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 B r i t i s h 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 p r e s e n t i n g t h i s thesis an advanced degree at further agree fulfilment of the requirements the U n i v e r s i t y of B r i t i s h Columbia, I agree the L i b r a r y s h a l l make i t I in p a r t i a l freely available for this thesis f o r s c h o l a r l y purposes may be granted by the Head of my Department of this thesis for It financial of g a i n s h a l l not C i v i l Engineering The U n i v e r s i t y o f B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date October 11 197? or i s understood that copying or p u b l i c a t i o n written permission. Department that reference and study. t h a t p e r m i s s i o n for e x t e n s i v e copying o f by h i s r e p r e s e n t a t i v e s . for be allowed without my ii MEASUREMENT OF SURFACE DISPLACEMENTS AND STRAINS BY THE DOUBLE APERTURE SPECKLE SHEARING CAMERA ABSTRACT In the testing of materials, structures and s t r u c t u r a l components i t i s often desired to determine the surface displacement and s t r a i n f i e l d s due to some external loading. Numerous o p t i c a l techniques have been developed for this purpose and successfully used i n p a r t i c u l a r applications. Unfortunate- l y , when the surface deformation i s quite large, as i s usually the case i n p r a c t i c a l testing, most of these methods f a i l and only a few suitable o p t i c a l interferometric techniques w i l l work. Two of the recently developed techniques that seem to work are based on laser speckle interferometry. Duffy [ l ] who The f i r s t technique was described i n 1972 by showed that a Double Aperture Speckle Camera (DASC) i s suitable for measurement of a reasonably large in-plane displacement having i t s d i r e c t i o n p a r a l l e l to the l i n e connecting the two apertures of the camera. A second technique was described i n 1973 by Hung [3] . He showed that a Double Aperture Speckle Shearing Camera (DASSC) may be used to measure both the i n and out-of-plane strains of planar surfaces. Duffy has not considered the fringe formation by DASC due to the d i s p l a cement normal to the surface and the displacement normal to the l i n e connecting the two apertures of DASC. Hung, i n turn, has not considered the e f f e c t on fringe formation of either the i n and out-of-plane or the in-plane s t r a i n , which i s the p a r t i a l derivative w, v displacements, (see F i g . 3.11 for the d e f i n i t i o n of w,y). Because of the great p o t e n t i a l 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 e f f o r t was made to determine the fringe formation of the two iii cameras due to a l l displacements and strains occuring i n a general deformation of a specimen surface. The t h e o r e t i c a l analysis of models of DASC and DASSC was performed resulted i n two "new" cameras. and equations describing the fringe formation by these The equations take into account the effect of a l l displacements , and strains on the fringe formation; i n addition, the equations are "symmet r i c " and the equation governing DASSC reduces to the one governing DASC for the l a t e r a l shear set equal to zero. The accuracy of these equations was then v e r i f i e d by a number of simple experiments. Various ways of using the two cameras were proposed so that the unknown displacements and strains i n the specimen surface may be calculated from the least number of fringe patterns. Computer programs based on these proposed methods were written and used i n 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 LIST OF TABLES iv viii LIST OF FIGURES NOTATION ix 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 1.5 1.6 2. 5 Measurement of Strains by Speckle Interferometry 9 Limits of Investigation 9 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 P r i n c i p l e 16 2.6 Point Source of Light 16 2.7 The P r i n c i p l e of Linear Superposition 18 2.8 Diffraction i 2.9 Huygen's P r i n c i p l e 20 8 V CHAPTER 3. 4. Page 2.10 Coherence 20 2.11 Imaging Properties of a Thin Lens 22 2.12 Aberrations i n Optical Systems 24 2.13 Generalized Coherent Interferometer 26 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 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 4.6 Use of DASSC to Measure General Deformation (Algebraic Solution) 4.7 73 76 Use of DASSC to Measure Plane Strain and Plane Stress Deformation (Algebraic Solution) 77 vi CHAPTER Page 4.8 Use o f DASSC t o Measure O u t - o f - P l a n e Bending (Algebraic Solution) 78 4.9 Use o f DASSC t o Measure G e n e r a l D e f o r m a t i o n 4.10 Use o f DASSC t o Measure u , u , ,v,,and v , y y ... from Two Photographs 5. 6. 78 81 4.11 Use o f DASSC t o Measure O u t - o f - P l a n e Bending 4.12 Use o f DASSC t o Measure P l a n e S t r e s s and . 85 Plane S t r a i n Deformation 87 EXPERIMENTAL APPARATUS AND PROCEDURE 89 5.1 The Camera 89 5.2 The R e c o r d i n g System 95 5.3 The F i l t e r i n g System 95 5.4 The Specimen L o a d i n g Systems 97 5.5 Specimens 104 5.6 Experimental Procedure 104 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, O u t - o f - P l a n e Translation of a P l a t e Specimen 6.3 R i g i d Body, I n - P l a n e R o t a t i o n o f a P l a t e Specimen 6.4 115 I n - P l a n e S t r e t c h i n g o f a T h i n Beam w i t h a Rectangular Cross-Section 6.6 HI O u t - o f - P l a n e Bending o f a T h i n Beam w i t h a Rectangular Cross-Section 6.5 107 119 I n - P l a n e S t r e t c h i n g o f a Beam w i t h a Variable Cross-Section 123 vii CHAPTER 7. ' Page 6.7 In-Plane Stretching of a Wooden Beam 126 6.8 Error Analysis 129 CONCLUSIONS 167 7.1 Summary and Conclusions 167 7.2 Suggestions f o r Future Research 169 BIBLIOGRAPHY 171 APPENDICES A DERIVATION OF EQUATION (2.9) 175 B DERIVATION OF EQUATIONS (2.18) AND (2.19) C DERIVATION OF EQUATION (3.5) D DERIVATION OF EQUATION (3.7) E DERIVATION OF EQUATIONS (3.10) AND (3.11) F ...... G DERIVATION OF r l AND r £ 2 .v. 181 182 ..... 184 DERIVATION OF EQUATION (3.12) - THE FIRST EXPOSURE I 187 r I DERIVATION OF EQUATIONS (3.14) AND (3.15) J DERIVATION OF r e l AND r e 2 188 FOR DASC DURING THE SECOND EXPOSURE K 183 FOR DASC DURING THE FIRST EXPOSURE H 176 178 . g ...... ••••• 1 8 9 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 O F - r j FOR DASSC DURING THE ei FIRST EXPOSURE 197 viii APPENDIX Page 0 DERIVATION OF EQUATIONS (3.23) AND (3.24) - THE P FIRST EXPOSURE I r DERIVATION OF r j FOR DASSC DURING e i •••• 203 THE SECOND EXPOSURE Q 206 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) U V ... 219 221 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 l i g h t 14 2.2 Optical path length of a ray 14 2.3 Point source 15 2.4 Fraunhofer d i f f r a c t i o n by an aperture 15 2.5 Fresnel d i f f r a c t i o n by an aperture 15 2.6 I l l u s t r a t i o n of the Huygen's p r i n c i p l e 19 2.7 Coherence of l i g h t 19 2.8 Longitudinal and l a t e r a l coherence of e l e c t r i c 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 31 3.3 Coordinate 3.4 D i f f r a c t i o n pattern of a single c i r c u l a r aperture 3.5 Double aperture camera 36 3.6 Diffraction 38 3.7 D i f f r a c t i o n pattern of two c i r c u l a r 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 3.11 i n a single aperture camera system of the c i r c u l a r aperture i n a double aperture camera 33 . 33 (1) and the second (2^,2g) exposure speckles 45 The schematic of DASSC 47 X Figure Page 3.12 D i f f r a c t i o n i n DASSC 48 3.13 Intensity d i s t r i b u t i o n I 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 r f o r DASSC with A y > D s s s .. 50 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 F i l t e r i n g 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 i n bending of the beam experiments .. 96 5.11 Schematic of the t e n s i l e 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 d i a l gages .. 100 5.15 Variable cross-section specimen 100 5.16 Central part of the beam used i n the beam 5.17 bending experiments 102 Tensile specimen with the uniform cross-section .... 102 xi 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 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 137 6.9 Rotation of a plate about x-axis 6.10 Measurement of the in-plane trace of Exp. 19 u - Exp. 19 138 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 , vs. experimental 144 6.22 Fringe pattern of Exp. 10IA 144 6.23 Fringe pattern of Exp. 101B 144 v u, v - Exp. 101C .... xii Figure Page 6.24 Predicted u vs. experimental u - Exp. 1 0 1 A @ 1 0 1 B ... 1 4 5 6.25 Predicted u , vs. experimental 6.26 Measurement o f t h e i n - p l a n e d e f o r m a t i o n 6.27 F r i n g e p a t t e r n o f Exp. 1 1 4 C 147 6.28 F r i n g e p a t t e r n o f Exp. 1 1 4 D 147 6.29 Predicted u vs. experimental 6.30 P r e d i c t e d u , vs. experimental 6.31 P r e d i c t e d v vs. experimental 6.32 Predicted v , vs. experimental 6.33 F r i n g e p a t t e r n o f Exp. 1 1 4 B 150 6.34 P r e d i c t e d u v s . e x p t l . u - Exp. 1 1 4 B @ 1 1 4 C @ 1 1 4 D 151 6.35 Predicted u, vs. exptl. u, 6.36 P r e d i c t e d v v s . 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 , v s . e x p t l . v , - Exp. 1 1 4 B @ 1 1 4 C @ 1 1 4 D . 152 6.38 Predicted u vs. experimental 6.39 Predicted u , v s . experimental 6.40 Predicted v vs. experimental 6.41 Predicted v , vs. experimental 6.42 T e n s i l e specimen o f Exp. 1 2 2 155 6.43 Normal v i e w o f t h e a p e r t u r e 156 6.44 F r i n g e p a t t e r n o f Exp. 1 2 2 S 1 157 6.45 F r i n g e p a t t e r n o f Exp. 1 2 2 S 2 157 6.46 Predicted u vs. exptl. 158 6.47 Predicted u, 6.48 P r e d i c t e d v v s . e x p t l . v - Exp. 1 2 2 S 1 @ 1 2 2 S 2 159 6.49 Predicted v , 159 6.50 F r i n g e p a t t e r n o f Exp. 1 2 2 S 3 u, y u, by D A S S C 114C@H4D . . . . - Exp. 1 1 4 C @ 1 1 4 D . 114C@114D v , - Exp. y u, vs. exptl. v , 148 148 .... 149 114C@114D. 149 u - Exp. 114C@114D u , - Exp. y 114C@114D. 153 153 v - Exp. 1 1 4 C @ 1 1 4 D .... 1 5 4 v, y - Exp. 1 1 4 C @ 1 1 4 D . 1 5 4 screen u - Exp. 1 2 2 S 1 @ 1 2 2 S 2 vs. exptl. 145 .... 1 4 6 y y y 101A@101B. - Exp. 1 1 4 B @ 1 1 4 C @ 1 1 4 D . 1 5 1 y y y y v - Exp. y y - Exp. u - Exp. y y y y y - Exp. 1 2 2 S 1 @ 1 2 2 S 2 - Exp. 1 2 2 S 1 @ 1 2 2 S 2 158 160 xiii Figure Page 6.51 F r i n g e p a t t e r n o f Exp. 122S3 6.52 The p a r t o f t h e specimen s u r f a c e where t h e displacements 6.53 160 and s t r a i n s were c a l c u l a t e d Contours o f c o n s t a n t d i s p l a c e m e n t 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 6.54 162 S t r a i n 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 v specimen 6.55 161 163 S t r a i n 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 v specimen 164 6.56 F r i n g e p a t t e r n o f Exp. 132D1 165 6.57 F r i n g e p a t t e r n o f Exp. 132D2 165 6.58 F r i n g e p a t t e r n o f Exp. 132S2 165 6.59 F r i n g e 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 a p e r t u r e camera 178 F.l Unit vectors of e l e c t r i c f i e l d s 183 V.l Out-of-plane 223 bending of a p r i s m a t i c beam NOTATION The meaning of symbols i s defined i n 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 l i k e to sincerely thank to his advisors, Dr. M. D. Olson and Dr. C. R. Hazell, for their h e l p f u l 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 h i s assistance during the research work. The author also wishes to thank to Mr. L. E. Dery, Mr. P h i l 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 ] f o r determining the displacements and strains i n materials or s t r u c t u r a l components subjected to various loads. Certain tests, for various reasons, must be. noncon- tacting, i . e . i t i s not possible to use stress coatings, photo-elastic coatings, s t r a i n gages, displacement gages or other contact probes. instances o p t i c a l interferometry may often be used successfully. In such Its pre- vious use i n p r a c t i c a l testing had been hindered by i t s excessive s e n s i t i v i t y , the need for complicated and expensive instrumentation and i t s s u s c e p t i b i l i t y to disturbing effects of environment. While some shortcomings remain, the use of o p t i c a l interferometry has spread dramatically since the invention of the laser i n the 1960's. Numerous o p t i c a l interferometric techniques using laser generated coherent l i g h t were developed and proved extremely valuable for s p e c i f i c applications. Most of the techniques that have been developed for the measurement of displacements to date are, unfortunately, only s u i table for the measurement of very small displacements. A l t e r n a t i v e l y some techniques have been developed to measure large displacements along the l i n e of sight. Therefore, there has existed a need to develop a technique for the measurement of the displacements and strains of the magnitudes encountered i n p r a c t i c a l testing. Duffy [ l , 2 ] described the double aperture camera and showed that i t i s 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 i n and out-of-plane s t r a i n s . Other researchers have proven the f e a s i b i l i t y of d i f f e r e n t methods for measuring displacements and strains, but these methods seemed less powerful and promising than Hung's. This thesis objective, i n 1974, was to examine the mechanism of f a i l u r e 2 of wooden beams subjected to various loads and, at that time, Hung's method appeared lable. to be the most suitable noncontacting experimental technique avaiSubsequently, DASSC was b u i l t 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 i n the way suggested by Hung. Unfortunately, the calculated strains and the actual surface strains (determined by s t r a i n gages and d i a l gages) were i n a considerable disagreement. made to f i n d the cause of this disagreement, A decision was and the subsequent theoretical analysis of DASSC resulted i n a new equation governing the fringe formation. The accuracy of the new equation was a l l y and by computer simulation. then thoroughly tested both experiment- 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 f a i l u r e 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 o p t i c a l interferometry and even coherent l i g h t i n t e r f e r o metry encompass such a great number of various techniques i t i s not possible to review a l l of them. Instead, only the best known techniques suitable for measurement of displacements and strains i n planar surfaces are investigated- with s p e c i a l emphasis i n the review attached to speckle interferometry. Hopefully, the review of a l l techniques relevant to the research presented i n this thesis i s complete. If an omission has been made, i t i s quite unintentional. The techniques are reviewed i n chronological order, starting 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] i n the early 1960's. conventional holography, The technique i s similar to except that two exposures (and, hence, two holo- grams) of the object are recorded on the same holographic plate; the surface of the object i s deformed or displaced between the two exposures. Upon recon- struction of the hologram, two three-dimensional images of the object are formed, i n t e r f e r e with each other, and produce a set of fringes. The fringes represent areas of the same change i n t o t a l o p t i c a l path length and, with knowledge of the parameters of the experimental setup, a component of the surface displacement along a l i n e of sight may be calculated. Since three displacement components must be calculated to determine the surface defor- mation, i n general, three holograms are needed. The surface strains may then be obtained by d i f f e r e n t i a t i o n of the displacements. Since 1965 holographic interferometry has been applied successfully to a study of transient and steady v i b r a t i o n [l2,...19] and i n a wide variety of materials testing [20,23,24,29,30,31,32] . Special techniques were deve- loped f o r the measurement of the wave propagation using pulsed lasers [25,26] . By illuminating the specimen surface by two beams i n c l i n e d 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 s t r a i n were developed as well [34,35,36] . The main advantage of holographic interferometry i s i t s a b i l i t y to work with an arbitrary three dimensional surface. Another advantage i s , i n many cases, i t s high s e n s i t i v i t y as the fringes usually represent - of the order of 1 x 10 displacements 5 i n . Unfortunately, such a high s e n s i t i v i t y makes this method unsuitable for ordinary engineering testing where displacements 4 — of the order of 1 x 10 3 — i n . and 1 x 10 2 i n . are commonly encountered. A l l holographic techniques are also quite sensitive to the disturbing effects of the environment. F i n a l l y , a major problem associated with using holography to obtain numerical measurements stems from the fact that the fringes may not be l o c a l i z e d [21,22] on the specimen surface and, consequently, the fringe positions cannot be c l e a r l y established. 1.3 Holographic Contouring Techniques There are several holographic interferometric techniques suitable f o r measurement of r e l a t i v e l y large s t a t i c or dynamic changes i n 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 producing an illuminated volume of space i n which the apparent illumination of any point i n that space i s some function of position alone. I f the function i s known, the shape of the illuminated area of the object can be determined. Most o p t i c a l f i e l d contouring techniques r e s u l t i n sinusoidal functions of position. Although i n theory the s e n s i t i v i t y of such a continuous function i s unlimited, i n practice a s e n s i t i v i t y of one-half of the period i s used. There are numerous ways i n 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 d i f f e r e n t r e t r a c t i v e index surrounding the object. A l t e r n a t i v e l y , 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 s h i f t i n 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 i s recorded while the object i s i n a medium of one r e f r a c t i v e index,and the hologram and the object then illuminated while the object i s i n another medium. A l l of these techniques generate Moire-type of fringes which are r e l a ted to the change i n shape of the object; the techniques are usually insensitive 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 i s f a m i l i a r with the speckle phenomenon which causes a grainy appearence of the laser illuminated surface. Laser speckle (or speckle pattern) i s formed when coherent l i g h t i s either scattered from a d i f f u s e l y r e f l e c t i n g surface or propagates through a medium with random r e f r a c t i v e index fluctuations. The speckle size i s 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. I f a d i f f u s e l y r e f l e c t i n g surface i s imaged by a lens on a - screen, the speckle size D i s related to the e f f e c t i v e numerical aperture s NA of the lens by D s b - .6 X — 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 i n the speckle pattern to the surface deformation. Speckle interferometers may be c l a s s i f i e d as those suitable for measurement of displacements smaller than speckle size and those suitable f o r measurement of displacements larger than speckle size. A l t e r n a t i v e l y , the speckle interferometers could be c l a s s i f i e d according to the type of l i g h t f i e l d s that the interferometers employ, i . e . , either Las interferometers combining speckle and uniform f i e l d s or as interferometers combining two speckle ' fields. The main advantage of speckle interferometers i s their a b i l i t y to vary their s e n s i t i v i t y by changing speckle size and other o p t i c a l parameters i t i s also claimed that these interferometers measure the in-plane d i s p l a cements of the tested surface independently of any displacement taking place i n the d i r e c t i o n 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 d i s p l a - cement measurements, steady state v i b r a t i o n analysis and q u a l i t a t i v e testing An excellent up-to-date review of existing techniques using speckle i n t e r ferometry for measurement of displacements A.E.Ennos [43] . and strains has been done by Several of the numerous techniques w i l l now be b r i e f l y 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 i s then imaged by a lens on a photographic plate. The two speckle patterns, one due to each illuminating beam, i n t e r f e r e coherently, producing fringes according to the rule 2u sinS^ = nX where u, 0 i 1 > n and X are defined as u ... i n plane displacement e ± ... angle of i l l u m i n a t i o n n . .. fringe number wavelength of i l l u m i n a t i n g l i g h t 7 The s e n s i t i v i t y o f the i n t e r f e r o m e t e r may the a n g l e 6 i . changing The maximum a l l o w a b l e i n - p l a n e d i s p l a c e m e n t u must be s m a l l e r than the apparent s p e c k l e d i a m e t e r . are of low be changed p r i m a r i l y by F r i n g e s produced by t h i s i n t e r f e r o m e t e r visibility. D u f f y [ l , 2 ] has proven the f e a s i b i l i t y of a double a p e r t u r e s p e c k l e interferometer. One beam i s needed to i l l u m i n a t e the o b j e c t s u r f a c e , and the s u r f a c e i s imaged on a p h o t o g r a p h i c p l a t e by a l e n s h a v i n g two opposed c i r c u l a r a p e r t u r e s i n i t s e n t r a n c e p u p i l . radially To each a p e r t u r e c o r r e s - ponds a p a r t i c u l a r s p e c k l e p a t t e r n , and the two p a t t e r n s i n t e r f e r e coher e n t l y i n the p h o t o g r a p h i c p l a t e e m u l s i o n to produce a f i n e g r i d over an a r e a of each s p e c k l e . grating-like I n a double exposure method two grids a r e formed over the a r e a of s p e c k l e , one g r i d c o r r e s p o n d i n g to the undeformed s u r f a c e and the o t h e r to the deformed s u r f a c e . 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 two g r i d s add The c o n s t r u c t i v e a d d i t i o n p r e s e r v e s the g r i d - l i k e s t r u c t u r e of the s p e c k l e whereas t h i s s t r u c t u r e v a n i s h e s i n destructive addition. The a d d i t i o n takes p l a c e over a l l s p e c k l e s c o m p r i s i n g the " s p e c k l e d " image of the o b j e c t , and g e n e r a t e s f r i n g e s r e l a t e d to the s u r f a c e i n - p l a n e d i s p l a c e m e n t i n the d i r e c t i o n of the l i n e c o n n e c t i n g the apertures. two However, o p t i c a l f i l t e r i n g i s n e c e s s a r y t o v i e w these f r i n g e s . T h i s i s done by i l l u m i n a t i n g the p h o t o g r a p h i c p l a t e w i t h p a r a l l e l l i g h t v i e w i n g i n the d i r e c t i o n of the f i r s t o r d e r d i f f r a c t e d beams. n o t a t i o n shown i n F i g . 3.11 U s i n g the and employed throughout t h i s t h e s i s D u f f y showed t h a t a s m a l l d i s p l a c e m e n t v a l o n g the y - a x i s causes f r i n g e s a c c o r d i n g to v(y,z) = Ax — s n(y,z) (1.1) D u f f y d i d not c o n s i d e r the e f f e c t on f r i n g e f o r m a t i o n by d i s p l a c e m e n t s u and w w h i c h may and occur i n a d d i t i o n to the d i s p l a c e m e n t v. T h i s method i s l e s s s e n s i t i v e than the p r e c e e d i n g one and the f r i n g e s a r e of h i g h 8 visibility. I t , too, i s limited to the measurement of displacements smaller than the object speckle size and the system i s further handicapped by i t s f a i l u r e to work when slope changes exceed a certain magnitude; i n addition, long exposures are necessary i f small apertures are used. Duffy has also described an alternative technique which uses one i l l u m i nating beam and a lens with a single aperture to image the surface on a photographic plate. A double exposure i s used to record surface deformation. This recording contains a l l s p a t i a l frequencies from zero to the highest frequency which the aperture limited lens i s capable of passing. Through the use of a double aperture screen o p t i c a l f i l t e r i n g of the recording i s used to produce fringes related to the displacements along the l i n e connecting the two apertures; the s e n s i t i v i t y i s determined by the separation of the apertures. The advantage of this method i s that the s e n s i t i v i t y can be chosen after the deformation of the object was recorded to obtain the most desirable fringe spacing. The disadvantage i s 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 l a t e r a l component of surface displacement ter than the object speckle s i z e . i s grea- The object i s illuminated by one beam and a double exposure photograph, one each before and a f t e r straining, i s recorded on high resolution f i l m . The o p t i c a l transform of this recording i s a pattern of p a r a l l e l fringes of angular d i s t r i b u t i o n a given as v — sina = nX m where m i s the demagnification factor. The examination of the recording on a point by point basis thus y i e l d s the magnitude and d i r e c t i o n of the l a t e r a l movement of the object surface. A l t e r n a t i v e l y , the recorded image may be s p a t i a l l y f i l t e r e d to y i e l d a 9 contour map showing the surface displacement along a chosen d i r e c t i o n . The technique i s i n s e n s i t i v e to displacements i n the d i r e c t i o n of the l i n e 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 i s 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 s t r a i n s . I t i s c a l l e d the Double Aperture Speckle Shearing Camera (DASSC) and i s described by Hung i n several papers [3,4,5,6]. The camera i s the same as DASC used by Duffy except f o r the l a t e r a l shear which i s produced either by placing i n c l i n e d glass blocks i n front of apertures or by defocussing, i . e . , positioning the photographic plate a small distance away from the f o c a l plane of the lens. Regardless of how the l a t e r a l shear i s produced, Hung showed that the fringes are formed according to A y ( l + cos9 )u, s x v + Ay cos9yV,y = - An(y,z) (1.2) s with the notation being that of F i g . 3.11. Hung has not considered the e f f e c t on fringe formation by the displacements u,v and w and the s t r a i n w,y a l l of which, i n general, are present i n a specimen deformation. He then solved equation (1.2) f o r u,y and v,y " a l g e b r a i c a l l y " from two fringe patterns. 1.6 Limits of Investigation The work presented i n this thesis i s devoted solely to the t h e o r e t i c a l and experimental investigation of the use of DASC and DASSC f o r the measurement of displacements and strains i n planar surfaces. The theoretical investigation of the fringe formation by DASC and DASSC i s r e s t r i c t e d to the 10 analysis of the simplest possible models of the two cameras. In the analysis the following assumptions ( r e s t r i c t i o n s ) are made: a) A small area of the specimen surface may be represented by a point source of l i g h t i n the analysis of DASC and by two point sources (reflectors) when DASSC i s considered. b) The imaging lens(es) i s n e g l i g i b l y thin, free of aberrations and coincident with the aperture c) plane. The only s i g n i f i c a n t d i f f r a c t i o n occurring i n the camera takes place i n the aperture(s). d) The specimen i l l u m i n a t i o n i s collimated, monochromatic and p e r f e c t l y coherent. e) The recording medium (photographic plate emulsion) i s n e g l i g i b l y thin and records the i n t e n s i t y of the incident l i g h t i n a l i n e a r fashion. f) The two cameras are used only i n the double exposure method. The results of the theoretical i n v e s t i g a t i o n are two equations bing the fringe formation by the models of DASC and DASSC. ways i n which the unknown displacements descri- Then, various 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 p a r t i a l d i f f e r e n t i a l 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 algebr a i c equations or to an ordinary f i r s t order d i f f e r e n t i a l equation with a variable c o e f f i c i e n t . The experimental work i s 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 d i f f u s e l y r e f l e c t i n g surface. Light of wavelength 5145 A provided by an argon gas laser was the only i l l u m i n a t i o n used. The f i e l d of view (defined as /y less than 1 : 10 . 2 + z 2 /x ) i s smaller than 1 : 5 and i n most cases g 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 i n the f i r s t part of this chapter to f a m i l i a r i z e the reader having l i t t l e or no background i n optics with those aspects of optics which are used l a t e r i n the analysis of DASC and DASSC. Those readers acquainted with optics and, i n p a r t i c u l a r , interferometry may wish to proceed d i r e c t l y to Section 2.13 where the basic concepts of a general coherent interferometer are developed, although a b r i e f review of the whole chapter might be h e l p f u l , as the terminology and notation introduced here i s used i n the subsequent chapters. The chapter starts with a discussion of l i g h t waves and their representation and behaviour, such as r e f l e c t i o n , d i f f r a c t i o n and interference. The coherence of l i g h t i s then defined and developed, an examination of some of the imaging properties of a thin lens follows and, i n addition, various types of specimen surfaces are defined. In the l a s t part of this chapter the properties of a general coherent interferometer are derived. Some of the material presented there i s o r i g i n a l and i s essential f o r the analysis of DASC and DASSC. So that this chapter may be kept to a reasonable length, many topics are discussed only b r i e f l y and, hence, the presentation may be at times overly s i m p l i s t i c . However, most topics presented here are well known and are discussed i n depth i n numerous textbooks and source books [50, 2.2 on optics 56,60,61]. Light V i s i b l e l i g h t i s a form of electromagnetic energy usually described 13 as electromagnetic waves. The behaviour of l i g h t i s governed by Maxwell's electromagnetic theory and quantum theory; Maxwell's theory describes the wave-like aspects of l i g h t , while quantum theory describes the p a r t i c l e l i k e nature o f . l i g h t . Even though l i g h t i s an electromagnetic nature i t w i l l be represented here, without loss of generality, by i t s e l e c t r i c component only. This i s done both to simplify the notation and because the photographic plate emulsion, ..used as a recording medium, i s sensitive only to the i n t e n s i t y of the e l e c t r i c f i e l d component of incident l i g h t . 2.3 Geometrical Optics There i s a class of o p t i c a l phenomena which may be described without taking into account any hypotheses concerning the wave nature of l i g h t or i t s i n t e r a c t i o n with material bodies. This d i v i s i o n of optics concerned with the image formation by o p t i c a l systems i s called geometrical optics since i t s description i s founded almost e n t i r e l y on geometrical r e l a t i o n s . The laws of geometrical optics may be stated as follows: 1. Light i s propagated i n straight l i n e s i n homogeneous medium. 2. Two independent beams of l i g h t may intersect each other and thereafter be propagated 3. as independent beams. The angle of incidence of l i g h t upon a r e f l e c t i n g surface i s equal to the angle of r e f l e c t i o n . 4. On r e f r a c t i o n , as i s shown i n F i g . 2.1, the r a t i o of the sine of the angle of incidence to the sine of the angle of r e f r a c t i o n i s constant, depending only on the nature of the media. given by This relationship i s known as Snell's law and i s Fig. 2.1 R e f r a c t i o n of light. F i g . 2.3 F i g . 2.4 Fraunhofer d i f f r a c t i o n by an aperture. Point source. F i g . 2.5 Fresnel d i f f r a c t i o n by an aperture. r^sincj)! = n sinc}> 2 where 2.4 and n 2 (2.1) 2 are the indices of r e f r a c t i o n of the media. The Ray The ray may be defined as the path along which l i g h t travels or, a l t e r n a t i v e l y , i t may be said that the ray i s the d i r e c t i o n i n which the wave motion propagates. The o p t i c a l length of a ray of length 1 i n a medium of index n i s defined as the product n l . For example, the o p t i c a l length SR shown i n F i g . 2.2, i s given as SR = n j l j + n l 2 2.5 2 + n 3 l 3 Fermat's P r i n c i p l e This p r i n c i p l e , sometimes c a l l e d the law of extreme path, states that the path taken by l i g h t i n passing between two points i s that which i t w i l l traverse i n the least time. 2.6 Point Source of Light It can be shown that, i n free space, the e l e c t r i c f i e l d component of l i g h t , E ( r , t ) , emitted by a point source S radiating uniformly i n 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 i s the speed of wave propagation. The point source S i s shown i n F i g . 2.3, with R being the receiving point where E(r,t) i s measured. Solution (2.2) s a t i s f i e s Maxwell's equations everywhere except at r = 0. This s i n g u l a r i t y i s unimportant since any r e a l 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 i s of the form f(r-.--ct). =a;:cos [ k ( r - ct) + where ty] (2.3) 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 and k,c,X, and cu are related as 2TT k =— (2.4) A co = kc Using equations (2.5) (2.3),(2.4), and (2.5) we may write the e l e c t r i c f i e l d at R(r) i n the form E(r,t) = ^ cos(kr - cot + ty) (2.6) Equation (2.6) describes a c i r c u l a r l y polarized e l e c t r i c f i e l d . If the f i e l d i s polarized i n any other way i t i s 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 i n the plane of polarization. With this notation a polarized e l e c t r i c 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, i n general, the polarized e l e c t r i c f i e l d may be described by E(r,t) = E(r) ecos(k«r - oJt + ip) (2.7b) where E(r) i s the amplitude of e l e c t r i c f i e l d at r . 2.7 The P r i n c i p l e of Linear Superposition The theory of o p t i c a l interference i s based e s s e n t i a l l y on the p r i n c i p l e of l i n e a r superposition of electromagnetic f i e l d s . According to this p r i n c i p l e , the e l e c t r i c f i e l d E produced at a point i n empty space due to n d i f f e r e n t sources i s equal to the vector E = Ej + E 2 + sum. (2.8) + E. 'n The same p r i n c i p l e holds for the magnetic f i e l d . In the presence of matter, however, the p r i n c i p l e of l i n e a r superposition i s only approximately 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 i s not d i r e c t l y exposed to the oncoming waves. This phenomenon i s called d i f f r a c t i o n . In the study of d i f f r a c t i o n i t i s customary to distinguish between two general cases known as Fraunhofer d i f f r a c t i o n and Fresnel d i f f r a c t i o n . Fraunhofer d i f f r a c t i o n , shown i n F i g . 2.4, occurs when the source of l i g h t and the screen on which the d i f f r a c t i o n pattern i s observed are e f f e c t i v e l y at i n f i n i t e distances from the aperture causing the d i f f r a c t i o n . If either the source or the screen, or both, are at f i n i t e distances from the aperture then Fresnel d i f f r a c t i o n occurs. An example of Fresnel 19 d i f f r a c t i o n i s shown i n F i g . 2.5. There i s no sharp l i n e of d i s t i n c t i o n between the two cases of d i f f r a c t i o n and, i f i t i s at a l l possible, the Fresnel d i f f r a c t i o n i s approximated by the Fraunhofer d i f f r a c t i o n as the Fraunhofer case i s much simpler to treat t h e o r e t i c a l l y . The approximation i s appropriate only i f the actual o p t i c a l path from the source to the screen and the o p t i c a l path given by the Fraunhofer approximation d i f f e r by much less than the wavelength of l i g h t . 2.9 Huygen's P r i n c i p l e This p r i n c i p l e says that each point on a wavefront may be considered as being instantaneously and continuously the o r i g i n of a new wavefront moving outward from that point. spherical 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 o r i g i n a l wavefront. principle This i s useful i n the explanation of d i f f r a c t i o n and the determination of d i f f r a c t i o n patterns of various apertures. The i l l u s t r a t i o n of the Huygen's p r i n c i p l e i s shown i n F i g . 2.6. 2.10 Coherence In discussing the idea of coherence of l i g h t i t i s convenient to consider two i d e n t i c a l point sources Sj and S each radiating 2 at different locations, harmonic t r a v e l l i n g waves of the same frequency co, as i s shown i n F i g . 2.7, and generating an e l e c t r i c f i e l d at point R — = — Ej = E 2 1 r 2 ej cos (krj e cos ( k r 2 2 cot + ^j) cot + \p ) 2 21 The resultant e l e c t r i c f i e l d at R i s given by the p r i n c i p l e of superposition as E = Ej + E 2 The instantaneous i n t e n s i t y at R i s given by I(t) = |E| = ( E + E ) . ( Ei + E ) = 2 : 2 2 and the i n t e n s i t y recorded over "exposure" I E | + 2E «E |E | + ? ? : 2 1 2 time T>>T, (T being the period of l i g h t wave), i s derived i n Appendix A as I -1 t lj f r 2 r = Ij + I + 2 7 2 a 2 l 2j r T 2 ai a l i j [r2J ( 2 e i . e ) cos ( k r 2 1 - kr + 2 - ty ) 2 r + 2 / I ^ a '( e ) cos (krj - k r + ty - ip ) 2 2 The term 2 / I I ' ( e ^ e ) cos ( k r - k r + 2 2 2 x (2.9) 2 - ty ) i s c a l l e d the 2 x interference term, and i t s presence causes the resultant i n t e n s i t y 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 ty - i> d i d not 1 change during the "exposure" time. z I f the two sources behave i n such a way, I f the phase difference ty - ty they are said to be mutually coherent. 1 change i n a random fashion with time during the "exposure", 2 does then the mean value of the cosine term would be zero and the two sources would be c a l l e d mutually incoherent. The product of the unit vectors ej»e p o l a r i z a t i o n of the two e l e c t r i c f i e l d s . depends on the r e l a t i v e 2 If the p o l a r i z a t i o n of these two f i e l d s are mutually orthogonal, then e • e x two propagation vectors, k : 2 = 0. In many instances the and k , are nearly p a r a l l e l (G i s very small) 2 and, i f the two f i e l d s are c i r c u l a r l y polarized or are polarized i n the same way, then e ^ e ^ = 1 and equation (2.9) i s reduced to the form I r = Ij + I 2 + 2 /ijl/cosCkr-j - k r + ^ 2 Since the argument of the cosine depends on r 1 - ij>) (2.10) 2 and r , periodic s p a t i a l 2 variations i n i n t e n s i t y occur; these variations are the f a m i l i a r fringes that are seen when two mutually coherent beams of l i g h t i n t e r f e r e . 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 r e l a t i v e phase difference ty - ty w i l l remain constant over a time of the 1 order of (Av) 2 This time i s usually referred to as the coherence time, and the distance that the radiation traverses i n the coherence time i s c a l l e d the coherence length. Often the two sources may be "locked" i n 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 i n a random manner i n time (Av) , where Av i s now the bandwidth of the common driving force, the r e l a t i v e phase difference w i l l remain constant. There are several more aspects to the coherence of l i g h t that need to be mentioned here. radiation. In F i g . 2.8, S i s a point source of monochromatic The two points, P and P , l i e i n the same d i r e c t i o n from the r 3 source; they d i f f e r only i n their distance from S . The e l e c t r i c f i e l d at P i s Ej and the f i e l d at P 1 and E 3 isE . 3 The coherence between the f i e l d s E 1 measures the longitudinal s p a t i a l coherence at the f i e l d . Point P„ i s at the same distance from S as Pj , but i t l i e s i n a d i f f e r e n t d i r e c t i o n . In this case the coherence between f i e l d s Ej at Pj and E 2 at P 2 measures the l a t e r a l s p a t i a l coherence of the f i e l d . 2.11 Imaging Properties of a Thin Lens A lens i s a most common element occuring i n o p t i c a l systems. It i s made of a transparent, o p t i c a l l y dense material, usually glass, having an 23 F i g . 2.11 L a t e r a l magnification by a thin lens. index of r e f r a c t i o n greater than one. are s p h e r i c a l . Usually the two surfaces of a lens In a simple thin lens the l i n e through the center of the lens j o i n i n g the centers of curvature of the lens surface i s c a l l e d the o p t i c a l axis. The imaging "power" of a lens i s defined by i t s f o c a l length f, which i s the distance from the lens at which a l l incident rays p a r a l l e l to the o p t i c a l axis w i l l meet a f t e r passing through the lens as i s shown i n F i g . 2.9 distance x s . If a thin lens i s used to image a source point at from the lens, then the image w i l l be formed at distance x-^ behind the lens, as i s shown i n F i g . 2.10 are related to the f o c a l length f by the . The two distances x s and x^ equation (2.11) An important imaging property of a thin lens i s that a l l rays emitted by S and passing through the lens to the receiving point R are of equal (or nearly equal) o p t i c a l path length. The image size of an object i s usually d i f f e r e n t from the actual size of the object. This imaging property of a thin lens i s called the l a t e r a l magnification m . By considering the geometry of F i g . 2.11 , equation (2.12) r e l a t i n g the object size and the image size to the lens parameters is obtained. (2.12) The minus sign i n the equation (2.12) means that the image of an object i s inverted. 2.12 Aberrations i n Optical Systems Optical systems i n which thin spherical lenses are used have a number of aberrations or f a u l t s which impair or l i m i t the imaging quality of the F i g . 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 , d i s t o r t i o n of the image, and chromatic aberration. A l l aberrations are analyzed i n great d e t a i l i n a number of optics textbooks and other. The only aberration that needs to be mentioned here i s curvature of the f i e l d . I t arises i f the object i s an extended plane; i n that case the astigmatic images w i l l not be planes but curved surfaces. For object points on or near the o p t i c a l axis, there w i l l be sharp point-to-point representa- tion i n the image plane, but, as the distance from the axis i s increased, the sharpness of the image w i l l decrease. be represented Each point of the object w i l l 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 l i n e about the axis. An example of this aberration i n a system using a lens with two r e l a t i v e l y small apertures i n i t s entrance p u p i l i s shown i n F i g . 2.12 . The curvature of f i e l d aberration may be corrected i f more than one thin lens i s used. 2.13 Generalized Coherent Interferometer We w i l l consider here a coherent interferometer with one point source of polarized monochromatic coherent l i g h t . meter i s shown i n F i g . 2.16 ... r n . The schematic of the i n t e r f e r o - S i s the point source of l i g h t , and r x ,r , 2 are the l i g h t rays passing through the interferometer and reaching the receiving point R. In Section 2.6 be described by equation (2.7a). , i t was shown that these rays may Our task i s to determine the expression for the i n t e n s i t y of l i g h t which would be recorded at the receiving point over some "exposure" time T much greater than the period of l i g h t radiated by the source. The analysis w i l l eventually be r e s t r i c t e d to the case where a l l rays reaching the receiving point are nearly p a r a l l e l so that the scalar description of l i g h t may be used. The resultant e l e c t r i c f i e l d at the receiving point i s given by the p r i n c i p l e of superposition as n E (t) = r I Y~ e i c o s ^ . ^ i - bit + i =1 ty) (2.13) 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 i s i n f i n i t e and, hence, the summation sign i n equation (2.13) must be replaced by the i n t e g r a l sign. E (t) = r Y7y~y Equation (2.14) r e s u l t s . 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 o p t i c a l path length of a ray, and the propagation vector are expressed as functions of the coordinates (y,z) of the aperture i n the entrance pupil of the interferometer. over "exposure" time T i s given by The intensity recorded at R 28 E ( t ) | dt (2.15) r 0 Let us now consider an interferometer where the l i g h t rays reaching the receiving point R are nearly p a r a l l e l and polarized i n the same way. In addition the o p t i c a l path lengths of the rays are almost equal so that the "mean" o p t i c a l path length r Q may be defined as the o p t i c a l path length of a t y p i c a l ray. With these assumptions, and using r approximate equation 0 , we may closely (2.14) by E (t) cos[kr(y,z) - cot +ty]dA r (2,16) It i s now convenient to introduce the path length v a r i a t i o n r„(y,z) defined as r ( y , z ) = r(y,z) - r . (2.17) e In Appendix B equations (2 .15),(2 .16), and (2.17) were used to derive equations (2.18) and (2.19) giving I as r r a ^ l oJ r T 2 j cos k r ( y , z ) dA "A j s i n k r ( y , z) dA ^A + g With the use of complex notation, equation a Ir = l oJ r T 2 ikr (y,z) e (2.18) may be written as ikr (y,z) dA e v (2.18) £ dA (2.19) A It should be noted that the i n t e n s i t y I r depends, i n general, on the positions of the source point S and the receiving point R, since any change i n their positions w i l l cause a change i n o p t i c a l path length v a r i a t i o n r (y,z). e The recorded intensity I r often varies i n some reasonable fashion 29 with y±> ± thus producing z a pattern of dark and bright fringes usually referred to as an "interference pattern". I f the image formation or c h a r a c t e r i s t i c s of the interferometer are known, i t may be possible to obtain some information about the source of l i g h t from the interference pattern, and indeed, most interferometers are used i n this way. used to produce the source interference patterns Various techniques are which provide information about (with respect to the object's size, p o s i t i o n , displacement, etc.). Reseachers usually s t r i v e to design an " i d e a l " interferometer which would be sensitive to only one p a r t i c u l a r aspect of the source or object behaviour which i s of interest, while being completely other aspects. i n s e n s i t i v e to the Unfortunately, i n most cases this cannot be achieved, and the object behaviour which i s of interest must be "extracted" from the interference pattern by an a d d i t i o n a l data processing, f i l t e r i n g , e t c . 3. ANALYSIS OF THE FRINGE FORMATION BY DASC AND DASSC 3.1 Preliminary Remarks In Chapter 2 the p r i n c i p l e s of physical optics were reviewed, and the equations describing the image formation by the generalized coherent interferometer were derived. Using these r e s u l t s , we w i l l now examine the image formation of three p a r t i c u l a r interferometers. In the study, the actual interferometers are represented by their mathematical models, and the object surface i s 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 r e a l interferometers with s u f f i c i e n t accuracy for p r a c t i c a l testing. The chapter s t a r t s with an analysis of image formation by a camera having one small c i r c u l a r aperture i n i t s entrance p u p i l . The basic p r i n c i p l e s , 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 i n the double exposure process are derived. 3.2 Image Formation by a Single Aperture Camera The model of the camera i s shown i n F i g . 3.1 . The cartesian coordinate system x,y,z i s set up with x-axis coincident with the o p t i c a l axis of the system and the y and z-axes i n the source plane. source S(yg,Zg) The point radiates a monochromatic, coherent l i g h t of wavelength X and amplitude a at the unit distance from S. The aberration free lens has a f o c a l length f, diameter D^, and i s n e g l i g i b l y 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 s z 11 ya A - 1 } 0,z z source plane lens a V aperture plane x s F i g . 3.1 ± z , R x (yic ic) z image plane x i S i n g l e a p e r t u r e camera. undiJ f r a c t e d r a y ( t c> R) d i f f i •acted r a y ( t o C)) : dw S W N\6^ y \\7 1 A P < s ( °> v ' Q(y , ± z } ' s z z x s F i g . 3.2 a x i D i f f r a c t i o n i n a s i n g l e a p e r t u r e camera. R X 1 ( y ic 32 are at approximately the same distance x s from the source plane. The aperture plane i s opaque, i n f i n i t e i n extent, and contains a c i r c u l a r aperture of a diameter d with i t s center at (y^,0). the image plane at R(y-^ > Z-j_ ) • c c The geometric image of S(yg,Zg) i s i n I t i s assumed that the aperture diameter d i s very much smaller that the lens diameter D-^ and, hence, the only s i g n i f i c a n t d i f f r a c t i o n i n the camera occurs i n the aperture i t s e l f . This d i f f r a c t i o n causes the image, R, of S to be "blurred" rather than being a point. We wish to determine the recorded intensity d i s t r i b u t i o n I r around the geometric image R. One of the imaging properties of a thin lens i s that the o p t i c a l path lengths of a l l undiffracted rays from a point source to i t s geometric image are equal. Thus i n F i g . 3.2, which shows the d i f f r a c t i o n i n the camera, a l l rays from S to R are of equal o p t i c a l length, that i s S-l-R = S-2-R = ... . Since the source wavefront sw reaching the aperture i s 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 where Q(y^,z^) would be the image of D. D-Q, Let us consider a d i f f r a c t e d spherical wavefront dw'of such curvature and orientation that i t would appear to originate from the point D. The o p t i c a l path lengths from the wavefront dw to the v i r t u a l image of D at Q are a l l equal, that i s 6-Q = 7-Q = ... . However, the d i f f r a c t e d wavefront dw i s derived from the spherical, constant phase , source wavefront sw and, therefore, i n general i s 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 i n terms of the system geometry, and the intensity at point Q due to the d i f f r a c t e d wavefront dw can be calculated according to the equation (2.18). The analysis i s r e s t r i c t e d to those systems where x s» i x > y yi>ys»yD' i' s> D» » »yA z z z d D (-) 3 1 33 F i g . 3.4 D i f f r a c t i o n pattern of a single c i r c u l a r aperture. 34 Let us now determine an approximate expression f o r the o p t i c a l path length v a r i a t i o n r ( y , z ) which was defined i n Chapter 2 as e (2.17) r ( y , z ) = r(y,z) - r. e Here r(y,z) i s the o p t i c a l path length from the point source S(y^,z^) i n the source plane, through a point (y,z) i n the aperture plane, to the image point Q(y^,z^) i n the image plane; i t i s given by (3.2) r(y,z) = S-A + dw-sw + A-Q where dw-sw i s the distance by which the d i f f r a c t e d wavefront source wavefront sw and A i s the aperture centre. dw leads the For example, r(y,z) f o r 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 (3.3) = S-A + A-Q then the substitution of equations (3.1) and (3.2) i n equation (3.3) y i e l d s r ( y , z ) = dw-sw (3.4) e It i s shown i n Appendix C that because of equation (3.1) the o p t i c a l path length v a r i a t i o n r r e =r 1 - 3ys^ e i s very closely approximated by A 2x"j Ay ys s Az cose + 3 z z — 1 s 2 \z 2 SYS A 2~~ Ay s i n t (3.5) The radius r and angle 8 of equation (3.5) are shown i n F i g . 3.3, and Ay and Az are defined as A y = yp_JLZS A z = ^ ^ (3.6) In Appendix F i t i s shown that a l l rays reaching a point i n the image plane 35 are nearly p a r a l l e l , thereby making i t possible to disregard the vector nature of l i g h t i n the calculation of the l i g h t intensity at that point and consequently, equation (2.19) may be used to calculate I r . The i n t e n s i t y i s calculated i n Appendix D from equation (2.19) with r •given by equation (3.3) and r ( y , z ) given by equation (3.5). The integration i s done over e the c i r c u l a r area A of the aperture with the result MP) I Q f2J (p)l 1 = i (3.7) c [al and p are given by 2 l oJ r kd P = 2x /(y± -- y ) i c ± T frrd ! 2 { * J 2 2 + 2 (3.8) - (3.9) H,? Equation (3.7) describes the recorded i n t e n s i t y d i s t r i b u t i o n I r as a function of the image plane coordinates y^ and z^ . J j i s the f i r s t order Bessel function of the f i r s t kind. The amplitude of the d i s t r i b u t i o n i s proportional to the square of the aperture area Tfd /4 and to the amplitude 'a' of the radiation. distance r Q .. I r I t decreases with the square of the source-image i s l i n e a r l y proportional to the exposure time T. The d i s t r i b u t i o n i s of the shape shown i n F i g . 3.4 and i s symmetric about the geometric image R ( y i > Z i ) of the source point; here i t also attains i t s c maximum value. c The f i r s t minimum i n the d i s t r i b u t i o n occurs when p = 3.83 , and the area within this perimeter minimum i s known as the Airy disk. I t s diameter D g i s given i n terms of the system geometry by equation (3.10) derived i n Appendix E . D s = 2.44A^ (3.10) The apparent diameter of the Airy disk i n the source plane i s given by (3.11) As i s shown i n F i g . 3.4 , the values of the maxima of the intensity d i s t r i b u t i o n decrease rapidly with increasing distance from the centre o the pattern and, hence, the d i f f r a c t i o n pattern of the c i r c u l a r 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 c i r c u l a r aperture i n i t s entrance pupil i s used to image a point source of monochromatic coherent l i g h t , the image the point source i s e s s e n t i a l l y a c i r c l e , sometimes referred to as a speckle, of diameter D F i g . 3.5 s . Double aperture camera. The model of DASC i s shown i n F i g . 3.5 . With the exception of the two apertures, i t has the same geometry as the single aperture camera. The two apertures are c i r c u l a r , of diameter d, and their centers are at (x ,-D/2,0) and (x ,+D/2,0) where D i s the separation of aperture centers. s s Again we wish to determine the recorded intensity d i s t r i b u t i o n around R ( y ^ , z ^ ) , the geometric image of S . c c The d i f f r a c t i o n process i n the two apertures i s shown i n F i g . 3.6 . The o p t i c a l path length v a r i a t i o n s , r for aperture 1 and r g 2 for aperture 2, are derived i n Appendix G as r ei r r e = e 2 i s given by equation (3.5) and r e = r e - D6 6 i s given by equation (3.13). Because of the assumptions stated i n equation (3.1), the e l e c t r i c f i e l d unit vectors are nearly p a r a l l e l and equation (2.19) may recorded i n t e n s i t y d i s t r i b u t i o n I . again be used to calculate the The calculations are done i n Appendix H with the result 2J (p)^| 1 I In equation (3.12) I0 L r = 4I n 2 kD6 cos —T- (3.12) i s given by equation (3.8), p by equation (3.9), and 6 by y± - y i c 6 = ~ - ^ (3.13) Equation (3.12) describes the recorded i n t e n s i t y d i s t r i b u t i o n i n the image plane as a function of the image coordinates (y-j^z-^) . The amplitude of the d i s t r i b u t i o n i s proportional to the square of the area of the two apertures 2 ^ and to the amplitude square of the distance r Q a' of the radiation. It diminishes with the from the source to i t s image and i s l i n e a r l y proportional to the exposure time T . An example of the t y p i c a l shape of F i g . 3.6 D i f f r a c t i o n i n a double aperture camera. 39 the d i s t r i b u t i o n i s shown i n F i g . 3.7 f o r the case of D/d = 4 . The envelope of the intensity d i s t r i b u t i o n i s of the same shape as that f o r the single aperture case shown i n F i g . 3.4 . However, because the intensity d i s t r i b u t i o n i s modulated by the cosine term of higher frequency, i t causes the g r i d - l i k e appearance of the speckle. This grid i s normal to the y£ coordinate and i s centered on and symmetric about the geometric image F i g . 3.7 D i f f r a c t i o n pattern of two c i r c u l a r apertures. 40 of the source point. The speckle diameter D s i s the same as that i n case of the single aperture, i . e . D The "grid p i t c h " G s = 2.44 A^p s (3.10) i n the image plane i s given by two successive zeros of the modulating cosine term and i s calculated i n Appendix I as G = A^ s The apparent "grid p i t c h " G G i n the source plane i s given by s s S S (3.14) =A^- (3.15) The result of this section may be summarized as follows: When a camera with 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 pupil i s used to image a point source of monochromatic coherent l i g h t , the image of the point source i s e s s e n t i a l l y a speckle of diameter D by a grid of pitch G g g modulated which i s perpendicular to the l i n e 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 i n the double exposure process are derived here. The equations r e l a t e 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 i s centered about the geometric image of S i n the image plane at the point R ( y > i ) where the coordinates y^ z l c c and z ^ are given c by equation (2.12) as v ys ic j The image i s a speckle of diameter D perpendicular to the y-axis. xc and i s modulated by a grid of p i t c h G s s The elevation of this speckle i s shown i n F i g . 3.8 . Between the f i r s t and second exposure of the double exposure recording method the specimen i s deformed i n a general manner. Point S , which repre- sents the specimen surface, i s therefore displaced both i n and out-of-plane and i t s new coordinates are (u,yg+v,Zg+w) as shown i n F i g . 3.9 . The three components comprising the displacement vector of S are u,v and w oriented along x,y and z-axes respectively. variations, r and r g 2 The o p t i c a l path length , occuring during the second exposure are derived to within an accuracy of A/30 i n Appendix J as r r where E ,E ,F 1 5 = u Xso 2 ys Xso ei e = E[( I e F)cos6 + E sin0] + 2 = l[( i " F)cos9 + E sin9] - D(6 - £) E 2 2 and E, are defined i n Appendix J with £ given as u1 Xso + Xso ys + w D X so Xso + Xso y| V Xso x soJ _w_ y z so Xso x s (3.16) I x r i s calculated i n Appendix K with the r e s u l t r = 4 I o JAP,) J^PPI , 2 + ,kD(6 r2J (p h 1 1 cos speckle g r i d term -O (3.17) s 43 where p x and p 2 kd / are given as P = 2 F) + E 2 —kd / ( E j - F) + E 2 2 2 ^ 2 The term Pi P 2 i s of much lower "frequency" J than the speckle grid term and i t s magnitude i s usually smaller than that of the term f2J (p )'| f 2 J ( p ) ] 1 1 1 2 Ps Pi pattern, I Since we are interested i n the recorded intensity , acting as a d i f f r a c t i o n grating, the former term may thought of as a low frequency be "background noise" which does not appreciably a f f e c t the d i f f r a c t i v e e f f i c i e n c y of the speckle grid described by the term 2J (P )i 1 2 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 " s h i f t e d " by the amount £ with respect to i t s f i r s t exposure position. This " s h i f t " and the s i m p l i f i e d plan view of the speckles recorded i n the two exposures i s shown i n F i g . 3.10 . In the double exposure method the images of the object i n the positions which i t occupied during the two exposures are both recorded the same photographic plate. on Usually the two exposures are of equal duration and, hence, the resultant image i n t e n s i t y d i s t r i b u t i o n i s the of the two d i s t r i b u t i o n s , as i s shown i n F i g . 3.10 . The two sum speckles overlap, and i n the overlaping area the two grids, each belonging to one speckle, add. This addition i s defined here as constructive when the high i n t e n s i t y regions of one speckle overlap the high i n t e n s i t y regions of the other speckle. This occurs when speckle grid s h i f t = nG c n = 0, ± 1, ± 2, 44 In a similar manner, the addition i s defined as destructive when the high i n t e n s i t y regions of one speckle overlap the low intensity regions of the other speckle; this destructive addition occurs when 1 n =+— speckle grid s h i f t = nG c 3 +— Thus, "n" indicates the type of addition which takes place i n a double exposure method. In general, n i s continuous and the product nG g i s equal to the s h i f t between the two grids caused by the deformation of the specimen surface. In p a r t i c u l a r we may write speckle grid s h i f t = y^ - y - , = nG„ an n n continuous CT (3.18) In equation (3.18) Y-^gj i s the speckle grid "center" position during the j - t h exposure. Equation (3.18) i s solved for n i n Appendix L with the result n = - 2.-1 D A.x c where X g Q ys 1- u S0 X J ry s x + u s ^ X S0 i s defined as X x S0 so 2x v y + wz , X s s s so = 4 + y| + ys s z v (3.19) + V so 4 yl + Equation (3.19) may be closely approximated by n D - A.x c ys ^ ys s z — u + v -— — w s s x (3.20) x The accuracy of equation (3.20) should be s u f f i c i e n t 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 t h i s , the displacements must be s u f f i c i e n t l y small. Equation (3.21), which r e s t r i c t s the size of the surface displacements, was derived i n Appendix M as The addition of the speckles recorded during the two exposures i s shown i n Fig. 3.10 . The out-of-plane component u of the displacement vector causes the second exposure speckle envelope to "divide" into two c i r c l e s 2^ and 2g . The Moire fringes are formed b y the addition of the speckle grids created during the f i r s t and second exposures. This addition takes place i n the area which i s common to speckles ( c i r c l e s ) and 2 B 1,2 . z w i5 2 x -w x i s z 1- u M") i i mum ic x s z ic second exposure speckle V f i r s t exposure speckle 1C x i y . - V •'IC Xe F i g . 3.10 . yic + U Xq D x ± -u x s 2x Elevation of the f i r s t speckles. i . yic i -v — + u + u — -— x x x zx x y s x D A i c s s s (1) and the second (2 ,2 ) exposure A g s 46 The r e s u l t of this section may be summarized by the following: When DASC i s used to record the displacement f i e l d of a specimen surface illuminated by a laser l i g h t , 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 s u f f i c i e n t l y small to s a t i s f y equation 3.5 (3.21). Image Formation by DASSC The double aperture speckle shearing camera (DASSC) i s similar to DASC except for the l a t e r a l shear of the images. This l a t e r a l shear may be achieved either by placing i n c l i n e d glass plates i n front of the apertures or by "defocusing". Both methods of producing the l a t e r a l shear were t r i e d and the defocusing method was found to be more convenient for p r a c t i c a l material testing, mainly because of the ease with which the size and the sign of the l a t e r a l shear may be adjusted to suit a s p e c i f i c test. The schematic of DASSC employing defocusing to produce the l a t e r a l shear i s shown i n F i g . 3.11 . With the exception of the position of the photographic plate, the geometry of DASSC i s the same as that of DASC shown i n F i g . 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 d i f f r a c t i o n process that actually occurs i n DASSC i s closely represented by the model shown i n F i g . 3.12 men . The schematic shows the speci- plane y,z and the image plane y±,z^ which are f a m i l i a r from the p r i o r analysis of DASC. There are, however, two more planes i n the system. One i s c a l l e d 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 i s c a l l e d the object plane, which i s an imaginary plane located at x = - y , where y i s such that the object plane would be focused by the system lens on the photo plane. The system lens(es) i s represented F i g . 3.11 The schematic of DASSC. F i g . 3.12 D i f f r a c t i o n i n DASSC. 49 by a thin lens so that the a x i a l distances are related according to equation (2.11) as 1 + 1_ 1= 1 __ x x f s where X g 1 + X Xi ± ,„ , ,x (2.11) s and X i are defined as X g = x s + y X ± = x ± - Y i Let us consider an imaginary point source S(-y,Yg,Zg) i n the object plane. The geometric image of S i s i n the photo plane at R ( Y i , Z i ) . c If c S were a r e a l point source, then the intensity d i s t r i b u t i o n around R at Q(Yi»Zi) would be found by considering an apparent source at D(-y,Y ,Z ), D the geometric image of which would be at Q . analysis of DASC previously. D This i s what was done i n the In the case of DASSC, the specimen surface i s represented by two r e a l point sources S j X C ^ y ^ j Z g ) and S. (0,yg 2 ,Zg ). 2 S l i e s where the l i n e from S to the center of aperture 1 (at y = y^ = D/2) 2 intersects the specimen plane, and S 1 l i e s where the l i n e from S to the center of aperture 2 (at y = - y^ = - D/2) The two r e a l point sources S Ay c S = y c o 2. ~ Yc • x and S 2 intersects the specimen plane. are separated by the distance The choice of positions of S and S 1 o 1 experience gained i n the analysis of DASC. i s based on the 2 I t i s anticipated (and con- firmed later) that the somewhat blurred image of S 2 made by l i g h t passing through aperture 1 is centered•about R; as is.the image of S -made by.light : x passing through aperture 2 . It i s reasonable to expect that the intensity d i s t r i b u t i o n around R w i l l depend greatly on the magnitude of the apparent speckle diameter D a n d the separation Ay gg S; 2 g of the two r e a l sources and therefore, two cases w i l l be considered. The f i r s t case occurs when Ay g > D gg . From the analysis of DASC i t i s known that the i n t e n s i t y d i s t r i b u t i o n around the geometric image of a point source i s of n e g l i g i b l e magnitude at distances greater than D /2 g from the 50 geometric image. The intensity d i s t r i b u t i o n around R w i l l therefore be produced e s s e n t i a l l y by the interference of the l i g h t radiated by S and 2 passing through aperture 1 with the l i g h t radiated by Sj and passing through aperture 2 . The interference pattern that i s produced i s shown i n F i g . 3.13 and i s similar to the one produced by DASC shown i n Fig.3.7 . modulated speckle unmodulated speckle unmodulated speckle 4 \ x 1C L Fig. 3.13 Intensity d i s t r i b u t i o n I r A i s f o r DASSC with Ay s > D s s Because of defocusing, the envelope of the pattern has changed; however, the speckle grid of pitch G through aperture 2 s i s s t i l l present. Light radiated by S and l i g h t radiated by S i 2 and passing and passing through aperture 1 are both imaged as "blurred" unmodulated speckles similar to the speckle shown i n F i g . 3.4 produced by a single aperture camera. These two speckles are not of interest i n as much as they do not d i f f r a c t l i g h t once the r e cording i s developed. In the c a l c u l a t i o n of the intensity d i s t r i b u t i o n I , computations r 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 nonl i n e a r terms representing " b l u r r i n g " of images may not be neglected. The calculation of the o p t i c a l path length v a r i a t i o n i s performed i n Appendix N, and the intensity d i s t r i b u t i o n s are determined i n Appendix 0 . The f o l l o - wing notation was used: sw^j . .. source wavefront'originating from and reaching aperture i dw^ ... wavefront D(-Y,YQ,ZQ) appearing to originate from caused by d i f f r a c t i o n of a source wavefront r £j and i n aperture i ... the o p t i c a l path-length v a r i a t i o n of rays radiated by e source j and passing through aperture i The mean o p t i c a l path length r r r e i 2 and r r 2 + dWj-Q 12 (3.22) are derived i n Appendix N to an accuracy of A/30 or better as g 2 j IE = ei2 r = S -sw Q was chosen a r b i t r a r i l y as E ^ c o s e + K sin0) = r 2+ 2 £ = q r + r ( K o s 9 + K sin6) - Dp 2 e 2 1 lC 2 where q,K ,K , and p are defined i n Appendix N and I • i s calculated i n 1 2 Appendix 0 as f X r = "l 2 rji r d/2 2TT . , d/2 2TT | / e.rdrd9 . / /e" f l k r 2 e L 0 0 0 i k l e r dr dO 4cos ^ 2 (3.23) 0 speckle envelope term speckle grid term 52 The integrals comprising the speckle envelope directly. term cannot be evaluated While some approximate solutions are certainly possible, just a one step integration by parts done i n Appendix 0 yields results from which i t appears that the speckle envelope term i s made up of terms having a much lower frequency than the speckle grid term. small defocussing. This i s especially true for a In as much as the speckle grid i s our main interest, equation (3.23) describes the recorded I s u f f i c i e n t l y well. shows 1^ for this case; the drawing i s only an approximation sole purpose i s to show graphically what I F i g . 3.13 of I , and i t s might be l i k e and to a i d i n the discussion of the image formation by DASSC. Let us now consider the situation where Ay < D s s s • ^ n this case l i g h t radiated by Sj and passing through apertures 1 and 2 and l i g h t radiated by S and passing through apertures 1 and 2 a l l interfere with one 2 another and contribute to I . r Since Ay < D s s s , the parameter q i s s u f f i c i e n t l y small so that i t may be neglected i n expressions for r r £ 2 i . In Appendix N r and r e i l are derived and the four o p t i c a l path e 2 2 length differences are given as r K = r cos0 + K sin£ 2 sJ l ei l r e i 2 r e 2 i = £( i K c o s - a +a A s 0 + K sin0) 2 = rtKjCOsB + K sin9) - Dp re22 = r i 2 D 1 cos0 + K sin0 2 V SJ Also define p as p = and kd / — /K + K 2 l 2 2 2 Dp + a A +a c 53 The parameters Pj, and P 1^ = 2 and a g are defined i n Appendix N, and the parameters f5 , are defined i n Appendix 0. I i s derived i n Appendix 0 as All 2rr d/2 2rr T i k r e 2 2 ? 161, 0 pi(p)l i P J emn r d r d I . / / ~ m = 1,2 o o n= 1,2 e 0 0 i k r e m n r dr d9 2 , Pj + P c o s - (Dp - a ) 2 2 (3.24) A speckle grid term From equation (3.24) i t i s evident that the speckle grid with the same frequency as i n the previous case i s again present. The speckle envelope and the actual i n t e n s i t y v a r i a t i o n are rather complicated i n shape; this fact i s not of great interest to us, however, since we are mainly interested i n the d i f f r a c t i v e properties of the recorded I term i s present, the i n t e n s i t y pattern I by a d i f f r a c t i o n grating of p i t c h G 3.6 s r r . Since the speckle grid may again be closely approximated over the speckle area. Formation of Moire Fringes by DASSC In this section an equation i s derived governing the formation of Moire fringes by the DASSC i n a double exposure process. This equation relates the Moire fringe number n to the i n 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 l i g h t . Even though this point source was i n fact i l l u - minated by a laser, the change i n 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- ... o p t i c a l path length from the laser to the point S on the specimen surface Ar ... increase i n r T due to the displacement of S ... o p t i c a l path length from S to point Q i n the image plane through aperture 1 r ... o p t i c a l path length from S to point Q i n the image plane 2 through aperture 2 r e i ... increase i n r, l due to the displacement of S c r^ ... increase i n r ? due to the displacement of S The difference i n the lengths of the two o p t i c a l paths (one' through each aperture) from the laser to the point Q i n the image plane i s given by Aj = ( r ^ + r ) - ( r + r ) L 2 ... difference during the f i r s t exposure difference during A 2 = (r + Ar + r L L : + - r ) - ( r + Ar^ + r ei L 2 + r e 2 ) ... the second exposure The r e l a t i v e change between the two o p t i c a l paths due to the displacement of S during the time between the two exposures i s given by A 1 2 = A 2 - A x Since i t i s A = 1 2 (r 1 + r e i ) - (r + r 2 £ 2 ) - (^ - r ) = r 2 e i -r £ 2 that determines the r e l a t i v e s h i f t 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 i n this d i s tance due to the displacement of S has no effect on the Moire Fringe formation of DASC. The s i t u a t i o n i s somewhat different i n the case of DASSC. The specimen 55 surface i s represented by two point driving force - the laser. of the point sources S S sources and S { illuminated by one 2 This time, the changes i n distances between each and S 2 and the laser, a r i s i n g from the surface d i s - placements and strains, w i l l have to be considered i n determining the fringe formation. Define the terms: ... o p t i c a l path length from the laser to S r^ on the specimen l surface L2 r '"' ° P t l c a P t h length from the laser to S on the specimen a l 2 surface Ar Li ... increase i n r due to the displacement of S Li Ar^ ... increase i n r ^ due to the displacement of S 1n 2 r^j 2 ... o p t i c a l path length from Sj to Q i n the image plane through aperture i r increase i n r . . due to the displacement of S. eij ij J It i s s u f f i c i e n t , for now, to look at the case where A y > D g g s and, there-c - fore, we need to consider only two o p t i c a l paths. One path i s from the laser to S and from S i through aperture 2 to Q i n the image plane. 2 path i s from the laser to S image plane. and from S 2 through aperture 1 to Q i n the 2 As before, l e t us define A and A these two paths i n the two exposures and A 2 as'the differences between as the r e l a t i v e change between J 2 the two o p t i c a l paths. A A ! = 2 = < A 1 2 ( r = A + L r 2 r l + *i> " A r L l - A x Li + ( r r = Ar L 2i L i + r i*> + r e2i> - 2 - Ar L 2 ( r + r L £ 2 i + 2 The other A r L - r + 2 r + r 1 ^ 2 e ) 1 2 56 A similar A 1 2 could be formulated f o r the case Ay < D g g g . I t i s not necessary to do so, however, as the only purpose of the above exercise was to show the need to consider the changes i n distances between points on the specimen surface and the laser i n the analysis of fringe formation by the DASSC. I t should be noted that the actual distances between the laser and points S 2 and S are of no importance and, hence, w i l l be a r b i t r a r i l y 2 defined as zero; therefore, equation (3.24) need not be altered. When a specimen i s deformed, i t s surface i s , i n general, displaced and strained both i n and out-of-plane. of the two points Sj and S 2 , representing the surface, as follows S^O.yg^Zg) S (0,y ,z ) 2 Fig. 3.14 S2 s The deformation changes the coordinates S * ( u , y + v , z + w) Si + s£(u + Su,y s g +v + 6v,z +w+Sw) g General deformation of the specimen surface. 57 where 6u , 6v , and Sw are defined as 8v 6v = ^ A y s du 97 A y Fig. s 6w = 8w ^ 3.14 shows the specimen surface with the two points S the two exposures. and S x 2 during The surface i s illuminated by a collimated laser l i g h t with i t s orientation defined by the " i l l u m i n a t i o n " vector . The i l l u m i - nation vector 1 i s a unit vector perpendicular to the illuminating plane wavefront and oriented from the specimen surface toward the l a s e r . makes angles 9 X , 9 y and 9 Z with the x, y and z axes respectively; It con- sequently 1 may be written as /\ S\ /N 1 = c o s 9 x i + cos9 y j + cos9 z k Due to the displacements and s t r a i n s , the increments Ar^ and Ar-^2 are given by ArLi /\ /\ /\ /\ = (ui + vj + wk) • ( - . ! ) = -ucos0 x - vcos8 y - wcos9 z A r L 2 = [(u .+ 6u)i + (v + 6v)j + (w + 6w)kJ • ( - 1 ) - (u + <5u)cos9x - (v + 6v)cos9y - (w + 6w)cos( Ar T - 6u c o s 9A„ - 6v cos9,7 - 6 w c o s 9 „ Li l y We are now i n a p o s i t i o n to calculate ^ ±j and I r e i s considered. Appendix P as r e i 2 a n d T ezi ' a c c u r a t e ; first, the case A y s > D g s to A/10 , are derived i n 58 " i e + r 2X 2 X g C + fZ D - Zs I X Z s u + 6u x c s x X g (1+3,) s w + 6wl X s sin8 > + + Ar-^ - 6u cos0 x - 6v cos0y - <5wcos6z "e2 i hi 2X + r I C ——- i _ _3 -J> X g X g (1 + 3 x ) X g Zc u w sin9 - — — - (l + 3 i ) XgJ x s s + ArLi - COS0 + X fz D - Zg I X c - D(p - £*) x where E,*' i s defined i n Appendix P . Note that now when we are considering DASSC the o p t i c a l path length variations ^ ±2 e 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 i s calculated i n Appendix Q as 1^ = R x + R cos y D(p - £ * ) + A* 2 + R,sin k D(p - C*) + I r + (3.25) A* consists of three terms containing Rj ,R and R 2 and A* i s defined i n Appendix P. 3 defined i n Appendix Q The f i r s t term R x may be thought of as a low frequency background noise; the second term R 2 i s l i k e l y to have the largest magnitude of the three terms and i s modulated by the cosine term producing the speckle grid; f i n a l l y , the t h i r d term R 3 i s l i k e l y to be of lower magnitude than the second term and i t i s also modulated by a grid 59 p r o d u c i n g term (but) of t w i c e t h e f r e q u e n c y T h e r e f o r e , , i n t h e sense o f I G g r than t h e second term. 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 , i t i s d e s c r i b e d by t h e term k cos — D(p - E*) 2 + A* = cos l*\ ,kD P - In t h e above s p e c k l e g r i d term, t h e presence e+ o f E* and A* i s due t o t h e specimen d e f o r m a t i o n t h a t took p l a c e between t h e two exposures. Through a comparison o f t h e s p e c k l e g r i d terms i n the two exposures i t i s obvious the second exposure s p e c k l e g r i d has been " s h i f t e d " by an amount E, w i t h r e s p e c t t o t h e f i r s t exposure g r i d . that D From t h i s ' ' s h i f t " o f t h e s p e c k l e g r i d t h e M o i r e number i s c a l c u l a t e d i n Appendix R as n(y,z) = - x — (1 + c o s 0 ) u , £ x v - c — Xo. Equation Ay yz u + v - cos9, - cost T y (3.26) w 'y (3.26) i s a c l o s e a p p r o x i m a t i o n o f a more c o m p l i c a t e d e q u a t i o n d e r i v e d i n Appendix R . "exact" The " e x a c t " e q u a t i o n i s o f l i t t l e interest to u s , though, as i t c o n t a i n s a number o f h i g h e r o r d e r terms w h i c h c o n t r i bute n e g l i g i b l y t o t h e e q u a t i o n ; t h e a c c u r a c y of e q u a t i o n (3.26) s h o u l d be s u f f i c i e n t f o r ordinary laboratory testing. As i n t h e case o f DASC, e q u a t i o n (3.26) i s v a l i d as l o n g as the s p e c k l e s r e c o r d e d d u r i n g t h e two exposures o v e r l a p . The e q u a t i o n s c o n s t r a i n i n g t h e s i z e of specimen de- f o r m a t i o n a r e d e r i v e d i n Appendix S as y - D/2 (u + 6u) + v + Sv Xc + (Xs (u + Su) + w + 6w 'ss (3.27a) 60 y + D/2 + u + v f \ Dss < z [X —— u + wJ (3.27b) . 2. s The addition of the speckles recorded i n the double exposure process i s of the same nature as i n DASC and therefore need not be examined again. Let us now consider the case A y g < D q and 3 X are so small that r £ 1 2 . g s and r e 2 1 , In this case the parameters accurate to A/10 and derived for the preceeding case (Ay s > D g s ) , may be approximated as Y r = r D - Y Xc Zg u + Xs r e21 - Yg-y^u+6u S Xc fiu X X fYp ~ Y S Xo ei i = r fY = £ D ~ Y S Y S Xc ~ YA XS rY - Y s Xc 3 + Ar-^ - 6u cos0 x - 6v cos0y - <5wcos8z _ Z . COS0 Xo Z + D - Z Z S S u x x Xc s s w sin0 Xe , re + and r £ 2 2 were derived i n Appendix P and Y s X s u v Xg Xg fzD COS0 + - z Zg _u_ s x x Xc s s w X s> n + y A u + 6u Xa Zg u + 6u Xs S (accurate to A/10) are presented here. D e22 Xo and r e 2 1 + ArLi - as+ r Z Xc J ^s D ~ - D(p - T ) In addition to r e i 2 the results sin0 Z COS0 + s _ YS + YA Xo + ArLi r w + 6w"| s v + Svl Xc Xc w + 6w' Xo J s m t v + Svl ••s Z COS0 + D ~ Z S Xc J + A r L i + a* + 3 2 2 - D ( P - E*) sine + 61 In the r i j above, the parameters 3 e I r r 2 2 are defined i n Appendix P i s derived i n Appendix Q as 16I I and 3 1X + Q cos - - D(p - £*) - a* + Q 2 t r 2 3n- 3 1 again contains the low frequency background noise term called here and another low frequency term Q grid i s produced again. Q 1 Q x which i s modulated so that the speckle 2 and Q (3.28) 22 2 are defined i n Appendix Q . 2 The Moire number i s obtained i n Appendix R by r e l a t i n g the s h i f t between the two exposure .speckle grid terms as yz n(y,z) = - u + v Y W + Ay c 2x„ u, + T V,y yz —2 Equations similar to equation (3.27) could be derived again. w > i (3.29) However, as four speckles would have to be considered, four equations of constraint would have to be derived. With the case Ay s < D ss of l i t t l e practical interest and because the system i s so much more sensitive to displacements than to strains, i t i s s u f f i c i e n t to use equation (3.17), derived for DASC, as a guide. The results of this section may be summarized by the following: When DASSC i s used to record the displacement and s t r a i n f i e l d s of a laser illuminated specimen surface the Moire fringes w i l l be produced according to equation (3.26) which i s applicable as long as the l a t e r a l shear Ay„ > D„„ . If Ay„ < D equation (3.29). oa , the Moire fringes w i l l be produced according to The surface displacements and strains must be s u f f i c i e n t l y small to s a t i s f y equation (3.27) f o r the case Ay i s used as a guide for the Ay s < D gs case. s > D ss , and equation (3.17) 62 3.7 Imaging of Real Surfaces by DASC and DASSC Equations (2.18) and (2.19) were derived f o r a coherent interferometer with one point source. When a r e a l , d i f f u s e l y r e f l e c t i n g surface i s considered, these, and other equations must be modified to take into a l l the l i g h t reaching a point (Q) i n the image plane. account This i s done by integrating over the illuminated area of the specimen, and consequently equation (2.14) becomes E (t) =/ / A A 5 where A a Z a ) a r ( y s ' cos [ k ( y , ) - r ( y , , y , z ) -cot+^(y, )]dA dA ' ' a' a e ( y a r Z y Z Z a Z a a Z s a (3.30) } i s the part of the area of the specimen surface illuminated by a g laser l i g h t , A a i s the aperture(s) area, y , z a and y,z are the specimen coordinates. the specimen surface coordinates. are the aperture coordinates a Note that f and ty are functions of As i n Section 2.13 we consider such an interferometer where a l l the rays reaching point Q are nearly p a r a l l e l , and r Q denotes some t y p i c a l object-image distance. Equation (3.30) i s then written as — cl E (t) = / /— r c o s [ k r ( y , z , y , z ) - cot + ty(y,z)] dA dA a a s £ (3.31) ^•s a A Assuming that a l l the points on the specimen surface r e f l e c t (radiate) l i g h t with nearly the same amplitude i s approximately I ^ / /e i k r edA dA s £ / Je~ s A A a a r (y,z,y ,z ) = r(y,z,y ,z ) - r a' "a' p , the recorded i n t e n s i t y given by A-s A where a - a a a a a ± k r e dA dA s a (3.32) 63 We know, from the analysis of the image formation by DASC, that the recorded intensity d i s t r i b u t i o n (due to a point source) i s of a s i g n i f i c a n t amplitude only within the speckle perimeter D g . I t i s therefore s u f f i c i e n t to confine the integration over the specimen surface to an area of approximately diameter D r„(y,z,y ,z) and o a. c gs about the p a r t i c u l a r point under consideration. iKy,z) may then be related to the (microscopic) surface a. geometry, and specimen illumination, and I (3.32). r i s calculated from equation Unfortunately, i n most instances the surface geometry i s not known, and since any r e a l surfaces are extremely rough on the scale of an o p t i c a l wavelength, no smooth-surface approximations necessary are possible and i t i s to rely on s t a t i s t i c s 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 i s moved" [43] . A l t e r n a t i v e l y , i t could be said that when the surface i s 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 v a r i a t i o n of the whole speckle then may be approximated by a phase v a r i a t i o n of a single " t y p i c a l " point - a point source(s) used i n this chapter. assume that equations We may therefore (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 r e a l d i f f u s e l y r e f l e c t i n g surfaces. The results of a number of experiments described i n Chapter 6. were found to support the v a l i d i t y of this assumption. The approximation of the speckle by a point source f a i l s , however, to explain the s e n s i t i v i t y of DASC or DASSC to the out-of-plane t i l t s of the specimen surface. I t 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 i s the aperture r a t i o of the lens, (f/d), and M i s 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. 4.1 CALCULATION OF DISPLACEMENTS AND STRAINS. 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 i s just the opposite, namely, the derivation of various methods of calculating the unknown surface deformation from a given pattern. When DASC or DASSC i s used to record the deformation of the specimen surface, the end product i s a photographic recording or recordings, showing a fringe pattern superimposed on the specimen surface. This fringe pattern i s i n fact "our" function n(y,z). When testing materials, a reseacher usually has some knowledge about the size of displacements and strains that are l i k e l y to occur during the testing. This knowledge may come from the theoretical considerations of the test, from d i r e c t 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 i s 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) i s desirable, a suitable curve f i t t i n g technique may be used. A l t e r n a t i v e l y , n(y,z) may be determined with reasonable accuracy at any point on the surface with the 2 knowledge that the fringe density v a r i a t i o n i s close to the cos variation. type of 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. I t w i l l be shown that usually i g . 4.1 F i g . 4.2 Normal view of the aperture plane. 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 i n certain cases of specimen deformation only one or two photographs are sufficient. photographs The c a l c u l a t i o n of the surface displacements from the 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 i n 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) = Ax ;7 U + V - (3.20) W s 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 l i k e equation (3.20) are required to provide a unique solution. varying the parameters D,X and x By we can obtain d i f f e r e n t patterns n(y,z); g however, these would a l l be l i n e a r l y dependent. One possible way of producing three independent patterns n(y,z) i s to "photograph" the specimen deformation with a camera having three aperture sets with d i f f e r e n t rotations i n the y,z plane-as i s shown i n F i g . 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 (y ,z ) + v (y ,z ) i i i 1 1 1 w ( y , z ) = - -^f~ i ( y i > i ) n i i 1 i = 1,2,3 z ( - ) 4 1 68 In equation (4.1) U j ,v , and w x vector along the x,yj , and z l displacement are the three components of the displacement x axes, and s i m i l a r l y u components along the x,y , and z 2 2 ,v 2 , and w are the 2 axes and so on. 2 i s the aperture separation D for the case where (j) = (j)^ . From F i g . 4.2 i t i s apparent that the coordinate systems y,z and y-^ , z^ are related as y^ = y costj)^ + z sincf)^ z^ = - y sincj)^ (4.2a) + 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 (y ,z ) i i = u(y,z) i (4.3a) v ( y , z ) = v(y,z)coscj) i w 1 1 i( i> i) v z = + w(y, z)sin<j> i (4.3b) ± ~ 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) i n equation (4.1) y i e l d s a u(y,z) + b v(y,z) + c w(y,z) = N (y,z) ± ± i ± i = 1,2,3 (4.4) where a^ ,b^ ,c^ , and N-^ are defined as a Yi =— s i (4.5a) A yi i = c o s ^ +—72- s i n ^ s z b ± yi i - -T72— cosa) s (4.5b) z c N i = sln< i y' ( z ) J>i = where, again, the the determinant x Xx s - ~tfT i ( y i > i ) i n (.y±>z±) z (4.5c) ± (4.5d) are related to (y,z) by equations (4.2). Provided of the c o e f f i c i e n t s of the set of equations (4.4) i s 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 p l a i n s t r a i n deformation where the out-of-plane displacement vanishes, i t may be possible to determine some displacement components from only one or two photographs. 4.3 The examination of some of these special cases follows. Use of DASC to Measure Plane Strain and Plane Stress Deformation Under these circumstances the out-of-plane displacement u of the specimen surface i s zero, or so small compared to the in-plane displacements that i t does not contribute s i g n i f i c a n t l y to the formation of fringes, and the term a^u i n 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 v(y,z) + c ( y , z ) = N (y,z) i i W i = 1,2 ± (4.6) If the magnitudes of the in-plane displacements v^ and w^ are of the same , 1 y-j photographed, the term ZA j , A 2 x over most of the specimen surface being W-j 1 w-^ may be neglected, and v. can be determined 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(yi.zi) = - n i(yi> i) z (4.7) It i s obvious that v^ i s given by equation (4.7) with the absolute error y± equal to the sum of the — A S y± ± z u- and ~— w,- terms. n A S If this absolute error i s F i g . 4.4 Rotated coordinate system. 71 acceptable equation (4.7) should be used, with the understanding that i t may r e s u l t i n large r e l a t i v e errors at those points of the specimen surface where v^(y£,z^) i s small or zero. While equation (4.7) relates ^y^> ^) v z t o ^(y^ ^) n nothing i s z determined about the other two displacement components u^(y^,z^) and w^(y^,z^). For the important case ofty^= 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° i s used and w(y,z) i s determined w(y,z) - - from 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 i s usually accompanied by small in-plane displacements. I f these displacements contribute n e g l i g i b l y |v (y ,z ) to the formation of fringe patterns, that i s i f - u ( y , z ) » x y±z± — 2 - w ( y , z ) over most of the specimen surface being photographed, the i i i i i i i i i two terms involving v^ and w^ may be neglected i n 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 The term i u ( y , z_ )N „ = -* — s n (y ,z ) x i y• — u±(y±,Zj) s i i r i i (4.9) i i s given by equation (4.9) with the absolute error x Y i -i z equal to the sum of the two neglected terms v^(y^,z^) and w^y^z^) , y* This may again cause large r e l a t i v e errors i n the calculated — u-^(y^,z^) s 2 x term i n 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 Ax , n,- (y-:, z-j ) ' i Yi 2 u (y ,z ) i i i lKyx y ± J + 0 (4.10a) D Equation (4.10a) relates the out-of-plane displacement f i e l d u^(y^,z^) to the fringe pattern function remain undetermined. n^y^jZ^). The in-plane displacements v^ and w^ Due to the peculiar form of equation (4.10a), a small error i n n^(y^,z^) may cause a large error i n the calculated u^ when y^ i s small. For the case of <j> = 0° equation (4.10a) becomes Ax u(y.z) = 2 - n(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). undetermined. The i n plane displacements v and w remain Similar considerations pertaining to accuracy that were made about equation (4.10a) apply to equation (4.10b). " s p e c i a l " cases where one or two photographs There are c e r t a i n l y other 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 s u f f i c i e n t l y the use of 73 DASC as an experimental testing device. To test the v a l i d i t y of equation (3.20) and get the " f e e l i n g " f o r DASC several experiments were performed a known specimen deformation. These experiments with and the numerical results obtained from the photographs taken by DASC are presented i n 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 l a t e r a l shear used. attention to the more p r a c t i c a l case of Ay g > D We s h a l l r e s t r i c t our , for which equation (3.26) s g i s applicable. n(y,z) = - y D A X [x S — U + y V X s — - COS0 xc Ay z 2~ w £ (1 + COS9 )u,y X S z COS0, V (3.26) w 'y y Equation (3.26) relates the surface displacements and strains to the fringe pattern function n(y,z) and f o r Ay = 0 , i t reduces to equation (3.20) . s Equation (3.26) may be written i n the form similar to equation ( 4 . 1 ) as y — x yz u + 5- v Ay X s w -I Ay x s (1 + cos0 )u,y - Ay X s y - YA - cost x £ x x| s s D XXc s v, s - cost 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 yi i " —X" w z x u s i + i v ± Ay i si + AysjXsi 7* n , ,„ (1 + c o s 0 ) u n n e f l x i s ' u y i Ay - s l X s i y i - yAi x s x COS0yi i > y i ^ si — — D? X - cost Z l WH- I J yi , . ni(yi,zi) This may be written i n a more compact form as A i i u + i i B v + i i + i i>yi c w D u + F i i»yi + i i > y i = i ( y i > i ) v H where the c o e f f i c i e n t s Ai = w N z (4.12) H-^ are defined as Yi B• =1 y-i1 C-; = - D F i = M i = S Hi = S z 1 + cos0 ) 1 xi i COS0 y ± i y - Ail y i cosfc) i - Xc z N-; = and S^ i s defined as Si = Ay i S x S i D? In Appendix T i t i s shown that, when a l l quantities i n equation (4.12) are transformed i n the y,z system, equation (4.12) becomes a u ± + b v ± + c-jw + d u, y + e . j U , + fj^v, + + 8iV, z + h w, + k w, = N (y,z) ± z ± y ± z In equation (4.13) the c o e f f i c i e n t s a , b , c i i the c o e f f i c i e n t s d^,....,^ are given by d-^ = S ^ ( l + c o s 0 i ) cos(j)£ x = S ^ ( l + c o s 0 i ) sincfi^ x ± i (4.13) are given by equation (4.5) and 75 = S ^ [ ( r - c o s 0 ) sintjj^ - (s - c o s 0 ) cosdpj coscj^ z i Si = S i [ ( " cos0 )sind) r z ± hj_ = ± : L - (s - c o s 0 ) coscjjj y ± sln<$> ± - S-^[(r - c o s 0 ^ ) coscj)-}. + (s - cosGy-^) sin<J>jJ c o s c ^ kj_ = - N y i z S±\_(* - cos0 zi )coscj)i + (s - c o s 6 ^ ) sind)-jj sind)^ y = - — 5 5 - niCyi.Zi) Di z w i t h r and s d e f i n e d as r = x i — s x s Due t o a l a c k o f equipment o r for o t h e r 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 t h e x,y p l a n e , i . e . 9 = 9 0 ° and c o s 0 Z z = 0 . For t h i s s p e c i a l , b u t i m p o r t a n t , case o f specimen i l l u m i n a t i o n t h e coefficients f^g^jh-j^k^ reduce t o t h e f o l l o w i n g s i m p l e r forms as i s shown i n Appendix U. f x = S-£(rsin<J>-£ - scoscj)^ - s i n 0 ^ ) c o s t j ) ^ x g^ = S^Crsincj)-^ - scoscf)^ - sin0 £)sincj^ x h± = - S£(rcos(j>i + ssincj)i)cosc})^ k^ = - S-^(rcos(J)£ + s.sind)^)sincj)^ Since Yi Zi < -jr , . i t i s obvious t h a t t h e c o e f f i c i e n t s h^ and k^ w i l l 5 p r o b a b l y be v e r y s m a l l ( e s p e c i a l l y near o r i g i n ) when compared t o t h e o t h e r c o e f f i c i e n t s , c o n s e q u e n t l y , t h e DASSC u s i n g t h e i n - p l a n e i l l u m i n a t i o n be r e l a t i v e l y i n s e n s i t i v e t o t h e s t r a i n s w, y and w, z will . There a r e t h r e e d i s p l a c e m e n t s and s i x s t r a i n components i n e q u a t i o n (4.13) and, i f t h e specimen d e f o r m a t i o n i s g e n e r a l i n n a t u r e , a l l n i n e terms contribute s i g n i f i c a n t l y . ways. One way independent Equation (4.13) may be solved i n two possible i s to regard strains and displacements as mutually and, consequently, nine equations (photographs) are needed to solve f o r the unknowns. The other way i s to take advantage of the fact that equation (4.13) i s r e a l l y a set of p a r t i a l d i f f e r e n t i a l equations and, i f the required boundary conditions are readily available, a set of three equations (photographs) i s s u f f i c i e n t to solve for a l l nine unknowns. The most advantageous approach to solving equation (4.13) depends greatly on a p a r t i c u l a r test s i t u a t i o n but, i n general, the f i r s t "algebraic" method of solving equation (4.13) should be undertaken only for special cases, and when there i s no point or l i n e of known displacements specimen surface. (boundary condition) on the When there i s a point or a l i n e of known displacements on the specimen surface the second method should be used. 4.6 Use of DASSC to Measure General Deformation Nine independent equations (photographs) (Algebraic Solution) could be obtained by using a number of aperture sets of d i f f e r e n t i n c l i n a t i o n to the y-axis and by using d i f f e r e n t illuminating beams associated with each aperture set, etc. p r i n c i p l e , the resultant set of nine independent In equations (4.13) could be solved f o r the nine displacements and strains; however, because of the unavoidable errors i n the determination of the camera parameters and i n the location of the fringes, this method of solution of equation (4.13) i s so inaccurate as to be of l i t t l e p r a c t i c a l use. The method i s reasonably accurate only when the specimen deformation i s of a special kind, or only i f the strains along a p a r t i c u l a r d i r e c t i o n are to be calculated, since under such circumstances fewer photographs need to be "taken" and processed. Using three aperture sets i n c l i n e d by the same angle cj) to the y-axis, three illuminating beams, each assoc iated with one aperture set, and two photographic plates i n series, one may calculate the three s t r a i n 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 i n these photographs s a t i s f y equation (4.12) as R(y ,z ) + i 1 D-jUj.yj F-jVj.yj + +H i W l , y i = N (y ,z ) i 1 i = 1,2,3,4 1 (4.14a) Equation (4.14a) may be solved for the three strains and the "displacement sum" R(y ,z ) defined as 1 R(y ,z ) = A 1 1 j U l + B J V J + C L W J When the illumination i s i n the x,y plane and the aperture sets are i n the d) = 0° position, the camera i s i n s e n s i t i v e to w and w, f o r small values y of z; i f the term H^w^,^ i s n e g l i g i b l e , equation (4.14a) reduces to R(y,z) + D u , ± y + F v, ± = N (y,z) y i = 1,2,3 ± (4.14b) Equation (4.14b) may be solved for the two strains u , , v,y and the y "displacement sum" R(y,z) from three independent photographs, with the absolute error being most l i k 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 s t r a i n where |u | << |v |,|w |, and V j x and Wj are such that | v | » y i x x z s i x x w. over most of the specimen surface being photographed, the displacement sum R ( y , z ) to v 1 1 Therefore,'in r addition to the three strains, a reasonably accurate v ( y , z ) i s also J calculated by solving the set of four equations (4.14). 1 1 I f the magnitudes of the in-plane strains are of similar order such that |v, | > |w,|, and y y i f the illuminating beams are i n the x,y plane and the aperture set rotations (J) a r e e q u a l to z e r o , the term H^w, i s so s m a l l t h a t i t may be n e g l e c t e d y and e q u a t i o n (4.14b) i s then c l o s e l y approximated by v + D , i U y + F v, ± In t h i s case, t h r e e independent of v , u , , and v , y . y = N (y,z) y 1=1,2,3 ± photographs (4.15) a r e s u f f i c i e n t f o r the s o l u t i o n A t those p o i n t s o f the specimen s u r f a c e where v , v , y or u,y a r e s m a l l o r zero, these v a r i a b l e s w i l l be c a l c u l a t e d from e q u a t i o n (4.15) w i t h a l a r g e r e l a t i v e e r r o r , but a r e l i k e l y to be determined every- where w i t h an a c c e p t a b l e a b s o l u t e e r r o r . 4.8 Use o f DASSC to Measure O u t - o f - P l a n e Bending In t h i s case o f specimen (Algebraic Solution) d e f o r m a t i o n the o u t - o f - p l a n e d i s p l a c e m e n t s and s t r a i n s a r e u s u a l l y much l a r g e r than the i n - p l a n e d i s p l a c e m e n t s and s t r a i n s , so t h a t Iv. —3- x s I U 1 » i1i vi »I ,y i -i . I v . zz . x x K'yil » s I (except f o r y w small) K'yil'K'yJ and e q u a t i o n (4.12) may be c l o s e l y approximated by A i i u + i i>yi D By u s i n g two independent equation u = NiCy^Zj) photographs, = 1» (4.16) 2 one may c a l c u l a t e • u and 1 U j j y j f r :om c (4.16) w i t h the a b s o l u t e e r r o r b e i n g p r o b a b l y o f the same magnitude as the terms t h a t were n e g l e c t e d . c o e f f i c i e n t A^, a l a r g e e r r o r i n u 4.9 1 Due t o the p e c u l i a r form of the t s h o u l d be expected where y Use o f DASSC to Measure G e n e r a l The main purpose x i s small. Deformation of d e r i v i n g e q u a t i o n (4.13) was to show t h a t a photo- graph taken through a s e t of r o t a t e d a p e r t u r e s d i s p l a y s a f r i n g e p a t t e r n which i s due to a combination of the three displacements u,v, .and w and the p a r t i a l derivatives of these displacements with respect to both y and z. By using three aperture sets, each with d i f f e r e n t rotation cf) , and by using three illuminating beams (at least one of them not i n the x,y plane), three independent photographs may be obtained. I f there i s a l i n e of known displacements u,v, and w (boundary condition) on the specimen surface i t i s possible, i n p r i n c i p l e , to solve the set of equations (4.13). For this the f i n i t e difference equations w i l l be used. The forward f i n i t e difference analogs f o r f i r s t order derivatives are u, (y,z) = u(y + A»z) ~ u(y,z) y u, (y,z) = u(y,z + A) - u(y,z) z A where A-is the grid spacing i n both y and z d i r e c t i o n s . By replacing the p a r t i a l derivatives with these f i n i t e difference analogs equation (4.13) becomes A ^i u(y,z) + u(y + A,z) + AJ A A u(y,z + A) + = N (y,z) ± i = 1,2,3 (4.17) If the displacements u,v, and w are known, f o r example, along the l i n e 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 ± " ( i a _ A i " e )u(y,z) - d ± : e i 3 §i k i = 1,2,3 (4.18) ] g, k § 3 k-3 2 *o Similar equations could be written f o r the case of the boundary condition being a l i n e y = y 0 or an i n c l i n e d straight l i n e . 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 l i n e . In contrast to t h i s , 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 p a r t i a l derivatives with respect to y only, the integration s t a r t s at a point on the boundary condition l i n e and progresses away along a l i n e z^ = constant as i s shown i n F i g . 4.4 . The p a r t i a l derivatives i n equation (4.12) are replaced by their backward difference analogs and, with these, equation (4.12) may be written as (A A + D ) u i ( y , z ) + ( B A + F ) v ( y , z ) + ± i 1 1 ± i 1 1 1 (C-jA + H ) w ( y , z ) = R i 1 1 1 ± i = 1,2,3 (4.19) where R^ i s defined as R i = N ( y , z ) A + D-jU^yj - A , ) + F y (y - A,z ) + H i 1 1 Z l i 1 x x (AjA + D j ) (BjA + F,) (C,A + H j ) (A A + D ) (B A + F ) (C A + H ) (A A + D ) (B A + F ) (C A + H ) 2 3 2 3 2 3 2 3 2 3 2 ± W l (y - A , ) * 0 : Z l (4.20) 3 To s a t i s f y equation (4.20) three illuminating beams are needed and the DASSC shutter must have three sets of apertures i n c l i n e d at the angle cf> with respect to the y axis. The advantage of this approach i s that the boundary condition l i n e C need not be straight and, since the f i n i t e differences are along a l i n e , i t i s possible to vary l i n e 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 c o e f f i c i e n t s D^, F^, and may be adjusted to be of the same magnitudes by a suitable choice of the aperture 81 rotation angle <$> . The programming of this scheme i s also simple. 1 If some of the displacement f i e l d s 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 i s then solved; for example, i f u i s known, equation (4.19) i s reduced to (B.jA + F ) v ( y , z ) + (C-jA + H ) w ( y , z ) = R i 1 1 1 i 1 1 1 1=1,2 ± (4.21) where R^ i s now defined as Ri = N ( y , z ) A i 1 1 - A - i A u j t y j . Z j ) - D Au , i 1 y i (y 1 (B A + F J (CjA + H ) (B A + F ) (C A + H ) : , zj + F y (y ,z ) i 1 1 1 + R y (y ,z ) i 1 1 1 x / 0 2 Once u ,v and w 1 2 2 2 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 d i f f e r e n t i a t i o n (using central f i n i t e differences f o r 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 s e n s i t i v i t y to the inaccuracies i n 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 i s i n the x,y plane and the aperture sets are at zero i n c l i n a t i o n to the y-axis ((J) = 0°), the terms i n 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 i s then possible to solve f o r u and v from two independent photographs as follows: ^- u + v - ^1 w + d \ j U , + f-jv, y hjV + y - 0 j = 1,2 = Nj y (4.22) • \ - 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 u 'yi i + l ~ A i u _ V 'y i + l ~ A v ± i v With these analogs equation (4.22) becomes —i y u ± +I v , i + l ~ +. d-j u ± i ,+ f i j + l ~ u v v f i „ = Nj Equation (4.23) i s solved f o r the unknowns U i + "i + 1 = U 1 ± + U , _ i n 3 = 1,2 and v^ x + ( , ^ (4.23) 0 x Ui + U .Vi 2 ± 3 (4.24) v i + i = i i + v V 2i i U where the c o e f f i c i e n t s U j i U 1 ± = - r(N f U 2 i = - r(f - f ) 1 + V 2 x 2 v V i are given by 3 - N f ) 2 3 i i V x Zi+ 1 v 1 ± 2 ± s U 3i = - with r defined as - f 2 ) V = r(N d x - N d ) 2 2 x = r(d - d ) | i s : 2 3 i= ( X - d ) + 1 r d 2 r = A/(f d x 2 In equation (4.24) the c o e f f i c i e n t s - f d ) 2 x V i are evaluated at y = y i . 3 I f , at some point y = yj , both displacements u and v are known, then u and v may be found elsewhere by s t a r t i n g at this point and evaluating the displacements at the point y = yj cedure, i s repeated are found. + by using equation (4.24). 1 This pro- until'.the displacements at a l l points to the r i g h t of To f i n d the displacements to the l e f t of yj one must solve equation (4.23) for u^ and v^ u i = Hii+ H i i + i + U i i + i u v 2 3 (4.25) v i = I i i+ li± ± + u l Isi i + l + v where the c o e f f i c i e n t s IJ ^ U are given by = s|N (A - f j ) - N,(A - f ) 1 ± 2 U i = d, - d, + s 2 v - N, A = s -ii 2 " 2 i = s(f - f ) 3 x V i 2 = s 3 with s defined as s = 1/ + d i " d - d, 0 ?" ( l " S d X l i x' d 2> S n " 2 In equation (4.25) the c o e f f i c i e n t s ( i f V_ ± a r ) f 2 evaluated at y = y^ e 3 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 s i t u a t i o n may a r i s e when the two displacements u and v are known at two d i f f e r e n t locations; u i s known at y = yj and v i s known at y = . for example, This case i s handled as follows: At point i + 2 equation (4.24) becomes U i + 2 = = U l i+ ( U n 1 + + i U + 2 i + U 2 i + l U i U i + 1 ± 1 + + 3 i + U u 3 i + l 1 v l V i ) i + 1 + ( u + ( 2 i U + 2 i + l U z i l 3 i U + u + 3 U i + l V 2 i 3 i + )u l 3i> V V 84 and a similar expression could be written for + . This process i s 2 repeated n times u n t i l i + n = k whereupon we get u i + n v i + n = k = Vi = u k = i k - U i + U i i + u - 2k U 3k i i v - (4.26) v The c o e f f i c i e n t s U _ l k k _ !+ V . V x 3 _ 2 k l U + V ± 2 l V i _ j are obtained by repeated use of the k recursion relationship between c o e f f i c i e n t s V £ _ 3 k + and U ^ > V \ ± x t^z± + l » j and so on. From equation (4.26) i t i s possible to solve for any + of u^ ,v^ , u , or v^ i f any two of these are known and, once two k 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 = U i+ i 'yi ~ i2A u i+ v i = 'y i 1 ~ i2A v i If the out-of-plane displacement u i s small, i t i s possible to solve equation (4.22) for v and v , as follows: y y — u + v + d l U , + f v, 7 y — u + v + d u, 2 = N 7 y + f v, 2 y = N (4.22) 2 By multiplying the f i r s t equation by d 2 and the second equation by (-d ) and x adding the two equations "we get (d 2 - ) d l s u + (d 2 d l ) v+ (f d x 2 - f 2 d l )v, By choosing the parameters of DASSC such that the term as possible and the term = N j d 2 -V 2 |d - d | i s as small 2 J | f d - f & \ i s as large as possible, we may 1 2 2 l closely approximate the above equation by (d 2 - d )v + (fjd x 2 which may be written as - f dj)v, 2 v, y = N d x - N d 2 2 t + p(y)v = q(y) d p(y) = f d with p(y) and q(y) defined as q(y) = f i d t 2 z d l N i d 2 - d 2 - f d, f l 2 - d - £ 2 x (4.27) " f N d, 2 2 2 i d * 0 Equation (4.27) i s an ordinary, f i r s t order d i f f e r e n t i a l equation with a variable c o e f f i c i e n t ; i t s solution [63] i s v(y) = 1 y(y) / y ( t ) q(t) dt + y ( y ) v ( y ) y 0 v (4.28) 0 0 where y(y) i s the integrating factor y(y) = exp / p ( t ) dt In equation (4.28) v ( y ) i s a known displacement v at one point on the Q specimen surface on the l i n e (z = z Q = constant) upon which v(y) i s sought. Once v(y) isfound, the s t r a i n v, (y) may be obtained by the numerical y d i f f e r e n t i a t i o n of v(y) or calculated from equation (4.27). v,y are determined, equations 4.11 After v and the two remaining quantities u and u,y are found from (4.22). 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) i s again applicable. Equation (4.16) may be written i n the standard form given by equation (4.29) and, since equation (4.29) i s of the same type as equation (4.27), i t i s also solved i n the same way, with the solution given by equation (4.30). 86 u i »yi Pi ( y i ^ i + u (4.29) ^ 1 <yi> = where P ^ y ^ and q ^ Y j ) are defined as PI(YI) = T> + 0 1 yi u^y,) = Ui(y ) x (4.30) / y ( t ) q ( t ) dt + y ( y ) u ( y ) 1 v 1 x 1 0 1 1 0 yio where the integrating factor P ^ Y j ) ^ 1 ^ 1 ) i s given by = e x Yi / Pi( ) dt P fc In equation (4.30) u ( y ) i s the known out-of-plane displacement u 1 on a l i n e ( z = z 1 0 x = constant) i n the rotated coordinate system y , z x 1 at y 1 0 where 1 u ( y ) i s being sought. 1 1 If the out-of-plane displacement at one point of the beam surface i s not available then two photographs are necessary for an approximate solution of U j and u . I f we use two sets of apertures at <j> = 0°, the two fringe 1 > y l patterns are given by equation (4.13) as y yz — 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 )u, 2 + (f j - f ) v , y 2 y + ( h - h )w, x 2 y = N x - N 2 By an appropriate choice of the DASSC parameters the term | (d - d ) u , | may 1 2 y be made much larger than the other two terms, and an approximate value of u,^ i s then given by equation (4.33): if |(dj - d ) u , | » 2 y \(f - f ) v , | , | ( h 1 2 y 1 - h )w, 2 ; (4.32) 87 d1 - d u »y ~ 2 (4.33) ± 0 Once u , i s calculated by using equation (4.33) i t i s substituted back i n y equation (4.31) and an approximate solution for u (except near the origin) and w i s found by neglecting small terms involving v,v »y >w, > w 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 i n the x,y, plane and the aperture sets rotated by a) = 0° the camera i s made i n s e n s i t i v e to w and w,y and, consequently, the terms involving w and w,y i n equation (4.12) are often n e g l i g i b l e . When this i s the case, equation (4.12) reduces to equation (4.22), the solution of which was derived and discussed i n 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 s t r a i n type, the out-of-plane displacement component u^ i s usually very small or zero. I f the terms involving u^ and i t s derivative(s) are so small that they do not contribute s i g n i f i c a n t l y to the fringe formation, these terms may be neglected i n 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 i n Section 4.9 allows v (4.21) with Uj and Ujjyj x and w x to be calculated from equation set equal to zero. As mentioned i n Section 4.10, when the specimen illumination i s i n the. x,y plane and the aperture sets are i n cj)^ = 0° position, the terms i n equation (4.12) involving w and are often n e g l i g i b l e ; and, when i t i s possible, equation (4.12) with the u, u , y ,w, and w, terms neglected y reduces to V which may be written as + f f V j y = Ni v , + p(y)v = q(y) (4.34) y p(y) = f~ where p(y) and q(y) are defined as w , -i i f E Equation (4.34) i s of the same type as equation (4.27) and, hence, i t i s solved i n the same way, with the solution given by equation (4.28) and the integrating factor given by y(y) = exp / P(t) dt Unfortunately i n most p r a c t i c a l testing, while i t i s often possible to neglect the terms involving w and w, y small and may not be neglected. , the term diU, y i s not s u f f i c i e n t l y I f this i s the case, equation (4.12) reduces to equation (4.22), the solution of which was discussed i n Section 4.10. 89 5. EXPERIMENTAL APPARATUS AND PROCEDURE In the f i r s t part of this chapter the various apparatus used i n the experimental work i s described and discussed i n 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 i s shown i n F i g . 5.1 and i t s schematic i n Fig. 5.2 . The camera may function as either DASC or DASSC, depending on the position of the photographic plate upon which the specimen i s imaged. The position of the photographic plate may be varied along the x-axis with the use of the adjustable s l i d e . The sub-assembly of the photographic plate holder and the s l i d e i s shown i n F i g . 5.3 and shutter assembly are each mounted on into holes i n the camera frame. . The s l i d e , lens assembly, i n . diameter s t e e l rods f i t t i n g These holes, spaced 1 i n . apart, are along the f u l l length of the frame, and are located precisely i n the centre of the frame to ensure the a x i a l 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 i n a tabular frame. positioned i n front of The shutter assembly shown i n F i g . 5.4 i s the lens assembly and u t i l i z e s a rotating shutter to open or close various aperture sets as required i n a p a r t i c u l a r experiment. The stationary part of the shutter assembly accomodates a number of interchangeable aperture plates each having apertures of various diameters. The camera enclosure i s used to prevent any unwanted l i g h t from reaching the photographic plate. F i g . 5.1 Double a p e r t u r e s p e c k l e s h e a r i n g camera (DASSC). enclosure ^ ^ h o t o g raphic plates ape r t u r e te plate holder lens ass 'y ad j u s t a b l e slide IT iii i Ml i 1 frame i • 11 | -M 1 W/////////A F i g . 5.2 Schematic of DASSC. Photographic plate holder assembly Fig. 5 . 6 Recording system. Fig. 5 . 7 F i l t e r i n g system. photographic recording made by DASSC beam expander mirror frosted glass diffracted light of f i r s t order f i l t e r i n g system camera laser mirror Fig. 5.8, Schematic of the f i l t e r i n g system. -p- 95 5.2 The Recording System A t y p i c a l recording system i s pictured i n F i g . 5.6 and i t s schematic i n F i g . 5.5 . The system consists of the camera, the o p t i c a l components, the laser providing the specimen illumination, and the loading apparatus with the specimen. The camera was described i n the previous section and the loading system w i l l be described i n Section 5.4 . The laser used i s an o argon laser providing a source of coherent l i g h t of wavelength 5140 A delivered at a maximum continuous power of approximately .6 watts. The laser beam of approximately .1 i n . diameter i s directed by the adjustable mirrors and divided by the beam s p l i t t e r s . 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 i n this way. The purpose of the recording system i s 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 i s developed, contains two speckle grids, each corresponding to one exposure; as described i n Chapter 3, the two speckle grids add and thus form a resultant grid of variable d i f f r a c t i v e e f f i c i e n c y . 5.3 The F i l t e r i n g System Once the photographic, double exposure recording of the specimen deformation i s made and subsequently developed, i t must be "processed" i n 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 i n Figs. 5.7 and 5.8, uses again the laser as a source of l i g h t and mirrors to d i r e c t the laser beam. After properly oriented, the laser beam i s expanded by the beam expander and, upon reaching a s u f f i c i e n t diameter, i s made to converge 97 by use o f a l a r g e l e n s . The p h o t o g r a p h i c p l a t e c o n t a i n i n g t h e r e c o r d i n g made by DASSC i s p l a c e d i n t h e c o n v e r g i n g l i g h t and, because i t c o n t a i n s a s p e c k l e g r i d , t h e r e 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 s p e c k l e 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 s c r e e n upon w h i c h t h e d i f f r a c t e d l i g h t orders a r e focused i s placed i n the f o c a l plane of the lens. A c i r c u l a r a p e r t u r e i n t h e s c r e e n p e r m i t s one o f these d i f f r a c t e d l i g h t o r d e r s t o pass through and be f o c u s e d by t h e l e n s o f t h e f i l t e r i n g system camera onto t h e f r o s t e d g l a s s where t h e specimen image may be viewed and r e c o r d e d . As was mentioned i n t h e p r e v i o u s s e c t i o n , t h e s p e c k l e 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 t h e a r e a o f t h e r e c o r d e d image o f the specimen, and t h i s causes t h e i n t e n s i t y o f t h e specimen image by t h e d i f f r a c t e d l i g h t ) t o v a r y i n some s y s t e m a t i c manner. (formed T h i s l a s t image o f t h e specimen ( a p p e a r i n g on t h e f r o s t e d g l a s s ) i s , o f c o u r s e , t h e f r i n g e pattern n(y,z). 5.4 The Specimen L o a d i n g I n t h e experiments Systems t o be d e s c r i b e d 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 o u t - o f - p l a n e r i g i d body t r a n s l a t i o n and t h e i n - p l a n e r o t a t i o n o f a p l a t e specimen r e s p e c t i v e l y , a K i n e m a t i c m i c r o t r a n s l a t i o n t a b l e (model TT-102) c o u p l e d w i t h a r o t a r y t a b l e (model RT-200) was used t o impose d i s p l a cements t o w i t h i n an a c c u r a c y o f ± .00025 i n . and r o t a t i o n s t o w i t h i n an a c c u r a c y o f ± .1 m i n . . The assembly w i t h t h e p l a t e specimen i s shown i n F i g . 5.9 . The schematic o f t h e e x p e r i m e n t a l apparatus used i n t h e o u t - o f - p l a n e beam bending experiments i n F i g . 5.10 . i s g i v e n i n F i g . 6.16 and i t s photograph i s shown Two C-clamps a t each end o f t h e beam were used t o clamp i t t o an aluminum c h a n n e l o f much g r e a t e r bending s t i f f n e s s t h a n t h e beam. A s m a l l c i r c u l a r h o l e was machined a t t h e c e n t e r o f t h e c h a n n e l so t h a t t h e -thrust bearing a x i a l displacement ^ f o t o n i c /-dial gage d i a l gage ^ sensorholder I T -load c e l l l a t e r a l displacement d i a l gage F i g . 5.11 load spring-^ 1 ~' L l a t e r a l adjustment screws Schematic of the tensile ^pin loading apparatus. load-\ nut \ load shaft-; Fig. 5.12 F i g . 5.13 Right side view of the loading apparatus. Left side view of the loading apparatus. F i g . 5.15 Variable cross-section specimen. 101 micrometer t i p c o u l d c o n t a c t and d i s p l a c e the c e n t e r of the beam out o f the p l a n e by a known amount t o w i t h i n an a c c u r a c y o f a p p r o x i m a t e l y ± .00025 i n . I n a l l t h e r e m a i n i n g e x p e r i m e n t s ( t o be d e s c r i b e d i n s e c t i o n s t h r o u g h 6.7) used. the l o a d i n g a p p a r a t u s shown i n F i g s . 5.11 6.5 , 5.12 and 5.13 was I t was d e s i g n e d so t h a t a d j u s t m e n t s o f the specimen shape and p o s i t i o n c o u l d be made w h i l e the specimen was b e i n g s u b j e c t e d t o as much as 6000 l b s . o f s t a t i c t e n s i l e a x i a l l o a d . The l o a d was imposed by t u r n i n g the l o a d i n g nut w h i c h i n t u r n compressed a c o i l s p r i n g . The l o a d was trans- m i t t e d t o the specimen t h r o u g h a l o a d s h a f t , p i n , and clamps w h i c h were c o n s t r a i n e d t o 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 o f the clamps. Such an arrangement p e r m i t t e d an a x i a l movement w i t h a minimum amount of " p l a y " i n o t h e r 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 t h e specimen. T h i s a x i a l t r a n s l a t i o n was m o n i t o r e d a t one p o i n t by a d i s p l a c e - ment t r a n s d u c e r ( F o t o n i c Sensor) mounted on a k i n e m a t i c t a b l e so t h a t t h e Sensor probe c o u l d be k e p t a t a c o n s t a n t d i s t a n c e from the specimen surface. By making a d j u s t m e n t s 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 o b s e r v i n g the F o t o n i c Sensor r e a d i n g , we c o u l d z e r o out the a x i a l d i s p l a c e m e n t o f t h i s particular point. The i n i t i a l crookedness o f the specimen o r a s m a l l m i s - a l i g n m e n t of the clamps a l s o caused a s m a l l l a t e r a l d i s p l a c e m e n t of the specimen. T h i s u n d e s i r a b l e d i s p l a c e m e n t was measured by a number of d i a l gages, shown i n F i g . 5.14, adjustment s c r e w s . and m i n i m i z e d by a d j u s t m e n t s made w i t h the l a t e r a l I n a d d i t i o n t o the a x i a l t e n s i l e l o a d , t h e specimen a l s o s u b j e c t e d t o some b e n d i n g caused by the asymmetry,of apparatus. was the l o a d i n g To m i n i m i z e t h i s b e n d i n g d e f o r m a t i o n , a d i a l gage was used t o measure the v e r t i c a l d i s p l a c e m e n t o f the c e n t r e o f the s u p p o r t i n g frame and t h i s d i s p l a c e m e n t was then zeroed w i t h t h e bending c o n t r o l screw. Once a l l t h e s e a d j u s t m e n t s were completed, the l o a d increment and the i n s t r u m e n t F i g . 5.17 T e n s i l e specimen w i t h uniform cross-section. the F i g . 5.18 Wooden beam specimen. 104 r e a d i n g s were r e c o r d e d and t h e second exposure o f t h e specimen s u r f a c e was made. 5.5 Specimens A l l specimens, i n c l u d i n g t h e wooden beam were c o a t e d w i t h a f l a t w h i t e enamel p a i n t t o p r o v i d e 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 c e . Detailed descript- i o n s o f t h e specimens a r e i n Chapter 6; t h e p l a t e specimen used i n t h e experiments o f s e c t i o n s 6.2 and 6.3 i s shown i n F i g . 5.9 and t h e r e m a i n i n g specimens a r e shown i n F i g s . 5.15 through 5.18 . 5.6 Experimental Procedure The work performed i n a l l t h e experiments d e s c r i b e d i n Chapter 6 f o l l o w e d , i n g e n e r a l , t h e same p a t t e r n whether DASC o r DASSC was used. The e x p e r i m e n t a l p r o c e d u r e c a n be o u t l i n e d i n t h e f o l l o w i n g s t e p s : a) U s i n g knowledge t h a t t h e camera forms f r i n g e s a c c o r d i n g t o e q u a t i o n (3.20) o r (3.26) and h a v i n g some i d e a about t h e s i z e o f t h e s p e c i men d e f o r m a t i o n i n t h e i n t e n d e d t e s t i n g , t h e parameters chosen so t h a t a d e s i r e d number o f independent o f t h e camera were 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 c o u l d be made. b) The specimen was c o a t e d w i t h w h i t e enamel p a i n t t o r e f l e c t t h e l a s l a s e r l i g h t d i f f u s e l y over t h e a r e a o f i n t e r e s t w i t h a p p r o x i m a t e l y t h e same intensity. I t was then clamped o r a t t a c h e d i n t h e l o a d i n g a p p a r a t u s which had been p r e v i o u s l y s e t up t o accommodate t h e specimen. c) V a r i o u s s e n s o r s and d i a l gages were p o s i t i o n e d and t h e i r ( f i r s t exposure) r e a d i n g s were r e c o r d e d . initial W i t h t h e use o f a t e l e s c o p e w i t h d u a l c r o s s - h a i r s o r some l a s e r a l i g n m e n t t e c h n i q u e t h e c e n t e r o f t h e frame of DASSC was made t o l i e i n t h e x,z p l a n e o f t h e specimen c o o r d i n a t e system. The o p t i c a l components t h a t make up t h e camera were t h e n mounted a t t h e 105 required height and orientation so that they were aligned along the x-axis. An aluminum r u l e r coated with f l a t black paint and having scratch marks made at 1 i n . 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 i n the plate holder of the camera and the specimen with the r u l e r was illuminated by one of the collimated laser beams. The image of the specimen and the r u l e r made by l i g h t passing through a p a r t i c u l a r aperture set was viewed on the frosted glass, and the size of the l a t e r a l shear Ayi was adjusted by positioning the plate holder with the adjustable s l i d e and observing the amount by which the bright images of the r u l e r scratch marks doubled. d) With both the room l i g h t s and the laser turned o f f and i n the near or t o t a l darkness one or, i f desired, two photographic plates were inserted i n the plate holder and the camera was then covered with the enclosure. The specimen was then illuminated by one illuminating beam at a time f o r the required exposure period at an appropriate shutter s e t t i n g . Once the desired number of f i r s t exposures was made the laser was turned o f f and the specimen was loaded. After a l l the specimen adjustments described i n Section 5.4 were made, the readings of the d i a l 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 i n the f i r s t exposures. e) A l l exposures were usually of the same duration. The laser was turned o f f and the photographic plates were removed from the holder and developed according to manufacturers s p e c i f i c a t i o n s . f) After development, the photographic plates were inserted i n the f i l t e r i n g system (described i n Section 5.3) where the fringe patterns responding to the specimen deformation could be viewed and permanently recorded on the f i l m . cor- 106 g) The l a t e r a l shear Ay^ was measured on a traversing microscope either from the photographic plate or preferably from the f i l m recording; using the f i l m recording was easier because each recording had only one l a t e r a l shear corresponding to a p a r t i c u l a r aperture set. In contrast to t h i s , 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 i d e n t i f y and measure the various l a t e r a l shears. h) With the a i d of a" microdensitometer the f i l m 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. 6.1 EXPERIMENTAL WORK Preliminary Remarks This chapter consists e s s e n t i a l l y of two parts. The f i r s t part i s devoted to the description and discussion of the results of a number of r e l a t i v e l y simple experiments, the purpose of which was to v e r i f y experimentally the v a l i d i t y 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 i n applications similar to those encountered testing of materials. i n the p r a c t i c a l In contrast to the other chapters, i n 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 v e r i f y 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 t r a n s l a t i o n of the specimen surface, that i s for u = u Q , and for other displacements and strains being i d e n t i c a l l y zero, both equations (4.4) and (4.13) governing the fringe formation by DASC and DASSC respectively reduce to a i u = 0 N i(y> ) z (6.1) By using the d e f i n i t i o n s (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 : — u Xx = - • — n ^ y c o s c j ) ^ + zsintj)^ , - ysincj)^ + zcoscj)^) s Q ycoscj)^ + zsinct^ XX u •= - — — n^ycoscj)^ + zsincj)^ , - ysinc})^ + zcostj)^) (6.2) g Q (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 of the plate. Q The plan view of the experimental setup f o r DASC i s shown i n F i g . 6.1 . One way to v e r i f y the fringe formation by the two cameras i s 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 DASC: DASSC D cj> = 0,o n(y,z) = - uy (6.4a) * = 90° n(y,z) = - ^ | u z (6.4b) cf) = 0° n(y,z) = - J ^ J - u y 0 0 Q (6.5) The DASC parameters f o r Experiment (Exp.) 19 (<$> = 0°) and for Exp. 22 (<j> = 90°) were x & = 38 i n . D = 2.5 i n . u Q = .025 i n . 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 p a r a l l e l to the z-axis and spaced .462 i n . apart, and for Exp. 22 the fringes are predicted to be p a r a l l e l to the y-axis and also spaced .462 i n . apart. The actual fringe patterns f o r these two experiments are shown i n F i g . 6.2 and F i g . 6.4 . From the microdensitometer traces of F i g . 6.2 and F i g . 6.4 the actual 109 fringe spacing was found to be .45 i n . i n both cases, which i s close to the predicted spacing of .462 i n . To show a t y p i c a l microdensitometer trace, we have presented i n F i g . 6.3 the trace of the fringe pattern of Exp. 19 shown i n F i g . 6.2 . The DASSC parameters for Exp. 17 and Exp. 18 (ty = 0°) were x g = 38 i n . u Q = .025 i n . Exp. 17: D = 1.25 in , Ay s = - .0257 i n . , X s = 37.22 i n . Exp. 18: D = 2.50 i n . , Ay g = - .0453 i n . , X g = 37.31 i n . 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 p a r a l l e l to the z-axis and spaced .905 in.apart; f o r Exp. 18 the fringes are also expected to be p a r a l l e l to the z-axis but spaced .454 i n . apart. The fringe patterns for these two cases, obtained with DASSC, are shown i n F i g . 6.5 and F i g . 6.6 Note the "doubling" or "shearing" of l i n e s and numbers i n these two figures caused by the defocussing of the system. From the microdensitometer traces and after an appropriate scaling, the actual fringe spacing i n Exp. 17 was found to be approximately .89 i n . , and i n Exp. 18 the actual fringe spacing was found to be approximately .46 i n . . Again the agreement between the predicted fringe spacing and the actual fringe spacing i s 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 F i g . 6.3 are given i n 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 - .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 on the above numerical data. .50 by a piecewise continuous cubic based For Exp. 19 this fringe function i s compared to Ill 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 e q u a t i o n (6.4a) i n F i g . 6.7 . The two f u n c t i o n s a r e i n good agreement and t h e a c t u a l d i f f e r e n c e between them i s shown by t h e t h i r d c u r v e . T h i s t h i r d c u r v e has i t s o r d i n a t e on t h e r i g h t s i d e o f t h e graph and t h e s c a l e o f t h e " 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 r e a s o n f o r t h e exaggerated s c a l e i s t h a t o f t e n t h e p l o t s of p r e d i c t e d and e x p e r i m e n t a l ( i . e . o b t a i n e d w i t h t h e use o f DASC o r DASSC) c u r v e s a r e so c l o s e t o g e t h e r t h a t i t would be d i f f i c u l t t o e s t a b l i s h t h e a c t u a l n u m e r i c a l d i f f e r e n c e between them. u = u 0 The p l o t o f t h e d i s p l a c e m e n t , 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 t h a t t h e two d i s p l a c e m e n t s a r e v e r y c l o s e except near t h e o r i g i n where even a v e r y s m a l l e r r o r i n t h e 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 t h e c a l c u l a t e d displacement. The r e a s o n f o r t h i s i s o b v i o u s from t h e form o f e q u a t i o n Ax? n ( y , z ) u ( y , z ) = - - ^ — ^ y^O (4.10b) T h e r e f o r e , i f DASC o r DASSC i s used t o measure t h e o u t - o f - p l a n e d i s p l a c e m e n t i t i s e s s e n t i a l t o a l i g n t h e camera p e r f e c t l y and t o e x t r a p o l a t e t h e d i s p l a c e m e n t a t t h e o r i g i n from the d i s p l a c e m e n t s c a l c u l a t e d nearby. The r e s u l t o f t h i s experiment agree v e r y w e l l w i t h t h e t h e o r e t i c a l p r e d i c t i o n s and c o n f i r m t h e v a l i d i t y o f t h e c o e f f i c i e n t a 6.3 i n e q u a t i o n s (4.4) and ( 4 . 1 3 ) . R i g i d Body, I n - P l a n e R o t a t i o n o f a P l a t e Specimen The purpose o f t h i s experiment was t o v e r i f y t h e f o r m a t i o n o f f r i n g e s by the two cameras due t o t h e i n - p l a n e d i s p l a c e m e n t s o f the specimen s u r f a c e . The i n - p l a n e d i s p l a c e m e n t s v and w a r e produced by t h e r i g i d body r o t a t i o n , a, of a p l a t e about t h e x - a x i s as i s shown i n F i g . 6.9 . A p o i n t which was i n i t i a l l y i n the y,z p l a n e a t (0,y,z) was d i s p l a c e d between t h e two exposures 112 to the (0,y*,z*) position. The displacements v and w are related to the coordinates (y,z) of an a r b i t r a r y point by u = u, = u, = 0 y (6.6a) z - Z SinCl (6.6b) w = z* - z = z(cosa - 1) + y s i n a (6.6c) = y * - y = y( OSa v - C 1) The p a r t i a l derivatives of these displacements are v, = w, y = cosa -1 z v,, - w, v (6.6d (6.6e) = - sma A l l p a r t i a l derivatives are constant and, for a small angle of rotation a as i n the case, the terms i n 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 y - 5.82 x 10 *z (6.7a) w(y,z) = - 5.82 x lO'^y - 1.7 x 1 0 z (6.7b) 7 _7 .-7 = v, = - w, z W, 1.7 x 10 V,y z v (6.7c) -4 = - 5.82 x 10 (6.7d) A schematic diagram for the experiments done i n this section i s shown i n F i g . 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 <j> = 45 v - /2 + yz x z 2 Xx c w n(y,z) (6.8a) c - y 2 2/2 x' V + 1 z 2 - y ^ 2 w 2/2 "s ' Xx, n(y,z) (6.8b) 113 o v z * = 90 : A x - — s 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 i n equations (6.8) and, using the fact that the viewing angle i n these experiments was 1 such that l neglect some small terms. Once this was To < x S ' w e m a y done, the equations were solved for the fringe functions n(y,z) as Exp. 24 (({> = 0° ) : n(y,z) - (5.82 x 10 Exp. 25 (ct = 45°) : n(y,z) = (5.82 Exp. 26 (<j) = 90°) : n(y,z) = (5.82 x 10 *) x 10 ") 4 Ax Z c D Z Ax., Y /2 ) -r^- y AXS Thus for Exp. Exp. 24 the fringes are predicted to be p a r a l l e l to the y-axis, for 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 p a r a l l e l to the z-axis. The DASC parameters used for the three experiments were x D s = 39 i n . = 2.5 i n . With these parameters the predicted fringe spacing i s .536 a l l three experiments. i n Figs. 6.11, 6.12 i n . - the same for The actual fringe patterns obtained by DASC are shown and 6.13 . The fringes are oriented as predicted from the microdensitometer traces of these figures, the fringe spacing found to be and, was .55 i n . , close to the predicted spacing of .536 i n . Let us now predict the positions and spacing of fringes formed by DASSC, 114 which for Exp. 2 considered here had the following x = 32 i n . s D e parameters: = 2.5 i n . = 20° , x Ay s 4> e y = no° , e = 9 0 0 z = .0608 i n . = o° The displacements and strains given by equations (6.7) are substituted i n equation (4.11); we can make use of the fact that the viewing angle was again < YQ such that 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 p a r a l l e l to the y-axis and spaced .429 i n . apart. The actual fringe pattern i s shown i n F i g . 6.14 . The fringes are p a r a l l e l to the y-axis and the fringe spacing was found from the microdensitometer trace to be .42 i n . which compares well with the predicted spacing of .429 i n . 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. I t 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, i s shown i n F i g . 6.15 . The results of these experiments confirm that the response of both DASC and DASSC to the in-plane displacements i s accurately described by equations (4.4) and (4.13). In p a r t i c u l a r , the accuracy of the c o e f f i c i e n t s b^ and c^ 115 of equations (4.4) and (4.13) was v e r i f i e d . 6.4 Out-of-Plane Bending of a Thin Beam with a Rectangular Cross-Section The purpose of this experiment was to v e r i f y the formation of fringes by DASSC due to the variable out-of-plane displacement of a specimen surface. A diagram of the experiment i s shown i n F i g . 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 i n the v i s i b l e surface of the beam. The strains i n the z d i r e c t i o n were not derived as the beam deformation w i l l be viewed through an aperture set(s) at zero i n c l i n a t i o n to the y-axis. The parameters of DASSC used i n the beam bending experiments considered i n this section were similar to those of Exp. 101C given here as beam: x s D 0 % x 2 x 50 i n . V = .25 =39.75 i n . , Ay = - .12 i n . , X s s = 37.84 i n . = 2.5 i n . X = 76.8° , 0 y = 166.8° , 0 Z = 90° With these parameters equation (4.11) becomes 39^75 U + 1.82 " iffo + V Z 3 9 7 5 w, y W " ' 2 2 3 U >y + U 8 2 ^ y - 1.25 - .919 v , 39.75 y + = - .00032 n(y,z) With the displacements and strains derived i n Appendix T the above equation becomes 116 2.5 x 10 2 1.9 x 10 7 y 1 - 3 yz 2 + 2 25 1 - 2 25 25 + 1.2 x 10 3 y •+ 2.1 x 10 1 - 1 - 2 + 1.1 x 10 1 + + 25 25 w 2.2 x 10 - 3 fy - 1.25 39.75 v, .919 25 6 z .00032 2 n w, y It i s obvious that, i n the case of out-of-plane bending of beams, the fringe equation may be approximated by only two terms involving u and u , accurate approximation would involve u,u, ,v and v , v v terms. v . A more 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 s u f f i c i e n t accuracy. u(y,z) Axf. n(y,z) (4.10b) y * 0 In Exp. 16 considered here the DASC parameters were beam: x s g- x 2 x 55 i n . = 29.5 i n . D = 2.5 i n . 6 = .50 mm = .019685 i n . The fringe pattern of this experiment i s shown i n F i g . 6.17 . Once the fringe centers are correctly numbered and their coordinates are read o f f the microdensitometer trace, the displacement u i s 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 i n F i g . 6.18. The two displacements correlate quite w e l l . When DASSC i s 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. and We s h a l l consider several of these and point out their advantages disadvantages. It was shown i n Section 4.8 that, f o r 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 f o r (j) = 0° becomes au + du, = N(y,z) (6.9) y In the case of beam bending, u and u , are e s s e n t i a l l y functions of y only y and, therefore, the p a r t i a l derivative i n equation (6.9) may be replaced by an ordinary derivative. du d 7 After some rearranging we obtain a + d u N(y) ^ <- > = 6 10 In Section 4.11 i s shown that, i f the out-of-plane displacement at one point, y = y 0 with p u, y i s known, the solution to equation (6.10) i s given by equation (4.32) = a/d and q i n Exp. 101C. = N(y)/d . This approach was used to solve f o r u and The DASSC parameters used i n Exp. 101C are given on page 118 and the fringe pattern i s shown i n F i g . 6.19 . A program, OUTlM.S, which was written solves,equation (6.10) according to , equation (4.30) r with <j> = 0° . The program accepts the system parameters, point, and fringe centers as data and calculates u and u , displacement at one y . The graphs of 118 predicted u and u , vs. the actual u and u, y are shown i n F i g s . 6.20 and 6.21 . y obtained by DASSC and OUTIM.S Note that i n both cases the agreement between predicted and actual values i s quite good. In the out-of-plane displacement u i s not known at any point of the v i s i b l e part of the beam, the two photographs must be taken and use made of the approach of Section 4.8 . Quantities u and u , equation (4.16) which for the aperture Au + D , ± i U are determined from set rotation (j) = 0° becomes = N (y,z) y y i = 1,2 ± (6.11) A program, BEAM2.S, which solves equation (6.11) was written to determine u and u,y for Exp. beam: x g 101; the system parameters were as follows: \ x 2 x 50 i n . = 39.75 i n . D = 2.5 in. D = 1.75 D = 2.5 in. in (illumination i n x,y plane) e x = 76.8° Ay s = - .012 in. Exp. 101A 9 X = 26° Ay s = - .097 in. Exp. 101B 0 X = 76.8° Ays•= - .12 i n Exp. 101C The two fringe patterns for Exp. and 6.23 u and u,y . 101A and Exp. The plots of the predicted u and u , 101B y are shown i n Figs. and the experimentally (calculated by BEAM2.S) are shown i n Fig.6.24 and 6.25 . important to note that i t i s not enough to know that well to obtain accurate 2 found Again the agreement between the predicted and the experimental r e s u l t s i s good. ,.|w',y.|. and to ensure that d j - d 6.22 It i s |u|,:|'u,y| >> | v"| , | v, y | , | w ^ 0 ; equation (4.32) must be s a t i s f i e d as solutions for u and u,y . For example, the combi- nation of Exp. 101A and Exp. 101B s a t i s f i e s this requirement while the combi- nation of Exp. 101B and Exp. 101C does not. The experiments described i n 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 v e r i f i e d the accuracy of c o e f f i c i e n t s 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 v e r i f y 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 i n F i g . 6.26 . A thin, f a i r l y wide, a c r y l i c beam with a rectangular cross-section was clamped at i t s ends, and a t e n s i l e load was applied between the two exposures i n 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 e f f e c t , and some misalignment of the system, the beam was also displaced outof-plane by a small amount. This out-of-plane displacement was monitored at the z = 0 l i n e by a number of d i a l gages. At the point (-t/2,0,h/2) there was also a d i a l gage which measured the displacement of this point i n the z d i r e c t i o n ; this l a s t displacement was caused mainly by the poisson e f f e c t . A s t r a i n gage was cemented to the illuminated surface of the beam and i t s reading served as a standard against which the s t r a i n determined experiment a l l y by DASSC was compared. The beam deformation i s e s s e n t i a l l y a special case of the plane stress, and the strains of the neutral surface are related to the imposed increase V i n the length L of the beam by £ £ _ V yo ~ L xo = " v e yo = " v l 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 v, - £ y y - £ y 0 " t u _ V t >yy ~ ^ ~ 2 ' y y U V £ v = y L " 2 'y v(0,z) = 0 u w »z ~ z ~ ~ W e rv = -V L V£y = ~ V 2 'yy u w(y,0) = 0 2 'yy u = Vz — 2 u, 'yyy w, u v The deformation of the beam was photographed by DASSC with the apertures at 0=0 , and the f i e l d of view was such that y x z x 9 ' s < -^jr . For these s parameters and the assumed form of the displacements and strains, and the specimen illumination i n x,y plane equation (4.11) reduces to V yz u + I v •y - Dj/2 t + z + 2'yy Ay X s i s i (1 + c o s 0 ) u , x i y - u w V - cost y i L t z t — Vz — u, yyy 2 'yy u 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 s i n u s o i d a l - l i k e with one half period over the beam length. Because of these c h a r a c t e r i s t i c s , and since the beam was thin (t = .25 i n . ) , 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 l a s t equation and f o r 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 n e g l i g i b l e , equation (4.22) reduces to the simple form v + f Equation (6.13) i s analogous the same manner. i V , y = N (6.13) ± to equation (4.34) and can be solved for v i n Unfortunately, i n the actual testing the term d^u,y found to be large i n magnitude and equation (6.13) could not be used. was How- ever, by a careful alignment of the loading mechanism i t was possible to l i m i t the out-of-plane displacement u to magnitudes less than . 0015 i n . and y for such a small value of u, the term — u i s very small, and i t i s them s x possible to solve for u,u,y,v, and v , discussed i n Section 4.10 . y from equation (4.22) i n the manner 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: x h x 6 x 32 i n . = 45 i n . s D 2.5 i n . Ay s = .07 i n . e = - D 1.75 i n . x Ay 6 s 21.2° = .14 i n . = - 21.2° X D X s = 46.26 i n . e = y x 68.8° Exp. 114B e z = 90° s = 48.6 i n . e = y 68.8° ' > Exp. 114C e z = 90° 2.5 i n . Ay s = .23 i n . e = x 23.7° ' x s = 49.1 i n . e = y v( - 3,0) = 0 u( - 1,0) = - . 00102 i n . 113.7° Exp. 114D e z = 90° 122 v, (-3.625,0) = 413 x y 10~ 6 The fringe patterns obtained by DASSC are shown i n F i g . 6.27 and Fig.6.28 . The computer plots of predicted and experimental displacements and strains, done by PLATE2.S are shown i n Figs. 6.29 through F i g . 6.32 . The value of v, (-3.625,0) calculated by the program from the two fringe patterns i s y - 6 419 x 10 - 6 , which compares favorably with the s t r a i n gage reading of 413 x 10 there. If we do not wish to neglect the out-of-plane displacement and s t r a i n , then three independent and properly chosen photographs are needed to solve for the displacements and s t r a i n s . A computer program, PLATE3.S, which solves equation (4.14b) for u,y and v,y i n the way discussed i n Section 4.6 (and for u and v by numerical intergration), was used to calculate these unknowns f o r Exp. 114B, 114B Exp. 114C and Exp. 114D. i s shown i n F i g . 6.33 The fringe pattern of Exp. and the displacements and strains calculated by PLATE3.S are shown i n Figs. 6.34 through F i g . 6.37 . - a l l y determined s t r a i n at y = -3.625 i n . i s 414 x 10 well with the s t r a i n gage reading of 413 x 10 6 there. This time the experiment6 , which again compares I t should be r e a l i z e d , however, that i t i s a coincidence that the two strains are this close at y = -3.625 i n . , as usually the two strains d i f f e r elsewhere by as much as f i v e percent or more. A l t e r n a t i v e l y , 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 i n 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 d i a l gages) and v(-3,0) were used. Figures 6.38 through 6.41 calculated displacements and s t r a i n s . show the comparison of the actual and The experimentally determined strain 123 at y = -3.625 i n . i s 420 x 10 6 vs. 413 x 10 6 measured by the s t r a i n gage there. A l l three approaches used to calculate the displacements and strains f o r this p a r t i c u l a r example y i e l d reasonably accurate solutions. I f the displacements u(y ,0) and v(y ,0) are available, the program FD2 should be Q Q used as i t i s , i n general, more accurate, easier to write and more e f f i c i e n t than the program PLATE2.S. I f the displacements u(y ,0) and v(y ,0) are not Q Q 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 i n testing involving primarily the plane stress deformation and i n p a r t i c u l a r the accuracy of the c o e f f i c i e n t f was v e r i f i e d . 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 i n a more " p r a c t i c a l " type of investigation and to test the accuracy of the two-dimensional solution scheme(s) for displacements and s t r a i n s . Unfortunate- l y , 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 i n the way which would allow accurate determination of a l l d i s p l a cements and s t r a i n s . two derivatives, w, v In p a r t i c u l a r , the effect of the displacement w and i t s and w, z , on the fringe formation by DASSC was not known. The dimensions of the a c r y l i c specimen used i n this experiment are shown i n F i g . 6.42. Two s t r a i n 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 l a t e r comparison with the strains obtained with DASSC. An a x i a l t e n s i l e load was imposed on the specimen by the use of the same loading apparatus as the one described i n the preceeding section and shown i n Fig.6.26. 124 The out-of-plane displacements of the beam were measured by a set of d i a l gages. The beam center, coincident with the coordinate o r i g i n , was kept i n the same position during the two exposures with the a i d of the Fotonic Sensor. DASSC used i n this experiment was equipped with the shutter shown i n F i g . 6.43 so that i t was possible to use two illuminating beams i n the x,y plane i n 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 i n the x,y plane was unfortunate, as i t makes DASSC i n s e n s i t i v e to the strains w, y and w, z . In this experiment DASSC was used to take four photographs with the following parameters: 45 i n . = 1.75 i n D Ay e s = .0914 i n . = -20.5° x <t> = 0° D = 2.50 i n Ay s * = 0° D = 1.75 i n . s 6 Y = 69.5° 6 = 90° Z = .162 i n . = 24° Ay Exp. 122S1 Exp. 122S2 9 Y = 114° e = 90° z = .104 i n = -20.5° Exp. 122S3 6 Y = 69.5° e = 90° z = 135° = 2.5 i n . D A 6 y x s = Exp. 122S6 .171 i n = 24° = 135° e y = ii4° 0 Z = 90° 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 i n 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 d i a l gages and also to the f i n i t e element solution of this problem. These comparisons are shown i n Figs. 6.46 through 6.49 and as can be seen from the graphs, the displacements and strains agree quite well. We would now l i k e to calculate u(y,z), v(y,z) and w(y,z). To do t h i s , we need three "independent" photographs taken by DASSC having three aperture sets and three illuminating beams with at least one of them not being i n the x,y plane. The three displacements may then be calculated with the use of the scheme discussed i n 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 i n 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 n e g l i g i b l e error as, obviously, u(y,z) ^ u(y,0) for z 4- 0 . The error i s expected to be small because u(y,z) i s 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 l i k e l y the same for a l l points on l i n e s y = constant. The v a r i a t i o n i n u(y>z), for y = constant, i s caused by the reduction of the specimen thickness due to the poisson e f f e c t , but this v a r i a t i o n i s 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 d i a l 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 i n F i g . 6.50 and F i g . 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 i n F i g . 6.52 . Since the specimen illumination was i n the x,y plane, w(y,z) was not calculated accurately enough and i s not shown. The displacement v(y,z) and i t s p a r t i a l derivative v,y(y,z) are compared to the f i n i t e element solution i n 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 d i a l gage readings. From the two plots i t i s obvious that either of the two boundary conditions y i e l d s reasonably accurate v(y,z) and v,y(y,z). The r e s u l t of this p a r t i c u l a r experiment confirm that the two dimensional computing schemes can produce accurate solutions f o r v(y,z) and v , ( y , z ) . y 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 p o s s i b i l i t y of using DASC or DASSC to measure the surface deformation of specimens made of materials such as wood. The wooden beam used i n this experiment shown i n Fig.5. ;was 4 in. wide, % i n . thick, and 48 i n . long, with a knot of approximately 1 i n . diameter located at the center of the beam. The beam was subjected to an a x i a l t e n s i l e load using the same experimental setup as i s shown i n F i g . 6.26. The beam center, which was coincident with the coordinate o r i g i n , 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 d i a l 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. Still, the photo- graphs which were taken provide useful information about the application of DASC and DASSC i n the testing of highly inhomogeneous materials l i k e wood. DASC used to produce the photographs of Exp. 132D1 and Exp. 132D2 (shown i n F i g . 6.56 and F i g . 6.57) had the following parameters: x D s = 45 i n . = 1.75 i n . 9 X = - 22.5° ej> = D = 2.5 i n . = e z = 90° Exp. (}y = 114.3 0 e z = Exp. 132D2 132D1 0° 0 X = 24.3° 4> (3 y = 67.5° 90° 0° a x i a l load increment between the two exposures was approximately from 1600 l b . to 1800 l b . ; this corresponds to the increase of 100 l b / i n t e n s i l e stress. in The a x i a l load caused some out-of-plane bending which was monitored by the d i a l gages with the results v(-.5,0,0) = .0 i n . v(0,16,0) = .00226 i n . v(0,-16,0) = -.00307 i n . u(-.5,-7,0) = .0 i n . u(-.5,-5,0) = -.0000787 i n . u(-.5,-2,0) = -.0000787 i n . . u(-.5';l,0) 2 = -.000315 i n . 128 u(-.5,4,0) = -.000394 i n . u(-.5,7,0) = -.000866 i n . w(-.25,0,2) = .0 i n . The fringe patterns i n photographs taken by DASC are f a i r l y simple and show that DASC works quite well i n testing specimens having highly nonuniform material properties. DASSC was used to produce the photographs shown i n F i g . 6.58 corresponding to Exp. x s D 132S2 and Exp. 132S1, for which i t had the parameters: = 45 i n . = 1.75 in. = .0488 i n . 9 x and Fig.6.59, = -22.5° 6 = 67.5° y e z Exp. 132S1 Exp. 132S2 = 90° = 0° D = 2.5 i n . = .0878 i n . 9x = 24.3° e y = 114.3° 0 Z = 90° = 0° The two fringe patterns obtained by DASSC are extremely complex, which suggest that the t e n s i l e loading of the wooden beam causes a s t r a i n f i e l d i n the illuminated surface of the beam. complicated This i s most l i k e l y due to the " a l i g n i n g " process of the wood f i b e r s involving large changes i n surface slopes to which DASSC i s most s e n s i t i v e . ment, the l a t e r a l shear Ay s was In this p a r t i c u l a r experi- obviously set too large, thus making DASSC too sensitive; the resultant fringe patterns were of such complexity as to be of no use i n the quantitive analysis, since there was no hope of successfully numbering the fringes i n these patterns. In future experiments of this type, the s e n s i t i v i t y 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 h e l p f u l . 6.8 Error Analysis The results of the experiments discussed i n this chapter have ascer- tained that the fringe formation of DASC and DASSC i s described with s u f f i c i e n t 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 a^u + b^v + ....+' k.jw, = z (4.4) (4.13) By using either of the two cameras i n a p a r t i c u l a r 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 i n 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 i n the derivation of equations (4.4) and (4.13). In the f i r s t approximation the r e a l cameras were replaced by the physical models shown i n F i g . 3.5 and F i g . 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 approximations i s very d i f f i c u l t . In p r i n c i p l e , 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 i n this thesis) of the two (not discussed cameras, with the results indicating that these errors are less s i g n i f i c a n t than those discussed i n the sub- sequent paragraphs. 2. Errors i n c o e f f i c i e n t s a-^,..., k-^ caused by the inaccuracies measurement of the parameters of the two cameras. were usually determined with the following i n the These parameters accuracies: D ... ±.005 i n . x ... ± .5, i n . s e ,6y,e ... ±i? x Ay 3. Errors i n z s ... ±.001 in. (related to the Moire fringe numbers n^) caused by a limited accuracy with which the location of the fringe centers be determined. may In the work presented i n this thesis the fringe centers were usually located within ±.02 i n . from the microdensito- meter traces. 4. Errors i n the calculated displacements and strains caused by the approximate nature of some of the solution schemes for the displacements and s t r a i n s . The size of these errors depends on the p a r t i c u l a r 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 i n paragraph 2. are the greatest source of errors i n the calculated d i s p l a cements and s t r a i n s . In p a r t i c u l a r , the inaccuracies i n the measurements 131 of Ay s and somewhat ambiguous parameter x s cause the largest errors. This problem could be a l l e v i a t e d by making a large number of measurements of Ay and then calculating and using i t s average value; the parameter x determined more accurately from a number of simple experiments, s s could be 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 f o r the lens, instead of using x s to approximate both of these coordinates. The simplest, and possibly the only p r a c t i c a l way of determining the error caused by the inaccuracies i n the camera parameters would involve the repeated use of the appropriate solution scheme, each time with the parameters being s l i g h t l y 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 i n the experiments described i n 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 i n 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 s t r a i n s , the upper and lower bounds on these quantities are determined. The errors are caused by inaccuracies i n the measurement of the camera parameters and fringe locations, and by using their extreme values i n the calculations the bounds may be established. This was done at one " t y p i c a l " point of the specimen surface for the experiments 26 and 114. 132 In the case of the experiment 26, the displacement w(0.,.547) at the point (0.,.547) i s 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 x D = .000020256 i n . s = .39.0 ± .5 i n . = 2.500 ± .005 i n . The centers of the fringes shown i n F i g . 6.13 were located with accuracy of 1 .02 i n . , 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 , D and n(0.,.547) g in equation (6.14), the displacement w(0.,.547) was found as -.000337 i n . < w(0.,.547) < -.000296 i n . With the mean value of w(0.,.547) equal to -.000316 i n . t h i s displacement may be written as w(0.,.547) = -.000316 ± .000021 i n . 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 f o r 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 i n . v(l.0,0.0) = ..00172 ± .00005 i n . 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 i n 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 l i k e l y be even smaller due to some cancellation of errors. The errors i n the other experiments were not calculated but should be similar to those i n the experiments 26 and 114. F i g . 6.2 F r i n g e p a t t e r n of Exp. 19. 135 F i g . 6.4 F r i n g e p a t t e r n of Exp. 22. Fig. 6.5 F i g . 6.6 Fringe pattern of Exp. Fringe pattern of Exp 137 in a F i g . 6.7 Predicted n (dashed line) vs. experimental n (solid line). F i g . 6.8 Predicted u (dashed line) vs. experimental u (solid line). 138 F i g . 6.9 F i g . 6.10 Rotation of a plate about x-axis. Measurement of the in-plane displacements v and w by DASC. 139 F i g . 6.11 F i g . 6.12 F r i n g e p a t t e r n of Exp. F r i n g e p a t t e r n of Exp. 25. 24. 140 Fig. 6.14 Fringe pattern of Exp. 2. 141 F i g . 6.16 Measurement of the out-of-plane displacement by DASSC. 142 F i g . 6.17 Fringe pattern of Exp. 16. < F i g . 6.18 Predicted u (dashed line) vs. experimental u (solid line). 143 F i g . 6.19 F i g . 6.20 F r i n g e p a t t e r n of Exp. P r e d i c t e d u (dashed l i n e ) v s . e x p e r i m e n t a l 101C u (solid line). 144 F i g . 6.23 Fringe pattern of Exp. 101B 145 146 F i g . 6.26 Measurement of the in-plane deformation by DASSC. F i g . 6.27 Fringe pattern of Exp. 114C F i g . 6.28 Fringe pattern of Exp. 114D 148 g. 6.29 Predicted u (dashed line) vs. experimental u (solid line). 149 F i g . 6.33 Fringe pattern of Exp. 114B 151 152 <3 F i g . 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 line). 154 F i g . 6.40 Predicted v (dashed line) vs. experimental v (solid line). a c r y l i c plate 7/16 i n . thick SCALE 1 : 4 156 i i y F i g . 6.43 Normal view of the aperture screen. F i g . 6.45 Fringe pattern of Exp. 122S2 158 I -6.0 I -3.6 I -3.2 I 1.2 I 3.6 I y(in.) 6 . 0 T < F i g . 6.46 Predicted u (dashed line) vs. experimental u (solid line). 159 F i g . 6.49 Predicted v , y (dashed line) vs. experimental v , y (solid line). F i g . 6.50 F i g . 6.51 F r i n g e p a t t e r n o f Exp. 122S3 F r i n g e p a t t e r n of Exp. 122S6 0 1 2 3 4 y(in.) 5 f i n i t e element s o l u t i o n e x p e r i m e n t a l s o l u t i o n u s i n g 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 .-. . . Fig. e x p e r i m e n t a l s o l u t i o n u s i n g the e x p e r i m e n t a l u(y,0) and v ( y , 0 ) as boundary c o n d i t i o n s 6.53 Contours of c o n s t a n t d i s p l a c e m e n t 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) i n the variable cross-section specimen. 1500 f i n i t e element ~ x experimental solution solution o Exp. 122S1 @ 122S2 '—•> 1000 500 4 F i g . 6.55 S t r a i n 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 y y(in.) specimen. 5 ON 165 F i g . 6.56 F i g . 6.57 F i g . 6.58 F r i n g e p a t t e r n of Exp. F r i n g e p a t t e r n of Exp. F r i n g e p a t t e r n of Exp. 132D1 132D2 132S2 F i g . 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 r e s u l t i n g equations r e l a t i n g the Moire fringe number to the deformed surface displacements and strains was v e r i f i e d by several simple and controlled experiments. Each experiment was set up to ascertain the accuracy of one or two c o e f f i 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 i n experiments similar to those encountered i n the t y p i c a l laboratory testing of materials or struct u r a l components. With some exceptions the agreement between the o p t i c a l l y 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 r e l a t i v e l y easy. of the fringes was In fact, i n some s p e c i a l cases the fringes are related d i r e c t l y to only one displacement component. When DASC i s used no boundary conditions are required to calculate the displacements, and the c a l c u l a t i o n involves the solution of a set of at most three algebraic equations, a process which i s straightforward and easy to program. That DASC i s i n s e n s i t i v e to the out-of-plane displacements of the specimen surface near the coordinate o r i g i n may be considered an advantage i f the measurement of in-plane d i s iplacements i s desired, but i t may make DASC p o t e n t i a l l y useless i f the outof-plane displacements near the coordinate o r i g i n are to be measured. The main disadvantage of DASC stems from i t s r e l a t i v e i n f l e x i b i l i t y as the s e n s i t i v i t y may be varied only by changing D,X or x , the p o s s i b i l i t i e s of g which exist only i n 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 l a t e r a l shear may be set by an appropriate 168 positioning of the photographic plate with the adjustable s l i d e . Thus, i f the specimen deformation i s approximately known before the test, the l a t e r a l shear may be chosen so that the density of the resultant fringe pattern i s suitable for processing. The camera i s sensitive to the surface displacements and strains (actually to p a r t i a l derivatives) and, hence, the fringe patterns could be used q u a l i t a t i v e l y to i d e n t i f y those areas of the surface where stress concentrations occur. The calculation of the displacements i n s p e c i a l cases often involves a solution of an ordinary l i n e a r d i f f e r e n t i a l equation or, i n the general case, a solution of a set of p a r t i a l d i f f e r - . / e n t i a l equations for which the f i n i t e difference approach was found suitable. The solution requires a point boundary condition for the p a r t i c u l a r d i s placement i n s p e c i a l cases and, i n the general case, a l i n e boundary condition for the three displacements. The solution schemes f o r 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 s t r a i n s . Which of the two cameras i s to be used depends greatly on the p a r t i c u l a r circumstances of the proposed test. If the boundary conditions are readily available, then DASSC should be used as i t i s l i k e l y 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 experimental work and computing e f f o r t , 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 l i n e z = z was found to be s a t i s f a c t o r y where solutions along a l i n e z = z sought. = constant Q Q were If a two-dimensional approximation of n(y,z) was necessary,.then i t was found to be s u f f i c i e n t to scan the fringe pattern along a number of l i n e s z = constant and to approximate n(y,z) along these "scan" l i n e s 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 l i n e y = constant through the points where this l i n e intersects already approximated scan l i n e s . 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 v e r i f i e d . 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 i n x,y plane would be used to take three independent photographs. A l i n e boundary condition would have to be available and the three displacements 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 i n this thesis was r e s t r i c t e d to specimens with one planar surface. There i s no reason why DASSC could not be used to measure displacements and strains i n specimens having shallow curved surfaces or surfaces consisting of planar and curved surfaces. be necessary to r e l a t e the l a t e r a l shear Ay dimension of the curved surface. g I t would only to the t h i r d , i . e . the x, This dependence of the l a t e r a l shear on the (y,z) coordinates could be e a s i l y incorporated i n the computer programs of the f i n i t e difference schemes used to calculate displacements and s t r a i n s . 170 The "depth" of the specimen surface along the x-direction would, of course, have to be reasonably small so that the l a t e r a l shear would not be excessively large. Numbering of the fringes could be made easier, i n some cases, i f one more photograph were processed than the minimum number required. 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Equations", 175 APPENDIX A DERIVATION OF EQUATION (2.9) 2 I(t) 2 = |EjI + |E | + 2E .E 2 X 2 a. cos ( k r - tot + ijjj) + a 2 (e . e ) c o s ( k r 2 l iJ r 1 cos ( k r - tot + i> ) + 2 2 x - tot + i(; )cos(kr 1 2 - tot + I|J ) 2 l 2j r / Kt) dt I>- = 0 / cos (kr 0 2 1 / cos (kr 0 - tot + ijij) dt + 2 ^2 ( e ^ e ) / cos(kr 2 1 2 - tot + i> ) dt + 2 - tot + ip )cos(kr - tot + ip ) dt 1 2 2 0 Let us now evaluate the three i n t e g r a l s and approximate the r e s u l t s by making use of the fact that T » T . T _T / cos ( k r - tot.+ ijjj) dt = 2 0 T_ 8TT z / cos (kr T 8TT 2 1 (kr - toT + i f ^ ) - s i n 2 ( k r x + I/J ) 1 z - tot + i | J ) c o s ( k r - tot + ty ) dt = -^cos ( k r - k r + i j ^ 1 sin(kr cos 1 T - tot + ty ) dt - — 2 / cos(kr 0 sin2(kr x 1 - 2 + kr + 2 kr 2 + With these i n t e g r a l s , I z - r x + i p - 2toT) - s i n ( k r j + k r + 1 ifjj 2 \p ) 2 i s given by equation (2.9) 2 2 ty 1 + ip ) 2 - ty ) 2 T 2 176 APPENDIX B DERIVATION OF EQUATIONS (2.18) AND (2.19) cos[kr(y,z) - wt + ty] = cos(kr Kt) - cot + 40coskr 2 |[cos(kr £ - cot + i(;)coskr - s i n ( k r - cot + iJO s i n k r ] dA 0 e With Ej and E = 'a^ 2 [E cos(kr 1 e E = / s i n k r ( y , z ) dA e - cot + ty) - E s i n ( k r Q 2 The recorded i n t e n s i t y I E = J I ( t ) dt = l oJ 0 r - 2E E / sin2(kr 0 2 T / cos (kr 0 2 Q - .cot +ty)dt With these integrals I + E 2 T / s i n ( k r - cot + ty) dt 0 2 0 - cot + ty) dt Q = r [cos2(kr dt - T — T r 2 J s i n ( k r - 0)t +ty)dt - T — t 2 J cos ( k r - cot + 0 Q and s i m i l a r l y i t could be shown that p - cot + ty)~\' - cot +ty)dt In Appendix A i t was shown that j sin2(kr Q i s given by r 2 a 2 e , I ( t ) becomes 2 1 0 Ej = / coskr (y,z) dA A and r Q r Let us define the integrals I - s i n ( k r - cot + ^ ) s i n k r E (t) a Kt) e ty) 0 0 - cot + ty) - cos2(kr 0 i s closely approximated by equation + tyj] « (2.18) ^ 177 With the use of complex notation equation (2.18) may be written as equation (2.19), as i s shown: j ikr (y,z) e e d A = A j c o s k r e ( y ) Z ) ^ + i / s i n k r ( y , z ) dA e A A - ikr (y,z) / e e -y' -' uAn =— / coskr (y,z) dA -if A A 1Kr ( e z d ikr (y,z) e e ikr (y,z) - dA e T 2 /e l k r e ( y ' z ) dA - ikr (y,z) dA /e ^ A e fcoskr„(y,z) dA ^A ^ ° d ' A Ir = s i n k r ( y , z) dA e / s i h k r ( y , z ) dA e A ' '•" - (2.19) 178 APPENDIX C DERIVATION OF EQUATION (3.5) The equations of the source (sw) and d i f f r a c t e d (dw) spherical wavefronts shown i n F i g . C.l are given as 2 dw: x sw: x 2 2 + (y - y ) + (z - z ) D 2 D + (y - y ) + (z 2 Z g ) g 2 2 2 2 = x s + (y - y ) = x s + (y - y ) + z D + z A 2 g D 2 A (C.l) (C.2) The equation of the d i f f r a c t e d ray i s the equation of the l i n e from the "apparent" source point D(0,yjj,Zjj) to the point ( x , y , z ) i n the aperture. d i f f r a c t e d ray: y^ — yjy = — x + y s s a a D x z = — ^ x + z n F i g . C . l D i f f r a c t i o n i n a single aperture camera. (C.3a) (C.3b) 179 Huygen's p r i n c i p l e permits us to assume that the d i f f r a c t e d ray originates from the source wavefront at the point ( x , y , z ) and reaches the d i f f r a c t e d 2 (x^y^Zj). wavefront at the point / e( a , . v a 2 (x e (y > z a a ) given by 2 2 x - (C.4) z) 2 that i s to be used are such that DASC > 30 i n . s D r 2 2 The magnitudes of the parameters of x the d i f f r a c t e d wavefront - x ) + (yj - y ) + (z x 2 Therefore "leads" the source wavefront by the distance r 2 < 2.5 in. A - 2 x 10~ |y , < 1.25 A in. in. 5 - y |,|z D y s - Zg | < D n Ax = 2.44-^— c ss hence Ay , Az < 4 x 10 The f i e l d of view i s such that Let us now define the following quantities: ya a = " v s - fyp b = - YA x s + J Y A ! *s c = z YD yp 2 . ( a " Z ^ Z D 2 - 2 - 2 a < 4 x 10 b < 4 x 10 c < 1.6 x 10 J » Z D Ay - — s A Az - it x r x„ < 2 x 10 |d*| < 1.6 x 10 d* = 3(Ay cos9 + Az sin9) 6 It i s convenient to introduce a polar coordinate system of the c i r c u l a r aperture, as i s shown i n F i g . 3.3 ya = y A + r cost" , so that we may z a = r smfc write y a and z a as 180 The equations equations (C.l) and (C.3) are evaluated at the point ( x , y , z ) and the 1 1 (C.2) and (C.3) are evaluated at the point ( x , y , z ) . 2 2 2 1 By combining these equations and using the quantities that have been defined we get a quadratic equation i n ( r / x ) e fr 1 l sJ x f - 2 c + d*l /¥ J l sJ x g (/G + 2 ^ = 0 (c + d*) - c (C.5) F = 1 + a where F and G are defined as G = 1 + b The solution of equation (C.5) f o r r e i s given by equation (3.5). I t s derivation i s quite tedious as Taylor's series f o r the square root and sever a l fractions must be used to determine r e with the desired accuracy; of By making use of equation (3.5) i t may be shown that | r / x | < 1.5 x 10 e With the magnitude of r £ known, we see that i f r e A./10' 5 s i s to be determined with Q A/10 accuracy (or r / x e s with 6.7 x 10 accuracy), the quadratic term i n equation (C.5) i s so small that i t may be neglected. This was done and i t was found, that the solution of the resultant l i n e a r equation i s s t i l l given by equation (3.5). The l a t t e r solution i s much less tedious and therefore this approach i s used i n the subsequent appendices whenever possible. 181 APPENDIX D DERIVATION OF EQUATION (3.7) Let us define the following terms: 1 - = 2 D 2x|j 3zsl 1 - kd 2x 2 J / Ay - 'sys Az - 2 (D.2) Ay 2~ /D, + D Using equations (D.l) (D.3) 0 (D.l),(D.2) and (3.5) we may write (D.4) k r ( r , 0 ) = kr(D cos8 + D sin6) e 2 The recorded intensity I d Ir = f 2 J ikr(D os0 e 0 lC f 2 0 x r = Using I Q 2 r d r d 6 . / " ikr(D os0 + D s i n 0 ) ^ lC 2 (D.5) d Q 0 2TT ,Z „ , r ± ikr(D cos0 + D„sin0) , ^d 2J!(p) /e r dr d0 = —; - 4 p 0 n 1 TTd d D sin0) e 0 Since + (2.19) and (D.4) as 0 d d/2 r J i s evaluated by using equations r J Q 2 N (D.6) 2,2 (D.7) 2 given by equation (3.8), I X r = f2J!(p) Z o I P J may be written as (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: 2 2 = Dj + D kd .2 - Ay 2 ,2 + Az With Ay and Az defined by equation (3.6), the above equation may be written as ( y _ D y s ) 2 + - s ( z z D 2xc kd ) 2 = (E.l) Equation (E.l) i s the equation of a c i r c l e of the radius the center at i n the specimen plane. (yg,Zg) f 2 x c kd , and with By making use of equation (2.12) we may put equation (E.l) i n the form (yi - y i c ) 2 + (z - z > ± 2 ic - f?a l (E.2) p kd p f2x. This time equation (E.2) i s the equation of a c i r c l e of the radius and with the center at (y± >z± ) c i t i s apparent that I R(y^c,Zic) as f2J. (p)l 1 r l the image plane. n c kd From equation (E.2) i s c i r c u l a r with i t s center at the geometric image of the source point As shown i n F i g . 3.4 , I S(yg,Zg). r varies ; i t has the maximum at p = 0 , and i t s f i r s t minimum at p = 3.832 . The area of the c i r c l e of the radius f2xi 3.832 kd 1.22 X-r d i s known as the Airy disk and i t w i l l be referred to as the '"speckle" . The diameter of the speckle i n the image plane w i l l be c a l l e d D s and the diameter of the apparent speckle i n the specimen plane w i l l be c a l l e d D The two diameters are given as D s = 2.44 Dss = 2 - 4 4 X X~f (3.10) "t (3.11) s s 183 APPENDIX F In this appendix i t i s shown that the unit e l e c t r i c f i e l d vectors of a l l rays reaching a point i n the image plane are nearly p a r a l l e l so that the vector nature of the rays may be neglected i n the calculation of I . r Image o r photo plane Fig. F . l Unit vectors of e l e c t r i c f i e l d s . y = tany - (y - y ) / x fc W ^ n l 1 ± = lemkl l^nllcosY = cosy = 1 - \ d -0625 For the single aperture camera Y < ~ < —^2— = + '0^52 and hence 2 with the error being smaller than .0052 /2 = 1.4 x 10 When DASC or DASSC i s considered, Y < D + d x± < - 2.5 + .0625 = 12 214 and this .023 which i s time e ^'e ^ - 1 with the error being smaller than .214 /2 (the dimensions of d,D and x^ are i n inches) k* nl ~ e m 5 2 s t i l l s u f f i c i e n t l y small. e 184 APPENDIX G DERIVATION OF r AND r e i FOR DASC DURING THE FIRST EXPOSURE. £ 2 The o p t i c a l path lengths from the apparent point source S to the point Q, i n the image plane , are given by through aperture 1 : S-Q = S-sWj + d W j - s W j + =r dWj-Q through aperture 2 : S-Q = S-sw + dw -sw + dw ~Q = r 2 2 2 2 2 In a focused system the o p t i c a l path lengths from the point D to i t s geometric image Q are a l l equal, and therefore we may write D-dw + 2 = D-dw + dw ~Q dWj-Q 2 dw -Q = 2 2 - D-dw + dw^Q D-dWj 2 Using the l a s t equation the o p t i c a l path length through aperture 2 i s given as S'-Q = S-sw + dw -sw + 2 2 2 D-dWj - D-dw + dw ~Q + ( S - s W j - S-sw ) We define the mean o p t i c a l path length r r With r Q = S-sWj Q + 2 Q x v 1 as dWj-Q now defined the t o t a l o p t i c a l path lengths are given as r = o i r r 2 dwj-swj + = r + dw -sw + ( D-dw - S-sw ) - ( D-dw - S-sw ) Q 2 2 x 1 2 2 The variations i n the two o p t i c a l path lengths are then given by r e i = r x - r 0 = dw -sw r £ 2 = r 2 - r Q = dw ~sw + ( D^dw~ - We must now express r £ 1 J 2 and r (G.l) 1 2 g 2 S-sWj ) - ( D-dw - S-sw ) 2 i n terms of system parameters. of the source and d i f f r a c t e d spherical wavefronts are given as 2 (G.2) The equations 185 2 X dw X 2 sw 2 y r + (Y 2 + (z - D z r = x; + + (y - y ) + (z - z ) = x 2 D x 2 x 2 2 D + (y - Y ) s 2 + (y - y ) + (z - z ) = x 2 2 2 2 A + <y + y ) + z + (y - y ) + 2 D A s A ( 2 s G > 4 ) (G.5) 2 + (y + y ) + z s (G.3) D 2 2 s s D 2 + (z - z ) = x 2 S (y - y )~ + z D 2 2 (G.6) A The equations of the d i f f r a c t e d ray are the same as before and are given by equations (C.3) . Note that the equations of dw and sw are the same those 2 of d W j 2 and sw except f o r the sign of y . 1 r e l A has been derived i n Appendix C (where r e i s called r ) , and i s l e given by equation (3.5) . An examination of that equation reveals that r i s not a function of y r r £ l e = dw -sw 1 2 = dw -sw + ( 2 and hence we may write A = dw -sw J 2 e 2 = r (G.7) e - S-sw ) - ( D-dWj D-dw 1 - S-sw ) 2 2 2 / = r e 2 + / x s " / s x + 2 2 + (y - y ) + z D (y D + y A A ) 2 + E> Z + n / s x 2 / - / x + ( y S + y s ~ 2 2 + (y - y ) + z A s ) 2 4 + A s ' (G 8) Let us now define the following parameters: X so = s A e x + y S + Z S + y A = ( y j " y| + 4 ~ 4>/ so |A| < 3 x 10" = ^A^D " V |e| < 3 x l O " X 2y y /X A s / X 4 sc 2 ; I ,1 f SO < 1.3 x 10 5 2 With these parameters and using the Taylor's series expansion f o r the square roots i n equation ( G . 8 ) and by neglecting terms smaller than A/30 we get AT X. 186 The l a s t equation may be accurately approximated by r e 2 = r e - D6 (G.9) where 6 i s given by 6= x _4.]zp_iys x so ' x .y^sfp^fs s o Note that a small change i n y Q x s b x ( G > 1 0 ) s o a l t e r s 6 much more than the same change i n would; because of t h i s , and using equation (2.12) 6 i s accurately approximated as 6 = - 7 1 " 7 1 0 (3.13) 187 APPENDIX H DERIVATION OF EQUATION (3.12) - THE FIRST EXPOSURE I ,2 |d/2 d/2 m / 27 TT ikr~ /e 0 0 d/2 2TT / /e 0 T . 2 d/2 2TT ; 0 - ikr ik.(r - D6) e r dr d0 + 0 1 + e 1 + e r d r d G 0 d/2 27T P r / - ik(r /e 0 0 d/2 2TT / /e 0 0 d/2 2TT / /e LkD.6 ikr e - D<5) r dr d9 e - ikr„ (H.l) 0 0 The values of the integrals i n the above equation are given by equation and with - ikDS 1 + e 1 + e ikD6 = 4cos kD6 equation (H.l) becomes 2 [a] T r !r = ~2[ k J ud i 2 4 2 r 2j (p>i 1 J i p cos J With I„ given by equation (3.8), I I r = 4I 1 (H.2) may be written as f2J (p)l 0 kD6 cos kD6 (3.12) (D.6) 188 APPENDIX I DERIVATION OF EQUATIONS (3.14) AND (3.15) The minima of I 2Jj(p) 2 given :by equation (3.12) occur when either of the terms r kD6 or cos — r — P i s zero. The f i r s t term has already been discussed i n 2 Appendix E where i t i s shown that I diameter D r may be approximated by the Airy disk of i n the image plane, and of diameter D g i n the specimen plane. s s The second, cosine, term modulates the f i r s t term..so that i n this case the resultant I r looks as i s shown i n F i g . 3.7 . The period of the modulation i s c a l l e d the speckle grid pitch G zeros of the cosine term. kDS ^ for y. - y = G : 2 i c since g kD(y s k = 2TT/A G = s and i t i s calculated from two consecutive - y 2x^ ± l c ) _ kDG 2x^ s Ax-£ (3.14) By using equation (2.12) the apparent speckle grid pitch G ' i s g g determined' from the l a s t equation as G s s Ax = ^ (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 i s displaced to S , and i t s coordinates^change between the two exposures as S(0,y ,z ) + S*(u,y + v , z s s s + w) s The equations of the d i f f r a c t e d wavefronts dw x and dw , as well as those 2 of the d i f f r a c t e d rays remain unchanged and are given by equations (G.3), (G.4) and (C.3) respectively. The equations of the source wavefronts sw x and sw must be modified to account for the altered, second 2 exposure position of S . (x - sw, + sw. u y - ( y + v) + g z - ( z + w) = (x, - u r s (x - u + r r+ + (y s + v) - y A + ( z + w) + (y s + v) + y A + ( z + w) (J.2) s (J. 1) y - ( y + y) s z - (z + w) = ( x - u) s g Calculations completely analogous to those presented i n Appendix C w i l l again be done. The d i f f r a c t e d 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 = V = W= u v xs w - U < 1.3 x 10 V < 4 x 10 W < 4 x 10 - k - 4 3 for u < 1 mm - .04 i n . for v < .012 i n . for w < .012 i n . 190 r , the o p t i c a l path length v a r i a t i o n of the rays passing through aperture 1 , w i l l be determined f i r s t . at the point (x ,y ,z ) 1 1 2 2 and equations ( J . l ) and (C.3) are evaluated at the 1 point ( x , y , z ) . A l l these equations are combined and we eventually get a 2 quadratic equation i n ( f ei^ r *s ) + /F - 2 *s The equations (G.3) and (C.3) are evaluated r 6 l / x s ) lA—ID J _ ( A y (Ay - V)cos9 + (Az - W)sin6 + 2 + - u + B (Ay - V)cos9 + (Az - W)sin0 2 V ZB Z4 _ 2W + 2U = _ fp_ v ) /G ( A z _ _ u w ) + y^LZa A y _V ) - ( ( 2 Ay ZP^24 + 2 A z Az - w) *D x c 0 (J.3) In Appendix C i t was shown that the quadratic term may be neglected when r i s to be determined to X/10 here too. the (or i n fact to X/30) accuracy and this was done Using Taylor's series expansions for square roots and fractions, solution of equation (J.3) (with the quadratic term neglected) i s found to accuracy of X/30 as r ei = r 1 - 1 - 3yi (Ay - V) - (Az - W) - 1 - y c Z o zx s , (Az - W) - x 1 - (Ay - V) - 3y£ + 3z 2 ys 2xT~ 3y + 3z 2 2 2x1 c - x yA COSD r -> x U sxnt c J Since r = dWj-sWj i s this time a function of y^ , we must replace y^ i n equation (J.4) by -y^ to get dw ~sw 2 r = ei(y r e 2 _ A y ) A + (J.4) ( D _ d w ! " S-sWj 2 , and r £ 2 i s then calculated as ) - ( D-dw - S-sw ) 2 2 + 191 r e 2 = r ( - y ) + A \ + (y - y ) + z 2 e i A D - / | + (y + y ) x D 2 + + ' < z A 2 A D - / ( x - u) + [ ( y + v) - y j + (zg + w) , 2 - > x u s 2 s 2 + g [ ( y + v) + y ] s 2 A + ( Z g + w) 2 (J.5) Using Taylor's series expansions for the square roots i n the above equation, r , accurate to A/30 was found as r = r ( - y ) - D«S - e 2 e i A where £ i s given by equation (3.16). (D.l) and (D.2) we define E 1 - E 2 = D 1 - 2 3y 2 _y_ 2x|j ys s z X S z s 3yf, + 3zS 1 D/2 1 - X X S 3y| + 3z| 2x| v 1 - 2 S X defined by equations 2 and F as 1 - y s s With Dj and D w S X w 2 2 + S 1 + 2x|J x 3 z ,E (J.6) K) S y X S Zg 2xi x u s s X S _u_ x s u 2x' We can now write equations (J.4) and (J.6) as r r = r [ ( E + F)cos0 + E sin9] e l e x = 2 2 E [ ( ! " )cos6 + E sin0] - D(6 - £) E F 2 (J.7) (J.8) 192 APPENDIX K DERIVATION OF EQUATION (3.17) - THE SECOND EXPOSURE I r The substitution of the second exposure o p t i c a l path length variations r and r e ei , given by equations ( J . 7 ) and ( J . 8 ) , i n equation (2.19) gives the 2 second exposure recorded i n t e n s i t y as d/2 2u _ d/2 | / e i r d9 dr + / 0 0 0 i k r 1^ = e d/2 / 0 We define p x 2TT d/2 / 0 . |e~ 0 ± k r e i r d6 dr + 2TT _ / i e k r e 2 r d6 dr 0 2TT (K.l) / e ' ^ r d B dr 0 and p as 2 Pi kd =~2 /(Ej + F) + E 2 kd p, = ~2 — 'z /(E (K.2) 2 (K.3) - F) f E 2 1 The integrals of equation (K.1) are evaluated as d/2 / 0 d/2 / 0 2TT T\d J ^ P , ) 2 Je 0 ± r < 2 i r d0 dr 2TT fe ± r e2 r d6 dr 0 (K.4) TTd^_ J ( p ) + ikD(6 - 5) 4 p, : 2 (K.5) e e Using these results equation (K.l) becomes f Trd 2 — r Trd 1 1Ri(Pi) , - i k D ( 6 - £ ) i ( P ) l f J i ( P i ) , ikD(6- g) J i ( P > l + e + e J 2 I Pi fJ^Pi) I Pi '2 JI(P )1 2 P 5 + 2 J '2Ji(Pi)1 f 2 J i ( P ) l 2 I Pi J 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 i t i s apparent that the speckle r grid has been shifted with respect to i t s f i r s t exposure position. This s h i f t produces Moire fringes and i t w i l l be now related to the camera parameters and the displacements u,v and w. Using equations (G.10) and (3.16) defining 6 and £ respectively we write kD(6 - Q kD 2 2 2 x ys r D X v X so so yp - s _ ys s y 1 - . z so 1 - yi Xso' s 1- w X X so' so D- s z X.2 so X '•so Z z so w + X so so y z s X IX X X X so so s + X so' (L.l) so During the f i r s t exposure the speckle grid "passed" through the center of the y^gj the speckle, and hence we may write Y g S ~ Yg > where Y g j l S l s analogous to ; i t denotes the y-coordinate of the "center", or the f i r s t maximum of modulating term, of the apparent speckle grid i n the j - t h exposure. We f i n d Y g S 2 > the speckle grid "center" during the second exposure, on the l i n e z = Z g from kD(6 - O where y 'Sg2 n i s replaced by Y g S y i - Yc + v + 2 = 0 (L.2) • The solution to equation (L.2) was found as x (Is X 1 X so so + u 0 v 2x s YS V + W Z S so so ys s z - w 'SO Equation (L.3) i s accurately approximated as y s g 2 = ys v H ys ys s z u - w (L.4) (L.3) 194 The Moire fringe number n i s found from equation (3.18) which by using equation (2.12) may be written as Ysg y 2 sgi v sg2 y S = - nG (L.5) ss n i s determined from equations (L.3) and (L.5) as 2 n = - Axc 1 - X -1 y X so-* s so x s + u 2x s vy + wzcO s so v so y z s - w s + V (3.19) *-so Equation (3.19) may be closely approximated as y Axc s u + v - ys s w z (3.20) 195 APPENDIX M DERIVATION OF EQUATION (3.21) The envelope of the speckle grid term of equation (3.17) i s given by the 2J (p.) 2J,(p ) 1 2 product . The two terms are c i r c u l a r l y symmetric about Pj P 2 their respective maxima (centers) which occur when P 1 P 2 = 0 (M.l) = 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 = D i = ^ : + y + v ^ u (M.3a) s Zc Z D D.i = = z z + + w s ~ u (M.3b) y P 2 = 0 : y Z = y D D = Z D = y D 2 = S + v + u g + y A (M.4a) x Z 2 ( M D l ' We denote the coordinates of the centers of the two speckles by and ( y , i z i c 2 C 2 ) respectively. centers are determined = - ~ z i c l =y = ic - z z D i = - ^ i c l z i C 2 w + f : + ic = 2 - ^ x y Z D = 2 = y D 2 z ici i c 1 C l (M.3) and (M.4) as - x y (y By making use of equation (2.12) these from equations y±ci = ~~ym 4 b ) - v ^ ~ u i + u ( -5b) M Yic ~ x u ^ i 2x; D ( M - 6 a ) < ' ) M 6b 5 i z c l ) 196 where y ^ c and z^ are the coordinates of the f i r s t exposure speckle centre. c The l i m i t on the magnitude of the displacements u,v and w i s 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 (y. ici V J - y. Y+ ic' J (yica - y i ) 2 + c v (z. ICI - z. ) i r <i - zic z LC2 Y -5 D -2 D c (M.7a) r (M.7b) ; Using equation (2.12) and by combining the two equations (M.7) we get v + u y s ± x D/2f c + w + u — x„ < D = 1.22 x c X-r (3.21) 197 APPENDIX N DERIVATION OF r_,-1 FOR DASSC DURING THE FIRST EXPOSURE With r defined by equation Q (3.22) as r = S -sw Q 2 + dWj-Q 12 path lengths r ^ from the two point sources Sj and S the four o p t i c a l to the point Q i n the 2 image plane are given as S -sw 2 S l r o " S W d w - s w 2 i 2 + d w + d W j + dw i" " d w s w n + (s i " n s w + 1 1 2 - s w 2 d w + 1 -Q = r 2 + dw Q -Q + ( S -sw 2 n s w In Appendix G i t was shown that r i l 11 + - s w d w + 12 " S -Q + 2 - ( S S W " (N. 1) 12 2 " S S 1 2 W ) (N. 2) 1 2 > W S 12 r - s w 2 1 2 " 2 " S S W 1 2 ) dw ~Q = D-dWj - D-dw + dWj-Q 2 and hence 2 may be written as 2l r r = r 2 1 22 = S 2 ~ = r + dw -sw Q Q 2 S W 2 2 + d w + ( D-dWj - S -sw 21 - S 2 W 2 + dw -sw 2 2 2 + d w 2 " ^ ^ + S 2 ~ + ( D-dWj - S -sw 22 2 ) - ( D-dw - Sj-sw 12 S 2 1 2 W ~ S 2 ~ S W 1 2 (N.3) ) (N.4) ^ ) - ( D-dw - S -sw 12 ) 21 2 2 22 The o p t i c a l path length v a r i a t i o n s r •• are then given by r eI2 r ei i r e2 l = = = r r r r o = dw^sw^ i l " r o = " r o = dw -sw j + ( D-dWj - d w - s w + ( D-dw - S 2 1 - r = r r e22 (N. 5) " ! 2 22 2 = 0 d W j - s w ^ +(S1-sw1x ' 2 2 22 1 - S -sw 1 2 S - s w 2 1 2 (N. 6) ) ) " ( D-dw ) " ( - s w 2 2 v 12 D-dw 2 2 - S -sw x - S 21 - s w 2 ) • (N. 7) ) (N. 8) 22 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 d i f f r a c t e d wavefronts have the center at the point D ( - Y » Y , Z Q ) and pass through the aperture centers at D ±y . A The d i f f r a c t e d ray appears to originate from D ( - Y , Y , Z ) and passes T ) R ) 198 through the point ( j y » ) x z s a l n t n e a aperture. The equations of the source and d i f f r a c t e d wavefronts and of the d i f f r a c t e d ray are given as dw : (x + Y ) dw : (x + Y ) sw x 2 x 2 : x 2 : x 2 x 2 : 12 s w n sw : 21 sw 2 2 + (Y + (y - Y ) 2 + (y - Y ) 2 " y + (y " S2> y S l ) 2 2 S l 22 + (y " y S2> + (z - 2 + (z - z + (z - *s> + (z - z y d i f f r a c t e d ray : g G G V s> = 2 = 2 =4 =4 =4 =4 2 2 2 4+ 4 + + ( y Y y 2 D y A 2+ - A + + y ) y ) 2 + 2 A ( y > + y A S2 (N.10) 2 A + + (N.9) 4 2 + (Y + y ) + Z - > S2 (x + y) + Y Z ( D - A> 2+ 4 4 (N.H) (N.12) + z| (N.13) 4 (N.14) Y y = z = s> D ) ~n x The case Ay > D Z •+ (z- 2 + (y - y > + (z - " ° (x + y) + Z s a (N.15a) n s (N. 15b) Z A i s considered f i r s t . n In Section 3.5 an argument was made that i n this case only the d i f f r a c t e d l i g h t from the source wavefronts sw and sw S 2 contributes s i g n i f i c a n t l y to I 21 at Q. r 12 The positions of SJDJSJ and are shown i n F i g . 3.12 , with the coordinates of D being such that |S-DI < D . gs The position of the imaginary point source S was chosen so that (hopefully) the blurred and sheared images of the r e a l point sources Sj and S would both be centered about the point R, the geometric image of S. 2 The subsequent analysis has shown this choice to be correct. The geometry of DASSC i s such that the positions of S related to the position of S by y y = s-^( s Y S l s = Y 2 s- Y A> +y < S " A> Y y x and S 2 are 199 Let us define (and redefine) the following parameters: a = 1 + fy b = 1+ 1 r a Y { D - x Y D1 2 - zDl + fa z x s y^ - x s 2 j a < 4 x 10- 2 b < 4 x 10 2 2 + s F = 1+ a G = 1+ b < 1 x 10 Ay Ay = Az = Z D ~ Z S 3 xl The distances dw^-sw^j < 4 x 10 l - h Az, < 4 x 10 r 3 - - >t < 2 x 10 are determined i n the same way as i n Appendix C. The d i f f r a c t e d ray originates from the source wavefront at the point ( x , y , z ) and reaches the d i f f r a c t e d wavefront at the point ( x , y , z ) and 2 2 2 1 therefore the distance Ll 1 by which the d i f f r a c t e d wavefront leads the source wavefront i s given by dw,- -sw,- = / ( x - x ) + ( y - y ) + (z - z ) l -"IJ 2 -elj : 2 2 2 2 x : 2 (N.16) We introduce the polar coordinate systems of the two apertures aperture 1 y A + r cos9 r sinG aperture 2 : - y A + r cost z, = r sin9 r ei2 w i ± x be determined f i r s t . The equations (N.9) and (N.15) are evaluated 200 at the point and the equations (x ,y ,z ) 1 1 1 at the point ( x , y , z ) 2 2 . 2 quadratic equation i n ( r ei 2 - 2 L. s J X r r ei2 I s ) r + 3 Ay + 3f Y Ay + + o + 2 Y /G + x s - y x ~A X c y D . A —JT~ g y + — /F ZP_ A X / X ) i s obtained as e i 2 Z 9 . . Y ^ y (s - 9 R X X sJ D Y Ay + 3i Y x X c x s - y 3 ^ A A Az + 3i A x D c /G + 2 /FJ COS0 + A y Xc - A X Az'+ = sin0 s; The small quadratic term i s neglected and by making an extensive use of Taylor's series we obtain an approximate solution to equation (N.17), accurate to A/30 r ei2 = £ q 2 as £( i + K where q,Kj and K c o s 9 + K sin0) = r 2 (N.18) £ are defined as 2 3, 2X K i = 1 - K = 1 - 2 We s h a l l now write 3Y ?N 2X* 3Z S 2X|J Y Ay - Az - S S Z xi Az YqZq xi determine r. "e2 A y Since r l ei2 i s not a function of y. we J dw-sw„, = dw-sw,„ = r e and r e2i i s then given as 2 2 i 1 1 2 r e2i = ^e + (P" d w i " S 2 _ s w i 2 ^ ^ ( D'^dw - Sj-sw 2 21 ) A D c (N.17) s i Y 3j v - y> + z z y )( D B l s - y' + 23 1 - sin0 + 3 Ay + 3^ Az + c D - A Az + 3 c s - y l i From these equations and equation (N.16) a COS0 + A (N.ll) and (N.15) are evaluated may 3 201 "e2 i = e + r + ( Y V D " -/x|+ (Y 2 + Z D - y ) + z 2 D+ / s x + ( y S " A y +/x|+ 2 A (y ) 2 + Z +y ) g + z 2 A (N.19) s Using the Taylor's series expansion f o r the square roots i n the equation above,,an approximate solution f o r r r ez 1 £ 2 i s found as 1 (N.20) ^ r - Dp e r where p i s defined as Y 1 - D " S xs o S Q v redefined as When the case Ay < D s — S S ^D so x so ZjS Z 2 A with X Y Y x Pi + so (N.21) X so (N.22) X „ = X . + Y| + Z , + y 2 2 2 i s considered the parameter 3! i s such that s g 3 |g | < 6 x 10 and consequently * i as r ei2 r £( ! = e 2 1 K and * e i z may be further approximated e 2 1 (N.23) + K sin9) C O S 0 2 (N.24) = rO^cosO + K sin9) - Dp 2 To determine r * ^ = dw ~sw 1 we use equation (N.16) again. 11 Equations (N.9) and (N.15) are evaluated at the point ( x , y , z ) and equations (N.12) and 1 1 1 (N.15) are evaluated at the point ( x , y , z ) . 2 2 From these equations, and 2 equation (N.16) we obtain a quadratic equation i n -ei i l x - 2 s /F Ay + + Az ^ /F (. s ) x Y + 3 -ei i + 3, S + A <D " V Y y A " A y Y X Az + 3X cos9 + ^ A - i P y 3 y + + Y S D Y + Z Y D Ay + B (r* /X ). 11 S + A1 y s Z * D z Az + x sl c X + 2 1 - sine 3 1 A y ^ + + c S D) S Ay + 3,-S? Y Z .+ 23 7? Xc + Az + 3 - sin9 sJ = 0 1 + y A c A cos9 + (N.25) 202 With the small quadratic term neglected an approximate solution to equation (N.25) accurate to X/30 was found as r ei i = i r r r I K 1 cos9 + K sin6 (N.26) 2 i s calculated from equation (N.6) as en = en r + S = r. n i" s M + " 2" ii w + S <y s w " s y i2 A ) : s + zl -/: -s x <y + - A> y s + Z S An approximate solution to the equation above accurate to X/30 i s given by r ei l = =r • where a A and a s - a. + as A cbs0 + K sin(: 2 (N.27) are defined as yi sl Y L S0 x X == 2y 3 OU r* r = dw ~sw 2 2 2 e 2 2 "'" ^ r- st e 2 2 i e n r o u n = r * (y, en^A n ( 22 y A A 1 YA Xs o i s obtained by replacing y d A l s o J 3 + X so 1 2 A i n equation (N.26) by -y^ , and from equation (N.8) -Y ) + ( D - d W j - S -sw A 2 -Y ) + A 2 A - A - S -sw 2 2 + (Y - y ) + Z j - /x* + ( y 2 D 2 ) - ( D-dw 12 A + (Y + y ) 2 D A + Z 2 + /x + (y 2 22 ) - y ) + z 2 s A + y ) + z 2 s A An approximate solution of the l a s t equation accurate to X/30 was found as D r e22 £ >xSJ cost 5 + K sin£ 2 Dp + a A + a s (N.28) 203 APPENDIX 0 DERIVATION OF EQUATIONS (3.23) AND (3.24) - THE FIRST EXPOSURE I The f i r s t exposure recorded i n t e n s i t y f o r the case Ay first. In this case I l i g h t radiated by S 2 r i s produced d/2 /e 0 d/2 / 0 ± k r e i 2 r d9 dr + 2TT / 0 l k r e Using equation (2.19) I / 0 /e 0 d/2 e i 2 r d9 dr + / 0 2 i r d6 dr ± k r e 2TT - ikr. je ^ 2 i r d 9 d r 0 ikDp 1+ e e i 2 and r e 2 are given by equations i s calculated as d/2 277 / 0 1 |e 0 ± k r e r d9 dr 277 / 0 /e" 0 ± d/2 277 fd/2 277 / 0 d/2 (0.1) e (N.18) and (N.20) and with these equations I - ikDp 1+e /e 0 ± k r / 0 erd9dr i s given by r 2TT The two o p t i c a l path length variations r 1^ = i s determined and passing through aperture 1 with the l i g h t radiated 2TT / 0 Ir = s g e s s e n t i a l l y by the interference of the by Sj and passing through aperture 2. d/2 > D g r /e" 0 i k r e r d0 dr 4cos' k r kDp r d9 dr e (3.23) Using the integration by parts equation (3.23) may be written as Ir- = 41, f2J (p)] 1 CO - 4• 2J (p)^|PrR^| 1 R R' co SOJ —5- dR + P . where R and CO are defined as 1 / Ipj o y f2J (R)^i f2J (R)'J 3 1 R 2J (R)'| 1 PJ sinco 1 - R^ sinco—5- dR R = = kDp cos . d/2 kqd< CO co dR+ 4- PR / L"0 P R R2N co < 2 IPJ (0.1b) 204 The first exposure r e c o r d e d i n t e n s i t y c a l c u l a t e d a c c o r d i n g to e q u a t i o n f o r the case A y (2.19), b u t t h i s the i n t e n s i t y d/2 2TT _ I f f e m= 1,2 0 n = 1,2 i k r kd to the r e s u l t a n t I I Je" emn l k r d6 d r + K 0 (0.3) 2 _ kd ~ ~~2~ 22 = ~2~ + (0.4) Ki 'l - l^r) * K B + The f o u r r •, a r e g i v e n by equations r „ | l k e + J~ ( N i k e r (N.18),(N.20),(N.26) and (N.16) we may w r i t e 2 ; e 2 2 + lk(r e a 2n + A ( e " °P) r d A a s " A a - + J~ Dp)^ + | " e / + i k ( e i i k e r 2 2 k r e + a e d ( en " a r A + a + /e A s A e " ^ d A + l k ( r + a ) s d A + j - - ^ ou) + J + Je- i^k D p + J, , eik(ou ~ s 11 u v J ik(a Using equation + a e^ "A v 2 2 A e d A + (0.6) i s e v a l u a t e d as 0 ikr - Dp) dA where A i s the a r e a o f the c i r c u l a r a p e r t u r e ( s ) . Ir=« By (0.2) as Ir = + (0.5) 2 (N.23),(N.24),(N.27) and (N.28). ex j making use of equations (0.2) as 2 2 2 £ kd equation r 2TT m= 1,2 0 n = 1,2 (K P 'i l I e m n r d9 dr 0 We now d e f i n e P s P i ! and p 5 " i s t h e r e f o r e c a l c u l a t e d as d/2 Ir = i s again g s time a l l the l i g h t r a d i a t e d by the two sources c o n t r i b u t e s s i g n i f i c a n t l y and < D g s - Dp) (D.6) I r 205 J n e - ikCa,, a.)+ J + J ikDp s - -A' e — + J e" 22 where J » J and J n 2 2 are defined as ± k ( a A + a s " P> (0.7) D Jj(p) J 11 Pi I J i(P 2> 2 p 22 r To interpret 2 2 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 l e t J n ^ k l ~ ^ m Equation (0.7) i s then put i n the form f l(p)] J I r - 16I where Pj and P 2 Q 2 , (3.24) + P cos - (Dp - a ) 2 p A are defined as k P 1 = 1 + coska coska P 2 = 4cos|-(a + a ) c o s ^ ( a s A s A - 2cos—(a A k A + a )cos—(a - a ) s s A - a ) s (0.8) (0.9) 206 APPENDIX P DERIVATION OF r„,, FOR DASSC DURING THE SECOND EXPOSURE As shown i n F i g . 3.14 , when a specimen i s deformed i t s surface i s , i n general, displaced and strained both i n and out-of-plane. The deformation occurs between the two exposures and i t changes the coordinates of the two point sources S t and S , representing the surface, as follows 2 s i ( » S i ' s ) ^ S * ( u , y + v , z + w) 0 v z Si s S ( 0 , y , z ) -»• S*(u+6u,y + v+6v,z +w+6w) 2 S 2 s S2 s The equations of the d i f f r a c t e d wavefronts d W j and dw , and of the d i f f r a c - ted ray(s) remain unchanged and are given by equations (N.9),(N.10) and 2 (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 sw n : (x - u ) + [y - ( y 2 (x sw i2 : sw 2i : s - (u + 6u)J 22 : 2 + [(y 2 s S 2 + v) - y ] + ( z + w) 2 A + v + 6v)] + [z - ( z + w + 6w)] 2 = 2 S z g A 2 + £ z + (w + 6w)] 2 s (P. 2) + v ) ] + [z - ( z + w ) ] = 2 g i (P.l) 2 s + v + 6v) - y ] 2 g - u) + [(y 2 s [x - (u + 6u)] 2 [x S l + [y - ( y (x - u ) + [y - ( y (x sw - u) + [(y 2 2 g 2 s [x - (u + S u ) ] [x + v)] + [z - ( z + w)J = 2 S i 2> S l + [y - ( y - (u + 6u)] 2 + [ ( y + v) + Y a ] + ( z + w) 2 + v + 6v)] + [z - ( z + w + 6w)] 2 = 2 g 2 g + v + 6v) + y ] 2 g 2 (P.3) 2 s A + [z + (w + 6w)] (P.4) 2 g As i s discussed i n Section 3.6 , when DASSC i s considered the changes i n the distances between each of the point sources and S 2 and the laser must be included i n the calculation of r j ; j , and hence using equations e (N.8) the second exposure r ^ are given by (N.5) through 207 d -sw -ei2 -ei r Wl dw 1 -sWj j x = dw -sw £ 2 1 r 12 2 2 r +(S 1 (P.5) u -sw +(D-dw 21 = dw -sw e 2 2 + +(D-dw 22 1 S -sw - 1 2 1 )+ 2 r (P.6) L l 1 - S ~sw 12 ) - ( D-dw - x - S -sw 12 ) - ( D-dw - S -sw 2 2 —sw ^ ) + 2 2 2 2 22 )+ rL i (P.7) rL2 (P. 8) The subsequent calculations of the second exposure ^ ± ^ are similar to the e calculations of the f i r s t exposure t e i j presented i n Appendix N. The d i f f r a c t e d ray originates from the source wavefront at the point (x ,y ,z ) and reaches the d i f f r a c t e d wavefront at the point (,-x. ,y ,z ) with the 1 2 1 2 2 distance r e*x•j ^ by which the d i f f r a c t e d wavefront leads the source wavefront given by equation (N.16). The case A y < D s g g i s considered f i r s t . To determine r * (N.9) and (N.15) are evaluated at the point ( >y > ) K z 1 l and (N.15) are evaluated at the point ( x , y , z ) . 2 2 2 a n 1 2 equations equations (P.2) d 1 From these equations and equation (N.16) we obtain a quadratic equation i n ( r * / X ) as 1 2 -ei2 I - 2 -ei 2 I Xc, J Xc - 6 - U + A " D Ay + 3 S - A - V y X /F + 2 1 + 2i /F s l x " / F j /G 7f Y A y Y D " A : + Pi S " A -V Y y Xc Ay + " V COS0 + where U,V and W are defined as COS0 + W s y Az + V = Az + + l x S " A U X Az + 3 , — - W sin9 s Xc y Y f y Xc 3 + u+ y Y Xc S " A Ay + B j ^ Y + 3,— " W g -w u + 6u v + 6v Xv c l x = 0 sin0 (P.9) U < 1.3 x 10 V < 4 x 10 W < 4 x 10 s w + 6w + Az + 3 , — - W s - - 4 3 208 The small qadratic term i n equation (P.9) i s neglected and with an extensive use of Taylor's series we obtain an approximate solution, accurate to for r r* + r fY - Yc, n — Yc, - y ^ X C + r* as e i 2 = -±r~ 2X -ei2 fZ - Zg D X Zg u + Su X e - Xc c x c (l v + Sv' u + 6u A -± + B,) - 3) (l + X coso + c w + Sw X (P.10) sine c i s obtained from equation (P.10) by replacing y 2 1 X/10, by - y A A , and u + Su , v + Sv and w + Sw by u,v and w respectively. Y r = 2X e2i Z + + r - 0 — x s y + (l x w' — 3) x 1 x r r r 3.) coso + Xc J / (P.11) sint = Ar 2 L i - Su cos9 - Sv cos6 x - Sw cos6 y z i s given by e i z ei2 = r ei ^ Li" ^ + r 2 u c o s ^ x -SvcosGy -5wcos9 (P.12) z i s given by equation (P.7) as e 2 1 e2 l + sJ In Section 3.6 i t i s shown that A r ^ and hence ^ V u A Xc ZQ u - — — -(1 + s s -•ZQ N s r = *2 l ( ^"dwj + - S2-sWj 2 ) - D-dw - S jsw ( 2 ( = e i + r 2 + (x + [(y r Li + - u) + 2 s )+ r L ( [s + ( D ~ y > Y X 2 + A [(y 2z + v) + y j + 2 s Z DJ " [ s + ( D + A> * + X ( z + w) Y 2 2 " I % - s ) y Z DJ f [x - (u + S u ) ] s 2 + V •» 1 + v + Sv) - y ] + 2 g A [ z + (w + Sw)] 2 s An approximate solution of equation (P.13), accurate to X/10, using the Taylor's series expansions (P. 13) i s obtained by f o r the square roots with the result 209 r where s = + r e21 + A r Li C) - D(p - (P.14) i s defined as rY x e e21 Xs o Xc Zg6w"\ s ^A s o D Y 1+ s ] 6u 2^1 o- x X DX so' S0 D X v 6v so x so Y x so so 1+ D _ X n so lx: 6w "I™ so DJ DYo 1 + 2 2D 2x soJ DYg> 2X w fYgZg D. "•so S - D/2 r D 6w 6v' 1 _ _2- so We s h a l l now consider the case A y s < D (P.15) + 2D s s and r * 2 1 may be obtained from equations . Since now j 3x j < 6 x 10 3 r*12 (P.10) and (P.11) by neglecting some small terms involving 3j with the r e s u l t Y D - Yg Yg - y A u + 6u -e 12 = r Xs rz n Xc r e21 = ri rei2 Y -Y *D *S Xs Y S -Xc and r £ 2 x + y Xc X A u Xc equation ( N . 1 6 ) . 6v~\ Xc cos6 + w + 6 w 3in8 (P.16) Xc s -v cose + X are given by equations given by equations Z D Z w' ^S _z_ S Xc x s x s x s- sin0 r^j[ = dw 1 -sw 1 1 i s determined from Equations (N.9) and (N.15) are evaluated at the point ( P . l ) and (N.15) are evaluated at the point From these equations and equation (N.16) we obtain a quadratic equation i n ( r * 1 1 / X s ) . •ei 1 - 2 (P.17) (P. 12) and (P.14) with r * 1 2 and r * 2 1 (P.16) and (P.17) . (x ,y , z ) and equations (x2,y2,z2). Xc Zc u + 6 u Zc v + 'T* r ei 1 Xc /F - U- x Ay+3. c Y s + yA Xa V Az + 210 + 3— - w IXe + 2 1 2 3 - X Az + 3 , — - W 3in6 X " V COS0 + 1 + X c S xc 1 "7fJ /G + Ay + 3, + Ay + 3 7f Y S A + X B -V " Ay + y v + Az + 3 , — - W Az + 3 , — - W s - V COS0 + Xc A 1 5 l x where this time U,V and W are defined as U= = 0 in6 s A V V = Xs" w w= xT + (P.18) - U < 1.3 x 10 V < 4 x 10 W < 4 x 10 - 4 - if 3 An approximate solution of equation (P.18), accurate to X/10, was found as Y e n r r £ 1 1 = £ D "S Y Y S ~A y Xq Xc V ' u Z COS0 + Xq D Z Xc S Xs X w• u (P.19) ;in0 Xs- s i s obtained from equation (P.6) as -ei i = e n r + r L i +(S -sw 1 - S -sw ) 11 2 + (z +w) : [x - (u + 6u)] + [ ( y +v+6v) - y ] + [z +(w+6w)] : = en + r r L (x -u) 12 + l s 2 + [(y + v ) - y ] s 2 A 2 s 2 s s A (P.20) s An approximate solution of equation (P.20), accurate to X/10, i s given by : where a a and $ =a + u A en I r ei i + r " * a L l (P.21) &n + are defined as XS O X; SO H SO f s z + Sw IX,so w6w 6u Xso yA s s Y X z so X X s + 6v s o 6 w 1 2X J S 0 IX S 0 + 6v 1 2X S 0 J + (P.22) 211 a ^11 , , - v c X r* x SQ Y + ou so yA s s - ov 3 Xso i s o b t a i n e d from e q u a t i o n 2 2 ZA_ ? x x y + so A S Y x y + so x A sl Y (P.23) so (P.19) by s u b s t i t u t i n g - y f o r y A A , and u + 6u, v + 6v and w + 6w f o r u,v and w r e s p e c t i v e l y . f D " S Y r = r r e 2 S + A Y y -e22 = r* = r e 2 2 2 2 x + Ar L + v 6 v l COS0 + Xs c w + 6w"| J (P.24) sinG L 2 (P.8) as + (D-dWj + X - S -sw 2 s + < D " A> Y (u+6u)] + 2 s [x u c + Ar [x - + 6 Zg u + 6u - Zg D i s g i v e n by e q u a t i o n 2 + u x Z + Y y ) - ( D-dw 1 2 2 - S -sw 2 + Z D " X 2 s <D + Y [(y +v+6v)+y ] + A 2 U s i n g T a y l o r ' s s e r i e s expansion A f o r the square where 3 l 2 2 2 2 =a = r*„„ + A r , + a* + 6 e22 Ll T and A 2y Y —x o A s + u A r y " so x A g u + 26u : (P.25) 2 r o o t s i n the above e q u a t i o n £ 2 2 as (P.26) - ?*) + A* 2 y + ou so + 6v P D a r e d e f i n e d as 2 A - D( 22 Z s we f i n d an approximate s o l u t i o n , a c c u r a t e to A/10, f o r r r-e22 _ + [z +(w+6w)J 2 s A^ s - (u + 6 u ) ] + [ ( y + v + 6 v ) - y ] + s y ) [z +(w+6w)] 2 g + 2 2 x y A S Y A X so so so y A S Y . 3 x 2 y p so = - 6u cos6 x - <5v cos0y - 6 w c o s 0 z + A Al so rs u so • so Y x Xc + so' (P.27) 1 X A S j (P.28) 212 APPENDIX Q DERIVATION OF EQUATIONS (3.25) AND (3.28) - THE SECOND EXPOSURE I The second exposure I equation (0.1). f o r the case Ay r Once r and r e i 2 e 2 : > D s s s r i s calculated according to given by equations (P.12) and (P.13) respectively are substituted i n equation (0.1) and the indicated m u l t i p l i c a t i o n i s done we get ikr ei2 Ir = - ikr dA /• - iA - i k re i 2 dA / + e je A f i2dA + /i A e2idA + e i k r f e e A = k D( where A i s defined as P ikr e 2 1 d A Je - ikr e 2 1 dA + A - ikr iA ikr. idA /e ei2dA / r 1 A (Q.D e 2 1 K r - hf) + A* Equation (Q.2) may be put i n the form equivalent to equation (3.25) .2 T 2 (R Ir = R R l5 and R 2 ,A + R c o s - + R sinA ) - 2 3 are defined as 3 r i k r , , ,. r - i k r l k i r ^ Q I T * d A Jer Rj = je dA Je e I 2dA + J e e iP 2 2 2 R 3 = 4 /coskrg A > 12 I dA - j dA - 2 / s i n k r g j dA / s i n k r g dA A A 2 2 dA/coskr* A 21 :2 equations en a n d r s dA + /sinkr* dA / s i n k r * dA A A x2 2 < D s s , I r 2 a r e §i v e n (Q.4) 2x 1 2 dA (Q.5) i s given by equation (0.2) with r (P.12) and (P.16) , and r e22 (Q.3) 2 x = 2 /coskrg d A / s i n k r * j dA - /coskr* j dA/sinkr* A A A A In the case Ay r "^"^•^ G2. e 2 1 2Jcoskr* j dA jcoskr* A A R (Q.2) z 1 by equations e 2 1 given by equations e i 2 given by (P.14) and (P.17) (P.21) and (P.26) respectively. When the indicated m u l t i p l i c a t i o n n J - ikD(p - V) e ik[a* + B J + A* - D ( - 5*)]] f j 2 2 P 2 i - ? * ) . , P e J where J m and p are defined as m n n r Y Y kd p fYD- S Y _ kd 2 21 x i: + A* - D ( - E*)V 2 2 J (Pmn) mn J mn 2 2 fZ - Z + n Zq u s X s X s Xc s x J s fZj) - Z g Zg Xc Xs + 1 w x s Ys-YA X (Q.6) P e s-yA x - ikA* ^ e + s u + 6u X s w + Sw^ Xc J s Y kd '22 I f D" S _ Y s _ 2 - ik[a* + B 2 2 i s defined as D" S ~1 , x 12 - i k C B ^ - a*) 2 kd 11 e + 11 22 ikD( J i n equation (0.2) i s done, we get - x Y Y S x s rz YA + D I s - Zg Zg u w x X Xg Xg s 2 Y 1x s + yA x s x s s J Z-Q - + I S Zg Zg X Xc u + Su Xg s To interpret equation (Q.6) we assume that J J k i - J w + 6V X (and hence p m n J s m n - p) within the speckle area, and equation (Q.6) may then be written as Ir - 161, where Q x Qi = fVP>l k Qj + Q cos — D(p - ?*) - a* + 2 ^ and Q 2 2 r - 22 are defined as 7 1 + cos I ( B n -g 2 2 - 2a*- 2A*)cos | ( B 1 X -B ) - 2cosk(a* + A*)cosk(B Q - 2 = cosk(a* + A*)cosk(B 11 -B ) 22 22 11 -B ) 22 (3.28) 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 i n each instance the speckle grid has been shifted with respect to i t s f i r s t exposure position. of By comparing the modulating (cosine) terms the f i r s t and the second exposure I the r e l a t i v e s h i f t of the speckle r grids may be related to the surface displacements and their increments, and to the camera parameters. The case Ay > D s i s considered f i r s t . s s The two modulating terms are given by equations (3.23) and (3.25) as f i r s t exposure : cos second exposure : cos kDp 2 D(p - E*) +• A * In a similar way as i n Appendix L, the apparent speckle grid centers Y g j S and Y g S 2 > i n the object plane, are found from the maxima of the two modula- ting terms. Y gi ^ s S f ° d on the l i n e Z = Z un kDp with Y s g l are s g 2 from (R.D = 0 substituted for Y D . Y D(p with Y g substituted f o r Y D E*) s g 2 i s found on the l i n e Z = Zg from (R.2) = 0 + A* . The solutions of equations (R.l) and (R.2) given by r Y sgi - S Y 1 - 3, 1 '2 . Ys g 2 = Y sgi + x 1 - Xso , 2 - l (R.3) X.so - 1 D The Moire fringe number n i s related to Y (R.4) "so s g l and Y s g 2 by 215 Y (R.5) sg2 ~ s g i ~ ^ D N Y Equation (R.5) was solved for n with the r e s u l t ,2 •>- i fY X s 1 - n = + Su IX DX so X f so cos9 + 1 H D ^•X £ Y s 6v s X X DX J S0 S1 — + so' 2 X X 6A 1 " " S0 s w ~ D ^ Y -D/2 so cos6 D fY Z Sv) 2 X s - v l + t X, "•so y 2 X 6w DY so cos6„ - — 1 + D 2X J X,so r z Yn + v SQ D Y x s + 6w Z s 6wl 2X S0 s + 6wl so D J 6v +2X s J so- + + 0 (R.6) so J Once the d e f i n i t i o n s of 6u,6v and 5w are substituted i n equation (R.6) i t may then be accurately approximated by equation (3.26) When the case Ay < D s s s i s 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 : cos — second exposure : cos ^D(p z (Dp - a ) A 3 2 - £*) - a* + Again the apparent speckle grid centers Y maxima of the two modulating terms. D( S substituted for Y D X + - B and Y g g 22 2 are found from the These maxima occur when a )=0 - ?*) - a* P In equation (R.7) Y g (Dp - s g l 11 1 1 (R.7) = 0 2 2 0 i s substituted for Y . In both equations Z D (R.7) and (R.8) are solved with the result n (R.8) and i n equation (R.8) Y g i s S 2 i s set equal to Zg . Equations 216 "Sg2 Y S " 1 - - 1 X;so ^i S Y ~ so X K D (R.10) 2D The Moire fringe number n i s obtained from equation (R.5) with Y g S Y s g 2 given by equations (R.9) and (R.10). given by equation (3.29). x and An approximate solution f o r n i s 217 APPENDIX S DERIVATION OF EQUATIONS (3.27a) AND (3.27b) The amplitudes from I of the speckle grid terms f o r the case Ay s > D are obtained s s recorded during the two exposures and described by equations (3.23) r and (3.25) . The two amplitudes are given by f i r s t exposure : /e A second exposure : R„ i k r e dA je~ A e dA ± k r e r 2 x When the e f f e c t of the small nonlinear term 2X on the shapes of the a speckle envelopes i s neglected the amplitudes are proportional to J Using equations Pi: kd 'D " S 2 Y (S.l) 2 Yg - y Y T A u + Su Xc Xc kd 2 1 Z - Zg Zg u+ Su D Y 2J „ k , (K + K ) 2 P i(Pkl> 12 kd + J for the second exposure, Pkl mn (N.18),(P.10) and (P.11) we define P,P and P as for the f i r s t exposure, and -r- i(Pmn) fVp)! D " S X Y Y c S A X + y c - (l + Bj) - (1 + B i ) ^ u v + Sv"! 2 w + Sw (S.2) Xc v 1 Xc X, + Z D Z x c S Zg u_ x W^l - (l + B ) X sJ x s s x (S.3) where K x and K are defined i n Appendix N. 2 The l i m i t on the size of displacements and strains i s obtained from the requirement speckles recorded i n the two exposures overlap. of the speckle centers (maxima) as (Y ,Z ) 1 1 Y 1 2 1 2 1 maximum when p m n = 0 , the coordinates Let us c a l l the coordinates for the f i r s t exposure, and Since f—ilP-ML_l is "mn J are found from the equation ( Y , Z ) and ( 2 > Z ) for the second exposure. 1 2 that a l l the (YJ,ZJ) 218 p = 0 , with Yj and Zj substituted f o r Y was and Z n n . An approximate solution found as (S.4a) (S.4b) When Y and Z 2 and p , and Y 1 2 and Z Z1 = 0 respectively, for Y are substituted i n the equation P '= 21 12 and n , the approximate 0, solutions of these equations are then found as Y • 1 2* Y Y,„ Z c = Z 1 2 s 2 i Z 2 i (u + 6u) + (1 + 3 ) ( v + 6v) + ^ (u + 6u) + (1 + 3 ^ (w + Q Z S + x + (S.5a) x y U + 6w) + ( 1 (S.5b) (S.6a) A s+-V^ s Y ~ A ~ Y Y s-y + < .3i)v 1 + + 3 1 ) (S.6b) W The distances between the speckle centers must be smaller than the speckle radius and hence we may write p and p 1 2 : (Y p and p„2 1 (Y V Y 1 , Z 2 1 2 Z 2 x - Y ) 2 2 - Y K 21 l' + (Z 1 2 + (Z V 21 - Z l ) - Z ) l SS : z < (S.7) (S.8) J given by equations (S.4a) through (S.6b) and the d e f i n i t i o n s of the displacement increments 6u,Sv and 6w are substituted i n equations (S.7) and (S.8). Since |3 | « closely approximated 1 , equations (S.7) and (S.8) may by equations (3.27a) and (3.27b). then be 219 APPENDIX DERIVATION T OF EQUATION (4.13). D A S S C using a set of apertures rotated by the angle CJJ^ forms fringes according to equation Yi ! (4.12). 2 u x i + i~ v 2 A W-! i w Ysi si A x y i s i fy±-yA x S D c A COS6y-L V y i si XXs i x s i » y i - C O S 0 i i»yi w x ni(yi,zi) z D - A l l terms i n equation ( 4 . 1 2 ) w i l l now be transformed s (4.12) into y,z coordinate system using the transformations y^ y = y^cosdK - z^sind)^ ycosd)^ + zsincj)^ = z = y^sincj)^ + z^coscj)^ Zj_ = - ysincj)^ + zcosd^ The transformations of the displacement (4.3), components are given by equations and the s t r a i n components transform as u±, ±(y±,z ) y = u, (y,z) = u, (y,z)y, ± yl y + u, (y,z)z, y i y y i (T.la) = u, cos(f>i + u^sincj)^ y v i>yi(yi» i) z = [v(y, z) cos<J>i + w(y, z) sin<J>i] , i = y = (v, y, y y i + v, z, )cos(j) z yi + (w, y, i y + w, z , yi z y i ) s±nty ± (T.lb) = v, cos c}).i + v, g.sincJ^coscfj-L + w, sincj)^cos())^ + w, sin cj)i 2 2 y w y z i » y i ( y i ' i ) ='•[ " v(y,z)sin(f) + w(y, z) coscjijj , z i = " ( v , y , i + v, z, )sind) y y z y± y i = + (w, y, ;L y yi + w, z, )cosd^ z yl = - v^sincj^costJj-L - v ^ s i n ^ + w, cos (j)^ + WjySincj^coscj)^ 2 2 y = (T.lc) 220 By substituting i n equation — the coordinate, displacement and s t r a i n transformations (4.12) i t becomes u + (vcosd)^ + wsincj)^) - ( - vsindp-L + wcosd)^) — Ay i si — 1 z xg ^ D (1 + c o s 9 i ) x x S •(u,yCOsa)^ + u, sind)^) z A + WjySincfi^cosd)^ + w, sin cj)^) 2 - cosOyi (v, cos d)-L + v , sin(J)iCOS(J)-L 2 y y i si x fZ-j s z v, sin z d)^ + w,yCOS d)^ + w,sin<j>-^cos(j)£) = With c o e f f i c i e n t s z a-j_,...,kj_ and may be written as equation (4.13). z - cost Z l XXsx ( - VjySincj^cosdpi n-j_(y.j_, z-j_) + - (T.2) defined i n Section 4.5, equation (T.2) 221 APPENDIX U The c o e f f i c i e n t s d^,...,^ for the case of the specimen illumination i n x,y plane ( 0 = 90°) are derived here. unit vectors i , j and ii»Ji are related Z 1 From F i g . 4.3 (where = <p^ the as = H j = jicosdpi - kisincJi-L For 8 z = 90° the angles 6 and 9 X are related by y cos6y = cos(90° + 0 ) X The = - sin0 x unit i l l u m i n a t i o n vector 1 i s written i n the two 1 = icos8 x + jcos0y + = icos0 x - jsin0 s\ z x /\ /\ = i^cos0 x - (j^cosvj)^ - k^sincj)^) s i n 0 = i^cos0 x - j ^ s i n 0 c o s o ) ^ + k£sin0 sincj)£ x s\ X\ = icos0 ^ x From the l a s t two kcos0 coordinate systems as equations we cos0 x i = + jcos0y-^ + x x /\ kcos0 ^ z get cos0 x cos0y-L = - sinGjjCOSc})-]^ COS0 -L = sinGxSintj)^ z By substituting these relationships i n the expressions for dj_,...,k-j_ i n Section 4.5 the simpler forms of these c o e f f i c i e n t s are obtained. 222 APPENDIX V DERIVATION OF THE DISPLACEMENTS AND STRAINS CAUSED BY THE OUT-OF-PLANE BENDING OF BEAMS Handbook o f S t e e l C o n s t r u c t i o n [62] g i v e s ( a f t e r change o f n o t a t i o n and the c o o r d i n a t e o r i g i n ) the d e f l e c t i o n o f a n e u t r a l s u r f a c e of a t h i n p r i s m a t i c beam w i t h -L/2 4 y 4 0 clamped ends and a p o i n t l o a d a t i t s c e n t e r as =6 u(y) 12 f v y 1 - 3 3- [L/ J =6 u(y) ' y i 1 - 3 (V.la) 1L/2J 2 0 4 y 4 L/2 y - 2 2 [L/2J + 2 V ^ 3- y L/2, (V.lb) where 6 i s the d e f l e c t i o n o f the beam c e n t r e and L i s the beam l e n g t h , u i s not a f u n c t i o n o f z, and s i n c e we c o n s i d e r o n l y a t h i n beam and s m a l l d e f o r m a t i o n s , i t may be assumed t h a t the o u t - o f - p l a n e d e f l e c t i o n of the i l l u m i n a t e d s u r f a c e o f the beam i s the same as that o f the n e u t r a l s u r f a c e . I t i s apparent from F i g . V . l the bending of the beam g i v e s r i s e to the i n - p l a n e displacement v as V Using -L/2 - | U,y e q u a t i o n s ( V . l a ) and (V.lb) v i s g i v e n as ^ y < 0 0 « y « L/2 The = v(y) t 6 y 12 - 1 + L/2 2 L L/2 (V.2a) v(y) t 6 y 12 2 L L/2 1 - L/2J (V.2b) s u r f a c e s t r a i n s E ( y ) and e ( y ) a r e g i v e n as y £y z ^'y £z = » z w = (V.3) 2 ^'yy " v e y = v ~2 >yy u (V.4) Fig. Out-of-plane bending of a prismatic beam. 224 where v i s Poisson's r a t i o . The displacement w i s such that w(y,z = 0) = 0 , and hence w(y,z) i s obtained from equation (V.4) as w(y,z) = Vz —2 'yy u, u and using equations (V.la) and (V.lb) w(y,z) may be written as 6 t 1 + 2y -L/2 4 y 4 0 w(y,z) = - 12vz 0 4 y 4 L/2 t 6 2yl 1 w(y,z) = - 12vz 2 LV2 L/2J The p a r t i a l derivatives u , , v , v -L/2 $ y 4 0 v 2 L /2 2 and w, - 12 V l 'y w, — L L/2 1 + 2 o - 48vt — r z L 6 y - 12 — L L/2 (V.6b) (V.7a) 172 t 6 2y 1 5 — i + — 7 2 L /2 L/2 12 (V.6a) are given as y 6 y i,y L/2, (V.7b) (V.7c) 3 0 4 y 4 L/2 1 V 'y w, 12 t 6 2 L /2 48vt 2 6 1 ^ " .41 (V.7d) (V.7e) L/2. (V.7f)
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Measurement of surface displacements and strains by the double aperture speckle shearing camer Brdicko, Jan 1977
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Title | Measurement of surface displacements and strains by the double aperture speckle shearing camer |
Creator |
Brdicko, Jan |
Date Issued | 1977 |
Description | In the testing of materials, structures and structural components it 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. Unfortunately, when the surface deformation is quite large, as is usually the case in practical testing, most of these methods fail and only a few suitable optical interferometric techniques will work. Two of the recently developed techniques that seem to work are based on laser speckle interferometry. The first 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 its 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 displacement 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 ability to measure displacements and strains over many orders of magnitude, a considerable effort was made to determine the fringe formation of the two cameras due to all 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 all displacements , and strains on the fringe formation; in addition, the equations are "symmetric" 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 all instances the actual and the calculated displacements and strains agreed quite well. |
Subject |
Photography -- Scientific application Strains and stresses |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-02-21 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0063025 |
URI | http://hdl.handle.net/2429/20647 |
Degree |
Doctor of Philosophy - PhD |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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