the errors discussed in paragraph 2. are the greatest source of errors in the calculated displa-cements and strains. In particular, the inaccuracies in the measurements 131 of Ay s and somewhat ambiguous parameter x s cause the largest errors. This problem could be alleviated by making a large number of measurements of Ay s and then calculating and using i t s average value; the parameter x s could be determined more accurately from a number of simple experiments, or the problem with this parameter could be avoided altogether by using a more elaborate models of the two cameras. Such models would include two x-coordinates, one for the aperture plane and the other for the lens, instead of using x s to approximate both of these coordinates. The simplest, and possibly the only practical way of determining the error caused by the inaccuracies in the camera parameters would involve the repeated use of the appropriate solution scheme, each time with the parameters being slightly changed within their range of accuracy. By examining the set of so calcula-ted numerical values of the displacements and strains an estimate of the accuracy of the results could be obtained. The comparison of the actual displacements and strains with those obtained through the use of the two cameras in the experiments described in this chapter indicates that a l l the errors discussed here are usually quite small and hence equations (4.4) and (4.13) need not be altered, although by implementing the suggestions made in this section, s t i l l more accurate results could presumably be obtained. To get some idea about the effect of experimental errors on the accuracy of the calculated displacements and strains, the upper and lower bounds on these quantities are determined. The errors are caused by inaccuracies in the measurement of the camera parameters and fringe locations, and by using their extreme values in the calculations the bounds may be established. This was done at one "typical" point of the specimen surface for the expe-riments 26 and 114. 132 In the case of the experiment 26, the displacement w(0.,.547) at the point (0.,.547) is given by equation (6.8c) as w(0.,.547) = - n(0.,.547) (6.14) The parameters of DASC that was used were measured as X = .000020256 in. x s = .39.0 ± .5 in. D = 2.500 ± .005 in. The centers of the fringes shown in Fig. 6.13 were located with accuracy of 1 .02 in., and from the plot of n(0,z) (not shown) i t was found that -n(.0.,.547) = 1.00 ± .05. By substituting the parameters A , j.x g, D and n(0.,.547) in equation (6.14), the displacement w(0.,.547) was found as -.000337 in. < w(0.,.547) < -.000296 in. With the mean value of w(0.,.547) equal to -.000316 in. this displacement may be written as w(0.,.547) = -.000316 ± .000021 in. The bound on w(y,z) was found to be of the same magnitude at the other points of the specimen surface. Similar calculations were done for experiment 114 which involved the use of DASSC. By varying the camera parameters and the fringe locations within their range of accuracy, and using the computer program FD2 the bounds on the displacements and strains were found at the point (1.0,0.0) as u(l.0,0.0) = -.00119 ± .00002 in. v(l.0,0.0) = ..00172 ± .00005 in. u, (1,0,0.0) = .000946 ± .000094 v, (1.0,0.0) = .000431 ± .000017 133 At other points on the y-axis the bounds on these displacements and strains were found to be similar to those at the point (1.0,0.0). The errors in the results of the two experiments that were examined are reasonably small and should provide some indication about the accuracy of the two cameras. In the actual calculations the errors would most likely be even smaller due to some cancellation of errors. The errors in the other experiments were not calculated but should be similar to those in the experiments 26 and 114. F i g . 6.2 Fringe p a t t e r n of Exp. 19. 135 F i g . 6.4 Fringe p a t t e r n of Exp. 22. Fig. 6.5 Fringe pattern of Exp. Fig. 6.6 Fringe pattern of Exp 137 in a Fig. 6.7 Predicted n (dashed line) vs. experimental n (solid line). Fig. 6.8 Predicted u (dashed line) vs. experimental u (solid line). 138 Fig. 6.9 Rotation of a plate about x-axis. Fig. 6.10 Measurement of the in-plane displacements v and w by DASC. 1 3 9 F i g . 6.11 Fringe pattern of Exp. 24. F i g . 6.12 Fringe pattern of Exp. 25. 140 Fig. 6.14 Fringe pattern of Exp. 2. 141 Fig. 6.16 Measurement of the out-of-plane displacement by DASSC. 142 Fig. 6.17 Fringe pattern of Exp. 16. < Fig. 6.18 Predicted u (dashed line) vs. experimental u (solid line). 143 F i g . 6.19 Fringe p a t t e r n of Exp. 101C F i g . 6.20 P r e d i c t e d u (dashed l i n e ) vs. experimental u ( s o l i d l i n e ) . 144 Fig. 6.23 Fringe pattern of Exp. 101B 145 146 Fig. 6.26 Measurement of the in-plane deformation by DASSC. Fig. 6.27 Fringe pattern of Exp. 114C Fig. 6.28 Fringe pattern of Exp. 114D 148 g. 6.29 Predicted u (dashed line) vs. experimental u (solid line). 149 Fig. 6.33 Fringe pattern of Exp. 114B 151 152 < 3 Fig. 6.36 Predicted v (dashed line) vs. experimental v (solid line). 153 g. 6.39 Predicted u, y (dashed line) vs. experimental u, v (solid l i n e ) . 154 Fig. 6.40 Predicted v (dashed line) vs. experimental v (solid line). acrylic plate 7/16 in. thick SCALE 1 : 4 156 i i y Fig. 6.43 Normal view of the aperture screen. Fig. 6.45 Fringe pattern of Exp. 122S2 158 I I I I I I T - 6 . 0 - 3 . 6 - 3 . 2 1 . 2 3 . 6 y(in.) 6 . 0 < Fig. 6.46 Predicted u (dashed line) vs. experimental u (solid line). 159 Fig. 6.49 Predicted v, y (dashed line) vs. experimental v, y (solid line). F i g . 6.50 Fringe p a t t e r n of Exp. 122S3 F i g . 6.51 Fringe p a t t e r n of Exp. 122S6 0 1 2 3 4 y ( i n . ) 5 f i n i t e element s o l u t i o n experimental s o l u t i o n using the f i n i t e element u(y,0) and v(y,0). as boundary c o n d i t i o n s .-. . . experimental s o l u t i o n using the experimental u(y,0) and v(y,0) as boundary conditions F i g . 6.53 Contours of constant displacement v(y,z) i n the v a r i a b l e c r o s s - s e c t i o n specimen. M 163 f i n i t e element solution — — experimental solution using the f i n i t e element u(y,0) and v(y,0) as boundary conditions .... experimental solution using the experimental u(y,0) and v(y,0) as boundary conditions Fig. 6.54 Strain v, (y,l) in the variable cross-section specimen. 1500 f i n i t e element s o l u t i o n x ~ experimental s o l u t i o n o '—•> Exp. 122S1 @ 122S2 1000 500 4 y ( i n . ) 5 F i g . 6.55 S t r a i n v, y(y,0) i n the v a r i a b l e c r o s s - s e c t i o n specimen. ON 165 F i g . 6.56 Fringe p a t t e r n of Exp. 132D1 F i g . 6.57 Fringe p a t t e r n of Exp. 132D2 F i g . 6.58 Fringe p a t t e r n of Exp. 132S2 Fig. 6.59 Fringe pattern of Exp. 132S1. £ ON 167 7. CONCLUSIONS 7.1 Summary and Conclusions A f a i r l y involved theoretical analysis of the image and fringe formation by DASC and DASSC has been undertaken. The accuracy of the resulting equations relating the Moire fringe number to the deformed surface displace-ments and strains was verified by several simple and controlled experiments. Each experiment was set up to ascertain the accuracy of one or two coeffi-cients of the fringe equations. The subsequent and more complicated experi-ments have proven the f e a s i b i l i t y of the two cameras in experiments similar to those encountered in the typical laboratory testing of materials or struc-tural components. With some exceptions the agreement between the optically determined and actual displacements and strains was good. It was easier.to use DASC than DASSC since the fringes formed by DASC are due to.the displacements only; thus, the numbering of the fringes was relatively easy. In fact, in some special cases the fringes are related directly to only one displacement component. When DASC is used no boundary conditions are required to calculate the displacements, and the calculation involves the solution of a set of at most three algebraic equations, a pro-cess which is straightforward and easy to program. That DASC is insensitive to the out-of-plane displacements of the specimen surface near the coordinate origin may be considered an advantage i f the measurement of in-plane dis-iplacements is desired, but i t may make DASC potentially useless i f the out-of-plane displacements near the coordinate origin are to be measured. The main disadvantage of DASC stems from i t s relative i n f l e x i b i l i t y as the sensitivity may be varied only by changing D,X or x g, the possibilities of which exist only in a rather narrow range. The attractiveness of DASSC stems from i t s great f l e x i b i l i t y since the sign and the size of the lateral shear may be set by an appropriate 168 positioning of the photographic plate with the adjustable slide. Thus, i f the specimen deformation is approximately known before the test, the lateral shear may be chosen so that the density of the resultant fringe pattern is suitable for processing. The camera is sensitive to the surface displacements and strains (actually to partial derivatives) and, hence, the fringe patterns could be used qualitatively to identify those areas of the surface where stress concentrations occur. The calculation of the displacements in special cases often involves a solution of an ordinary linear differential equation or, in the general case, a solution of a set of partial differ-./ ential equations for which the f i n i t e difference approach was found suitable. The solution requires a point boundary condition for the particular dis-placement in special cases and, in the general case, a line boundary condition for the three displacements. The solution schemes for DASSC are usually more complicated than those for DASC. If the required boundary conditions are available, the results obtained by DASSC are usually superior to those obtained by DASC. Both DASC and DASSC were found to be useful and reasonably accurate instruments for measurement of displacements and strains. Which of the two cameras is to be used depends greatly on the particular circumstances of the proposed test. If the boundary conditions are readily available, then DASSC should be used as i t is likely to provide a more accurate solution. When a l l displacements and strains are to be determined the illumination not coincident with the x,y plane must be provided and the researcher should be ready to do a rather large amount of programming to interpret the fringe patterns obtained by DASSC. On the other hand, i f the boundary conditions are not available or i f a researcher wishes to minimize the amount of experi-mental work and computing effort, then the use of DASC should be considered. The approximation of the fringe function n(y,z) by a continuous cubic 169 based on fringe centers (multiplies of .50) along a line z = z Q = constant was found to be satisfactory where solutions along a line z = z Q were sought. If a two-dimensional approximation of n(y,z) was necessary,.then i t was found to be sufficient to scan the fringe pattern along a number of lines z = constant and to approximate n(y,z) along these "scan" lines by a continuous cubic. The fringe number n(y,z) anywhere else was then found by f i t t i n g a continuous cubic along a line y = constant through the points where this line intersects already approximated scan lines. 7.2 Suggestions for Future Research Mainly due to a lack of time two solution schemes were not tested and, hence, their accuracy remains yet to be experimentally verified. The f i r s t experiment should be such that none of the displacements u,v or w may be neglected and that these three displacements should be determined from three independent fringe patterns obtained with the use of DASC. The second experiment would be similar, but DASSC with at least one illuminating beam not in x,y plane would be used to take three independent photographs. A line boundary condition would have to be available and the three displace-ments u,v and w could then be calculated from the three fringe patterns by using the proposed f i n i t e difference scheme. The experimental work done in this thesis was restricted to specimens with one planar surface. There is no reason why DASSC could not be used to measure displacements and strains in specimens having shallow curved sur-faces or surfaces consisting of planar and curved surfaces. It would only be necessary to relate the lateral shear Ay g to the third, i.e. the x, dimension of the curved surface. This dependence of the lateral shear on the (y,z) coordinates could be easily incorporated in the computer programs of the f i n i t e difference schemes used to calculate displacements and strains. 170 The "depth" of the specimen surface along the x-direction would, of course, have to be reasonably small so that the lateral shear would not be excessi-vely large. Numbering of the fringes could be made easier, in some cases, i f one more photograph were processed than the minimum number required. With the knowledge of the deformations on the line boundary the fringe numbers at the boundary could be calculated and the fringe numbers at the neighbouring points could, in principle, be determined by finding that fringe number which would satisfy the overdetermined system of equations at those points. This process would be repeated until the fringe numbers of a l l points would be known; in fact, numbering of the fringe centers would probably be sufficient. 171 BIBLIOGRAPHY Duffy "Moire Gauging of In-plane Displacement Using Double Aperture Imaging" Applied Optics, 1778-1781, Aug., 1972. Duffy, D., "Measurement of Surface Displacement Normal to the Line of Sight", Experimental Mechanics, Vol. 14, No. 9, pp. 378-384 (1974). Hung, Y. Y. and Taylor, C. 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B., "Advanced Calculus for Applications", Prentice-Hall, New Jersey, 1962, p. 7 175 APPENDIX A DERIVATION OF EQUATION (2.9) 2 2 I(t) = |EjI + |E2| + 2E X.E 2 cos (kr x - tot + ijjj) + a. cos (kr 2 - tot + i>2) + a 2 l r i J l r 2 j (e . e 2)cos(kr 1 - tot + i(; 1)cos(kr 2 - tot + I|J 2) I>- = / Kt) dt 0 / cos 2(kr 1 - tot + ijij) dt + 0 ^ 2 / cos 2(kr 2 - tot + i>2) dt + 0 (e^ e 2) / cos(kr 1 - tot + ip 1)cos(kr 2 - tot + ip2) dt 0 Let us now evaluate the three integrals and approximate the results by making use of the fact that T » T . T / cos (kr z - tot.+ ijjj) dt = 0 _T T _ 2 8TT sin2(kr 1 - toT + i f^) - sin2(kr x + I/J 1) T 2 T / cos 2(kr 2 - tot + tyz) dt - — / cos(kr 1 - tot + i | J 1)cos(kr 2 - tot + ty2) dt = -^cos (kr x - k r 2 + i j ^ - ty2) -0 T 8TT sin(kr 1 + k r 2 + 1 + i p z - 2toT) - sin(krj + k r 2 + ty1 + i p 2 ) cos (kr x - kr 2 + ifjj - \p2) With these integrals,I r is given by equation (2.9) 176 APPENDIX B DERIVATION OF EQUATIONS (2.18) AND (2.19) cos[kr(y,z) - wt + ty] = cos(kr - cot + 40coskr e - sin(kr Q - cot + ^ )sinkr £ K t ) E r(t) 2 a |[cos(kr 0 - cot + i(;)coskre - sin(kr 0 - cot + iJO sinkr e] dA Let us define the integrals Ej = / coskr e(y,z) dA A and E = / sinkr e(y,z) dA With Ej and E 2 , I(t) becomes K t ) = ' a ^ 2 [E 1cos(kr Q - cot + ty) - E 2sin(kr Q - cot + ty)~\' The recorded intensity I r is given by I r = J I(t) dt = 0 2 a l r o J T T E 2 / cos 2(kr Q - cot + ty) dt + E 2 / s i n 2 ( k r 0 - cot + ty) dt 0 0 - 2E 1E 2 / sin2(kr Q - cot + ty) dt 0 In Appendix A i t was shown that t 2 T J cos (kr Q - cot + ty) dt - — 0 T r 2 T and similarly i t could be shown that J sin (kr 0 - 0)t + ty) dt - — j sin2(kr p - .cot + ty) dt = [cos2(kr 0 - cot + ty) - cos2(kr 0 + tyj] « ^ With these integrals I r is closely approximated by equation (2.18) 177 With the use of complex notation equation (2.18) may be written as equation (2.19), as is shown: j e i k r e ( y , z ) d A = j c o s k r e ( y ) Z ) ^ + i / sinkr e(y,z) dA A A A - ikr e(y,z) u n — A A / e 1 K re (-y' z-' dA = / coskr e(y,z) dA -if sinkr e(y, z) dA ikr e(y,z) dA ' A i k r e ( y , z ) d - fcoskr„(y,z) dA ^ A ^ ° /sihkr e(y,z) dA A ' ' • " -Ir = T 2 / el k r e ( y ' z ) d A /e ^ A - ikr e(y,z) dA (2.19) 178 APPENDIX C DERIVATION OF EQUATION (3.5) The equations of the source (sw) and diffracted (dw) spherical wavefronts shown in Fig. C.l are given as 2 2 2 2 2 2 dw: x + (y - y D) + (z - z D) = x s + (y D - y A) + z D (C.l) sw: x 2 + (y - y g ) 2 + (z - Z g ) 2 = x s + (y g - y A ) 2 + z 2 (C.2) The equation of the diffracted ray is the equation of the line from the "apparent" source point D(0,yjj , Z j j ) to the point (x s,y a,z a) in the aperture. y^ — yj-diffracted ray: y = — x + y D (C.3a) x s z = — ^ x + z n (C.3b) Fig. C.l Diffraction in a single aperture camera. 179 Huygen's principle permits us to assume that the diffracted ray originates from the source wavefront at the point (x 2,y 2,z 2) and reaches the diffracted wavefront at the point ( x ^ y ^ Z j ) . Therefore the diffracted wavefront "leads" the source wavefront by the distance r e ( y a > z a ) given by r e ( v a /2 2 2 , a. (xx - x 2) + (yj - y 2) + (z x - z 2) (C.4) The magnitudes of the parameters of D A S C that is to be used are such that x s > 30 in. D < 2.5 in. , y A < A - 2 x 10~ 5 in. 1.25 in. Axc |yD - y s|,|z n - Z g | < D s s = 2.44-^— hence Ay , Az < 4 x 10 The f i e l d of view is such that Let us now define the following quantities: a = y a " Y D vs J + z a " Z D a < 4 x 10 - 2 b = f y p - Y A ! 2 . ( Z ^ 2 *s J b < 4 x 10 - 2 Y A - y p » Z D A c = Ay - — Az x s x s c < 1.6 x 10 - it r x„ < 2 x 10 d* = 3(Ay cos9 + Az sin9) |d*| < 1.6 x 10 6 It is convenient to introduce a polar coordinate system of the circular aperture, as is shown in Fig. 3.3 , so that we may write y a and z a as y a = y A + r cost" z a = r smfc 180 The equations (C.l) and (C.3) are evaluated at the point (x 1,y 1,z 1) and the equations (C.2) and (C.3) are evaluated at the point ( x 2 , y 2 , z 2 ) . By combining these equations and using the quantities that have been defined we get a quadratic equation in (r e/x g) fr 1 - 2 f l x s J l x s J c + d*l /¥ J + 2 (/G ^ (c + d*) - c = 0 (C.5) where F and G are defined as F = 1 + a G = 1 + b The solution of equation (C.5) for r e is given by equation (3.5). Its derivation is quite tedious as Taylor's series for the square root and seve-ra l fractions must be used to determine r e with the desired accuracy; of A./10' By making use of equation (3.5) i t may be shown that |r e/x s| < 1.5 x 10 5 With the magnitude of r £ known, we see that i f r e is to be determined with Q A/10 accuracy (or r e / x s with 6.7 x 10 accuracy), the quadratic term in equation (C.5) is so small that i t may be neglected. This was done and i t was found, that the solution of the resultant linear equation is s t i l l given by equation (3.5). The latter solution i s much less tedious and therefore this approach i s used in the subsequent appendices whenever possible. 181 APPENDIX D DERIVATION OF EQUATION (3.7) Let us define the following terms: 1 - 2x|j Ay - (D.l) D 2 = 1 -3z s l 2x2J Az -'sys Ay (D.2) kd / 2 2~ /D, + D0 (D.3) Using equations (D.l),(D.2) and (3.5) we may write kr e(r,0) = kr(D cos8 + D 2sin6) (D.4) The recorded intensity I r is evaluated by using equations (2.19) and (D.4) as Ir = d f 2J eikr(D l Cos0 + D 2 s i n 0 ) r d r d 6 . 0 0 d f 2/ e" ikr(D l Cos0 + D 2 s i n 0 ) ^ d Q 0 0 d/2 2TT , Z „ , N r r ± ikr(D ncos0 + D„sin0) , J Q ^d 2J!(p) Since J / e - 1 2 r dr d0 = —; - - 4 p 0 0 (D.5) (D.6) x r = d 2 TTd 2,2 (D.7) Using I Q given by equation (3.8), I may be written as X r = Z o f2J!(p) I P J (3.7) 182 APPENDIX E DERIVATION OF EQUATIONS (3.10) AND (3.11) Using equations (D.l),(D.2) and (D.3) we may write the following: kd 2 2 . 2 , 2 = Dj + D2 - Ay + Az With Ay and Az defined by equation (3.6), the above equation may be written as ( y D _ y s ) 2 + ( z D - z s ) 2 = 2xc kd Equation (E.l) is the equation of a circl e of the radius f 2 x c kd (E.l) , and with the center at ( y g , Z g ) in the specimen plane. By making use of equation (2.12) we may put equation (E.l) in the form (yi - y i c) 2 + (z± - zic>2 - f?a p l kd p ( E . 2 ) f 2 x . kd This time equation (E.2) is the equation of a circl e of the radius and with the center at (y±c>z±c) l n the image plane. From equation (E.2) i t i s apparent that I r is circular with i t s center at the geometric image R ( y ^ c , Z i c ) of the source point S ( y g , Z g ) . As shown in Fig. 3.4 , I r varies as f2J . 1 (p)l ; i t has the maximum at p = 0 , and i t s f i r s t minimum at f2xi kd 3.832 1.22 X - r d p = 3.832 . The area of the circl e of the radius is known as the Airy disk and i t w i l l be referred to as the '"speckle" . The diameter of the speckle in the image plane w i l l be called D s and the diameter of the apparent speckle in the specimen plane w i l l be called D s s The two diameters are given as D s = 2.44 X~f (3.10) Dss = 2 - 4 4 X " t (3.11) 183 APPENDIX F In this appendix i t is shown that the unit electric f i e l d vectors of a l l rays reaching a point in the image plane are nearly parallel so that the vector nature of the rays may be neglected in the calculation of I r . Image o r photo p l a n e Fig. F.l Unit vectors of electric fields. y = tany - (y f c - y 1)/x ± W ^ n l = lemkl l^nllcosY = cosy = 1 - \ + d -0625 For the single aperture camera Y < ~ < —^2— = '0^52 and hence e m k * e n l ~ 2 - 5 with the error being smaller than .0052 /2 = 1.4 x 10 D + d 2.5 + .0625 When DASC or DASSC is considered, Y < < = x± 12 time e ^ 'e ^ - 1 with the error being smaller than .2142/2 s t i l l sufficiently small. 214 and this .023 which i s (the dimensions of d,D and x^ are in inches) 184 APPENDIX G DERIVATION OF r e i AND r £ 2 FOR DASC DURING THE FIRST EXPOSURE. The optical path lengths from the apparent point source S to the point Q, in the image plane , are given by through aperture 1 : S-Q = S-sWj + d W j - s W j + d W j - Q = r through aperture 2 : S-Q = S-sw2 + dw2-sw2 + dw2~Q = r 2 In a focused system the optical path lengths from the point D to i t s geometric image Q are a l l equal, and therefore we may write D-dw2 + d W j - Q = D-dw2 + dw2~Q dw2-Q = D - d W j - D-dw2 + dw^Q Using the last equation the optical path length through aperture 2 is given as S'-Q = S-sw2 + dw2-sw2 + D - d W j - D-dw2 + dwx~Q + ( S - s W j -vS-sw1 ) We define the mean optical path length r Q as r Q = S - s W j + d W j - Q With r Q now defined the total optical path lengths are given as r i = r o + dwj-swj r 2 = r Q + dw2-sw2 + ( D-dwx - S-sw1 ) - ( D-dw2 - S-sw2 ) The variations in the two optical path lengths are then given by r e i = r x - r 0 = dwJ-sw1 (G.l) r £ 2 = r 2 - r Q = dw2~sw2 + ( D^ dw~ - S - s W j ) - ( D-dw2 - S-sw2 ) (G.2) We must now express r £ 1 and r g 2 in terms of system parameters. The equations of the source and diffracted spherical wavefronts are given as 185 dw2 sw2 2 X 2 X + (Y - y D r + (z - z D r = x; + (y D - y A ) ~ + z D (G.3) + (y - y D ) 2 + (z - z D ) 2 =x 2 s + <yD + y A ) 2 + z 2 ( G > 4 ) x 2 + (y - Y S ) 2 + (z - z s ) 2 = x 2 + (y s - y A ) 2 + 2 2 (G.5) x 2 + (y - y s ) 2 + (z - z s ) 2 = x 2 + (y s + y A ) 2 + z 2 (G.6) The equations of the diffracted ray are the same as before and are given by equations (C.3) . Note that the equations of dw2 and sw2are the same those of d W j and sw1 except for the sign of y A . r e l has been derived in Appendix C (where r e l is called r e ) , and is given by equation (3.5) . An examination of that equation reveals that r e is not a function of y A and hence we may write r £ l = dw1-swJ = dw2-sw2 = r e (G.7) r e 2 = dw2-sw2 + ( D - d W j - S-sw1 ) - ( D - d w 2 - S-sw2 ) / 2 2 2 / 2 ~ 2 2 = r e + / x s + (y D - y A) + z n - / x s + (y s - y A) + z s " / x s + ( yD + y A ) 2 + ZE> + / x s + ( yS + y A ) 2 + 4 (G'8) Let us now define the following parameters: Xso = x s + yS + ZS + yA A = (yj " y| + 4 ~ 4>/Xso |A| < 3 x 10" 4e = ^ A^D " V / X s c |e| < 3 x l O " 5 2y Ay s/X ; 2 I ,1 2 SO f < 1.3 x 10 With these parameters and using the Taylor's series expansion for the square roots in equation (G.8) and by neglecting terms smaller than A/30 we get AT X. 186 The last equation may be accurately approximated by r e 2 = r e - D6 (G.9) where 6 is given by 6 = x _ 4 . ] z p_iys . y ^ s f p ^ f s ( G > 1 0 ) x s o ' x s o x s b x s o Note that a small change in y Q alters 6 much more than the same change in would; because of this, and using equation (2.12) 6 is accurately approximated as 6 = - 7 1 " 7 1 0 (3.13) 187 APPENDIX H DERIVATION OF EQUATION (3.12) - THE FIRST EXPOSURE I r ,2m|d/2 27T d/2 2TT T . 2 d T/ /e 0 0 d/2 2TT / /e 0 0 1 + e ikr~ - i k r P ;ik.(r e- D 6 ) r d r d G 0 0 d/2 27T r dr d0 + / /e 0 0 - i k ( r e - D<5) r dr d9 1 + e LkD.6 d/2 2TT / /e 0 0 d/2 2TT / /e 0 0 i k r e - ikr„ (H.l) The values of the integrals in the above equation are given by equation (D.6) and with 1 + e - ikDS 1 + e ikD6 = 4cos kD6 equation (H.l) becomes !r = [a] 2 T r ud2i 2 r 2 j 1 ( p > i k J ~2 [ 4 J i p J kD6 cos (H.2) With I„ given by equation (3.8), I may be written as I r = 4I 0 f2J 1(p ) l kD6 cos (3.12) 188 APPENDIX I DERIVATION OF EQUATIONS (3.14) AND (3.15) The minima of I r given :by equation (3.12) occur when either of the terms 2Jj(p) 2kD6 or cos — r — is zero. The f i r s t term has already been discussed in P 2 Appendix E where i t i s shown that I r may be approximated by the Airy disk of diameter Dg in the image plane, and of diameter D s s in the specimen plane. The second, cosine, term modulates the f i r s t term..so that in this case the resultant I r looks as is shown in Fig. 3.7 . The period of the modulation is called the speckle grid pitch G g and i t is calculated from two consecutive zeros of the cosine term. for y. - y i c = G s : kDS ^ kD(y ± - y l c ) _ kDGs 2 2x^ 2x^ Ax-£ since k = 2TT/A G s = (3.14) By using equation (2.12) the apparent speckle grid pitch G g g ' i s determined' from the last equation as Ax G s s = ^ (3.15) 189 APPENDIX J DERIVATION OF r e i AND r e 2 FOR DASC DURING THE SECOND EXPOSURE Due to a .general deformation of a specimen, the point S representing the specimen surface is displaced to S , and i t s coordinates^change between the two exposures as S(0,y s,z s) + S*(u,ys + v,z s + w) The equations of the diffracted wavefronts dwx and dw2 , as well as those of the diffracted rays remain unchanged and are given by equations (G.3), (G.4) and (C.3) respectively. The equations of the source wavefronts swx and sw2 must be modified to account for the altered, second exposure position of S . sw, (x - u r + y - (y g + v) + z - (z s + w) = (x, - u r + (y s + v) - y A + (z s + w) (J. 1) sw. (x - ur + y - (y s + y) + z - (z + w) = (x s - u) + ( y s + v) + y A + (z g + w) (J.2) Calculations completely analogous to those presented in Appendix C w i l l again be done. The diffracted wavefronts lead the source wavefronts.by the distance r„(y„,z_) given by equation (C.4). 6 a- d We now define U,V and W as U = u U < 1.3 x 10 - 3 for u < 1 mm - .04 in. v V = x V < 4 x 10 - k s w W = W < 4 x 10 - 4 for v < .012 in. for w < .012 in. 190 r , the optical path length variation of the rays passing through aperture 1 , w i l l be determined f i r s t . The equations (G.3) and (C.3) are evaluated at the point (x1,y1,z1) and equations (J.l) and (C.3) are evaluated at the point ( x 2 , y 2 , z 2 ) . A l l these equations are combined and we eventually get a quadratic equation in ( r 6 l/ x s) f r e i ^ *s ) - 2 *s J lA—ID ( A y _ v ) _ fp_ ( A z _ w ) _ u + + /F (Ay - V)cos9 + (Az - W)sin6 + 2 /G y^ LZa ( A y _ V) - (Az - w) - u + B (Ay - V)cos9 + (Az - W)sin0 + 2Ay Z P ^ 2 4 + 2 A z *D x c 2 V ZB Z4 _ 2W + 2U = 0 (J.3) In Appendix C i t was shown that the quadratic term may be neglected when r is to be determined to X/10 (or in fact to X/30) accuracy and this was done here too. Using Taylor's series expansions for square roots and fractions, the solution of equation (J.3) (with the quadratic term neglected) is found to accuracy of X/30 as r = r ei 1 -1 -3yi (Ay - V) - (Az - W) -zx s , y c Z o (Az - W) - (Ay - V) -x c 1 -1 -3y£ + 3z 2xT~ 3y2 + 3z 2x1 2 2 -> ys - yA x r U COSD + x c sxnt J(J.4) Since r = d W j - s W j is this time a function of y^ , we must replace y^ in equation (J.4) by -y^ to get dw2~sw2 , and r £ 2 is then calculated as r e 2 = r e i ( y A _y A) + ( D _ d w ! " S-sWj ) - ( D-dw2 - S-sw2 ) 191 r e 2 = r e i ( - y A ) + A \ + (y D - y A) 2+ z 2 - / ( x s - u) 2 + [(y g + v) - y j + (zg + w) 2, - / x | + (y D + y A ) 2 + z D + ' < x s - u > 2 + [(y s + v) + y A ] 2 + ( Z g + w) 2 (J.5) Using Taylor's series expansions for the square roots in the above equation, r , accurate to A/30 was found as r e 2 = r e i ( - y A ) - D « S - K) (J.6) where £ is given by equation (3.16). With Dj and D2 defined by equations (D.l) and (D.2) we define E ,E 2 and F as 1 - 3y 2 2x|j _y_ y s z s w + 2 X S X S X S 1 -3y| + 3z| 2x| y s u X S X S E 2 = D2 1 -3 zS 1 w 2x|J x s + y s z s v 2 X S X S 1 - 2xi Z g _u_ x s x s 1 -3yf, + 3zS 1 D/2 u 2x' We can now write equations (J.4) and (J.6) as r e l = r [ ( E x + F)cos0 + E 2sin9] r e 2 = E [ ( E ! " F)cos6 + E 2sin0] - D(6 - £) (J.7) (J.8) 192 APPENDIX K DERIVATION OF EQUATION (3.17) - THE SECOND EXPOSURE I r The substitution of the second exposure optical path length variations r ei and r e 2 , given by equations ( J.7) and ( J . 8 ) , in equation (2.19) gives the second exposure recorded intensity as 1^  = d/2 2u _ d/2 2TT _ | / e i k r e i r d9 dr + / / e i k r e 2 r d6 dr 0 0 0 0 d/2 2TT . d/2 2TT / |e~ ± k r e i r d6 dr + / / e ' ^ r d B dr 0 0 0 0 (K.l) We define px and p 2 as kd Pi =~2 /(Ej + F) 2 + E 2 kd 'z ~2 p, = — / ( E 1 - F ) 2 f E (K.2) (K.3) The integrals of equation (K.1) are evaluated as d/2 2TT / Je± r < 2 i r d0 dr 0 0 d/2 2TT / fe± r e 2 r d6 dr 0 0 T\d2 J ^ P , ) TTd^ _ J : ( p 2 ) e+ ikD(6 - 5) 4 p, e (K.4) (K.5) Using these results equation (K.l) becomes f Trd 1 2 — r Ri(Pi) , -ikD(6-£) J i ( P 2 ) l f Ji(Pi) , ikD(6- g) Ji(P 2>l I Pi + e '2 J + e Trd 1 f J ^ P i ) J I ( P 2 ) 1 I Pi P5 + '2Ji(Pi)1 I Pi J f2Ji(P 2)l cos ,kD(6 - g) 2 (3.17) 193 APPENDIX L DERIVATION OF EQUATIONS (3.19) AND (3.20) From the form of the second exposure I r i t i s apparent that the speckle grid has been shifted with respect to i t s f i r s t exposure position. This shift produces Moire fringes and i t w i l l be now related to the camera para-meters and the displacements u,v and w. Using equations (G.10) and (3.16) defining 6 and £ respectively we write kD(6 - Q 2 2 xr kD 2 1 - '•so yp - ys _ ys zs Z D - zs .2 X so X so X so X so X so D X so ys . zs v 1- w X so X so' X so 1 - yi X + w y s z s so' X s o X s o IX + so X so' (L.l) During the f i r s t exposure the speckle grid "passed" through the center of the speckle, and hence we may write YSgl ~ Yg > where Y Sgj l s analogous to y^gj ; i t denotes the y-coordinate of the "center", or the f i r s t maximum of the modulating term, of the apparent speckle grid in the j-th exposure. We find Y Sg 2 > the speckle grid "center" during the second exposure, on the line z = Z g from kD(6 - O = 0 (L.2) where y n is replaced by Y Sg 2 • The solution to equation (L.2) was found as 'Sg2 Yc + v + y i - 1 X so (Is x 0 + u 2x s VYS + W Z S X so vso so - w ys zs 'SO (L.3) Equation (L.3) is accurately approximated as y s g 2 = ys ys yszs v H u - w (L.4) 194 The Moire fringe number n is found from equation (3.18) which by using equation (2.12) may be written as Y s g 2 y s g i v sg2 y S = - nG ss (L.5) n is determined from equations (L.3) and (L.5) as n = - Axc 1 -2 X - 1 so-* y s x s + u 2xs vy s + wzcO y s z s X so so vso - w *-so + V (3.19) Equation (3.19) may be closely approximated as Axc y s u + v - yszs w (3.20) 195 APPENDIX M DERIVATION OF EQUATION (3.21) The envelope of the speckle grid term of equation (3.17) is given by the 2J 1(p.) 2J,(p 2) product . The two terms are circularly symmetric about Pj P 2 their respective maxima (centers) which occur when P1 = 0 (M.l) P 2 = 0 (M.2) Using equations (K.2) and (K.3) defining p and p 2 the approximate solutions of equations (M.l) and (M.2) are given by Pi = 0 : YD = yDi = ^  + v + u ^ (M.3a) s Z c ZD = zD.i = z s + w + u ~ (M.3b) yS + yA P 2 = 0 : y D = y D 2 = y g + v + u x (M.4a) ZD = ZD 2 = Z D l ( M ' 4 b ) We denote the coordinates of the centers of the two speckles by ( y 1 C l 5 z i c l ) and ( y i c 2 , z i C 2 ) respectively. By making use of equation (2.12) these centers are determined from equations (M.3) and (M.4) as y±ci = ~~ym = y i c - + zi c l = - ~ z D l = z i c - w f : + u ~ (M-5b) x i x i Yic x i D y i C 2 = - ^  y D 2 = y i c - v ^ + u ~ u ^ 2x; ( M - 6 a ) z i c 2 = - ^  ZD 2 = z i c i <M'6b) 196 where y^ c and z^ c are the coordinates of the f i r s t exposure speckle centre. The limit on the magnitude of the displacements u,v and w is obtained from the requirement that the f i r s t and the second exposure speckles must overlap so that Moire fringes may be formed. We write -5 Dc (M.7a) - z- Y L C 2 i c ; -2 Dr (M.7b) (y. - y. Y+ (z. - z. ) V J i c i J i c ' v I C I i r (yica - y i c ) 2 + < zi Using equation (2.12) and by combining the two equations (M.7) we get y s ± D/2f v + u x c + w + u — x„ D xc < = 1.22 X-r (3.21) 197 APPENDIX N DERIVATION OF r_,-1 FOR DASSC DURING THE FIRST EXPOSURE With r Q defined by equation (3.22) as r Q = S2-sw12 + dWj-Q the four optical path lengths r ^ from the two point sources Sj and S 2 to the point Q in the image plane are given as S 2 -sw 12 + d w l - s w i 2 + d w i -Q = r Q + dw r S W 1 2 (N. 1) S l " S W 1 1 + dw1 " s w n + d W j -Q + ( S 2-sw 1 2 " S 2 " S W 1 2 ) ro + d w i " s w n + (s i " s w n " S 2 - S W 1 2 > (N. 2) - s w 2 1 + d w 2 - s w 2 1 + d w 2 -Q + ( S 2 - s w 1 2 " S 2 " S W 1 2 ) In Appendix G i t was shown that dw2~Q = D-dWj - D-dw2 + dWj-Q and hence r2l may be written as r 2 1 = r Q + dw2-sw21 + ( D-dWj - S 2-sw 1 2 ) - ( D-dw2 - Sj-sw 2 1 ) (N.3) r 2 2 = S 2 ~ S W 2 2 + d w 2 - S W 2 2 + d w 2 " ^ + ^ S 2 ~ S W 1 2 ~ S 2 ~ S W 1 2 ^ = r Q + dw2-sw22 + ( D-dWj - S 2-sw 1 2 ) - ( D-dw2 - S 2-sw 2 2 ) (N.4) The optical path length variations r •• are then given by r e I 2 = r ! 2 " r o = dw^sw^ (N. 5) r ei i = r i l " r o = d W j - s w ^ +(S 1-sw 1 x - S2-sw 1 2 ) (N. 6) r e 2 l = r 2 1 " r o = dw2-sw2 j + ( D - d W j - S 2 - s w 1 2 ) " ( D - d w 2 - S x-sw 2 1 ) • (N. 7) r e 2 2 = r 2 2 - r 0 = d w - s w ' 2 2 2 + ( D - d w - S - s w 1 2 12 ) " ( D - d w v 2 - S - s w 2 2 2 ) (N. 8) The four spherical source wavefronts have the centers at the points S (0,y ,z ) and S (0,y ,z ) and pass through the aperture centers at y and -y^ respectively. The two spherical diffracted wavefronts have the center at the point D ( - Y » Y D , Z Q ) and pass through the aperture centers at ±y A. The diffracted ray appears to originate from D ( - Y , Y T ) , Z R ) ) and passes 198 through the point ( x sjy a» z a) l n t n e aperture. The equations of the source and diffracted wavefronts and of the diffracted ray are given as dwx : (x + Y) 2 + (y - Y ) 2 + (z - Z D ) 2 = 4 + ( Y D - yA> 2 + 4 (N.9) dw2 : (x + Y) 2 + (y - Y ) 2 + (z - V 2 = 4 + (Y D + y A) 2+ Z2 (N.10) sw : 12 x 2 + (Y " y S 2 > 2 •+ (z - = 4 + ( y S 2 - yA>2 + 4 (N.H) s w n : x 2 + (y " yS l ) 2 + (z - zs>2 = 4 + - yA ) 2 + 4 (N.12) sw21 : x 2 + (y - y S l> 2 + (z - *s>2 = 4 + + y A ) 2 + z| (N.13) sw : 22 x 2 + (y " y S 2 > 2 + (z - zs>2 = 4 + ( y S 2 + yA>2 + 4 (N.14) y ~ Y n diffracted ray : y = (x + y) + Y n (N.15a) x s z = Z a " Z ° (x + y) + Z n (N. 15b) A s The case Ay g > D G G is considered f i r s t . In Section 3.5 an argument was made that in this case only the diffracted light from the source wavefronts sw12 and sw 2 1 contributes significantly to I r at Q. The positions of SJDJSJ and S 2 are shown in Fig. 3.12 , with the coordinates of D being such that |S-DI < D g s. The position of the imaginary point source S was chosen so that (hopefully) the blurred and sheared images of the real point sources Sj and S 2 would both be centered about the point R, the geometric image of S. The subsequent analysis has shown this choice to be correct. The geometry of DASSC is such that the positions of Sx and S 2 are related to the position of S by yS l = Y s - ^ ( Y s + yA> y s 2 = Ys - <YS " yA> 199 Let us define (and redefine) the following parameters: a = 1 + b = 1 + F = 1 + a fy a - Y D1 2 + f za - z D l 2 1 x s x s j r Y D - y^  2 + 2 { x s a < 4 x 10 b < 4 x 10 - 2 G = 1 + b Ay = 3 Az = Z D ~ Z S r x l < 1 x 10 - l Ay < 4 x 10 - h Az, < 4 x 10 3 < 2 x 10 - >t The distances dw^-sw^j are determined in the same way as in Appendix C. The diffracted ray originates from the source wavefront at the point (x 2,y 2,z 2) and reaches the diffracted wavefront at the point (x 1,y L l,z 1) and therefore the distance by which the diffracted wavefront leads the source wavefront i s given by dw,- -sw,- = /(x : - x 2 ) 2 + (y 2 - y 2 ) 2 + (zx - z 2 ) : -elj l - " I J We introduce the polar coordinate systems of the two apertures (N.16) aperture 1 aperture 2 : z, = y A + r cos9 r sinG - y A + r cost r sin9 r e i 2 w i ± x be determined f i r s t . The equations (N.9) and (N.15) are evaluated 200 at the point (x1,y1,z1) and the equations (N.ll) and (N.15) are evaluated at the point (x 2,y 2,z 2) . From these equations and equation (N.16) a quadratic equation in ( r e i 2 / X g ) is obtained as ei 2 L. Xs J r r - 2 ei2 I x s ) /G + — /F + y A - D Ay + 3i Y s - yA' xc Xc Az + 3i Xc + 3 Ay + 3f rs - y A xc COS0 + Az + 3X X sJ sin0 + 23x 1 -/G /FJ + 2 y A - Y D Ay + Y s - y A l i Xc ZP_ X c Az + 3 ^ + 3 Ay + 3 ^ s - y A COS0 + Az'+ 3j Xc vs; sin0 + o Y D ~ yA A . 9 Z D . . 9 R ( Ys - y A ) ( Y D - yA> + z D z s  + 2 —JT~ y ^ B l x i = (N.17) The small quadratic term is neglected and by making an extensive use of Taylor's series we obtain an approximate solution to equation (N.17), accurate to A/30 as r e i 2 = q £ 2 + £ ( K i c o s 9 + K 2sin0) = r £ (N.18) where q,Kj and K2 are defined as 3, 2X K i = K2 = 1 -1 -3Y ?N 2X* 3Z S Ay -YS ZS xi Az 2X|J YqZq Az - Ay xi We shall now determine r. "e2 l Since r is not a function of y. we may ei2 J A 3 write dw-sw„, = dw-sw,„ = r and r is then given as 2 2 i 1 1 2 e e2i r e 2 i = ^e + ( P " d w i " S 2 _ s w i 2 ^ ^  ( D'^ dw2 - Sj-sw 2 1 ) 201 "e2 i = re + + ( Y D " V 2 + ZD - / x s + ( yS " y A ) 2 + Z - / x | + (Y D + y A ) 2 + z 2 +/x|+ (y g + y A ) 2 + z s (N.19) Using the Taylor's series expansion for the square roots in the equation above,,an approximate solution for r £ 2 1 is found as r ^ r - Dp ez 1 e r (N.20) where p i s defined as 1 -x 2  A s o YD " YS vso YS ZS D^ ZjS x s o x s o + Pi X so (N.21) with X S Q redefined as X 2„ = X2. + Y| + Z2, + y (N.22) When the case Ay s < D s g is considered the parameter 3! is such that — 3 |gi| < 6 x 10 and consequently * e i z and * e 2 1 may be further approximated as r e i 2 = £ ( K ! C O S 0 + K 2sin9) r e 2 1 = rO^cosO + K 2sin9) - Dp (N.23) (N.24) To determine r * ^ = dw1~sw11 we use equation (N.16) again. Equations (N.9) and (N.15) are evaluated at the point (x 1,y 1,z 1) and equations (N.12) and (N.15) are evaluated at the point ( x 2 , y 2 , z 2 ) . From these equations, and equation (N.16) we obtain a quadratic equation in (r*11/Xs). - e i i l x s - 2 - e i i (. x s ) /F 3 + y A - Y D P i X c Ay + B x YS + y A 1 Z D * Az + z s l + /F YS + y A Ay + 3 ^ cos9 + Az + 3X Xc sine + 2 1 - 3 1 + A y ^ + + Az ^  + 3, <YD " V y A " y A + YS YD + ZS ZD) X c .+ 23 7? YS + y A Ay + 3,-S? cos9 + 1 A c + Az + 3 - sin9 = 0 (N.25) sJ 202 With the small quadratic term neglected an approximate solution to equation (N.25) accurate to X/30 was found as r = r ei i i K I1 cos9 + K 2sin6 (N.26) r i s calculated from equation (N.6) as r e n = r e n + S i " s w i i " S 2 " s w i 2 = r. n  + M + <ys " y A ) : + zl -/: s - x s + <ys - yA> + ZS An approximate solution to the equation above accurate to X/30 is given by r = r ei l =• cbs0 + K2sin(: - a. + a A s (N.27) where a A and a s are defined as yiYsl L X S 0 x s o J O U = 2y A3 1 YA X s o l + 3 1 X 2 A s o r * 2 2 = dw2~sw22 is obtained by replacing y A in equation (N.26) by -y^ , and r e2 2 "'"s t^ i e n r o u n d from equation (N.8) r = r* - e 2 2 (y, n ( y A e n ^ A -YA) + ( D - d W j - S 2-sw 1 2 ) - ( D - d w 2 - S 2-sw 2 2 ) -YA) + A 2 + (Y D - y A ) 2 + Zj - /x* + (y s - y A ) 2 + z - A 2 + (Y D + y A) 2 + Z 2 + /x 2 + (y s + y A ) 2 + z An approximate solution of the last equation accurate to X/30 was found as r e22 £ D 5>x SJ cost + K2sin£ Dp + a A + a s (N.28) 203 APPENDIX 0 DERIVATION OF EQUATIONS (3.23) AND (3.24) - THE FIRST EXPOSURE I r The f i r s t exposure recorded intensity for the case Ay g > D s g is determined f i r s t . In this case I r is produced essentially by the interference of the light radiated by S2 and passing through aperture 1 with the light radiated by Sj and passing through aperture 2. Using equation (2.19) I r is given by Ir = d/2 2TT d/2 2TT / / e ± k r e i 2 r d9 dr + / / e ± k r e 2 i r d6 dr 0 0 0 0 d/2 2TT / / e l k r e i 2 r d9 dr + / je ^ e 2 i r d 9 d r d/2 2TT - ikr. 0 0 0 0 (0.1) The two optical path length variations r e i 2 and r e 2 1 are given by equations (N.18) and (N.20) and with these equations I is calculated as 1^ = 1 + e - ikDp 1 + e ikDp fd/2 277 / / e ± k r e r d 9 d r 0 0 d/2 277 / | e ± k r e r d9 dr 0 0 d/2 277 / /e" i k r e r d0 dr 0 0 d/2 277 / /e" ± k r e r d9 dr 0 0 4cos' kDp (3.23) Using the integration by parts equation (3.23) may be written as Ir- = 41, f2J1(p)] f2J1(R)^i R - 4 • R' -co SOJ —5- dR P . CO 2J1(p)^|PrR^|3 f2J1(R)'J / yo Ipj R + PJ 2J1(R)'| R sinco R^  R 2N 1 - dR+ 4-co sinco—5- dR P . kDp cos P / L"0 R IPJ (0.1b) where R and CO are defined as R = d/2 kqd< CO = co < 2 204 The f i r s t exposure recorded i n t e n s i t y f o r the case Ay g < D g s i s again calculated according to equation (2.19), but th i s time a l l the l i g h t " radiated by the two sources contributes s i g n i f i c a n t l y to the resultant I r and the i n t e n s i t y i s therefore calculated as Ir = d/2 2TT _ I f f e i k remn r d9 dr m= 1,2 0 0 n= 1,2 d/2 2TT I I Je" l k remn £ d6 dr 0 m= 1,2 0 n = 1,2 (0.2) We now define PsPi! and p 2 2 as kd P ( K 2 + K 2 (0.3) _ kd ' i l ~ ~~2~ kd 522 = ~2~ + Ki K ' l - B l ^ r ) + *2 (0.4) (0.5) The four r •, are given by equations (N.23),(N.24),(N.27) and (N.28). By ex j making use of equations (N.18),(N.20),(N.26) and (N.16) we may write equation (0.2) as r „ N 2 I r = ;elk(r2n " a A + + / e i k r e d A + / e l k ( r e " ^ d A + + | e l k ( r e 2 2 + a A + a s - D p ) ^ Je~ i k ( r e n " a A + a s ) d A + j - i k r e d A + + Je~ i k ( r e " ° P ) d A + | e " i k ( r e 2 2 + a A + a s - Dp) dA (0.6) where A i s the area of the c i r c u l a r aperture(s). Using equation (D.6) I r i s evaluated as I r = « 0 ik(ou - ou) J, , e s ~ ^  + J + Je ^ u v + J 2 2 e ^ v " A --ikDp i k ( a A + a s - Dp) 11 205 J n e - ikCa,, - a.) s -A' + J + J e — + J e" ± k ( a A + a s " DP> ikDp 2 2 (0.7) where J»J n and J 2 2 are defined as J 11 2 2 Jj(p) P i I J i ( P 2 2 > p r 2 2 To interpret equation (0.7) we assume that a l l products J m r l J k i have approximately the same value over the speckle area, I.e. we let J mn^kl ~ ^  Equation (0.7) is then put in the form I r - 16IQ fJl(p)] 2 , p + P 2cos - (Dp - a A) (3.24) where Pj and P 2 are defined as k k P 1 = 1 + coska scoska A - 2cos—(a A + a s ) c o s — ( a A - a s) (0.8) P 2 = 4cos|-(a A + a s ) c o s ^ ( a A - a s) (0.9) 206 APPENDIX P DERIVATION OF r„,, FOR DASSC DURING THE SECOND EXPOSURE As shown in Fig. 3.14 , when a specimen i s deformed i t s surface i s , in general, displaced and strained both in and out-of-plane. The deformation occurs between the two exposures and i t changes the coordinates of the two point sources St and S2, representing the surface, as follows s i ( 0 » v S i ' z s ) ^ S*(u,y S i+v,z s + w) S 2(0,y S 2,z s) -»• S*(u+6u,y S 2 + v+6v,zs+w+6w) The equations of the diffracted wavefronts d W j and dw2 , and of the diffrac-ted ray(s) remain unchanged and are given by equations (N.9),(N.10) and (N.15) respectively. The equations of the source wavefronts sw^ ^ must be modified to account for the changed, second exposure positions of Sj and S 2 > sw n : (x - u) 2 + [y - ( y S i + v)] 2 + [z - (z g + w)J 2 = (x s - u) 2 + [(y S l + v) - y A ] 2 + (z s + w) 2 (P.l) sw i 2 : [x - (u + Su)] 2 + [y - ( y S z + v + 6v)] 2 + [z - (z g + w + 6w)] 2 = [x s - (u + 6u)J 2 + [ ( y S 2 + v + 6v) - y A ] 2 + £z s + (w + 6w)]2 (P. 2) sw 2 i : (x - u) 2 + [y - ( y g i + v ) ] 2 + [z - (z g + w)] 2 = (x s - u) 2 + [(y S l + v) + Y a ] 2 + (z s + w) 2 (P.3) sw 2 2 : [x - (u + 6u)] 2 + [y - ( y g 2 + v + 6v)] 2 + [z - (z g + w + 6w)] 2 = [x s - (u + 6u)] 2 + [ ( y g 2 + v + 6v) + y A] 2 + [z g + (w + 6w)] 2 (P.4) As i s discussed in Section 3.6 , when DASSC is considered the changes in the distances between each of the point sources and S 2 and the laser must be included in the calculation of r ej; j , and hence using equations (N.5) through (N.8) the second exposure r ^  are given by -ei2 -ei 1 d W l-sw 1 2 + r u d w x - s W j j + ( S 1 - s w 1 1 - S 2 - s w 1 2 )+ r L l r £ 2 1 = dw2-sw21 +(D-dw1 - S2~sw12 ) - ( D-dw2 - —sw2 ^  )+ r L i r e 2 2 = dw2-sw22 +(D-dwx - S2-sw12 ) - ( D-dw2 - S2-sw22 ) + r L2 207 (P.5) (P.6) (P.7) (P. 8) The subsequent calculations of the second exposure ^ e±^ are similar to the calculations of the f i r s t exposure t e i j presented in Appendix N. The diffracted ray originates from the source wavefront at the point (x 2,y 2,z 2) and reaches the diffracted wavefront at the point (,-x. ,y1,z1) with the distance r*•^ by which the diffracted wavefront leads the source wavefront e x j given by equation (N.16). The case Ay s < D g g is considered f i r s t . To determine r * 1 2 equations (N.9) and (N.15) are evaluated at the point (K1>yl>z1) a n d equations (P.2) and (N.15) are evaluated at the point ( x 2 , y 2 , z 2 ) . From these equations and equation (N.16) we obtain a quadratic equation in (r* 1 2/X g) as -ei2 I X c - 2 -ei 2 + 3 , — " W l x s I Xc, J /F /F - 6X - U + y A " YD YS " yA Ay + B j ^ - - V Xc Xc COS0 + Ay + 3 f S - yA Az + 3 , — - W l x s - V sin9 X Az + s + + 2 + 2i 1 " / F j 3 : + u + YD " yA X c Ay + /G 7f YS " yA A y + Pi Y " V COS0 + YS " yA Az + X c Az + 3 , — - W l x s - w s i n 0 = 0 + (P.9) where U,V and W are defined as U u + 6u v + 6v V = X c W vs w + 6w U < 1.3 x 10 - 3 V < 4 x 10 W < 4 x 10 - 4 208 The small qadratic term in equation (P.9) is neglected and with an extensive use of Taylor's series we obtain an approximate solution, accurate to X/10, for r e i 2 as r* = -±r~ + r -ei2 2XC fY n - Yc, Yc, - y A u + 6u v — ^ - ± ( l + 3 X) -+ Sv' Xc x c coso + + f Z D - Zg Zg u + Su Xe X c Xc - ( l + B,) w + Sw Xc s i n e (P.10) r * 2 1 is obtained from equation (P.10) by replacing y A by -y A , and u + Su , v + Sv and w + Sw by u,v and w respectively. r = + r e2i 2X0 - Xc Ys + y A u ( l + 3.) V J / Xc coso + + Z N - • Z Q Z Q u w' — - — — - ( 1 + 3) — x s x s x s 1 x s J s i n t (P.11) In Section 3.6 i t is shown that Ar^ 2 = A r L i - Su cos9 x - Sv cos6 y - Sw cos6 z and hence ^ e i z i s given by r e i 2 = r e i 2 + ^ r L i " ^ u c o s ^ x -SvcosGy -5wcos9 z (P.12) r e 2 1 is given by equation (P.7) as + r e 2 l = r * 2 l + ( ^ "dwj - S 2 - s W j 2 ) - ( D-dw2 - S jsw2 z ) + r L ( ( = r e 2 i + r L i + [ Xs + ( Y D ~ y A> 2 + Z D J " [Xs2 + ( Y D + yA> * + Z D J "I % f (x s - u) 2+ [(y s + v) + y j 2 + ( z s + w) 2 - [x s - (u + Su)] 2 + ) V •» 1 + [(y g + v + Sv) - y A] 2+ [z s + (w + Sw)] 2 (P. 13) An approximate solution of equation (P.13), accurate to X/10, is obtained by using the Taylor's series expansions for the square roots with the result 209 re21 re21 + A r L i - D(p - C ) (P.14) where i s defined as e = rY sx s X s o ^ As o D X S 0 Zg6w"\ DX v so' X so 1 _ _ 2 - _ "•so 6v' D . w X so fYgZg 6w n "I™ D J lx: so Xc + 1 + D Y s ] 6u 6v Y S - D/2 r DYo 1 2^1 o- xso xso D 2 2 xsoJ + 2D 6w xso D 1 + DYg> 2X so + 2D (P.15) We shal l now consider the case Ay s < D s s . Since now j 3 x j < 6 x 10 3 r * 1 2 and r * 2 1 may be obtained from equations (P.10) and (P.11) by neglecting some small terms involving 3j with the result -e 12 = r Y D - Yg Yg - y A u + 6u v + 6v~\ X s Xc r z n - Zc Zc u + 6 u w + 6 w r = r e21 i X c Y - Y *D *S X s X s Y S + y A u -X c Xc X c -v X Xc cose + Xc 3in8 Z D Z S cos6 + Xc ^S _z_ x s x s (P.16) w ' xs-sin0 (P.17) r e i 2 and r £ 2 x are given by equations (P. 12) and (P.14) with r * 1 2 and r * 2 1 given by equations (P.16) and (P.17) . r^j[ = dw1-sw1 1 is determined from equation (N.16). Equations (N.9) and (N.15) are evaluated at the point (x ,y ,z ) and equations (P.l) and (N.15) are evaluated at the point ( x 2 , y 2 , z 2 ) . From these equations and equation (N.16) we obtain a quadratic equation in ( r * 1 1 / X s ) . •ei 1 - 2 ' T * r e i 1 X c / F - U - x c Ay+3. Ys + yA X a V Az + + 3 — - w IXe + Ay + 3, - " V1 X c COS0 + Az + 3 , — - W XX S 3in6 210 + + 2 1 " 7 f J 1 xc Ay + B 1 - V A " v + Az + 3 , — - W + /G + 2 3 7 f Ay + 3X YS + yA X c - V COS0 + Az + 3 , — - W l x s 5in6 = 0 (P.18) where this time U,V and W are defined as U = U < 1.3 x 10 - 3 V = w = A s V Xs" w xT V < 4 x 10 - 4 W < 4 x 10 - if An approximate solution of equation (P.18), accurate to X/10, was found as r e n = £ YD " YS X c YS ~ yA u X q X q V ' COS0 + ZD ZS u Xs X s Xc w • Xs-;in0 (P.19) r £ 1 1 is obtained from equation (P.6) as -ei i = r e n + r L i +(S 1-sw 1 1 - S 2-sw 1 2) ( x s - u ) 2 + [(y s + v ) - y A ] 2 + (z s+w) : [x s - (u + 6u)] 2+ [(y s +v+6v) - y A ] 2 + [z s+(w+6w)] : = r e n + r L l + (P.20) An approximate solution of equation (P.20), accurate to X/10, is given by : e n r e i i + r L l " a * + &n (P.21) where a and $ are defined as a = a A + u I X S O X ; S O S O w6w X s H 6u + 6v Xso X s o 6v 1 + I X S 0 2X S 0 J + + Sw f z s yA Y s z s 6 w 1 IX, so X so 2X S 0 J (P.22) 211 ^11 a c - v X S Q x s o , , y A Y s x s ? + ou 3 - ov Xso ZA_ + y A Y S + y A Y s l xso xso xso (P.23) r * 2 2 i s obtained from equation (P.19) by su b s t i t u t i n g - y A for y A , and u + 6u, v + 6v and w + 6w for u,v and w res p e c t i v e l y . f YD " Y S Y S + y A u + 6 u v + 6 v l r = r xc X COS0 + s J + Z D - Zg Zg u + 6u w + 6w"| xc sinG (P.24) r e 2 2 i s given by equation (P.8) as -e22 = r * 2 2 + A r L + ( D - d W j - S 2-sw 1 2 ) - ( D-dw2 - S 2-sw 2 2 ) = r e 2 2 + A r L 2 + X s + <YD " yA> 2 + ZD " X s + <YD + y A ^ + ZD + [ x s - (u+6u)] 2+ [ ( y g + v + 6 v ) + y A ] 2 + [z s+(w+6w)] : [x s - (u + 6u)] 2+ [ ( y s + v + 6 v ) - y A ] 2 + [z s+(w+6w)J 2 (P.25) Using Taylor's ser i e s expansion f or the square roots i n the above equation we f i n d an approximate s o l u t i o n , accurate to A/10, for r £ 2 2 as r _ = r*„„ + Ar T , + a* + 6 2 2 - D ( P - ?*) + A* -e22 e22 L l (P.26) where 3 2 2 and A are defined as 2y AY g u + 26u l 2 2 = a s + u — o + ou + 6v x 2  Aso xso r y A y A Y S " xso Aso X + so 2 y A Y S rs u Aso • xso Xc so' + y A Y S . p 2 y A l 3 1 xso X s o j (P.27) A = - 6u cos6 x - <5v cos0y -6wcos0 z (P.28) 212 APPENDIX Q DERIVATION OF EQUATIONS (3.25) AND (3.28) - THE SECOND EXPOSURE I r The second exposure I r for the case Ay s > D s s is calculated according to equation (0.1). Once r e i 2 and r e 2 : given by equations (P.12) and (P.13) respectively are substituted in equation (0.1) and the indicated multiplication i s done we get Ir = ikr ei2 dA /• - ikr e i 2 d A + / i A ikr e 2 1dA Je A - ikr e 2 1dA + - iA f - ikr + e je A e i 2 f dA / e i k r e 2 i d A + e 1 A / e 1 K r e i 2 d A / i A r ikr. - ikr e 2 i d A (Q.D where A i s defined as A = k D( P - hf) + A* Equation (Q.2) may be put in the form equivalent to equation (3.25) . 2 Ir = T , A 2 -(R 1 + R 2cos z- + R 3sinA ) (Q.2) R l 5R 2 and R3 are defined as r i k r P , , ,. r - ikr Rj = je e i 2dA J e e I 2 l k i r ^ Q I T * r " ^ " ^ • ^> G 2 . I dA + J e e 2 1dA Je dA -2Jcoskr* j 2 dA jcoskr*2 j dA - 2/sinkrgj 2 dA /sinkrg 2 x dA A A A A R2 = 4 R3 = 2 /coskrg 1 2 dA/coskr* 2 1 dA + /sinkr* x 2 dA /sinkr* 2 x dA A A A A /coskrg : 2 dA/sinkr* 2 j dA - /coskr* 2 j dA/sinkr* 1 2 dA A A A A (Q.3) (Q.4) (Q.5) In the case Ay s < D s s , I r is given by equation (0.2) with r e i 2 given by equations (P.12) and (P.16) , and r e 2 1 given by equations (P.14) and (P.17) r e n a n d re22 a r e § i v e n by equations (P.21) and (P.26) respectively. When the indicated multiplication in equation (0.2) i s done, we get J n e - ikD(p - V) + ik[a* + B 2 2 + A* - D( P - 5*)]] f j e- ikCB^ - a*) J 2 2 1 1 i : ikD( P - ? * ) . , - ik[a* + B 2 2 + A* - D( P - E*)V J 2 i e J 2 2 e - ikA* ^  e + (Q.6) where J m n is defined as and p m n are defined as J (Pmn) Jmn mn 1 1 1 2 p 2 1 '22 kd r YD" YS Y s - y A ~1 _ , x s x s kd fYD- YS Ys-YA 2 _ I x s X s kd f YD" YS YS + YA 2 _ - x s x s kd Y s + yA 2 1 x s x s 2 2 fZ n - Z s Zq u w 1 + x s X s X s Xc x s J + f Z j ) - Z g Z g u + 6u w + Sw^ Xs X s X c r z D - Z g Z g u w I x s 2 x s J + I X c X s X g X c J X S X g X g Z-Q - Z g Z g u + Su w + 6V X s J To interpret equation (Q.6) we assume that J m n J k i - J (and hence p m n - p) within the speckle area, and equation (Q.6) may then be written as Ir - 161, f V P > l 2k Qj + Q2cos — D(p - ?*) - a* + ^ - r -22 (3.28) where Qx and Q2 are defined as Q i = 7 1 + cos I ( B n - g 2 2 - 2a*- 2A*)cos | ( B 1 X - B 2 2) - 2cosk(a* + A*)cosk(B 1 1 - B 2 2) Q2 = cosk(a* + A*)cosk(B 1 1 - B 2 2) 214 APPENDIX R DERIVATION OF EQUATIONS (3.26) AND (3.29) The forms of the second exposure I r , for the two cases of Ay s considered, reveal that in each instance the speckle grid has been shifted with respect to i t s f i r s t exposure position. By comparing the modulating (cosine) terms of the f i r s t and the second exposure I r the relative shift of the speckle grids may be related to the surface displacements and their increments, and to the camera parameters. The case Ay s > D s s is considered f i r s t . The two modulating terms are given by equations (3.23) and (3.25) as f i r s t exposure : cos 2 second exposure : cos kDp D(p - E*) +• A * In a similar way as in Appendix L, the apparent speckle grid centers Y Sgj and Y Sg 2 > in the object plane, are found from the maxima of the two modula-ting terms. Y Sgi ^ s f° u nd on the line Z = Z g from kDp = 0 (R.D with Y s g l substituted for Y D . Y s g 2 is found on the line Z = Zg from D(p - E*) + A * = 0 (R.2) with Y s g 2 substituted for Y D . The solutions of equations (R.l) and (R.2) are given by Y s g i - YS 1 - 3, r2 , 1 - X. so - l Y = Y + x s g 2 sgi '2 . 1 - X so - 1 D "so (R.3) (R.4) The Moire fringe number n i s related to Y s g l and Y s g 2 by 215 Y sg2 ~ Y s g i ~ N ^ D (R.5) Equation (R.5) was solved for n with the result n = ,2 •> 1 -- i fY sX s Y s6v Zs6wl IX so DX S 0 DX S QJ + v Yn Sv) 1 " 2 " ~ X S 0 D ^ w fY sZ s 6wl + so D J + + Su + 6w X£ D X f Xso D YS 1 cos9 x + 1 H — •^Xs 2 X s o ' + 6A X so D Ys-D/2 cos6 v -y X, "•so l + t 2 X s 0 J 6v + -2X so-+ so D cos6„ - — z X, so DYr 1 + 2X S 0J 6w 2X s oJ (R.6) Once the definitions of 6u,6v and 5w are substituted in equation (R.6) i t may then be accurately approximated by equation (3.26) When the case Ay s < D s s is considered the modulating terms for the two exposures are obtained from equations (3.24) and (3.28) as k f i r s t exposure : second exposure : cosz— (Dp - a A) cos 2^D(p - £*) - a* + 3 1 1 - B22 Again the apparent speckle grid centers Y s g l and Y gg 2 are found from the maxima of the two modulating terms. These maxima occur when - (Dp - a ) = 0 D( P - ?*) - a* + 1 1 0 2 2 = 0 (R.7) (R.8) In equation (R.7) Y Sg X is substituted for Y n and in equation (R.8) Y Sg 2 is substituted for Y D . In both equations Z D is set equal to Zg . Equations (R.7) and (R.8) are solved with the result 216 "Sg2 Y S " 1 - X; so - 1 ^ i Y S ~ X s o K D 2D (R.10) The Moire fringe number n is obtained from equation (R.5) with Y Sg x and Y s g 2 given by equations (R.9) and (R.10). An approximate solution for n is given by equation (3.29). 217 APPENDIX S DERIVATION OF EQUATIONS (3.27a) AND (3.27b) The amplitudes of the speckle grid terms for the case Ay s > D s s are obtained from I r recorded during the two exposures and described by equations (3.23) and (3.25) . The two amplitudes are given by f i r s t exposure : second exposure : /e i k r e dA je~ ± k r e dA A A R„ When the effect of the small nonlinear term e x r 2 2Xa on the shapes of the speckle envelopes is neglected the amplitudes are proportional to Ji(Pmn) Ji(Pkl> fVp)! P for the f i r s t exposure, and mn Pkl for the second exposure, Using equations (N.18),(P.10) and (P.11) we define P,P 1 2 and P 2 J as kd , „ k - r - (K2 + K 2) 2 (S.l) Pi: 2 1 kd T + kd 'YD " YS Yg - y A u + Su v + Sv"! 2 Xc Z D - Zg Zg u+ Su Xc - (l + B j ) - (1 + B i ) w + Sw Xc (S.2) YD " YS Xc YS + yA u ^ v 1 Xc X, Xc + ZD ZS xc Zg u_ x s x s - (l + B x ) W^l X sJ (S.3) where Kx and K2 are defined i n Appendix N. The limit on the size of displacements and strains i s obtained from the requirement that a l l the speckles recorded in the two exposures overlap. Let us c a l l the coordinates of the speckle centers (maxima) as (Y1,Z1) for the f i r s t exposure, and (Y 1 2,Z 1 2) and ( Y2 1>Z 2 1) for the second exposure. Since f—ilP-ML_l "mn J i s maximum when p m n = 0 , the coordinates ( Y J , Z J ) are found from the equation 218 p = 0 , with Yj and Zj substituted for Y n and Z n . An approximate solution was found as (S.4a) (S.4b) When Y 2 and Z 1 2 , and YZ1 and Z21 are substituted i n the equation P12'= 0 , and p = 0 respectively, for Y n and , the approximate solutions of these equations are then found as Y s - y A Y,„ * Y c + ~ (u + 6u) + (1 + 3 x)(v + 6v) • 1 2 Z 1 2 = Z s + ^ (u + 6u) + (1 + 3^ (w + 6w) Y Q + y A Y 2 i s Y s + - V ^ + < 1 + . 3 i ) v Z 2 i ~ Z S + x U + ( 1 + 3 1 ) W (S.5a) (S.5b) (S.6a) (S.6b) The distances between the speckle centers must be smaller than the speckle radius and hence we may write p and p 1 2 : p and p„ 2 1 (Y 1 2 - Y 2 ) 2 + ( Z 1 2 - Z l ) : (Y - Y K + (Z - Z ) z < V 2 1 l' V 2 1 lJ S S (S.7) (S.8) Y 1 , Z 2 Z 2 x given by equations (S.4a) through (S.6b) and the definitions of the displacement increments 6u,Sv and 6w are substituted in equations (S.7) and (S.8). Since |3 | « 1 , equations (S.7) and (S.8) may then be closely approximated by equations (3.27a) and (3.27b). 219 A P P E N D I X T D E R I V A T I O N OF EQUATION ( 4 . 1 3 ) . DASSC using a set of apertures rotated by the angle CJJ^  forms fringes according to equation ( 4 . 1 2 ) . x c u i + v i ~ 2 w i Y i 2 ! A Y s i x s i W-! A y S i x s i fy±-yA D - x s COS6y-L V A y s i x s i i » y i D - COS0 zi XX wi»yi s i ni(yi,zi) ( 4 . 1 2 ) A l l terms in equation ( 4 . 1 2 ) w i l l now be transformed into y,z coordinate system using the transformations y^ = ycosd)^ + zsincj)^ Zj_ = - ysincj)^ + zcosd^ y = y^cosdK - z^sind)^ z = y^sincj)^ + z^coscj)^ The transformations of the displacement components are given by equations ( 4 . 3 ) , and the strain components transform as u±,y±(y±,z±) = u, y l(y,z) = u, y(y,z)y, y i + u, y(y,z)z, y i = u,ycos(f>i + u^sincj)^ (T.la) vi>yi(yi» zi) = [v(y, z) cos<J>i + w(y, z) sin<J>i] , y i = = ( v , y y , y i + v, zz, y i)cos(j) i + (w, yy, y i + w, z z , y i ) s±nty± = v,ycos2c}).i + v, g.sincJ^ coscfj-L + w,ysincj)^cos())^ + w, zsin 2cj)i (T.lb) w i » y i ( y i ' z i ) ='•[ " v(y,z)sin(f) i + w(y, z) coscjijj , y i = = " ( v , y y , y i + v, zz, y ±)sind) ; L + (w, yy, y i + w, zz, y l)cosd^ = = - v^sincj^costJj-L - v ^ s i n 2 ^ + w,ycos2(j)^ + WjySincj^coscj)^ (T.lc) 220 By substituting the coordinate, displacement and strain transformations in equation (4.12) i t becomes — u + (vcosd)^ + wsincj)^) - ( - vsindp-L + wcosd)^) — z 1 ^ (1 + cos9 xi) xg D A y S i x s i •(u,yCOsa)^ + u, zsind)^) — - cosOyi (v, ycos 2d)-L + v, zsin(J)iCOS(J)-L + + WjySincfi^cosd)^ + w, zsin2cj)^) A y s i x s i fZ-j - cost Z l ( - V j y S i n c j^cosdpi -XX sx v , z s i n d)^  + w,yCOS d)^  + w, zsin<j>-^ cos(j)£) = n-j_(y.j_, z-j_) (T.2) With coefficients a-j_,...,kj_ and defined in Section 4.5, equation (T.2) may be written as equation (4.13). 221 APPENDIX U The coefficients d^,...,^ for the case of the specimen illumination in x,y plane ( 0 Z = 90°) are derived here. From Fig. 4.3 (where = <p^ the unit vectors i , j and ii»Ji are related as 1 = H j = jicosdpi - kisincJi-L For 8 z = 90° the angles 6 X and 9 y are related by cos6y = cos(90° + 0 X) = - s i n 0 x The unit illumination vector 1 is written in the two coordinate systems as 1 = i c o s 8 x + jcos0y + kcos0 z = i c o s 0 x - j s i n 0 x s\ /\ /\ = i ^ c o s 0 x - (j^cosvj)^ - k^sincj)^) s i n 0 x = i ^ c o s 0 x - j ^ sin0 xcoso)^ + k£sin0xsincj)£ s\ X\ / \ = i c o s 0 x ^ + jcos0y-^ + kcos0 z^ From the last two equations we get c o s 0 x i = cos0 x cos0y-L = - sinGjjCOSc})-]^  COS0z-L = sinGxSintj)^ By substituting these relationships in the expressions for dj_,...,k-j_ in Section 4.5 the simpler forms of these coefficients are obtained. 222 APPENDIX V DERIVATION OF THE DISPLACEMENTS AND STRAINS CAUSED BY THE OUT-OF-PLANE BENDING OF BEAMS Handbook of St e e l Construction [62] gives ( a f t e r change of notation and the coordinate o r i g i n ) the d e f l e c t i o n of a neutral surface of a t h i n prismatic beam with clamped ends and a point load at i t s center as f v 1 2 3 --L/2 4 y 4 0 u(y) = 6 1 - 3 y [L/ 2 J - 2 y 1 L / 2 J ' y i 2 V ^ 3-0 4 y 4 L/2 u(y) = 6 1 - 3 [L/2J + 2 y L/2, (V.la) (V.lb) where 6 i s the d e f l e c t i o n of the beam centre and L i s the beam length, u i s not a function of z, and since we consider only a thin beam and small deformations, i t may be assumed that the out-of-plane d e f l e c t i o n of the illuminated surface of the beam i s the same as that of the neutral surface. It i s apparent from F i g . V . l the bending of the beam gives r i s e to the in-plane displacement v as V = - | U , y Using equations (V.la) and (V.lb) v i s given as -L/2 ^ y < 0 v(y) 0 « y « L/2 v(y) t 6 y 12 - -2 L L/2 t 6 y 12 2 L L/2 1 + 1 -L/2 L/2J (V.2a) (V.2b) The surface s t r a i n s E y ( y ) and e z ( y ) are given as £y ^'y 2 ^ 'yy £z = w » z = " v e y = v ~2 u>yy (V.3) (V.4) Fig. Out-of-plane bending of a prismatic beam. 224 where v is Poisson's ratio. The displacement w i s such that w(y,z = 0) = 0 , and hence w(y,z) is obtained from equation (V.4) as w(y,z) = Vz — u, 2 u'yy and using equations (V.la) and (V.lb) w(y,z) may be written as -L/2 4 y 4 0 w(y,z) = - 12vz t 6 2 L2/2 1 + 2y 0 4 y 4 L/2 w(y,z) = - 12vz -t 6 2 LV2 1 -L/2, 2 y l L/2J (V.6a) (V.6b) The partial derivatives u, v , v, v and w,y are given as -L/2 $ y 4 0 0 4 y 4 L/2 i,y 'y w, V 'y 6 y - 12 — V l L L/2 t 6 12 5 — 2 L2/2 1 + 1 7 2 2y 1 i + — 7 -L/2 o - 48vt — r z L 3 6 y - 12 — 1 L L/2 1 " ^ w, 12 48vt t 6 2 L2/2 6 . 4 1 L/2. (V.7a) (V.7b) (V.7c) (V.7d) (V.7e) (V.7f) 

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