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

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MEASUREMENT OF SURFACE DISPLACEMENTS AND STRAINS BY THE DOUBLE APERTURE SPECKLE SHEARING CAMERA by JAN BRDICKO B.Sc, University of I l l i n o i s , 1970 M.A.Sc., University of 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. With the knowledge of the deformations on the l i n e boundary the fringe numbers at the boundary could be calculated and the fringe numbers at the neighbouring points could, i n p r i n c i p l e , be determined by finding that fringe number which would s a t i s f y the overdetermined system of equations at those points. This process would be repeated u n t i l the fringe numbers of a l l points would be known; i n fact, numbering of the fringe centers would probably be sufficient.  171 BIBLIOGRAPHY Duffy "Moire Gauging of In-plane Displacement Using Double Aperture Imaging" Applied Optics, 1778-1781, Aug., 1972. Duffy, D., "Measurement of Surface Displacement Normal to the Line of Sight", Experimental Mechanics, Vol. 14, No. 9, pp. 378-384 (1974). Hung, Y. Y. and Taylor, C. E., "Speckle-Shearing Interferometric Camera - A Tool for Measurement of Derivatives of Surface-Displacements Proceedings of Society of Photo-Optical Instrumentation Engineers, Vol. 41, pp. 169-175 (1973). 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Hovanesian and Varner, "Methods for Determining the Bending Moments i n Normally Loaded Thin Plates by Hologram Interferometry", Strathclyde Symposium on the Engineering Uses of Holography (1968).  173 30.  S o l l i d , J . E., "Holographic Interferometry Applied to Measurements of Small S t a t i c Displacements of D i f f u s e l y Reflecting Surface", Appl. Opt., 8, 1587 (1969).  31.  S o l l i d , J . E., "Translation Displacement Versus Deformation i n Holographic Interferometry", Opt. Commun. 2 (6)  32.  Boone, P. M. and Verbiest, R., "Application of Hologram Interferometry to Plate Deformation and Translation Measurement", Opt. Acta. 16 (1969).  33.  Wilson, A. D., "In-plane Displacement of a Stressed Membrane with a Hole Measurement by Holographic Interferometry", Appl. Opt., 10, 908-912 (1971).  34.  Ennos, A. E., "Measurement of In-plane Surface Strain by Hologram Interferometry", J. of S c i e n t i f i c Instruments, (Journal of Physics E), Series 2, 1.  35.  Aleksandrov, E. B. and Bonch-Bruevich, A. M., "Investigation of Surface Strains by the Hologram Technique", Soviet Physics - Technical Physics 12 (2) (1967).  36.  Schumann, W. , "Some Aspects of the Optical Techniques for Strain Determination", Experimental Mechanics, 13 (8), 225-231 (1973).  37.  Hovanesian, J . D. and Hung, Y. Y., "Moire Contour-Sum Contour-Difference, and Vibration Analysis of Arbitrary Objects", Applied Optics, Vol. 10, No. 12, 1971.  38.  Meadows, D. M., et a l . "Generation of Surface Contours by Moire Patterns", Appl. Opt., 7, 942 (1970).  39.  Hildebrand, B. P. sources Holography 155 (1967). Brooks, R. E. and ference Patterns,  40.  Displacement  and Haines, K. A., "Multiple-wavelength and MultipleApplied to Contour Generation", J. Opt. Soc. Am., 57, Heflinger, L. 0., "Moire Gaging Using Optical InterAppl. Opt., 8, 935-939 (1969).  41.  Post, D., "New Optical Methods of Moire-fringe M u l t i p l i c a t i o n " , Experimental Mechanics, 8 (12), 63-68 (1968).  42.  Sciammarella, C. A., "Moire-fringe M u l t i p l i c a t i o n by Means of F i l t e r i n g and a Wave-front Reconstruction Process", Experimental Mechanics, 9 (4), 179-185 (1969).  43.  Dainty, J . C , "Laser Speckle and Related Phenomena", Springer-Verlag, New York, (1975).  44.  Fernelius, N. and Tome, C , "Vibration-Analysis Studies Using Changes of Laser Speckle", J . Of Opt. Soc. Am. Vol. 61, No. 5, 566-572, 1971.  45.  T i z i a n i , H. J . , "Application of Speckling for In-Plane Vibration Analysis", Optica Acta, 18, 1971.  174 46.  Archbold, E., Burch, J. M. and Ennos, A. E., "Recording of In-Plane Displacement by Double-Exposure Speckle Photography", Optica Acta, 17, 883-898 (1970).  47.  Groh, g., "Engineering Uses of Laser-produced Speckle Patterns", Strathclyde Symposium on the Engineering Uses of Holography, Cambridge University Press (1969).  48.  Archbold, A. E. Ennos: J. Strain Analysis 9, 10 (1974)  49.  Leendertz, J . A., "Interferometric Displacement Measurement on Scattering Surface U t i l i z i n g Speckle E f f e c t " , J. of Phys. E. Sc. Inst.,3 (1970)  50.  Monk, G. S., "Light-Principles and Experiments", Dover, New York,  51.  F. A. Jenkins and H. E. White., "Fundamentals of Optics", McGraw-Hill, New York, 1957.  52.  Stroke, G. W., "An Introduction to Coherent Optics and Holography", Academic Press, New York, 1969.  53.  Williams, C. E. and Becklund, 0. A., "Optics: A Short Course for Engineers and S c i e n t i s t " , Wiley-Interscience, New York, 1972.  54.  Yu, F. T. S., "Introduction to D i f f r a c t i o n , Information Processing, and Holography", The MIT Press, Massachusetts, 1973.  55.  Cathey, W. 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Hildebrand, F. B., "Advanced Calculus for Applications", Prentice-Hall, New Jersey, 1962, p. 7  1963.  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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