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Radiation from tensile fractures. 1962

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RADIATION FROM TENSILE FRACTURES by LALATENDU MANSINHA B.Sc, Indian Institute of Technology, 1957 M.Tech., Indian Institute of Technology, 1959 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE, DEGREE OF DOCTOR qF 'PHILOSOPHY in the Department of PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1962 In presenting t h i s thesis i n p a r t i a l f u lfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis f o r scholarly purposes may be granted by the Head of my Department or by his representatives. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of f HVS 1 C >̂ The University of B r i t i s h Columbia, Vancouver 8, Canada. Date 2 q ^ O O u W \Q 6? The University of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of LALATENDU MANSINHA B.Sc,, Indian In s t i t u t e of Technology, 1957 M.Tech., Indian In s t i t u t e of Technology, 1959 MONDAY, OCTOBER 29, 1962, at 3:30 P.M. IN ROOM 303, PHYSICS BUILDING COMMITTEE IN CHARGE Chairman: F.H. Soward M.A. Chinnery W.F. Slawson J.A. Jacobs R.W. Stewart D.L. Livesey E. Teghtsoonian J.C. Savage W.H. White External Examiner: Jean-Claude De Bremaecker Rice University RADIATION FROM TENSILE FRACTURES ABSTRACT The geographical d i s t r i b u t i o n of the sense of the f i r s t motion of P waves (and to a very limited extent, S waves) has been studied by seismologists to provide information on the focal mechanism of earthquakes. In this, thesis we investigate the.inverse problem: knowing the type and form of displacement at the focus at the focal instant, we study the azimuthal d i s t r i b u t i o n of the sense of f i r s t P and S motion, using model seismic technique. The source of e l a s t i c energy i s a thermally induced tensile fracture i n a glass plate. Two types of fractures have been studied: I n i t i a l ( b i l a t e r a l l y propagating) Fractures and Extended ( u n i l a t e r a l l y propagating) Fractures. The azimuthal d i s t r i b u t i o n of P and S wave amplitudes i s indicated. The experiments reported i n this thesis constitute a p a r t i a l test of a recent theory by Knopoff and Gilbert (1960) on f i r s t motions from seismic sources. The type of fracture studied corresponds to Case 3 of Knopoff and Gilbert. Our results show sig n i f i c a n t discrepancies with the theory. The sense of the measured f i r s t S motion i s opposite to that predicted by the theory, for both I n i t i a l and Extended Fractures. The ratios P e/P q o and S e/P Q d i f f e r s i g n i f i c a n t l y in magnitude from the theory i n many azimuths. I t i s suggested that the discrepancies are possibly due to the neglect in the theory of non-linear e l a s t i c effects near the t i p of the fracture. GRADUATE STUDIES F i e l d of Study: Seismology Introduction to Quantum Mechanics J. Grindlay Theory of Measurements J.R. Prescott Nuclear Physics J.B. Warren Advanced Geophysics J.A. Jacobs Radioactive and Isotopic Processes i n Geophysics R.D. Russell Modern Aspects of Geophysics J.A. Jacobs Related Studies: Applied Electromagnetic Theory Mineral Deposits G.B. Walker W.H. White - i i - ABSTRACT The g e o g r a p h i c a l d i s t r i b u t i o n o f the sense of the f i r s t motion of P waves (and to a very l i m i t e d e x t e n t , S waves) has been s t u d i e d by s e i s m o l o g i s t s to provide i n f o r m a t i o n on the f o c a l mechanism of earthquakes. I n t h i s t h e s i s we i n v e s t i g a t e the i n v e r s e problem; knowing the type and form o f displacement a t the focus a t the f o c a l i n s t a n t , we study the a z i m u t h a l d i s t r i b u t i o n of the sense of f i r s t P and S motion, u s i n g model s e i s m i c technique. The source of e l a s t i c energy i s a t h e r m a l l y induced t e n s i l e f r a c t u r e i n a g l a s s p l a t e . Two types of f r a c t u r e s have been s t u d i e d : I n i t i a l ( B i l a t e r a l l y propagating) F r a c t u r e s and Extended ( U n i l a t e r a l l y propagating) F r a c t u r e s . The a z i m u t h a l d i s t r i - b u t i o n o f the P and S wave amplitudes i s i n d i c a t e d * The experiments r e p o r t e d i n t h i s t h e s i s c o n s t i t u t e a p a r t i a l t e s t of a r e c e n t theory by Knopoff and G i l b e r t (1960) on f i r s t motions from s e i s m i c sources. The type of f r a c t u r e s t u d i e d corresponds to Case 3 of Knopoff and G i l b e r t . Our r e s u l t s show s i g n i f i c a n t d i s c r e p a n c i e s w i t h the t h e o r y . The sense of the measured f i r s t S motion i s opposite to t h a t p r e d i c t e d by the theory, f o r both I n i t i a l and Extended F r a c - t u r e s . The r a t i o s P Q / P g 0 and S @ / P Q d i f f e r i n magnitude from the theory i n many azimuths. I t i s suggested t h a t the d i s c r e p a n c i e s are p o s s i b l y due to the n e g l e c t i n the theory o f n o n - l i n e a r e l a s t i c e f f e c t s near the t i p o f the f r a c t u r e . TABLE OF CONTENTS Page ABSTRACT i i ACKNOWLEDGMENT i l l LIST OP FIGURES i v LIST OF SYMBOLS v i LIST OF TABLES v i i CHAPTER I - INTRODUCTION 1.1 General 1 1.2 Pr e v i o u s Works 2 1.3 Method o f I n v e s t i g a t i o n 7 CHAPTER I I - FRACTURE PHENOMENON 2.1 D e f i n i t i o n 10 2.2 V e l o c i t y C o n s i d e r a t i o n 11 2.3 C r i t e r i o n f o r Crack I n s t a b i l i t y 14 2.4 S t r e s s D i s t r i b u t i o n Around a S t a t i c Crack .. 15 2.5 S t r e s s e s Around a Propa g a t i n g F r a c t u r e 19 2.6 R a d i a t i o n from a Propagating F r a c t u r e 23 CHAPTER I I I - INSTRUMENTATION AND EXPERIMENTAL TECHNIQUE 3.1 S c a l i n g C o n s i d e r a t i o n s 30 3.2 Method of Inducing F r a c t u r e 31 3.3 E l e c t r o n i c D i s p l a y and Recording System .... 36 3.3.1 Ceramic Transducer 37 3.3.2 F i l t e r System 39 Page 3.3.3 A m p l i f i e r System 40 3.3.4 D i s p l a y and Recording 41 3.3.5 T r i g g e r i n g System 41 3.4 M e c h a n i c a l Arrangement 42 CHAPTER IV - PROCEDURE 4.1 C a l i b r a t i o n o f De t e c t o r s 44 4.2 V e l o c i t y Measurements 46 4.3 I d e n t i f i c a t i o n of P and S waves 46 4.4 Determination o f the R a d i a t i o n P a t t e r n 47 4.5 Dete r m i n a t i o n o f the F a r - F i e l d Region 50 4.6 Symmetry of I n i t i a l F r a c t u r e s 52 CHAPTER V - RESULTS 5.1 General 54 5.2 I n i t i a l F r a c t u r e s 57 5.3 Extended F r a c t u r e s 58 CHAPTER VI - DISCUSSION 6.1 General 60 6.2 High Frequency 61 6.3 D i f f r a c t i o n E f f e c t s 62 6.4 Correspondence of F r a c t u r e s to Other D i s l o c a t i o n Models 63 6.5 V e l o c i t y o f F r a c t u r e Propagation 64 6.6 Non-Linear E f f e c t s 65 6.7 A c t u a l Correspondence of F r a c t u r e i n P l a t e to F r a c t u r e i n Three Dimensions .... 66 CHAPTER V I I - CONCLUSIONS 70 Page FIGURES 1 - 28 72 TABLES I - VII 91 APPENDIX I 97 APPENDIX I I 101 BIBLIOGRAPHY 108 - i i i - ACKNOWLEDGEMENT I t i s a p l e a s u r e to acknowledge the help I have r e c e i v e d from Dr. James C. Savage. Dr. Savage suggested that I i n v e s t i g a t e problems of e l a s t i c i t y and seismology w i t h model s e i s m i c technique, and a f t e r t h a t devoted a c o n s i d e r a b l e amount of h i s own time i n d i s c u s s i n g i d e a s and performing experiments. I have l e a r n e d a l o t from working w i t h htm, and I take t h i s opportunity- to express my indebtedness to him. I a l s o thank Dr. R. D. R u s s e l l , who ac t e d as my s u p e r v i s o r d u r i n g Dr. Savage's absence, and Dr. J . A. Jacobs f o r h i s constant i n t e r e s t and encouragement d u r i n g the exp e r i m e n t a l work. This work was f i n a n c i a l l y supported by g r a n t s from the American Petroleum I n s t i t u t e . - i v - LIST OF FIGURES Page F i g . 1. Method of a p p l y i n g gas flame t o g l a s s p l a t e . 72 F i g . 2. I n f i n i t e l y l o n g f r a c t u r e i n t h r e e - d i m e n s i o n a l body 72 F i g . 3. I n f i n i t e s i m a l p a r t o f f r a c t u r e that c o n t r i b u t e s to f i r s t motion a t any p o i n t on the X - Z plane 72 F i g . 4. Coordinate system and d i r e c t i o n of p o s i t i v e u n i t v e c t o r s , 73 F i g . 5. D i s t r i b u t i o n of 6£ w i t h © ( a f t e r J o f f e , 1951) : 74 F i g . 6. F i g u r e shows volume V bounded by s u r f a c e s S'. 74 F i g . 7. D i s t r i b u t i o n of temperature w i t h r a d i a l d i s t a n c e on a g l a s s p l a t e 75 F i g . 8. S t r e s s d i s t r i b u t i o n i n the g l a s s p l a t e because of the temperature d i s t r i b u t i o n shown i n f i g u r e 7 76 F i g . 9. Method of i n d u c i n g I n i t i a l F r a c t u r e s 77 F i g . 10. Method of extending s h o r t f r a c t u r e s 77 F i g . 11. Method of extending l o n g f r a c t u r e s 77 F i g . 12. Schematic c i r c u i t f o r measuring P and S wave v e l o c i t i e s i n g l a s s p l a t e s 78 F i g . 13. Schematic c i r c u i t f o r s t u d y i n g r a d i a t i o n from f r a c t u r e s i n g l a s s p l a t e s 79 F i g . 14. Transducer and f r a c t u r e p o s i t i o n s f o r de t e r m i n a t i o n of P Q /PgQ r a t i o 80 F i g . 15. Transducer and f r a c t u r e p o s i t i o n s f o r d e t e r m i n a t i o n of Sg / P ^ r a t i o 80 F i g , 16. Transducer and f r a c t u r e p o s i t i o n s f o r the d e t e r m i n a t i o n o f the f a r - f i e l d r e g i o n 81 - V - Page jTlg# 17. A t t e n u a t i o n o f P waves a t 90° from I n i t i a l f r a c t u r e s 81 F i g . 18. A t t e n u a t i o n o f P waves a t 0° from I n i t i a l f r a c t u r e s 82 F i g . 19. T y p i c a l r e c o r d f o r the d e t e r m i n a t i o n o f p $ / p90 r a t i o f r o m I n i t i a l F r a c t u r e s 85 F i g . 20. T y p i c a l r e c o r d f o r SQ /P@ r a t i o from I n i t i a l F r a c t u r e 85 F i g . 21. T y p i c a l r e c o r d f o r P Q / P Q Q r a t i o from Extended F r a c t u r e 86 F i g . 22. Record showing d i f f i c u l t y of i d e n t i f y i n g S waves from Extended F r a c t u r e s i n the r e a r quadrant 86 F i g . 23. P l o t of measured P Q / P q n r a t i o s from I n i t i a l F r a c t u r e s .V 87 F i g . 24. P l o t of measured S0 /P© r a t i o s from I n i t i a l F r a c t u r e s 88 F i g . 25. P l o t o f measured P e/PqQ r a t i o s from Extended F r a c t u r e s 87 F i g . 26. P l o t o f measured S@/Pg r a t i o s from Extended F r a c t u r e s 88 F i g . 27. F i g u r e showing change o f Q w i t h p r o p a g a t i o n o f the f r a c t u r e 89 F i g . 28. F i g u r e shows c o n t r i b u t i n g segment of i n f i n i t e f r a c t u r e f r o n t 90 - v i - LIST OF SYMBOLS Fo r convenience a p a r t i a l l i s t o f symbols f r e q u e n t l y used i n the t e x t i s g i v e n here. Other symbols are d e f i n e d wherever they are i n t r o d u c e d . oC L o n g i t u d i n a l wave v e l o c i t y . |3> Transverse wave v e l o c i t y . 7^ D i r e c t i o n c o s i n e . ^ V e l o c i t y o f Pr o p a g a t i o n o f f r a c t u r e , ( r , 0 ) Plane p o l a r c o o r d i n a t e s . Pg (A) P wave recorded a t ( r , 6 ) w i t h transducer A. P Q (r) P wave amplitudes a t angle & , f o r d i f f e r e n t r . E P § P (R, 9 ) from Extended F r a c t u r e s . I P Q P (R, 6 j from I n i t i a l F r a c t u r e s . P@ / P g o R a t i o o f P amplitude a t (R, 6 ) to P amplitude a t (R, 90°). SQ/TQ R a t i o o f S amplitude a t (R, 6 ) to P amplitude a t (R, 9 ) . S Q (B) S wave recorded a t ( r , 9 ) w i t h transducer B. - v i i - LIST OF TABLES Page I . Table showing y , d and f f o r d i f f e r e n t 0 . 91 I I . P 9 / P G O R a t i o f o r I n i t i a l F r a c t u r e s 92 I I I . S @ / P @ R a t i o s f o r I n i t i a l F r a c t u r e s 93 IV. P Q / P 9 0 R a t i o f o r Extended F r a c t u r e s 94 V. s e / p 0 R a t i o s f o r Extended F r a c t u r e s 95 V I . C a l c u l a t e d S Q / P g o R a t i o s f o r I n i t i a l F r a c t u r e s 96 V I I . C a l c u l a t e d S Q / P 9 q R a t i o s f o r Extended F r a c t u r e s 96 - 1 - CHAPTER I INTRODUCTION 1.1 G e n e r a l . A r e c e n t paper by Knopoff and G i l b e r t (1960) has p r e d i c t e d the r a d i a t i o n p a t t e r n of e l a s t i c waves a s s o c i a t e d w i t h f r a c t u r e s of v a r i o u s types. T h i s theory takes as a model o f a f r a c t u r e a d i s l o c a t i o n i n a c e r t a i n component o f displacement or s t r a i n ; l i n e a r e l a s t i c i t y t heory i s then used to p r e d i c t the e a r l i e s t l o n g i t u d i n a l and t r a n s - verse motions a s s o c i a t e d w i t h the d i s l o c a t i o n . The work r e p o r t e d here c o n s i s t s of an experimental t e s t of t h i s theory, as w e l l as a g e n e r a l experimental study o f e l a s t i c waves from f r a c t u r e s of c e r t a i n types. The p r i n c i p a l i n t e r e s t i n the study o f the r a d i a t i o n of e l a s t i c waves from a f r a c t u r e i s to a p p l y the r e s u l t s to the i n v e r s e problem o f i n f e r r i n g the p r o p e r t i e s of the f r a c t u r e from the r a d i a t i o n p a t t e r n . T h i s problem i s of p a r t i c u l a r importance i n geophysics where the d i r e c t i o n and nature of f r a c t u r e i n earthquakes i s not adequately known; indeed, i t i s p o s s i b l e that f r a c t u r e does not p l a y a r o l e i n some earthquakes (e.g., are deep f o c u s earthquakes the r e s u l t o f a d i s l o c a t i o n i n s t r a i n r a t h e r than i n d i s p l a c e m e n t ? ) . Other questions of i n t e r e s t concern the p a r t i t i o n i n g of energy between l o n g i t u d i n a l - 2 - and t r a n s v e r s e waves and the q u e s t i o n of whether f r a c t u r e tends to propagate i n ,a s i n g l e d i r e c t i o n ( u n i l a t e r a l f r a c t u r e ) o r s i m u l t a n e o u s l y i n two op p o s i t e d i r e c t i o n s ( " b i l a t e r a l f r a c t u r e ) . The experiments r e p o r t e d here have a b e a r i n g on a l l of these q u e s t i o n s . 1,2 P r e v i o u s Works. The l i t e r a t u r e on s e i s m i c sources i s not exten- s i v e . Kasahara (1952 to 1955) p u b l i s h e d a s e r i e s o f papers on h i s experimental s t u d i e s on the g e n e r a t i o n o f e l a s t i c waves, u s i n g w i d e l y d i f f e r e n t experimental techniques. I n the f i r s t paper, he used the t r a n s d u c i n g p a r t s o f a loudspeaker as a source on a b l o c k of agar-agar (72 x 33 x 13 cms). Surface motions a t v a r i o u s d i s t a n c e s from the source were s t u d i e d w i t h a v a r i a b l e c a p a c i t o r type of d e t e c t o r . L o n g i t u d i n a l , t r a n s v e r s e and s u r f a c e waves were i d e n t i f i e d . He a l s o v e r i f i e d a r e s u l t o f Lapwood, that a p u r e l y downward motion a t a p o i n t on the s u r f a c e i s accompanied by upward motion i n the immediate ar e a surrounding the p o i n t . I n the f i f t h paper of t h i s s e r i e s , Kasahara simulates an e x p l o s i v e source. A c y l i n d r i c a l c a v i t y l i n e d w i t h t h i n rubber f i l m was embedded i n a b l o c k of agar-agar 90 x 30 x 30 cms i n dimensions. The dimensions of the c a v i t y were 0.6 cms i n r a d i u s and 20 cms i n l e n g t h . E l a s t i c - 3 - waves were e x c i t e d by an i m p u l s i v e i n c r e a s e of pressure i n the c a v i t y w i t h an a i r pump. Motion a t the e p i c e n t e r , and a t v a r i o u s p o i n t s away from i t , were s t u d i e d . Only l o n g i t u d i n a l and s u r f a c e waves were i d e n t i f i e d . The form, as w e l l as the amplitude, o f the displacement changed as one moved away from the e p i c e n t e r . Kasahara and a number of other i n v e s t i g a t o r s (Howes e t a l (1953) , T a t e l (1954), Evans et a l (1954), O l i v e r et a l (1954), Press et a l (1954), Knopoff (1955), L e v i n et a l (1955), O'Brien (1955), Knopoff (1956), Kato et a l (1956)), have p e r f e c t e d the theory and technique of s e i s m i c m o d e l l i n g . Most of the e a r l y works were on thtfee d i m ensional models, which a re cumbersome and expen- s i v e to use. O l i v e r , P r e s s and Ewing (1954) were the f i r s t to propose and use two dimensional models to study s e i s m i c phenomena. They e s t a b l i s h e d the correspondence between c e r t a i n p l a t e v i b r a t i o n modes and body waves. The group v e l o c i t i e s of symmetric and a n t i - symmetric p l a t e waves have a pronounced dependance on kh, where k i s the wave number and h the t h i c k n e s s of the p l a t e (see appendix I I ) . F o r wavelengths long i n comparison w i t h the p l a t e t h i c k n e s s , kh i s s m a l l and the d i s p e r s i o n of the f i r s t symmetric mode i s s m a l l . I t has been e s t a b l i s h e d (appendix I I ) that the f i r s t symmetric mode i n the two dim e n s i o n a l model corresponds to l o n g i t u d i n a l - 4 - body waves i n three dimensions. F o r s m a l l k'h the v e l o c i t y of the asymmetrical ( f l e x u r a l ) mode i s much l e s s than symmetric or shear modes and thus does not i n t e r f e r e w i t h these modes i n seismic m o d e l l i n g . Shear waves w i t h displacement i n the plane of the two dimensional model show no d i s p e r s i o n and have v e l o c i t y independent o f p l a t e t h i c k n e s s . O l i v e r , P r e s s and Ewing used a p l a t e t h i c k - ness of 1/16 i n c h and a frequency o f the order o f I O 2 k c / s . The f i r s t attempt to study the e l a s t i c wave r a d i a t i o n from non-propagating f a u l t s i n two-dimensional models was by Press (1956, 1957) and independently by Kato and Takagi (1957). A h i g h v o l t a g e , s h o r t d u r a t i o n p u l s e a p p l i e d to a p i e z o - e l e c t r i c bimorph transducer produces a s h o r t d u r a t i o n mechanical displacement. T h i s s i m u l a t e s a s i n g l e t source. A p r o p e r l y o r i e n t e d combination of such bimorph tran s d u c e r s can simulate any f o r c e or displacement systems a c t i n g at the focus a t the f o c a l I n s t a n t . At the ce n t e r of a t h i n c i r c u l a r sheet combinations o f benders served to simulate s i n g l e t , doublet and quadruplet sources. Two o p p o s i t e l y phased benders, separated by a s h o r t d i s t a n c e , form a do u b l e t source; f o u r form a quadruplet source. By i n t e r p o s i n g a s l i t between the doublet benders, Press was able to simu- l a t e a f a u l t . With s i l i c o n e as a l u b r i c a n t he was a b l e to study a l u b r i c a t e d f a u l t . The arms of the doublet - 5 - p o i n t e d i n the 0° and 180° d i r e c t i o n s , r e p r e s e n t i n g a f a u l t o r i e n t e d i n the same d i r e c t i o n . The P wave d i s t r i b u t i o n p a t t e r n had v a n i s h i n g amplitudes a t 0° and 90°, both w i t h and without s l i t s . But the S wave p a t t e r n changed s i g n i f i c a n t l y i n the presence o f a s l i t . With no s l i t , S amplitude i s a t a minimum a t 0 ° , i n c r e a s i n g to a maximum a t 90°. I n the presence of a s l i t , the S wave n u l l i s s h i f t e d 25° to 30° to the s i d e from t h a t of a simple d i p o l e . The sense of shear motion i s c l o c k w i s e , i n c o n t r a s t to the c o u n t e r c l o c k w i s e motion i n the 30° to 90° zone. L a t e r work by Healy and Press (1959) suggests t h a t the anomalous shear d i s t r i b u t i o n i s due t o R a y l e i g h waves t r a v e l l i n g along the s l i t face and being converted at the t e r m i n a t i o n i n t o shear waves. T h i s i s supported by the work o f J o f f e (1951), who shows that a system of s u r f a c e waves (not n e c e s s a r i l y R a y l e i g h waves) can be found which s a t i s f y the boundary c o n d i t i o n s of a moving G r i f f i t h c r a c k . P r e s s , Ben-Menahem and Toksoz (1961) r e p o r t e d a method to r e c o v e r two parameters o f the earthquake focus from the f a u l t plane s o l u t i o n . The methods are based on the a n a l y s i s o f f r e e o s c i l l a t i o n of the e a r t h and o f l o n g p e r i o d mantle R a y l e i g h waves and G waves. The t h e o r e t i c a l b a s i s f o r the method has been d e s c r i b e d by B e n i o f f , Press and Smith (1961), and Ben-Menahem (1961). Free s p h e r o i d a l o s c i l l a t i o n s o f the e a r t h f o r h i g h mode numbers are - 6 - e q u i v a l e n t to s t a n d i n g wave i n t e r f e r e n c e p a t t e r n s o f R a y l e i g h waves pro p a g a t i n g i n opposite d i r e c t i o n s . The phase s h i f t between v e r t i c a l and h o r i z o n t a l displacement f o r p r o pagating waves i s 90°, and f o r s t a n d i n g waves 180°. F o r a p o i n t source e x c i t a t i o n , R a y l e i g h waves are e x c i t e d e q u a l l y i n a l l d i r e c t i o n s , and phase s h i f t s of 0° or 180° are produced. I f the energy i s r a d i a t e d i n o n l y one d i r e c t i o n , the phase s h i f t w i l l be 90°. For cases where r a d i a t i o n i n o p p o s i t e d i r e c t i o n i s unequal, the phase s h i f t w i l l assume some in t e r m e d i a t e v a l u e . T h i s i s the case f o r f a u l t i n g w i t h a f i n i t e r u pture v e l o c i t y . To t e s t t h i s theory, P r e s s , Ben-Menahem and Toksoz used a c i r c u l a r aluminum p l a t e 12" i n r a d i u s and 1/16" i n t h i c k n e s s . A s i n u s o i d a l f u n c t i o n was used to e x c i t e the ceramic source t r a n s d u c e r . The frequency was v a r i e d u n t i l resonance o c c u r r e d . Nodes were counted by r e c e i v e r a t the r i m of the p l a t e . The r e c i p r o c i t y theorem permits the use of a moving r e c e i v e r Instead o f a moving source. The source was kept f i x e d w h i l e the r e c e i v e r was moved i n d i s c r e t e i n t e r v a l s of 2 ° . The s o l u t i o n s were superimposed a f t e r time s h i f t i n g by an amount equal to the t r a n s l a t i o n time o f the source f o r a 2° i n t e r v a l . Such a composite r e c o r d i s approximately e q u i v a l e n t to a f i n i t e moving source. A more p r e c i s e method i s mentioned by P r e s s , Ben- - 7 - Menahem and Toksoz, but to date no r e s u l t s have been p u b l i s h e d . A moving source i s simulated by a s e r i e s of s t a t i c ceramic source t r a n s d u c e r s p u l s e d i n sequence, w i t h s u i t a b l e time d e l a y s between p u l s i n g of adjacent t r a n s - ducers. The net impression i s a source moving i n d i s c r e t e s teps w i t h an e q u i v a l e n t v e l o c i t y to a c o n t i n u o u s l y moving source. 1.3 Method of I n v e s t i g a t i o n . Thermal s t r e s s e s are s e t up i n a g l a s s p l a t e by a p p l y i n g an open flame to the s i d e o f the p l a t e a t a p o i n t about l-ir cm from, and a l o n g the e x t e n s i o n o f , a 1/2 cm l i n e a r s c r a t c h p r e v i o u s l y i n s c r i b e d on the p l a t e (see f i g . l ) . When s u f f i c i e n t t e n s i l e s t r e s s b u i l d s up a c r o s s the s c r a t c h , a f r a c t u r e i n i t i a t e s at the s c r a t c h and propagates a l o n g i t s p r o l o n g a t i o n . The sudden displacement a s s o c i a t e d w i t h the f r a c t u r e g i v e s r i s e to symmetric p l a t e waves (here- a f t e r termed P', to d i s t i n g u i s h from P waves i n three dimensions) and t r a n s v e r s e S waves. I t should be emphasized here that the experiment i s by no means a d i r e c t analogue of the problem o f an i n i t i a t e d f r a c t u r e p r opagating i n a three d i m e n s i o n a l body. To e s t a b l i s h the co n n e c t i o n between the two we w i l l have to c o n s i d e r the f o l l o w i n g a n a l o g i e s . F i r s t , we note that the waves i n the t h i n p l a t e are r e l a t e d to waves i n a three dimensional body s u b j e c t to the c o n s t r a i n t o f - 8 - plane s t r a i n . ( i . e . , v a n i s h i n g components of s t r a i n a c r o s s the s i d e s o f the p l a t e ) . We note that i n e i t h e r case the s i g n i f i c a n t motions are i n the plane o f the p l a t e . I n the un c o n s t r a i n e d p l a t e , however, s m a l l motions p e r p e n d i o u l a r to the p l a t e do ocour i n the symmetrlo p l a t e wave. These motions permit a v a r i a t i o n o f volume per u n i t a r e a o f the p l a t e , whereas a l l v a r i a t i o n s of volume per u n i t area o f the c o n s t r a i n e d p l a t e come about through compression o f matter. Thus the u n c o n s t r a i n e d p l a t e may be thought o f as a c o n s t r a i n e d p l a t e having an u n u s u a l l y h i g h compressi- b i l i t y . T h i s analogy has been f o r m a l l y e s t a b l i s h e d by O l i v e r , P r e s s and Ewing (1954). The second stage o f the argument i n v o l v e s the J u x t a p o s i t i o n of a very l a r g e number of the c o n s t r a i n e d p l a t e s to form a s o l i d three dimensional body i n plane s t r a i n . In order t o be d e f i n i t e l e t us choose a s e t of c a r t e s i a n axes o r i e n t e d so that a l l motion i s p a r a l l e l to the X-Z pla n e . We imagine that the f r a c t u r e propagates i n the X d i r e c t i o n . The three dimensional analogue of a f r a c t u r e i n a p l a t e i s then a f r a c t u r e whloh I n i t i a t e s s i m u l t a n e o u s l y a t every p o i n t on the Y a x i s and propagates i n the X d i r e c t i o n so t h a t every s e o t l o n p a r a l l e l to the X-Z plane I s a t a l l times i d e n t i c a l (see f i g . 2 ) . On the other hand the theory o f Knopoff and G i l b e r t (1960) c o n s i d e r s a f r a c t u r e which has onl y an i n f i n i t e s i m a l l e n g t h along the Y a x i s and propagates along - 9 - the X a x i s ( f i g . 3 ) . I t i s the l a t t e r problem which we w i s h to model. T h i s l e a d s us to the t h i r d and f i n a l s t e p i n our argument. We w i l l r e s t r i c t o u r s e l v e s to observa- t i o n s i n the X-Z plane (see f i g . 3) and c o n s i d e r o n l y the f i r s t motions o f the l o n g i t u d i n a l and t r a n s v e r s e waves. We see from f i g u r e 2 that o n l y an i n f i n i t e s i m a l l e n g t h of the f r a c t u r e centered upon the o r i g i n may c o n t r i b u t e to the f i r s t motions observed on the X-Z plane, the d i s t u r b a n c e from the remainder o f the p o i n t s on the i n f i n i t e f r a c t u r e not having had time to reach the p o i n t of o b s e r v a t i o n . (There w i l l be, to be sure, a d i f f e r e n c e i n the geometric a t t e n u a t i o n of the wave i n the two cases, the amplitude being p r o p o r t i o n a l to r 8 i n our experiments and propor- t i o n a l to r " 1 i n the t h e o r y . T h i s d i f f e r e n c e need not concern us here.) On the b a s i s of these arguments we may conclude that the experiments d e s c r i b e d i n t h i s t h e s i s should g i v e the c o r r e c t r a d i a t i o n p a t t e r n i n the X-Z plane f o r the f r a c t u r e d e p i c t e d i n f i g u r e 2. I t i s t h e r e f o r e an experimental t e s t of the theory of Knopoff and G i l b e r t . - 10 - CHAPTER I I FRACTURE PHENOMENON 2.1 D e f i n i t i o n . We d e f i n e f r a c t u r e as a process by which a f r e e s u r f a c e i s c r e a t e d i n the i n t e r i o r o f an e l a s t i c medium under the a c t i o n o f an i n t e r n a l or e x t e r n a l s t r e s s d i s t r i b u t i o n . The f r e e s u r f a c e may expand a t a f i n i t e v e l o c i t y ; t h i s v e l o c i t y i s presumably l e s s than the l o n g i t u d i n a l or t r a n s v e r s e wave v e l o c i t i e s i n the medium. I n g e n e r a l the f r e e s u r f a c e w i l l enclose a penny shaped volume. As the two plane bounding s u r f a c e s o f the penny shaped volume approach each o t h e r , the volume tends to zer o , and i t i s p o s s i b l e to speak of a f r a c t u r e p l a n e . We assume that there i s no r e l a t i v e motion of p a r t i c l e s immediately adjacent t o , and on both s i d e s of the f r a c t u r e plane b e f o r e the passage of the f r a c t u r e f r o n t . The passage o f the f r a c t u r e f r o n t then corresponds to a suddenly a p p l i e d d i f f e r e n c e i n displacement or s t r a i n a c r o s s the f r a c t u r e p l a n e . I f we assume a step f u n c t i o n time depend- ence, a f r a c t u r e would correspond to a pro p a g a t i n g step f u n c t i o n i n the d i f f e r e n c e of displacement and/or s t r a i n a c r o s s the f r a c t u r e p l a n e . I t f o l l o w s that i n an i d e a l f r a c t u r e o f the type d e s c r i b e d above, a l l e l a s t i c wave r a d i a t i o n i s due to a c t i o n a t the f r a c t u r e f r o n t . - 11 - In l i t e r a t u r e the terms crack and f r a c t u r e have been used as in t e r c h a n g e a b l e terms to d e s c r i b e the e x i s t e n c e of f r e e s u r f a c e i n the medium. I n t h i s t h e s i s we use the term ' f r a c t u r e ' and 'crack' f o r s t a t i c case and 'propagating f r a c t u r e ' f o r the dynamic case. When r e f e r r i n g to works of other authors i n the f i e l d t h e i r o r i g i n a l terminology has been maintained. 2.2 V e l o c i t y C o n s i d e r a t i o n s . l o f f e (1951) has shown that the s t r e s s d i s t r i b u - t i o n around the head of a pro p a g a t i n g f r a c t u r e can be determined by c o n s i d e r i n g the f r a c t u r e as a system of moving s u r f a c e d i s t u r b a n c e s . The s t r e s s d i s t r i b u t i o n v a r i e s w i t h v e l o c i t y . For a s t a t i o n a r y f r a c t u r e (see f i g . 4 f o r c o o r d i n a t e system) i s a maximum some w Ir-R d i s t a n c e ahead of the crack i n the d i r e c t i o n o f propaga- t i o n ( 0 - 0 ° ). I t i s due to t h i s maxima of t e n s i l e s t r e s s t h a t the f r a c t u r e propagates ahead i n a s t r a i g h t l i n e . With i n c r e a s i n g v e l o c i t y the v a r i a t i o n o f 6~e Co©) 8 decreases u n t i l a t about 0.6 o f the t r a n s v e r s e wave v e l o c i t y <5~Q i s independent o f 0 . I t i s expected t h a t a t t h i s v e l o c i t y the f r a c t u r e w i l l have no p r e f e r r e d d i r e c t i o n o f pr o p a g a t i o n and w i l l form branches, l e a d i n g to a d i v i s i o n o f a v a i l a b l e energy o f f r a c t u r e f o r m a t i o n . With f o r m a t i o n o f branches the v e l o c i t y should continue to be - 12 - maintained a t l e s s than 0 .6 (3 . Therefore we can c o n s i d e r 0.6 p as the l i m i t i n g or maximum v e l o c i t y o f pro p a g a t i o n of a t e n s i l e f r a c t u r e . There has been no other t h e o r e t i c a l work on other types of propagating d i s c o n t i n u i t i e s . However, from i n t u i t i v e r e a s o n i n g one cannot see a method by which a f r a c t u r e can propagate a t v e l o c i t i e s h i g h e r than the l o n g i - t u d i n a l wave v e l o c i t y . P o i n t s away from the f r a c t u r e cannot know of the e x i s t e n c e of the f r a c t u r e before a time equal to the t r a v e l time of the l o n g i t u d i n a l wave e l a p s e s . Cases have been observed i n which f r a c t u r e s have t r a v e l l e d a t apparent v e l o c i t i e s i n excess o f the l o n g i t u d i n a l wave v e l o c i t y . Presumably the e x p l a n a t i o n i n such cases i s tha t the h i g h t e n s i l e s t r e s s has caused the f r a o t u r e to o r i g i n a t e a t more than one nu c l e u s , r e s u l t i n g i n a h i g h apparent v e l o c i t y . S c h a r d i n and S t r u t h (1937) have e x p e r i m e n t a l l y v e r i f i e d t h at the maximum t e n s i l e f r a c t u r e v e l o c i t y i n g l a s s i s 1500 m/sec, about 0 .5 of the tr a n s v e r s e wave v e l o c i t y . T h i s maximum v e l o c i t y i s p r a c t i c a l l y independent o f temperature and o f the o v e r a l l s t r e s s at the moment of f r a c t u r e ( Schardin, 1959) . Glass p l a t e s w i t h h i g h i n t e r - nal s t r e s s e s gave i d e n t i c a l maximum v e l o c i t y as those from which s t r e s s e s have been removed. Measurement o f f r a c t u r e v e l o c i t y i n p l a t e s under h i g h compressive loads a l s o gave - 13 - no v e l o c i t y v a r i a t i o n w i t h l o a d . We can thus e s t a b l i s h an e m p i r i c a l r e l a t i o n between , the maximum f r a c t u r e v e l o c i t y and the shear wave v e l o c i t y . T h i s i s 1n -- 0-6/J = o-6 JjjT From the f o r e g o i n g i t would appear t h a t a f r a c t u r e may propagate a t any v e l o c i t y between 0 and 0.6 p> • During the experiments d e s c r i b e d i n t h i s t h e s i s i t was not uncommon to see f r a c t u r e creep forward a t v e l o c i t i e s l e s s than 1 mm/sec. However, there i s evidence t h a t t h i s may o n l y be an apparent v e l o c i t y , being made up o f i n t e r m i t t e n t f r a c t u r e s each o f which advanced o n l y a s h o r t d i s t a n c e but a t the f u l l v e l o c i t y o f 0.6 j3 . The major evidence support- i n g t h i s view i s the o b s e r v a t i o n o f S c h a r d i n (1959) t h a t no f r a c t u r e has been observed t o move a t a lower v e l o c i t y once i t a t t a i n s the maximum v a l u e . A f r a c t u r e may stop suddenly a f t e r moving a t a v e l o c i t y o f 0.6 (3> , but w i t h i n the l i m i t s o f r e s o l u t i o n o f measuring instruments (1 mm, 1 ̂  sec) a t r a n s i t i o n to a lower v e l o c i t y c ould not be observed. I t i s of some importance to note t h a t the f r a c t u r e v e l o c i t y does not depend on the t h i c k n e s s o f the p l a t e b e i n g used. This i s a l s o seen from the e m p i r i c a l r e l a t i o n between $^ and |2> . The t r a n s v e r s e wave v e l o c i t y |3 i s - 14 - Independent of p l a t e t h i c k n e s s . 2.3 C r i t e r i o n f o r Crack I n s t a b i l i t y . To e x p l a i n the d i s c r e p a n c y between the t h e o r e t i c a l estimate o f s t r e n g t h of s o l i d s and actual" measurements G r i f f i t h (1920, 1922) suggested t h a t the d i f f e r e n c e might be due to the presence of l a r g e number of s m a l l c r a c k s i n s o l i d s . The cracks c o u l d c o n c e n t r a t e the s t r e s s l o c a l l y such that the t h e o r e t i c a l s t r e n g t h might be l o c a l l y exceeded a t a comparatively low mean s t r e s s . I t i s p o s s i b l e to c a l c u l a t e the s t r a i n energy due to the presence of a crack i n a p l a t e . The r e l i e f of s t r a i n i s opposed by the energy r e q u i r e d to c r e a t e two new s u r f a c e s . I t was proposed by G r i f f i t h t h a t the crack would spread o n l y i f the decrease i n s t r a i n i s g r e a t e r than the i n c r e a s e i n s u r f a c e energy. about an e l l i p t i c a l hole i n a s t r e s s e d p l a t e . He suggested that the case i n which the e c c e n t r i c i t y approaches u n i t y (very f l a t e l l i p s e ) the e l l i p t i c a l hole should be a good model f o r a c r a c k . The change i n s t r a i n energy i n a l a r g e t h i n p l a t e due to the presence o f an e l l i p t i c a l c rack i s g i v e n by I n g l i s (1913) c a l c u l a t e d the s t r e s s d i s t r i b u t i o n ( 3 - P ) * c V where T = t e n s i o n W = - (2-1) 2c = l e n g t h of crack ^ = Po i s s o n ' s r a t i o . ^if*- s Lame's constants - 15 - I n the above e x p r e s s i o n i t i s assumed t h a t the c r a c k i s v e r y narrow; i . e . the angle between the two s i d e s o f the crack i s 0 ° . As the c r a c k propagates s u r f a c e energy i s c r e a t e d . P o t e n t i a l energy of the s u r f a c e of the crack, per u n i t t h i c k n e s s o f the p l a t e i s R = 4cS where S = s u r f a c e t e n s i o n . The t o t a l decrease of the p o t e n t i a l energy of the system due to the presence of the c r a c k i s g i v e n by W — R = - 4 C 5 (2-2) G r i f f i t h ' s c o n d i t i o n that the crack may extend i s g i v e n by or ( 3 - ! = > ) A C T 2 r 1 6 / A S 2.4 S t r e s s D i s t r i b u t i o n Around a S t a t i c Crack. A treatment of the s t a t i c s t r e s s d i s t r i b u t i o n around a s t a t i o n a r y c r a c k was done by Westergaard (1939) u s i n g a complex s t r e s s f u n c t i o n . T h i s technique was l a t e r e x p l o i t e d by W i l l i a m s (1956) i n a more complete a n a l y s i s - 16 - o f the problem. We g i v e below a b r i e f account of the method of W i l l i a m s . The s o l u t i o n o f the equations V 4" F - O (2-4) w i l l g i v e the s t r e s s f u n c t i o n of the type -I- C 3 s \ n ( > - 0 > + C«. cos ( * - 0 > ] ( 2 - 5 ) The above s t r e s s f u n c t i o n w i l l s a t i s f y c o n d i t i o n s o f s t r e s s f r e e edges a t ^ -.0 and ^ = oi i f the A are chosen as the p o s i t i v e r o o t s of S i n ( a o(. ) - •+ 7\ Siri oC F o r the case where o( = 27\ , corresponding to the case of a c r a c k w i t h f l a n k angle <p - - 2.7!; = 0 the above equation takes the form s i n A = 0, thus r e q u i r i n g > = n/^ , where n = 1,2,3, and g i v i n g the s t r e s s f u n c t i o n n/2 + i |T ( r , > , r y 2 ) - r [ C , Sin (n/a-r M- c^co^Cya + O"^ (2-6) - 17 - S o l v i n g the equations W i l l i a m s obtained the s o l u - t i o n s f o r the s t r e s s as 07 00 e) 1 4 T a a , j - 5 c o s I ^ c o s ^ ] where 0 = ^ - K + b, ( - S S ^ | + ^ ^ j (2-7) 0^ Cr ;B) = 1 4ri a . 3 c o s I c o s 33 2 4 D , _ I _ 3 s u n 3 § 1 V 4. 4 a 2 SAW © + O C R A J + . . . . (2_e) - 18 - 4-r a ^ cos & -t 3 cos - 2a2 ^28 + o Cr*")* . . . (2.9) F o r convenience the s o l u t i o n can be t r e a t e d i n two p a r t s , the symmetric case when bj_ = 0 and the a n t i - symmetric case when a ^ = 0 . I n both cases a c h a r a c t e r - i s t i c square r o o t s i n g u l a r i t y i s o b t a i n e d . The symmetric crack i s more common, be i n g observed i n p l a t e s under t e n s i o n . Some i n t e r e s t i n g o b s e r v a t i o n s can be made from the r e s u l t s by W i l l i a m s . F i r s t l y , a t Q = 0 , alo n g the a x i s o f the cra c k the shear s t r e s s v a n i s h e s , making 0> and 0~e the p r i n c i p a l s t r e s s e s . The v a l u e s of <rr and CTQ are g i v e n by <rr (0') = cr^ ( o ° ) = - c x t r Hi « ^ » * * * T h i s means that a s t a t e o f two di m e n s i o n a l h y d r o s t a t i c t e n s i o n e x i s t s near the end of a crack, along i t s a x i s - 19 - 2.5 S t r e s s e s Around a P r o p a g a t i n g F r a c t u r e . The work of J o f f e (1951) on the s t r e s s d i s t r i b u - t i o n around the head o f a propagating f r a c t u r e has been mentioned before i n t h i s c h a p ter. We propose to g i v e here a b r i e f account of her method and r e s u l t s . Consider a crack of l e n g t h 2c i n a t h i n i n f i n i t e p l a t e . I f the p l a t e i s under a s u f f i c i e n t l y l a r g e t r a n s v e r s e t e n s i o n T, the c r a c k may be expected to propagate. For l a r g e 2c the s t r e s s d i s t r i b u t i o n s at the ends of the crack w i l l be independent o f each o t h e r . By keeping the c r a c k of constant l e n g t h the problem reduces to that o f a p l a t e under u n i f o r m t e n s i o n a c r o s s which a d i s t u r b a n c e (the crack) passes a t a constant speed. We d e f i n e our c o o r d i n a t e system such t h a t the X-Y plane i s the plane o f the p l a t e , and the c r a c k propagates i n the p o s i t i v e X d i r e c t i o n . The t e n s i o n T a c t s i n the Y d i r e c t i o n . The boundary c o n d i t i o n s are (T^ = = 0 on y = 0, -c < x < c - 20 - C o n s i d e r i n g s e p a r a t e l y the moving p a r t of the system, w i t h T removed we have on y = 0, -o <^x <^o C T ^ = 0 The symmetry permits us to c o n s i d e r o n l y the upper h a l f p l a n e . D e f i n i n g u x and Uy. as the displacements, and x = x - ^ t where ^ i s the v e l o c i t y of f r a c t u r e p r o p a g a t i o n the boundary c o n d i t i o n s f o r a dynamic e l a s t i c system are = -T a t y = 0 -c <^x<^c ^ = 0 a t y = 0 a l l x' = 0 a t y = 0 lx'\ y c These boundary c o n d i t i o n s have to be s a t i s f i e d by s u r f a c e d i s t u r b a n c e s p r o p a g a t i n g a t a v e l o c i t y ^ without displacement i n the Z d i r e c t i o n . Such d i s t u r b a n c e s can be r e p r e s e n t e d by U - v * . = A S T exp ( - T s ^ ) cos sx' (2-10) - 21 - = fe£ e x p ( - ^ s ^ ) c o s s x ' ( 2 _ i o ) i 2 Displacements of the above type vanish r a p i d l y w i t h the i n c r e a s e o f y, i f s i s p o s i t i v e . U s i n g the c o n d i t i o n = P ( aft*" + T* 1 ) = 0 ° n y = ° we o b t a i n the r e l a t i o n The g e n e r a l form o f the displacement i s g i v e n by i n t e g r a t - i n g the e x p r e s s i o n s over a l l p o s i t i v e v a l u e s of S • Here I - \ _ T - 1 - • - 22 - Thus s x ' OiS and OO c o s - s x e l s S u b s t i t u t i o n of the above e x p r e s s i o n i n t o (T% = ( * + 2 / M ^ -r * e+c. w i l l g i v e e x p r e s s i o n s f o r the v a r i o u s s t r e s s components. The unknown Ag i s determined from the boundary c o n d i t i o n s . From the r e l a t i o n o~~v s i n 2 © 4 az cos 2 " e ~_ 1 (T^y S \ n 9 COS0 one can determine the hoop s t r e s s a t constant r and d i f f e r e n t 8 . The o r i g i n s of the p o l a r c o o r d i n a t e system ( r , 0 ) are a t x = c and y = 0. I n f i g u r e 5 we r e p r o - duce from J o f f e (1951) the v a r i a t i o n of <7© w i t h 6 f o r v a r i o u s ^ • F o r low ^ t n e n ° o p s t r e s s d i s t r i b u t i o n shows a maximum i n the d i r e c t i o n of the c r a c k a t Q = 0 ° . The v a r i a t i o n o f 0~e w i t h © decreases w i t h i n c r e a s e of ^ u n t i l a t about 0.6^3 j (TQ does not change w i t h © . For <̂ 0.6|3> t h e r e f o r e the f r a c t u r e propagates i n a - 23 - s t r a i g h t l i n e . But when ^ approaches 0.6/3 the f r a c t u r e has no p r e f e r r e d d i r e c t i o n of p r o p a g a t i o n as n e a r l y equal s t r e s s e s e x i s t over a wide range of a n g l e s ahead of the f r a c t u r e . 2.6 R a d i a t i o n from a P r o p a g a t i n g F r a c t u r e . An i n t e g r a l o f the e l a s t i c wave equation g i v e n by Knopoff (1956) and de Hoop (1958) has been used by Knopoff and G i l b e r t (1960) to determine the f i r s t motion r a d i a t i o n p a t t e r n from v a r i o u s types of p r o p a g a t i n g d i s - l o c a t i o n s . Consider a volume V bounded by two s u r f a c e s S' as shown i n f i g u r e 6. I n the l i m i t i n g case the outer s u r f a c e i s expanded u n t i l i t goes to i n f i n i t y and the i n n e r s u r f a c e i s shrunk u n t i l i t reduces to the f a u l t p l a n e . The i n n e r s u r f a c e , the f a u l t p l a n e , i s o r i e n t e d so that the normal c o i n c i d e s w i t h the Z a x i s i n a c a r t e s i a n system. I t should be noted here that f o r the s u r f a c e s s ' the outwardly drawn normal i s the p o s i t i v e d i r e c t i o n . The + and - s u p e r s c r i p t and s u b s c r i p t a f t e r the s t r e s s and displacement tensors r e p r e s e n t the d i f f e r e n c e s i n these terms between the p o s i t i v e and negative Z s i d e s of the f a u l t . We assume t h a t , 1) there are no body f o r c e s i n s i d e the volume V; 2) that the o n l y s u r f a c e p e r t i n e n t to the problem i s i n t e r i o r to the i n f i n i t e e l a s t i c medium - 24 - and 3) we are i n t e r e s t e d o n l y i n the f i r s t mptions (the h i g h frequency s o l u t i o n ) from the im p u l s i v e e x c i t a - t i o n o f the s u r f a c e s ' b y suddenly a p p l i e d d i s c o n t i n u i t i e s i n the s t r e s s tensor t ^ j or the displacement t e n s o r | \ j . We f o l l o w here the terminology and d e f i n i t i o n s of Knopoff and G i l b e r t (1960). For o l a r i t y we d e f i n e here c e r t a i n q u a n t i t i e s : Displacement = S t r e s s tensor t ^ j = (°C- 2pv) - ^ ( a ^ j 4 "-Ji Displacement t e n s o r r\j R e t a r d a t i o n w i t h P v e l o c i t y R e t a r d a t i o n w i t h S v e l o c i t y C oordinates o f the p o i n t o f o b s e r v a t i o n C o o r d i n a t e s o f a p o i n t on the f a u l t plane D i s t a n c e r D i r e c t i o n c o s i n e s T; - [utco] = u. Ct - %0 - < u . W > = V i a ) - 25 - In the 2-Z plane the d i r e c t i o n = - (5L - ^^/^ = COS 0 cos i n e s o f r Tz - (J- - *'Vr - s v / n 6 The d i f f e r e n t i a l equation o f motion of a homo- geneous i s o t r o p i c e l a s t i c medium i s ct grad d i v U - |3 c u r l c u r l U - = f/p where f / p i s the body f o r c e d e n s i t y . The s o l u t i o n o f t h i s e q u a t i o n by Knopoff and de Hoop i s (2-12) where the o p e r a t o r Gj_j i s d e f i n e d by (2-13) In the absence o f body f o r c e s , and n o t i n g that the u n i t normal v e c t o r on S f p o i n t i n g i n t o V i s negative we have - j^M^J- d s' - J fx<Gti C*s' (2~14) - 26 - The op e r a t o r G-.y i s approximated f o r l a r g e r by n e g l e c t - i n g terms i n v o l v i n g 1/r and 1/r Slj - Oj^Ti + ^ r < * > ( ^ l "UTS) (2-15) and jp^j i s The motion a t a remote p o i n t can be w r i t t e n as the sum of P wave motion and the S wave motion. Thus (2-16) F o r f i r s t motions a t l a r g e r we may w r i t e (2-17) J J t t " J (2-18) as the second term i n the d i f f e r e n t i a t i o n f a l l s o f f as r 2 . The d i s t u r b a n c e a t l a r g e r may be co n s i d e r e d as o r i g i n a t i n g from a p o i n t , even though the p o i n t propagates along the f a u l t p l a n e . For f i r s t motions we are i n t e r e s t e d i n a very s m a l l range o f time, and w i t h i n t h i s i n t e r v a l the d i s c o n t i n u i t i e s may be c o n s i d e r e d t o be s u b s t a n t i a l l y u n i f o r m over the s u r f a c e S , e l i m i n a t i n g the need f o r 27 - i n t e g r a t i o n over d s f . I t must be noted here that the s t r e s s and d i s p l a c e - ment t e n s o r s used here d i f f e r from the standard d e f i n i t i o n s of these q u a n t i t i e s by c e r t a i n dimensions o f v e l o c i t y and d e n s i t y . We change the dimensions of t ^ j and p±j to now i n c l u d e the elemental area ds'. Since these are constant f a c t o r s they w i l l not a f f e c t the c a l c u l a t e d r a d i a t i o n p a t t e r n . The e x p r e s s i o n s f o r u^ and u ^ s then become (2-20) In p o l a r c o o r d i n a t e s the above equations become 4 ^oc^CL - n 21 p r (2-21) where - 28 - The v e c t o r r i s a u n i t v e c t o r i n the r a d i a l d i r e c t i o n and 0j i s a u n i t v e c t o r i n a d i r e c t i o n a t r i g h t angles to 17 and l i e s i n a plane c o n t a i n i n g e~j and r, , and p o i n t i n g i n the p o s i t i v e d i r e o t i o n from e^ to r, . The u n i t v e c t o r i n the j d i r e c t i o n i s e^. The s i n e of the angle between e j and r^ i s g i v e n by a. i 83 = (1 - Tj )« An a c t u a l f a u l t or f r a c t u r e may be a combination o f a l l or some of the terms i n the e x p r e s s i o n Wy Knopoff and G i l b e r t have determined the f i r s t motion r a d i a t i o n p a t t e r n hy c o n s i d e r i n g each o f the terms of w. i n t u r n , and J assuming the other seven to be z e r o . F o r a s t e p f u n c t i o n time dependance we can w r i t e and The f a u l t propagates i n the x d i r e c t i o n w i t h v e l o c i t y ^ , HCCp) i s the u n i t step f u n c t i o n . For a t e n s i l e f r a c t u r e of the type c o n s i d e r e d i n t h i s t h e s i s a l l terms i n ŵ  except u a are z e r o . Then ŵ i s equal to W i : C**- 2£) ^± , (*x'2f?)£±± , **^» - 29 - N o t i n g t h a t L H ( t - r/oL ) « *&- 6 (t - v/di ) , we o b t a i n , ox r on the X-Z plane f LoC o O f o ( » J (S-S2) r I ( i P*s Knopoff and G i l b e r t appear to have overlooked the d i r e c t i o n o f the normal to S T . T h i s changes the s i g n o f the f i r s t i n t e g r a l i n t h e i r equation (2) and the s i g n o f (T^j g i v e n j u s t below t h a t e q uation. There appear to be some f u r t h e r e r r o r s i n s i g n which p a r t l y , but not completely, c a n c e l the p r e v i o u s e r r o r . Our equations f o r the d i s p l a c e - ments caused by a type 3 d i s l o c a t i o n ( i n Knopoff and G i l b e r t ' s nomenclature) d i f f e r from t h e i r equation (12) In the s i g n o f a l l terms c o n t a i n i n g and i n the i n c l u s i o n o f s__ i n the f i r s t term o f the second o f the two equations. T h i s l a t t e r i s c l e a r l y o n l y a t y p o g r a p h i c a l e r r o r . (2-23) - 30 - CHAPTER I I I •INSTRUMENTATION AND EXPERIMENTAL TECHNIQUE 3.1 S c a l i n g C o n s i d e r a t i o n s . Knopoff (1955) shows t h a t i n sei s m i c models the r e l a t i o n s h i p g = t and p = v = 1 should h o l d . Here g i s the g e o m e t r i c a l s c a l e f a c t o r , p the P o i s s o n ' s r a t i o s c a l e f a c t o r , t the time s c a l e f a c t o r , and v the v e l o c i t y s c a l e f a c t o r . I n t h i s t h e s i s our primary aim i s to study the f a r f i e l d r a d i a t i o n o f a moving source. Therefore we do not "have to conform to the r i g i d s c a l i n g c o n s i d e r a t i o n s . We must, however, s a t i s f y the t h i n p l a t e r e s t r i c t i o n s and a l s o r e c o r d s u f f i c i e n t l y f a r from the f r a c t u r e so t h a t o n l y the f a r f i e l d i s d e t e c t e d . The t h i n p l a t e and the f a r f i e l d requirements oppose each o t h e r . The f a r f i e l d c o n d i t i o n means that o b s e r v a t i o n s be made a l a r g e number of wavelengths away from the source. F o r f i n i t e p l a t e dimensions t h i s c o n d i t i o n can be achieved o n l y w i t h s m a l l wavelengths. On the other hand, t h i n p l a t e theory r e q u i r e s t h a t the wavelength be lo n g compared w i t h the p l a t e t h i c k n e s s . I n the s e r i e s o f experiments d e s c r i b e d here we have used a p l a t e t h i c k n e s s o f 3 mm, w i t h measured P f wave - 31 - v e l o c i t y o f 5500 m/sec and an S wave v e l o c i t y of 3700 m/sec. At a frequency of 100 kc/s the wavelengths are ^ t= 5 e5 cms and ^(a = c m s » Thus the wavelengths are much g r e a t e r than the p l a t e t h i c k n e s s . Most o f the o b s e r v a t i o n s have been made a t d i s - tances g r e a t e r than 25 cms from the s o u r c e . I n o r d e r to v e r i f y t h a t the measurements were not i n f l u e n c e d by the near f i e l d , an attempt was made to measure the geometric a t t e n u a t i o n o f the e l a s t i c waves from the f r a c t u r e . For i two dimensional waves the amplitude should vary as r - 8 i n the f a r f i e l d , but i n the near f i e l d the amplitude should f a l l o f f more r a p i d l y . D e t a i l s of the experiment are g i v e n i n s e c t i o n 4.5. The measured exponent of r came out to be - 0.53 ± 0.08 (Standard E r r o r ) . I t i s f e l t t h a t the experimental r e s u l t v e r i f i e s t h a t measurements at d i s t a n c e s g r e a t e r than 20 cms do not i n v o l v e near f i e l d to any a p p r e c i a b l e e x t e n t . 3.2 Methods of Inducing F r a c t u r e . An inhomogeneous thermal s t r e s s f i e l d i s s e t up i n the g l a s s p l a t e by the a p p l i c a t i o n o f heat to a s m a l l r e g i o n (see f i g . 1 ) . Because of the thermal g r a d i e n t , there i s d i f f e r e n t i a l expansion, and r a d i a l and hoop s t r e s s e s are developed i n the p l a t e . As i s w e l l known, - 32 - f r a c t u r e i n g l a s s i s r e a d i l y induced by t e n s i l e s t r e s s , and the hoop s t r e s s e s are thus admirably s u i t e d to i n i t i a t e f r a c t u r e s . I f the temperature i n the c e n t r a l heated r e g i o n becomes too h i g h , the p r o p e r t i e s of the g l a s s w i l l be a l t e r e d , and the p r o p a g a t i o n of e l a s t i c waves i n the p l a t e may be a f f e c t e d . I n a d d i t i o n , p l a s t i c y i e l d i n g may take p l a c e . F o r the purpose of t h i s e x p e r i - ment, t h e r e f o r e , i t i s important that the temperature of the heat source be s u f f i c i e n t l y low to prevent any s i g n i f i c a n t a l t e r a t i o n of the e l a s t i c p r o p e r t i e s o f g l a s s . Temperature measurements on the g l a s s p l a t e a f t e r h e a t i n g w i t h a gas flame f o r 60 seconds gave a c e n t r a l heated zone of 5 mm r a d i u s a t 200° C. To determine whether t h i s heated zone would s u f f i c e to d i s t o r t the r a d i a t i o n p a t t e r n and a r r i v a l times of an u l t r a s o n i c wave, an experiment was d e v i s e d . An u l t r a s o n i c s i g n a l was t r a n s m i t t e d along a g l a s s p l a t e from a r e p e t i t i v e source to a pick-up t r a n s d u c e r . The s i g n a l was f i l t e r e d i n e x a c t l y the same way as the s i g n a l s from an a c t u a l f r a c t u r e . The flame was then a p p l i e d 2 cm away from the source along the l i n e j o i n i n g source and r e c e i v e r . No change i n the s i g n a l was observed i n the two minutes that the flame was a p p l i e d to the p l a t e . There was a l s o no d e t e c t a b l e change i n the time o f a r r i v a l . I t i s concluded t h e r e f o r e t h a t the e l a s t i c p r o p e r t i e s of the g l a s s were - 33 - not m o d i f i e d by h e a t i n g i n a s u f f i c i e n t amount to d i s t o r t the r a d i a t i o n p a t t e r n or the a r r i v a l time o f the e l a s t i c waves. T h i s i s f u r t h e r supported by the f a c t t h a t the e l a s t i c constants of the type of g l a s s used do not change more than 3 per cent over a temperature range from 0° to 200° C. In Appendix I c a l c u l a t i o n s are shown f o r computing r a d i a l and hoop s t r e s s e s s e t up by such a heated zone. The approximate temperature d i s t r i b u t i o n a f t e r 45 sec of the a p p l i c a t i o n of the flame i s shown i n f i g u r e 7. The r a d i a l and hoop s t r e s s d i s t r i b u t i o n s e t up by the temperature g r a d i e n t i s shown i n f i g u r e 8. I t i s seen that the compressive r a d i a l s t r e s s e s i n c r e a s e r a p i d l y towards the c e n t e r o f the heated zone, w h i l e the t e n s i l e hoop s t r e s s reaches a maximum value a t about 1.5 cm from the c e n t e r o f the heated zone. We have observed t h a t the f r a c t u r e does i n f a c t o r i g i n a t e a t about 1.5 cm from the p o i n t o f a p p l i c a t i o n o f heat. To i n i t i a t e a f r a c t u r e a v e r y shallow s c r a t c h about OS cm lo n g i s made on the s u r f a c e o f the g l a s s sheet. Heat Is a p p l i e d 1.5 cm from the end o f the s c r a t c h by a gas flame. A f t e r a p e r i o d of b u i l d - u p of s t r e s s (about 30 to 60 seconds) a f r a c t u r e forms spontaneously and propa- gates i n the d i r e c t i o n o f the s c r a t c h . The s c r a t c h a c t s as a f l q w i n the g l a s s and thus determines where the f r a c t u r e - 34 - begins and a l s o i n f l u e n c e s the d i r e c t i o n i n which i t propagates. O c c a s i o n a l l y , l o n g e r s c r a t c h e s were made to ins u r e a s t r a i g h t f r a c t u r e . A f r a c t u r e o r i g i n a t i n g i n a p r e v i o u s l y unbroken medium i s , f o r convenience, termed here as an I n i t i a l F r a c t u r e . When a p r e v i o u s l y e x i s t i n g f r a c t u r e i s extended i t i s termed as an Extended F r a c t u r e . These two types of f r a c t u r e s are d i f f e r e n t i n many aspects and d i f f e r e n t techniques have to be employed to produce them. The procedure f o r producing I n i t i a l f r a c t u r e s i s , as s t a t e d b e f o r e , by the a p p l i c a t i o n o f a gas flame about 1.5 cms away from the end of a 0.5 cm lo n g s c r a t c h . I n the beginn i n g i t was assumed that I n i t i a l f r a c t u r e s propagate u n i l a t e r a l l y , as i s observed v i s u a l l y . There was a p o s s i - b i l i t y however, that the I n i t i a l f r a c t u r e s might be prop a g a t i n g b i l a t e r a l l y . To r e s o l v e the doubt, an e x p e r i - ment was performed. The gas flame was a p p l i e d a t d i f f e r e n t d i s t a n c e s from the edge of the s c r a t c h . At a d i s t a n c e o f 10 cms there was no f r a c t u r e f o r m a t i o n even a f t e r h e a t i n g f o r 5 minutes. At a d i s t a n c e o f 5 cms, a f r a c t u r e was observed t o form a f t e r 4.5 minutes. The f r a c t u r e remained s t a t i c f o r a s h o r t w h i l e and then propagated both ways. The end moving towards the flame came to a stop a f t e r r e a c h i n g the c e n t r a l heated zone. The h e a t i n g time r e q u i r e d f o r f r a c t u r e f o r m a t i o n decreased w i t h the d i s t a n c e o f the - 35 - p o i n t of a p p l i c a t i o n of flame from the edge of the s c r a t c h . The minimum d i s t a n c e and time r e q u i r e d f o r f r a c t u r e f o r m a t i o n was found to be 1.5 cm and 30 sees. The end of the f r a c t u r e that moves away from the flame u s u a l l y proceeds a c o n s i d e r a b l e d i s t a n c e (about 10 cms or more) before coming to a s t o p . I t i s t h i s end which g i v e s the erroneous v i s u a l i m pression o f u n i l a t e r a l p r o p a g a t i o n . To s t a n d a r d i z e o b s e r v a t i o n s , and to p r e s e r v e the g l a s s p l a t e f o r f u r t h e r use, a spot A i s heated ( f i g . 9 ) i n a d d i t i o n to spot B. The h e a t i n g a t spot B a c t u a l l y causes the I n i t i a l f r a c t u r e to form, w h i l e h e a t i n g spot A causes the other end of the f r a c t u r e to s t o p . The technique f o r producing Extended f r a c t u r e i s more i n v o l v e d . As i n the case o f the I n i t i a l f r a c t u r e , the c e n t r a l zone, w i t h zero or compressive hoop s t r e s s i s used to stop f r a c t u r e a t a c e r t a i n p o i n t . The c e n t r a l heated zone i s a l s o used to l o c k i n s u f f i c i e n t energy. Once a f r a c t u r e i s formed, a l a r g e c o n c e n t r a t i o n o f s t r e s s occurs a t the t i p (the w e l l known r " ^ s i n g u l a r i t y ) . T herefore i t r e q u i r e s very l i t t l e energy to extend a pre- e x i s t i n g f r a c t u r e . Our problem here i s to l o c k i n enough energy around the e x i s t i n g f r a c t u r e which w i l l be r e l e a s e d a t the time the f r a c t u r e extends. T h i s i s done by h e a t i n g spot A. The hoop s t r e s s s e t up by h e a t i n g spot B ( f i g . 1 0 ) w i l l then have t o overcome the compressive s t r e s s a t A. - 36 - I n a d d i t i o n a spot G has to be heated to prevent the f r a c t u r e from extending i n d e f i n i t e l y . I t i s a l s o p o s s i b l e to extend the f r a c t u r e i n the d i r e c t i o n B D by h e a t i n g a t E . I n t h i s case the spot C has to be heated ( f i g u r e 11), to prevent the f r a c t u r e from extending i n the wrong d i r e c t i o n . Of course the spot D has to be heated to prevent the f r a c t u r e from running away, and the spot B has to be heated to l o c k i n energy. Thus the technique f o r extending l o n g f r a c t u r e s i s d i f f e r - ent from that of extending s h o r t f r a c t u r e s . Extended f r a c t u r e s are not as sudden as I n i t i a l f r a c t u r e s . The passage o f the f r a c t u r e through the zone of compression i s slow and proceeds i n s h o r t s p u r t s . Only a f t e r the c r o s s i n g of the zone of compression does the f r a c t u r e 'go w i t h a bang'. Extended f r a c t u r e s are always preceded by a number of 'fo r e shocks'. These are d i s c r e t e a u d i b l e 'tInks' which u s u a l l y do not have enough energy to s e t o f f the t r i g g e r . The f o r e shocks' are probably due to some process along the f r a c t u r e i n t e r f a c e , and not a t the t i p of the f r a c t u r e . 3.3 E l e c t r o n i c D i s p l a y and Recording System. Two d i f f e r e n t but b a s i c a l l y s i m i l a r e l e c t r o n i c c i r c u i t s have been used. The f i r s t system ( f i g . 1 2 ) i s used f o r v e l o c i t y measurements and c a l i b r a t i o n of d e t e c t o r s . - 37 - The second system i s used f o r s t u d y i n g e l a s t i c wave r a d i a t i o n from propagating f r a c t u r e s ( f i g . 13). The f i r s t system i s e s s e n t i a l l y the same as t h a t used by O l i v e r , P r e s s and Ewing (1954), w h i l e the second i s the m o d i f i c a t i o n of the same to f i t the s p e c i a l requirements of n o n r e p e t i t i v e nature of i n d i v i d u a l o b s e r v a t i o n s . The source o f e l a s t i c energy f o r the f i r s t c i r c u i t i s a ceramic d i s c transducer a c t i v a t e d by a h i g h v o l t a g e sho r t d u r a t i o n pulse from a pulse g e n e r a t o r . The p u l s e generator i s t r i g g e r e d by a low frequency (10 - 20 c y c l e s / s e c ) square wave generator. The d e t e c t i n g p a r t of the c i r c u i t c o n s i s t s o f ceramic transducers - p r e a m p l i f i e r - v a r i a b l e band pass f i l t e r - o s c i l l o s c o p e - camera. I n a d d i t i o n a t r i g g e r t a k e - o f f c i r c u i t a m p l i f i e s a t r i g g e r i n g s i g n a l from the p u l s e generator and t r i g g e r s the 502 o s c i l l o s c o p e sweep. The second system i s s i m i l a r to the above except that no p r e a m p l i f i e r i s used because of the h i g h e r energy of e l a s t i c waves from f r a c t u r e . The t r i g g e r i n g s i g n a l comes from a d e t e c t o r near the expected o r i g i n of the f r a c t u r e . The v a r i o u s components of both the systems are d i s c u s s e d i n d e t a i l i n the f o l l o w i n g s e c t i o n s . 1 3.3.1 Ceramic Transducers. Barium T i t a n a t e t r a n s d u c e r s were used to p i c k up e l a s t i c waves and convert them to e l e c t r i c a l s i g n a l s . Two - 38 - types of tran s d u c e r s were used: c y l i n d e r s and benders. The c y l i n d e r s are 0,25 cm i n t h i c k n e s s and 0.64 cm i n diameter, and are mai n l y s e n s i t i v e t o motion along the a x i s . The bender c o n s i s t s o f two t h i n s t r i p s o f barium t i t a n a t e fused t o g e t h e r to form a p a r a l l e l e p i p e d o f dimensions 0.05 x 0.16 x 1.27 cm. They are p o l a r i z e d such t h a t the v o l t a g e generated by the two s t r i p s add up and i n c r e a s e the s e n s i t i v i t y . The benders are most s e n s i t i v e to d i f f e r e n t i a l displacement p e r p e n d i c u l a r to the plane of adhesion, and are very d i r e c t i o n s e n s i t i v e . C y l i n d e r s can o n l y be used a l o n g the edge of a p l a t e t o d e t e c t P o s c i l l a t i o n s , w h i l e benders can be used on the s u r f a c e of the p l a t e to d e t e c t e i t h e r P o r S waves. I f the bender Is o r i e n t e d w i t h f l a t s i d e a l o n g the d i r e c t i o n o f propa- g a t i o n , m a i n l y S waves are p i c k e d up. A change i n o r i e n t a t i o n by 90°, so t h a t f l a t s i d e i s p e r p e n d i c u l a r to the d i r e c t i o n o f p r o p a g a t i o n , means that mainly P waves are d e t e c t e d . The d i s c s and benders were mounted i n low v e l o c i t y f i b r e rods to a v o i d r e f l e c t i o n s from end of the r o d s . To f u r t h e r prevent r i n g i n g both benders and d i s c s were clamped w i t h foam rubber. The frequency response o f the t r a n s - ducers i s not known-, and can be expected t o be non-uniform over the range of f r e q u e n c i e s used. To a v o i d t h i s d i f f i c u l t y , most o b s e r v a t i o n s were made i n a v e r y narrow - 39 - frequency range. Both h i g h and low c u t - o f f d i a l s o f the f i l t e r were s e t a t the same frequency (10 c p s ) ; the 3db a t t e n u a t i o n p o i n t then occurs a t 0.77 and 1.30 times the c u t - o f f frequency. The v o l t a g e output of the t r a n s d u c e r s ranges up to 4 v o l t s a t the extreme low frequency range, but i s lower a t h i g h e r f r e q u e n c i e s and some amount o f a m p l i f i c a t i o n i s n e c e s s a r y . I n the p r e s e n t study the energy o f waves r a d i a t e d from a crack i s s u f f i c i e n t to g i v e a transducer output of the order of m i l l i v o l t s . An a m p l i f i c a t i o n o f 100 i s u s u a l l y s u f f i c i e n t to a l l o w the s i g n a l t o he p r o p e r l y d i s p l a y e d on the o s c i l l o s c o p e s c r e e n . 3.3.2 F i l t e r System. A f i l t e r system i s necessary to e l i m i n a t e e x t e r n a l e l e c t r o m a g n e t i c and a c o u s t i c n o i s e t h a t i s p i c k e d up and a m p l i f i e d . Major sources of e l e c t r o m a g n e t i c n o i s e are f l u o r e s c e n t lamps, switches, and normal 60 c y c l e hum. A o o u s t i c n o i s e i s caused by movement i n the l a b o r a t o r y , machinery i n the b u i l d i n g , p a s s i n g t r u c k s , and o c c a s i o n a l l y l o u d c o n v e r s a t i o n . As d i s c u s s e d e a r l i e r , the c h o i c e o f wave l e n g t h i s r a t h e r c r i t i c a l ; i t must be s m a l l enough so that the r e c e i v e r to source d i s t a n c e i s a t l e a s t s e v e r a l wavelengths. On the o t h e r hand the wavelength must be l a r g e i n - 40 - comparison w i t h the p l a t e t h i c k n e s s . I n order to f u l f i l 5 these c o n d i t i o n s a frequency c l o s e to 10 cps i s r e q u i r e d , both higher and lower f r e q u e n c i e s being u n d e s i r a b l e . A commercial v a r i a b l e band pass f i l t e r (Kron-Hite 310 AB) was used. The f i l t e r c o n s i s t s o f s e v e r a l s e c t i o n s of r e s i s t a n c e c a p a c i t a n c e combinations. The g a i n between output and input t e r m i n a l s i s u n i t y . I n any v a r i a b l e band pass f i l t e r , some amount o f phase d i s t o r t i o n i s unavoidable. The d i s t o r t i o n i s zero i n the mid-frequency range, g r a d u a l l y i n c r e a s i n g as one moves away on both s i d e s o f the mid-frequency. When the h i g h and low c u t - o f f are se t a t the same frequency, the phase d i s t o r t i o n i s n e g l i g i b l e . A s e t t i n g o f the l o w - c u t - o f f d i a l a t 2 kc/s e f f e c t i v e l y e l i m i n a t e s o r d i n a r y e l e c t r o m a g n e t i c and acous- t i c n o i s e . (This i s s t r i c t l y t r ue f o r a s i g n a l a m p l i f i c a - t i o n o f 100; the s e t t i n g i s l i a b l e to vary w i t h h i g h e r a m p l i f i c a t i o n ) , 3.3.3 A m p l i f i e r System, The h i g h i n t e r n a l g a i n o f the T e k t r o n i x 502 o s c i l l o s c o p e makes an e x t e r n a l a m p l i f i e r unnecessary. For c a l i b r a t i o n and f o r v e l o c i t y measurements i t i s sometimes necessary to d e t e c t s i g n a l s from a pulsed transducer as a source o f e l a s t i c energy. The energy of waves i s low and - 41 - a m p l i f i c a t i o n of the order o f 10 i s e s s e n t i a l . I n such a case a commercial b a t t e r y powered a m p l i f i e r (Tektronix) or a l a b o r a t o r y designed instrument i s used. Both types have s u i t a b l e a m p l i f i c a t i o n and low n o i s e c h a r a c t e r i s t i c s . 3.3.4 D i s p l a y and Recording. The f i l t e r e d and a m p l i f i e d s i g n a l from the t r a n s - ducer i s d i s p l a y e d on the s c r e e n of a T e k t r o n i x 502 or 535 o s o i l l o s c o p e . A DuMont camera w i t h a P o l a r o i d back i s used to photograph the d i s p l a y . The oamera back Is movable, p e r m i t t i n g s e v e r a l exposures to be made on the same p r i n t . 3.3.5 T r i g g e r i n g System. To r e c o r d the d i s t u r b a n c e s from the f r a c t u r e s s p e c i a l i n s t r u m e n t a t i o n was developed. I n the u s u a l two-dimensional model s e i s m i c experiment ( O l i v e r , P r e s s and Ewlng, 1954) a s h o r t d u r a t i o n , h i g h v o l t a g e p u l s e i s a p p l i e d to a transducer to p r o v i d e a source of e l a s t i c energy. T h i s pulse a l s o t r i g g e r s the o s c i l l o s c o p e sweep so that f i r s t motions a t d i f f e r e n t p o i n t s i n the medium can be observed. I n the present i n v e s t i g a t i o n s the source of e l a s t i c energy i s a p r o p a g a t i n g f r a c t u r e w i t h u n p r e d i c t - a b l e time of o r i g i n . To p r o v i d e a t r i g g e r i n g s i g n a l , a d e t e o t o r i s plaoed very near the expected p o i n t of o r i g i n o f the f r a c t u r e . The d e t e c t o r i s a Barium T i t a n a t e bender. The s i g n a l from the bender i s a m p l i f i e d , c r u d e l y f i l t e r e d - 42 - of low frequency hums, and f e d i n t o the single-sweep c i r c u i t o f a T e k t r o n i x 535 o s c i l l o s c o p e . A g a t i n g s i g n a l from t h i s o s c i l l o s c o p e i s used to t r i g g e r the sweep of a T e k t r o n i x 502 o s c i l l o s c o p e on which the P o l a r o i d camera i s mounted. Because o f the h i g h energy o f the e l a s t i c waves generated by f r a c t u r e s , a c o u s t i c 'noise' p e r s i s t s i n the g l a s s p l a t e f o r times ranging up to tens of m i l l i s e c o n d s . T h i s i s l o n g compared w i t h the beam sweep time a c r o s s the o s c i l l o s c o p e s c r e e n . To prevent c o n f u s i o n o f the f i r s t r ecorded motions w i t h succeeding s i g n a l s i t i s imperative that the r e c o r d i n g stop a f t e r the beam has moved a c r o s s the o s c i l l o s c o p e s c r e e n once. The s i n g l e sweep c i r c u i t o f the T e k t r o n i x 535 generates one g a t i n g s i g n a l a f t e r r e c e i v i n g the f i r s t of a t r a i n o f t r i g g e r i n g s i g n a l s from the t r i g g e r a m p l i f i e r , and then shuts o f f u n t i l r e s e t . Hence, the beam o f T e k t r o n i x 502 o s c i l l o s c o p e sweeps acro s s the s c r e e n once, u n t i l the s i n g l e sweep c i r c u i t Is r e s e t . 3.4 M e c h a n i c a l Arrangement. Glas s p l a t e s of dimensions 45 x 45 cms or 60 x 60 cm were used i n the experiments. The p l a t e s are used commercially as window g l a s s . The co s t f o r the l a r g e r p l a t e s i s approximately 60 c e n t s . No s p e c i a l treatment - 43 - was g i v e n to the g l a s s p l a t e s . The t h i c k n e s s was a p p r o x i - mately 3 mm, although p l a t e s o f other t h i c k n e s s have o c c a s i o n a l l y been used. The r e c o r d i n g time was too s h o r t f o r an e l a s t i c wave to be r e f l e c t e d - a s u f f i c i e n t number o f times to s e t up f r e e o s c i l l a t i o n o f the p l a t e . Therefore e l a b o r a t e mounting and s u p p o r t i n g arrangements f o r the p l a t e were unnecessary. The p l a t e was supported h o r i z o n t a l l y by two wooden bars 1.2 cm wide. These were p l a c e d a l o n g the edges of the g l a s s p l a t e . The ceramic t r a n s d u c e r s were mounted i n f i b r e rods and clamped w i t h rubber or foam p l a s t i c . The f i b r e rods are clamped onto movable benches made of s t e e l r o d s . The c o u p l i n g between the benches and f i b r e rods was l o o s e , making the p o s s i b i l i t y o f the whole system r i n g i n g remote. - 44 - CHAPTER IV PROCEDURE 4.1 C a l i b r a t i o n o f D e t e c t o r s . The c h a r a c t e r i s t i c s of each transducer v a r i e s s i g n i f i c a n t l y w i t h the type of mounting, and w i t h i n d i - v i d u a l ceramic p i e c e s . A procedure has been worked out f o r choosing from a group o f d e t e c t o r s o n l y those which 1) show a response i n the 100 kc/s r e g i o n , 2) e x h i b i t no r i n g i n g a t t h a t frequency and 3) show d e f i n i t e d i r e c t i o n a l s e n s i t i v i t y to l o n g i t u d i n a l and t r a n s v e r s e o s c i l l a t i o n s o f the e l a s t i c p l a t e . There should be a minimum o f p i c k up o f t r a n s v e r s e waves when the d e t e c t o r i s o r i e n t e d f o r l o n g i t u d i n a l o s c i l l a t i o n s , and v i c e v e r s a . Without a standard and convenient method of o b t a i n i n g the frequency response of d e t e c t o r s one can use - on l y an e m p i r i c a l system of c r o s s checking. The p r i n c i p l e i s simple. Let us d e f i n e as n o i s e any s i g n a l other than a d i r e c t P and S wave from a p o i n t source (such as a p u l s e d transducer) on an e l a s t i c p l a t e . The d i r e c t P and S v i b r a t i o n s are termed s i g n a l s . I f we use a d e t e c t o r w i t h unknown c h a r a c t e r i s t i c s we w i l l observe i n a d d i t i o n to the true s i g n a l a c o n s i d e r a b l e amount of n o i s e . I f the transducer i s improperly damped then a r i n g i n g w i l l be observed a f t e r the P o r S s i g n a l i s r e c e i v e d . As the - 45 - exact wave form o f the e l a s t i c s i g n a l a t 100 kc/s i s unknown, one can compare the s i g n a l s from two d e t e c t o r s . I f they are not s i m i l a r i n form and phase, then one o f the d e t e c t o r s i s u n s u i t a b l e . T e s t i n g o f a l a r g e number of t r a n s d u c e r s w i l l p r o v i d e a group whose responses are rea s o n a b l y i d e n t i c a l . The e l a s t i c source used i n p r e l i m i n a r y t e s t i n g i s a p u l s e d transducer which g i v e s a mechanical impulse to the p l a t e on r e c e i v i n g a 1 kv spike p u l s e from a pul s e g e n e r a t o r . The amplitude of the e l a s t i c s i g n a l s r e c e i v e d from such a source i s s m a l l , b e i ng about 100 times s m a l l e r than those r e c e i v e d from f r a c t u r e s . Hence another t e s t i s done to determine i f the d e t e c t o r s m a i n t a i n t h e i r response a t s i g n a l s of h i g h amplitudes, and are not over- d r i v e n . Two d e t e c t o r s are p l a c e d a t the same p o s i t i o n w i t h r e s p e c t to an I n i t i a l f r a c t u r e . The r e s u l t i n g s i g n a l s should be i d e n t i c a l a t l e a s t up to 200 micro seconds a f t e r the r e c e i v i n g o f the f i r s t s i g n a l . Instrument S e t t i n g s - The f i l t e r h i g h and low c u t - o f f d i a l s were set a t 100 k c / s . The s i g n a l amplitude a f t e r f i l t e r i n g i s around 1 to 10 m i l l i v o l t s . The o s c i l l o s c o p e s e t t i n g s were from 1 mv/cm to 10 mv/cm, and 20 ^Ls/cm. - 46 - 4.2 V e l o c i t y Measurements. Measurement o f P and S wave v e l o c i t i e s i n g l a s s p l a t e s was made u s i n g a p u l s e generator f o r a source. The c i r c u i t f o r t h i s measurement i s g i v e n i n f i g u r e 12. A h i g h v o l t a g e (1 kv) s p i k e was a p p l i e d to a ceramic d i s o t r a n s d u c e r p l a c e d a t the edge of a g l a s s p l a t e . P and S waves were recorded a t 10 cm i n t e r v a l s . The v e l o c i t y was c a l c u l a t e d from the slope of the s t r a i g h t l i n e of the p l o t o f time of a r r i v a l a g a i n s t the d i s t a n c e . An a l t e r n a t i v e method of measuring the v e l o c i t y o f P and S waves i s by p l a c i n g two d e t e c t o r s about 40 cms a p a r t and r e c o r d i n g P and S waves from a f r a c t u r e . As the e l a s t i c d i s t u r b a n c e from a f r a c t u r e i s of much h i g h e r amplitude i t i s e a s i e r to p i c k out the d e s i r e d P or S s i g n a l s . However, o n l y two d e t e c t o r s can be used f o r each f r a c t u r e . Thus, the method Is l e s s a c c u r a t e than the f i r s t one and i s used o n l y f o r rough measurements. Moreover, the f i r s t method permits an estimate of the e r r o r . 4.3 I d e n t i f i c a t i o n o f P and S Waves. The P phase i s the f i r s t a r r i v a l and i t s f i r s t motion can be p i c k e d up w i t h a good d e a l o f c o n f i d e n c e . U n f o r t u n a t e l y , t h i s i s not the case f o r the S phase which a r r i v e s i n the midst of s p u r i o u s p i c k up o f the P phase by the S d e t e c t o r . I n order to s o r t out the f i r s t motion - 47 - of the S wave i t i s necessary to study s e v e r a l records and to employ such a i d s as the c a l c u l a t e d a r r i v a l time f o r S waves. A f t e r s u f f i c i e n t S s i g n a l s had been i d e n t i - f i e d i t was n o t i c e d t h a t the S waveform was q u i t e s i m i l a r to the P waveform i n the frequency range i n v e s t i g a t e d . Thus an a d d i t i o n a l check upon the i d e n t i f i c a t i o n of the f i r s t S motion c o u l d be obtained by matching peaks and troughs of the P and S waves. 4.4 D e t e r m i n a t i o n o f the R a d i a t i o n P a t t e r n . The n o n - r e p e t i t i v e nature of each o b s e r v a t i o n c o n t r o l l e d the e x p e r i m e n t a l procedure. I d e a l l y , to observe the r a d i a t i o n p a t t e r n from a source l i k e a f r a c t u r e , one should have a b a t t e r y of i d e n t i c a l instruments surrounding the source. However, f o r t h i s s e r i e s o f experiments o n l y two r e c o r d i n g channels were a v a i l a b l e . Moreover, the v a r i - a t i o n o f the c o u p l i n g o f i n d i v i d u a l d e t e c t o r s to the g l a s s makes i t seem u n l i k e l y that more than two channels c o u l d be used s u c c e s s f u l l y . The energy r e l e a s e d by each f r a c t u r e i s extremely v a r i a b l e . So f o r the d i r e c t p l o t t i n g of e i t h e r the P or S r a d i a t i o n p a t t e r n a m o n i t o r i n g d e t e c t o r i s necessary. T h i s monitor i s p l a c e d a t a f i x e d d i s t a n c e and a t a f i x e d angle w i t h r e s p e c t to the f r a c t u r e . The other d e t e c t o r i s p l a c e d a t v a r i o u s a n g l e s Q w i t h r e s p e c t to the f r a c t u r e . - 48 - The measured amplitudes are normalized w i t h r e s p e c t to the monitor. The c o o r d i n a t e system, as d e f i n e d b e f o r e , i s as f o l l o w s : the f r a c t u r e propagates i n the p o s i t i v e X d i r e c t i o n ; the d i r e c t i o n normal to the f r a c t u r e , i n the plane of the p l a t e , i s Z. The o r i g i n o f the c a r t e s i a n system i s a l s o the o r i g i n o f the p o l a r system. Angle Q i s measured i n an a n t i c l o c k w i s e manner from the p o s i t i v e X a x i s . I n the p o l a r system we d e f i n e ( f o l l o w i n g Knopoff and G i l b e r t ) outward r a d i a l motion as p o s i t i v e , and a n t i c l o c k - wise t a n g e n t i a l motion as p o s i t i v e , on the X - Z p l a n e . F i g u r e 4 shows the d e t a i l s of the co o r d i n a t e system and the d i r e c t i o n of p o s i t i v e u n i t v e c t o r s . The amplitude of P waves at 0 azimuth i s expressed as a non-dimensional r a t i o w i t h r e s p e c t to amplitude o f P a t 9 0 ° . For S waves the S @ / P Q amplitude r a t i o has been determined a t v a r i o u s azimuths. Knowing the P @ /Pgo r a t i o a t the p a r t i c u l a r azimuth i t i s p o s s i b l e to determine S @ / P g o f o r any 0 . A major problem has been the v a r i a b l e c o n t a c t between the g l a s s p l a t e and the two measuring t r a n s d u c e r s . Without any compensation f o r such unequal c o u p l i n g , a l a r g e s c a t t e r i n the values of P Q / P Q Q and S Q / P q r a t i o s was observed. A method has been de v i s e d which e l i m i n a t e s the e f f e c t o f unequal c o u p l i n g . L e t two transducers be designated A and B. F i r s t A i s used f o r r e c o r d i n g P Q , - 49 - and B f o r r e c o r d i n g P g o , from a f r a c t u r e F^. Next, without d i s t u r b i n g A or B, a d i f f e r e n t f r a c t u r e l o c a t i o n Fg i s used so that A now r e c o r d s P g o> and B r e c o r d s Pg> . Th i s c o n s t i t u t e s a s e t o f r e a d i n g s . The r a t i o s are P6 CA)/ Pqo (60 and P6 Ceo/ p q o(A) The mean 4 r [ P e W / P ^ C B ) + PeU»/P,.CA)] = P s / P , 0 i s now independent of unequal c o u p l i n g a t the c o n t a c t s o f tran s d u c e r s A and B and the g l a s s p l a t e . Each measured r a t i o , t h e r e f o r e , r e p r e s e n t s the mean o f two o b s e r v a t i o n s from two d i f f e r e n t f r a c t u r e s . The arrangement i s shown i n f i g u r e 14. Transducers A and B are o r i e n t e d so as to r e c o r d o n l y the cos y component of P r a d i a t i o n . The r a t i o o f course i s independent of y . The angles 0 , y and the s e p a r a t i o n s r , f , and d are shown i n the f i g u r e . The d i s t a n c e r between the f r a c t u r e o r i g i n and the tra n s d u c e r s i s 30 cms. From geometry / F j C D = ^_CF-jA + ICAF1 o r Y 2 - T - ® + y o r 1^ z * / 2 - 9 t h e r e f o r e f = % - ® / 2 - 50 - and j = r sin y and d = r «>s7 The q u a n t i t i e s ^ , d and f are c a l c u l a t e d f o r r = 30 cms and d i f f e r e n t Q and are shown i n Table 1 . As the S amplitude i s s m a l l a t 9 0 ° , the same arrangement cannot be used f o r S wave measurements. Besides the o r i e n t a t i o n o f the tr a n s d u c e r s a t an angle to the l i n e j o i n i n g the p o i n t of o b s e r v a t i o n to the f r a c t u r e o r i g i n would cause a s i g n i f i c a n t amount of p i c k up of P waves which w i l l i n t e r f e r e w i t h S wave o b s e r v a t i o n . However, i f we study the S © / P 0 r a t i o then the interchange of t r a n s - ducers i s a g a i n p o s s i b l e . D e s i g n a t i n g the trans d u c e r s as A and B, as be f o r e , we use A to r e c o r d P © and B to r e c o r d SQ , from the f r a c t u r e F-^. Next without d i s t u r b i n g A or B, a d i f f e r e n t f r a c t u r e l o c a t i o n F g i s used so t h a t A r e c o r d s S Q and B r e c o r d s P Q . We have the r a t i o s V B V P e C A ) and S e C A ) / p e ( B ) The mean g i v e s S Q / P Q , which i s independent of unequal c o u p l i n g . The arrangement f o r t h i s i s shown i n f i g u r e 1 5 . The s e p a r a t i o n r i s equal to 25 cms. 4 . 5 Determination of the F a r - F i e l d Region. The f i r s t motion theory of Knopoff and G i l b e r t i s v a l i d o n l y i n the f a r f i e l d r e g i o n , i . e . , a s u f f i c i e n t l y - 51 - l a r g e number of wavelengths away from the s o u r c e . I t was cons i d e r e d n e c e s s a r y t h e r e f o r e to determine whether the d i s t a n c e s o f 25 and 30 cms from the source are indeed w i t h i n the f a r f i e l d r e g i o n f o r P waves w i t h wavelengths of 5,5 cm. Because of the geometric spreading of energy the amplitude of two-dimensional waves i s p r o p o r t i o n a l to r " ^ i n the f a r f i e l d r e g i o n . Here r i s the d i s t a n c e o f the p o i n t o f o b s e r v a t i o n from the source. I n the near f i e l d r e g i o n the value of the a t t e n u a t i o n constant i s hi g h e r . of t r a n s d u c e r s A and B to e l i m i n a t e the e f f e c t o f unequal c o u p l i n g . Transducer A i s f i r s t used t o r e c o r d P G O a t 20 cms,and B i s used t o r e c o r d P Q Q a t r cms d i s t a n c e from f r a c t u r e F-j_. Next, without d i s t u r b i n g e i t h e r A or B , a new f r a c t u r e l o c a t i o n Fg i s used such that A now records PQQ a t r cms and B r e c o r d s PQQ a t 20 cms. The arrangement i s shown i n f i g u r e 16. Let us assume the amplitude i s p r o p o r t i o n a l to r 3 1 . Then, n o r m a l i z i n g our r e s u l t s to 20 cms we have the r a t i o s As b e f o r e , the arrangement i n v o l v e d interchange P,oCr) W D CO (A) r Pa0C=0)CA) 20 - 52 - or In n*c a < 0 20 A l e a s t square a n a l y s i s gave the value of n to be - 0.53 ± 0.08 ( standard e r r o r ) The p o i n t s are p l o t t e d on l o g l o g s c a l e i n f i g u r e 17. The value of n i n d i c a t e s t h at the r e g i o n r ^ 20 cms i s d e f i n i t e l y w i t h i n the f a r f i e l d r e g i o n . A s i m i l a r study, w i t h i d e n t i c a l arrangement, was made f o r P Q . The v a l u e f o r n i n t h i s case came to be - 0.45 i 0.09 ( standard e r r o r ) T h i s confirms the p r e v i o u s measurements. The data i s p l o t t e d on l o g l o g s c a l e i n f i g u r e 18. 4.6 Symmetry o f I n i t i a l F r a c t u r e s . As mentioned i n s e c t i o n 3.2, a t e s t determined t h a t I n i t i a l f r a c t u r e s are b i l a t e r a l i n n a t u r e . The f r a c t u r e i s i n i t i a t e d by i n d u c i n g thermal s t r e s s mostly at one end ( f i g u r e 9 ) . The o t h e r end i s heated o n l y to stop the f r a c t u r e from p r o p a g a t i n g too f a r . Therefore we had reason to suspect that there might be an unequal r a d i a t i o n between the A and the B s i d e s . We t e s t e d f o r symmetry by s i m u l t a n e o u s l y measuring amplitudes a t 45° - 53 - and 135° S waves from I n i t i a l f r a c t u r e s . The rough measurements of the r a t i o of ^^5/^4.5 came to be 0.96 1 0.08 (standard d e v i a t i o n o f the mean). Therefore we f i n d no evidence of b i l a t e r a l asymmetry of I n i t i a l f r a c t u r e s induced by our method. - 54 - CHAPTER V RESULTS 5.1 Gen e r a l . Energy from I n i t i a l and Extended f r a c t u r e s ranges from the audio i n t o the 200 kc/s r e g i o n . The eigenfrequency of the g l a s s p l a t e s d i d not permit us to observe and r e c o r d the audio f r e q u e n c i e s . But each f r a c t u r e i s s i g n a l l e d by a c l e a r l y a u d i b l e 'ping', i n d i - c a t i n g measurable energy i n the audio range. There i s some r e l a t i o n s h i p between the sharpness of the 'ping' sound and energy content i n the h i g h frequency 100 kc/s r e g i o n . F r a c t u r e s w i t h 'pings' o f h i g h audio f r e q u e n c i e s u s u a l l y show high e r amplitudes i n the 100 kc/s r e g i o n . Almost a l l f r a c t u r e s that produced very l o u d , but low audio frequency n o i s e ('bangs') showed no a p p r e c i a b l e amplitude i n the 100 kc/s r e g i o n . The P 'waveform' observed i s produced by e l i m i n - a t i n g a l l o f the s i g n a l except a narrow frequency band around the 100 kc/s r e g i o n . I t i s remarkable t h e r e f o r e to observe the r e p r o d u c i b i l i t y of the 'waveform'. The u s u a l P waveform c o n s i s t s of three extrema, two peaks and a trough. A l l f i r s t motion measurements have been made by a) measuring the hei g h t of the f i r s t extremum, or b) mea- s u r i n g the v e r t i c a l s e p a r a t i o n between the f i r s t and the - 55 - second extrema. The procedure (b) has been used o n l y when the f i r s t extremum i s very s m a l l or when (as f o r S waves) the exact a r r i v a l i s i n d i s t i n c t . The s i g n a l to n o i s e r a t i o i s h i g h e r i f procedure (b) i s employed, and the e r r o r s of measurement are l e s s . Measurements of the f i r s t S motion have not been so easy. I t i s not too d i f f i c u l t to f i n d the S wave, but d i f f i c u l t y i s experienced i n the i d e n t i f i c a t i o n of f i r s t S motion. The exact beginning of S i s obscured by p i c k up of s p u r i o u s P o s c i l l a t i o n s o f v a r i a b l e amplitude. I t appears t h a t the S waveform a t 100 kc/s resembles the P waveform a t t h i s frequency. Then the i d e n t i f i c a t i o n of f i r s t S motion i s simply a q u e s t i o n of l o o k i n g f o r the three extrema c h a r a c t e r i s t i c waveform of P wave. The f i r s t S motion, then, i s the f i r s t o f three extrema. I f the S waveform i s d i f f e r e n t from the P waveform by having a s m a l l f i r s t motion p r e c e d i n g the three extrema, then the f i r s t S motion i s extremely d i f f i c u l t to i d e n t i f y and measure. I f such i s indeed the case, our measurements, based on the three extrema waveform, w i l l be a l t e r e d i n magnitude as w e l l as s i g n . Such a p o s s i b i l i t y i s u n l i k e l y , as can be seen from some t y p i c a l P and S three extrema waveform shown i n f i g u r e s 19 and 20. D e f i n i t e a r r i v a l of S waves can be i d e n t i f i e d by the k i n k s that appear i n some t r a c e s . The u s u a l three extrema waveform f o l l o w s the k i n k s . However, - 56 - a l l S a r r i v a l s are not so d e f i n i t e and c l e a r , g i v i n g r i s e to the ambiguity of f i r s t motion. We mention t h i s here because the sense of the observed f i r s t S motion i s oppo s i t e to that p r e d i c t e d by Knopoff and G i l b e r t . The r a t i o s P 0 / P Q Q and S 0 / P ^ have been measured f o r v a r i o u s Q f o r both I n i t i a l and Extended f r a c t u r e s . The S r a d i a t i o n p a t t e r n can be determined from S 0 / P 9 O = P 9 / P90 x S © / P 0 The e r r o r s being compounded are much h i g h e r than f o r e i t h e r P e / P Q Q or SQ / P 0 . However, we c o u l d f i n d no d i r e c t means of d e t e r m i n i n g S e/PgQ r a t i o which would a l s o permit us to interchange transducers to e l i m i n a t e the e f f e c t of unequal c o u p l i n g . S c a t t e r i s observed i n s u c c e s s i v e measurements of the r a t i o s a t the same 0 . We use the standard d e v i - a t i o n f o r an i n d i v i d u a l o b s e r v a t i o n , and the standard d e v i a t i o n o f the mean as measure of s c a t t e r . The f o l l o w - i n g d e f i n i t i o n s o f these q u a n t i t i e s have been used Mean X Standard D e v i a t i o n (f x l n - 1 Standard D e v i a t i o n o f the mean 07.- £1 in - 57 - where Xj_ r e p r e s e n t s the n values of P Q /Pgo o r S@ / P Q a t the same 0 • I n g e n e r a l the standard d e v i a t i o n ranged 10 to 20 per cent o f the mean. The standard d e v i a t i o n o f the mean i s of course much lower. The data i s presented i n Tables I I to VII and F i g u r e s 23 to 26. T h e o r e t i c a l values a t each azimuth are g i v e n f o r comparison. 5.2 I n i t i a l F r a c t u r e s . The measured P ^ / P G O and S Q / P Q r a t i o s are shown p l o t t e d i n f i g u r e s 23 and 24 and t a b l e s I I and I I I . The standard d e v i a t i o n of the mean i s a l s o p l o t t e d as v e r t i c a l l i n e s as a measure of u n c e r t a i n t y . . The t h e o r e t i c a l curves, c a l c u l a t e d from the formulae o f Knopoff and G i l b e r t are shown f o r comparison. F o r P Q / P Q Q the agreement i s good f o r 30° ^ 0 < 9 0 ° . I n the 0 < 3 0 ° range the observed P 0 / P Q Q i s almost constant i n v a l u e . The S Q / P 0 measure- ments show reasonable agreement w i t h theory i n magnitude up to 0 = 45°. F o r & ^ 45° the measured magnitudes are s m a l l e r . The sense of S motion i s opp o s i t e that p r e d i c t e d by theory. C o n s i d e r i n g the disagreement i n both P © / P 9 Q r a t i o and the SQ / P 0 r a t i o we can conclude t h a t the S Q / P Q Q r a t i o s w i l l d i f f e r from the t h e o r e t i c a l values i n the 0 <^45° range. - 58 - 5 . 3 Extended F r a c t u r e s . We b e l i e v e that Extended f r a c t u r e s are u n i l a t e r a l d i s l o c a t i o n s of the type d e s c r i b e d by Knopoff and G i l b e r t as t h e i r model I I I . I f so, then the measured r a t i o s of P Q / P Q Q and S Q / P Q d i f f e r e d markedly from theory. There i s no r e v e r s a l of d i r e c t i o n o f motion a t P Q w i t h r e s p e c t to PQQ* as p r e d i c t e d by theory. A l l measured r a t i o s of P Q / P Q Q are p o s i t i v e , i n d i c a t i n g motion away from the source. By theory,' the motion should be towards the source i n the Q <̂  2 5 ° r e g i o n . The measured P Q / P Q Q maxima f a l l s i n the forward quadrant i n s t e a d of around Q - 100°. There- f o r e , except f o r Q = 90°, the n o r m a l i z i n g p o i n t , the measured va l u e s do not agree w i t h t h e o r e t i c a l v a l u e s . R e f e r r i n g to f i g u r e 2 6 , we see t h a t the S Q / P ^ r a t i o a l s o shows l a c k o f agreement w i t h theory. As P amplitude does not go to zero or become n e g a t i v e i n the f i r s t quadrant, there i s no s i n g u l a r i t y i n the measured value o f S 0 / P Q i n the f i r s t quadrant. As i n the I n i t i a l f r a c t u r e , the sense of S motion i s o p p o s i t e that g i v e n by t h e o r y . S e v e r a l d i f f i c u l t i e s were encountered i n measure- ments from the Extended f r a c t u r e s . F i r s t l y the amplitudes of waves was 2 - 5 times lower than the amplitude of waves from I n i t i a l f r a c t u r e s . T h e r e f o r e measurements had g r e a t e r s c a t t e r , due to the d e c r e a s i n g s i g n a l to n o i s e r a t i o . The second d i f f i c u l t y was that Extended f r a c t u r e s - 59 - tended to creep a s h o r t d i s t a n c e before going w i t h a 'bang', as i t were. The r e s u l t was that both P and S waves were emergent, and not sharp. In the 90°<C© <^180° quadrant the frequency was about h a l f of the frequency i n the forward quadrant, i n s p i t e o f the narrow f i l t e r bandwidth ( f i g u r e 22,) . The measurements i n the back quad- r a n t t h e r e f o r e are of dubious v a l u e . T h i s i s p a r t i c u l a r l y true o f S Q / P Q measurements. We have not presented any data f o r SQ/TQ measurements i n the 9O ° < ^ 0<^18O° r e g i o n . - 60 - CHAPTER VI DISCUSSION 6.1 G e n e r a l . Prom the r e s u l t s presented In the l a s t chapter i t can be r e a l i z e d t h a t our measured values d i f f e r c o n s i d e r a b l y from those of Knopoff and G i l b e r t . The disagreement c o u l d be due to v a r i o u s f a c t o r s , the major ones being whether our model corresponds to the type o f d i s l o c a t i o n contemplated by Knopoff and G i l b e r t i n t h e i r Model 3 d i s l o c a t i o n , and whether we have taken care to s a t i s f y the v a r i o u s assumptions i n the theory. From the d e t e r m i n a t i o n o f the a t t e n u a t i o n f a c t o r , we can be c o n f i - dent that our measurements have been made i n the f a r f i e l d r e g i o n , as envisaged i n the t h e o r y . The I n i t i a l F r a c t u r e d e f i n i t e l y corresponds to Model 3 of Knopoff and G i l b e r t f o r b i l a t e r a l p r o p a g a t i o n . The Extended F r a c t u r e should correspond to the u n i l a t e r a l p r o p a g a t i o n o f a d i s c o n t i n u i t y i n the d i f f e r e n c e of u z . But we can o n l y s t a t e w i t h c e r t a i n t y that the f r a c t u r e propagates o n l y one way, w h i l e there might be a b i l a t e r a l p r o p a g a t i o n of d i s l o c a t i o n . However, from our measurements i t appears that most of P and S energy i s r a d i a t e d i n the forward d i r e c t i o n from Extended F r a c t u r e s . The extreme asymmetry makes the p o s s i b i l i t y o f a b i l a t e r a l d i s l o c a t i o n - 61 - prop a g a t i o n from a u n i l a t e r a l f r a c t u r e u n l i k e l y . 6.2 High Frequency. The theory o f Knopoff and G i l b e r t i s o n l y v a l i d f o r f i r s t motions, i . e . the h i g h frequency s o l u t i o n s . But they have not g i v e n any c r i t e r i o n f o r determining whether a c e r t a i n frequency i s s u f f i c i e n t l y h i g h . Therefore we are f o r c e d to use our own i n t e r p r e t a t i o n o f what i s h i g h frequency. A most e m p i r i c a l c r i t e r i o n , v a l i d f o r s t a t i c sources, i s that the frequency i s h i g h i f the wavelength i s much s m a l l e r than the source dimensions. I n the present case the theory assumes a propagating p o i n t source. Hence any a r b i t r a r y wavelength can be shown to be l o n g e r than the source dimension, and t h i s c r i t e r i o n has no meaning. The i n t e r p r e t a t i o n of f i r s t motion i s f u r t h e r complicated by the f a c t that the theory assumes an i d e a l s t e p - f u n c t i o n displacement a t the source. I n the case of the a c t u a l f r a c t u r e the displacement probably r e t u r n s to zero a f t e r a shor t time. Our c r i t e r i o n f o r h i g h frequency i s as f o l l o w s . L e t X be the time over which f i r s t motion amplitude measurements are made on a s i g n a l . In our usage % would mean the time i n t e r v a l between a r r i v a l of the s i g n a l (P or S wave) and the second extrema. In f i g u r e 27 00*B i s the d i r e c t i o n o f f r a c t u r e p r o p a g a t i o n and A i s the p o i n t o f o b s e r v a t i o n . I n the i n t e r v a l the t i p of f r a c t u r e - 62 - moves from 0 to 0 ' , where 00' . The angle AOB i s 9 and the angle AO'B i s 0+ 60 . We contend that X> i s s u f f i c i e n t l y s m a l l i f the angle £9 i s n e g l i g i b l y s m a l l . I n the present case r- 3 0 T h i s <o0 i s an o u t e r l i m i t and c o n s t i t u t e s a s m a l l f r a c - t i o n of the t o t a l 0 . I f the time range i s c o n s i d e r e d a rough measure o f the time p e r i o d of the h i g h frequency, we get the corresponding frequency as 100 k c / s . 6.3 D i f f r a c t i o n E f f e c t s . The P@ /Pgo r a t i o shows marked disagreement w i t h theory o n l y i n the © ^ 3 0 ° range. T h i s c o u l d be due to r e f r a c t i o n through the hot spots or d i f f r a c t i o n a t the edges of the f r a c t u r e s . The e f f e c t o f hot spot has been shown to be s m a l l i n s e c t i o n 3.3. The r e f r a c t i o n e f f e c t s due to the hot spot c o u l d not be s i g n i f i c a n t as the e l a s t i c p r o p e r t i e s of t h i s type of g l a s s change on l y 3 per cent over the temperature range of 0° to 200°C. To determine i f the disagreement i s due to the d i f f r a c t i o n a t the edges, we measured the a t t e n u a t i o n constant f o r P a t 0 = 0 ° . The procedure and experimental s e t up i s s i m i l a r to that d e s c r i b e d i n s e c t i o n 4.5. The l e a s t squares d e t e r m i n a t i o n o f the exponent o f r gave the value - 63 - 0.45 ± 0.09 (standard e r r o r ) . T h i s i s equal to the a t t e n u a t i o n constant f o r P measured at 0 - 90°, w i t h i n the l i m i t s of e r r o r . Had the anomalous value of P © / P q Q f o r © < ^ 3 0 ° been due to d i f f r a c t i o n a t the edges the a t t e n u a t i o n would d i f f e r s i g n i f i c a n t l y from 1 / 2 . We can conclude t h e r e f o r e , that d i f f r a c t i o n p l a y s no s i g n i f i c a n t r o l e i n the measured value of P g / P g o and SQ / P Q r a t i o s . 6.4 Correspondence of F r a c t u r e s to Other D i s l o c a t i o n Models We have so f a r assumed that the f r a c t u r e s Induced by thermal s t r e s s i n g l a s s p l a t e s correspond to Model 3 of Knopoff and G i l b e r t . I t i s p o s s i b l e that the f r a c t u r e s s t u d i e d correspond i n s t e a d to a combination o f Model 3 and one or more of the other models. The theory s t a t e s that there must be a suddenly a p p l i e d d i f f e r e n c e i n the s t r a i n or displacement components on both s i d e s of the f r a c t u r e p l a n e . A d i s c o n t i n u i t y i n the d i f f e r e n c e o f u would correspond to Knopoff and G i l b e r t ' s Model 3. The symmetry o f c o n d i t i o n s on both s i d e s o f f r a c t u r e plane p r e c l u d e s a d i f f e r e n c e i n the other components of d i s p l a c e - ment. The same a p p l i e s to s t r a i n components. Any asymmetry t h a t e x i s t s i s due to random inhomogeneity i n the g l a s s and would be one of the reasons f o r the observed s c a t t e r i n the measured r a t i o s . We f a i l to see the p o s s i - b i l i t y o f a d i s c o n t i n u i t y i n the d i f f e r e n c e of any o t h e r - 64 - s t r a i n or displacement component. 6 . 5 V e l o c i t y of F r a c t u r e P r o p a g a t i o n . I n s e c t i o n 2 . 2 we contend that the v e l o c i t y of prop a g a t i o n o f f r a c t u r e i s a constant w i t h a va l u e of 0 . 5 . T h i s i s the maximum measured v a l u e . F r a c t u r e s have been observed to move a t slower v e l o c i t i e s , but we contend that t h i s c o u l d be the apparent v e l o c i t y due to the f a c t t h at the f r a c t u r e propagates i n short s p u r t s . We must examine the consequences of our assumption being i n c o r r e c t i n that the value of ̂  can range anywhere from 0 to oc . F o r ~$ —> 0, the term i n v o l v i n g ^ 3-N i s preponderant. The r a t i o P Q / P g Q — > 0 0 f o r a 1 1 ® except Q—> 9 0 ° . O b v i o u s l y t h i s i s not the r i g h t s o l u t i o n . The ex p r e s s i o n f o r u g i s Here a g a i n the term i n v o l v i n g f i s preponderant f o r —> 0 ° . The r a t i o S Q / P Q w i l l then vary as tan 9 Thus S Q / P @ r a t i o w i l l have a s i n g u l a r i t y around 0 - ? 9 O ° , c o n t r a r y to our measurements. - 65 - There I s a p o s s i b i l i t y t h a t the v e l o c i t y o f pro p a g a t i o n o f f r a c t u r e may momentarily r e a c h the value of the l o n g i t u d i n a l wave v e l o c i t y . I f so, the P^ / P g 0 curve f o r j 5 = cC w i l l f a l l between the curves f o r ^ = _^/S and ^ = oo . I t can be seen that the P^ / P g o curve s t i l l i s a very poor f i t f o r the measured p o i n t s f o r Extended f r a c t u r e s , even w i t h 5 = oC. 6.6 N o n - l i n e a r E f f e c t s . The theory o f Knopoff and G i l b e r t assumes that l i n e a r e l a s t i c i t y theory i s a p p l i c a b l e i n the r e g i o n immediately adjacent to the f r a c t u r e . There i s evidence to doubt t h i s . F i r s t l y the e x i s t e n c e o f the well-known r ~ s s i n g u l a r i t y i n the c o n c e n t r a t i o n o f s t r e s s a t the t i p of a cr a c k s t r o n g l y suggests the p o s s i b i l i t y o f a p l a s t i c y i e l d zone i n the immediate neighbourhood of the t i p of the c r a c k . Secondly experimental photographs presented i n Schardin (1959) show a s e t of propagating f r a c t u r e s w i t h 'coronas' a t the t i p . I n the same photographs there i s a l s o a s e t of s t a t i c f r a c t u r e s , but i n these the coronas are absent. The corona i s i n d i c a t i v e of a p l a s t i c zone a t the t i p . The r a d i a t i o n p a t t e r n from a pr o p a g a t i n g f r a c t u r e would t h e r e f o r e be complicated by the zone of p l a s t i c i t y . I t i s c o n c e i v a b l e that the i n i t i a l d i s t u r b a n c e i s a p l a s t i c wave which i s converted i n t o an e l a s t i c wave some d i s t a n c e - 66 - from the f r a c t u r e . We are c e r t a i n that our measurements are i n the f a r f i e l d r e g i o n of the source, where l i n e a r e l a s t i c theory i s a p p l i c a b l e . But we have no i d e a o f the extent to which the a n - e l a s t i c zone a f f e c t s our measure- ments. I t i s p o s s i b l e that the disagreement between our measurements and the theory of Knopoff and G i l b e r t c o u l d be e x p l a i n e d by t a k i n g i n t o c o n s i d e r a t i o n the a n - e l a s t i c e f f e c t s i n the neighbourhood of the f r a c t u r e . 6.7 A c t u a l Correspondence of F r a c t u r e i n P l a t e to F r a c t u r e i n Three Dimensions. I n s e c t i o n 1.3 we mentioned that on the X - Z plane the f i r s t motion comes from an i n f i n i t e s i m a l p a r t of the f r a c t u r e f r o n t . I n a c t u a l case the l e n g t h of the s e c t i o n i s not i n f i n i t e s i m a l , and we s h a l l now d i s c u s s - the e f f e c t o f f i n i t e l e n g t h on our measurements. F i g u r e 28 shows the i n f i n i t e f a u l t f r o n t p r o p a g a t i n g i n the X d i r e c t i o n and extending to i n f i n i t y a long the Y d i r e c t i o n . Because of symmetry, we s h a l l o n l y d i s c u s s the p o s i t i v e Y d i r e c t i o n . The plane o f measurement (which i n our case i s the plane of the p l a t e ) i s X - Z. The p o i n t o f obser- v a t i o n i s A, and OA makes angle 0 w i t h the X a x i s . AC and AD are p r o j e c t i o n s o f AO on the X and Z axes r e s p e c t i v e l y . D u r i n g the time o f measurement ( X ) o f f i r s t motion a t A, o n l y the element 00' c o n t r i b u t e s to the - 67 - f i r s t motion a t A. Hence f o r P waves, A O >/o< = X Now (AO' - AO) = ^OC I n our case OC = 5500 m/s AO = 30 cms. T h e r e f o r e , cLX - 2.75 cms. .2 v.* and 00' = (AO' - AO )' = (33 - 30 ) s = 13.7 cms, By the argument presented i n s e c t i o n 1.3 our measurements, though made on a p l a t e 3 mm t h i c k , are e q u i v a l e n t to s i g n a l s from a p a r t o f the f a u l t f r o n t 28 cm l o n g , which i s c e r t a i n l y not i n f i n i t e s i m a l . At f i r s t s i g h t i t appears as i f a l l our measurements are i n v a l i d . T h i s i s not so. R e f e r r i n g to formula (2 - 22) we see that the o n l y f a c t o r s that c o u l d a f f e c t the amplitudes of P (or S) are changes i n , and r . We can n e g l e c t the change i n l e n g t h r , because the a t t e n u a t i o n depends on , which i s s m a l l . We have to determine Tx and f o r AO', w i t h r e s p e c t to a new c o o r d i n a t e system at 0'. D e f i n i n g these as Tx. and Tz we have, from the f i g u r e 28 - 6 8 - O C OC AO T 1 £ A O AO' ^ A O ' OD - OD x AO_ _ ̂  2>p_ AO AO' * 3?> AO Thus the end of the c o n t r i b u t i n g segment o f the f a u l t f r o n t has a d i r e c t i o n c o s i n e d i f f e r i n g from that o f AO by 9 per cen t . T h i s i s l a r g e : but i t must be remembered t h a t the s i g n a l s from the end of the c o n t r i b u t i n g segment of the f a u l t f r o n t a r r i v e at the t a i l end of the time of measure- ment £ and have a r e l a t i v e l y minor e f f e c t on the f i r s t motion. Therefore i n the a c t u a l case the e r r o r i s much lower. The P Q / P Q Q r a t i o f o r 0 ° from Extended F r a c t u r e s g i v e s t h e o r e t i c a l value of 0 . 0 1 , when the d i r e c t i o n c o s i n e s i n the formula (2 - 22) are reduced by 9 per cen t . I t s t i l l d i f f e r s c o n s i d e r a b l y from the measured r a t i o of 0 . 5 6 . The e f f e c t of summation of the c o n t r i b u t i o n o f the whole segment w i l l be s m a l l . The value f o r t f o r S wave measurements i s 1 0 / * 5 , and s i m i l a r c a l c u l a t i o n s can be done. In earthquake seismology, the time p e r i o d of e l a s t i c waves i s o f the order o f 1 second. T h i s corresponds to our p e r i o d o f the order o f 10 s. The s c a l i n g f a c t o r 5 i s thus 10 . T h e r e f o r e , 1 cm on the model corresponds to 1 km i n the f i e l d . Our r e s u l t s , then, are e q u i v a l e n t to - 69 - r e c o r d i n g on the s u r f a c e at 30 km d i s t a n c e from a f a u l t 26 km deep. During the time of r e c o r d i n g of f i r s t motion the f a u l t advances about 3 km. We r e a l i z e that 30 km i s too c l o s e to the source, s p e c i a l l y i n earthquake seismology: but as we have recorded i n the f a r f i e l d our r e s u l t s should be v a l i d over much l o n g e r d i s t a n c e s , a t l e a s t up to 100 km. Another i n t e r e s t i n g i n t e r p r e t a t i o n of our r e s u l t s i s p o s s i b l e . The c o n t r i b u t i n g segment 00', ( F i g . 28) elongates w i t h time a t i n f i n i t e v e l o c i t y . I n stead of c o n s i d e r i n g the t i p of the f r a c t u r e i n the X-Z plane as the moving source, i t i s p o s s i b l e to c o n s i d e r the t i p of the c o n t r i b u t i n g segment i n the Y % plane as the moving source. The p r o p a g a t i o n of the f r a c t u r e i n the p l a t e , then, simply means t h a t the source i s widening w i t h i n c r e a s i n g X, • The r a t e of f r a c t u r e p r o p a g a t i o n i s s m a l l compared w i t h the v e l o c i t y of e l o n g a t i o n o f the c o n t r i b u t i n g segment. An interchange of c o o r d i n a t e s i s now necessa r y . R e f e r r i n g to f i g u r e s 2 and 28, we now c o n s i d e r the d i r e c t i o n of elonga- t i o n o f the c o n t r i b u t i n g segment 0'00'' as the X d i r e c t i o n , and the plane o f the p l a t e as the Y-Z p l a n e . The f r a c t u r e i n the p l a t e now propagates i n the Y d i r e c t i o n . Our r e s u l t s can be r e i n t e r p r e t e d as measurements of the f i r s t motion on the Y-Z plane from a d i s c o n t i n u i t y i n u z , pr o p a g a t i n g a t i n f i n i t e v e l o c i t y i n the X d i r e c t i o n . The t h e o r e t i c a l e x p r e s s i o n s f o r u and u are unchanged i f we take ^ = 6 0 - 69a - and measure 6 from the Y a x i s . The curves f o r P d /Pg 0 and S Q / P & are the same as shown i n f i g u r e s 23 to 26, except t h a t they now r e p r e s e n t v a l u e s on the Y-Z pla n e . The e f f e c t of widening source (due to the p r o p a g a t i o n of f r a c t u r e i n the p l a t e , now i n the Y-Z plane) w i l l be s m a l l . However, even w i t h t h i s i n t e r p r e t a t i o n , our measurements s t i l l show c o n s i d e r a b l e d i s c r e p a n c y w i t h the theory. - 70 - CONCLUSIONS We have s t u d i e d the l o n g i t u d i n a l and t r a n s v e r s e e l a s t i c wave r a d i a t i o n i n the 100 kc/s range from propagating t e n s i l e f r a c t u r e s i n g l a s s p l a t e s . For I n i t i a l ( B i l a t e r a l ) F r a c t u r e s the P r a d i a t i o n i s a maximum i n a d i r e c t i o n normal to the f r a c t u r e ; the P wave amplitude drops s t e a d i l y u n t i l 9 = 50°, a f t e r which the amplitude i s rea s o n a b l y constant a t about 35 per cent of the P amplitude a t 90°. For Extended ( U n i l a t e r a l ) F r a c t u r e s , the P amplitude maxima i s i n the forward quadrant; the P wave amplitude f o r 0 = 0° ( i n the d i r e c t i o n o f propagation) i s h i g h , being about h a l f the P amplitude a t 90°. I n a l l cases the f i r s t P motion i s away from the source. For S waves, the f i r s t motion i s away from the f r a c t u r e plane, towards the normal to the f r a c t u r e . The experiment e s s e n t i a l l y c o n s t i t u t e s a p a r t i a l t e s t o f the F i r s t M otion Theory of Knopoff and G i l b e r t (1960), f o r t h e i r Case 3. For I n i t i a l F r a c t u r e s the measured f i r s t P motion amplitude d i f f e r s from the t h e o r e t i c a l v alue i n magnitude o n l y f o r 9 <̂  30° range, For Extended F r a c t u r e s the measured magnitudes d i f f e r from the p r e d i c t e d values a t a l l p o i n t s , except 6 = 90° „ F o r S waves, the most s i g n i f i c a n t d i f f e r e n c e i s i n the sense of the f i r s t motion. The measured f i r s t S motion - 71 - lias a sense opposite that p r e d i c t e d by theory. We suspect that the d i s c r e p a n c y between the experiment and theory may be due to the n e g l e c t of the n o n - l i n e a r e l a s t i c e f f e c t s a t the source of the d i s t u r b a n c e ( i . e . the f r a c t u r e t i p ) . I t i s impossible to make a p r e c i s e comparison of experiment and theory s i n c e the d u r a t i o n o f what i s d e s c r i b e d as f i r s t motion i n the theory i s not p r e c i s e l y d e f i n e d . N e v e r t h e l e s s , i t seems l i k e l y that the d i s c r e p a n c i e s between the experiment and theory d e s c r i b e d above are r e a l d i f f e r e n c e s i n what s e i s m o l o g i s t s would c a l l f i r s t motions. I f t h i s i s so, we conclude that the theory i s not adequate to d e s c r i b e a l l the o b s e r v a t i o n s . Although we have t e s t e d o n l y one p a r t i c u l a r case out of e i g h t independent d i s l o c a t i o n models proposed by Knopoff and G i l b e r t , we f e e l the d i s c r e p a n c i e s are so s e r i o u s as to cadt doubt upon the u s e f u l n e s s o f the theory i n other cases. - 72 - GAS FLAME SCRATCH HORIZONTAL JLASS PLATE F i g . 1. Method of a p p l y i n g gas flame to g l a s s p l a t e , The g l a s s p l a t e i y h o r i z o n t a l . F i g . 2. I n f i n i t e l y l o n g f r a c t u r e i n t h r e e - d i m e n s i o n a l body. The f r a c t u r e f r o n t i s p a r a l l e l to the Y a x i s and propagates i n the X d i r e c t i o n . The displacement i s i n the Z d i r e c t i o n . F i g . 3. I n f i n i t e s i m a l p a r t of f r a c t u r e that c o n t r i b u t e s to f i r s t motion a t any p o i n t on the X - Z pl a n e . - 73 - Z X D I R E C T I O N O F F R A C T U R E P R O P A G A T I O N F i g . 4. Coordinate system and d i r e c t i o n o f p o s i t i v e u n i t v e c t o r s . - 74 - 1.5 k*0.9/3 5=0.8/9 - / / A / = 0.5/3 0.5 1 1 0 30 60 90° —> e F i g . 5. D i s t r i b u t i o n of <5Q w i t h 6 ( a f t e r J o f f e , 1 9 5 1 ) . F i g . 6. F i g u r e shows volume V bounded by s u r f a c e s S'. The p o s i t i v e normal n to S' i s outward. I n the l i m i t i n g case the outer S' goes to i n f i n i t y and the i n n e r s u r f a c e s h r i n k s to the f a u l t s u r f a c e . - 75 - O O CM L_ o O O O F i g . 7. D i s t r i b u t i o n o f temperature w i t h r a d i a l d i s t a n c e on a g l a s s p l a t e . The c e n t r a l r e g i o n of 0.5 cm r a d i u s i s suddenly r a i s e d to a temperature of 200° C a t time t = 0. The f i g u r e shows temperature d i s t r i b u t i o n a f t e r 45 seconds, - 76 - E •Si ( S i D q ) F i g . 8. S t r e s s d i s t r i b u t i o n i n the g l a s s p l a t e because of the temperature d i s t r i b u t i o n shown i n f i g u r e 7 . - 77 - S C R A T C H F i g . 9. Method of i n d u c i n g I n i t i a l F r a c t u r e s . The average l e n g t h o f I n i t i a l F r a c t u r e s i s 3 cms. E X I S T I N G B F R A C T U R E F i g . 10. Method of extending s h o r t f r a c t u r e s , C E B D , i i. c a * E X I S T I N G F R A C T U R E F i g . 11. Method of extending l o n g f r a c t u r e s . - 78 L O W F R E Q . O S C I L L A T O R P R E - A M P L I F I E R TRIGGER INPUT P U L S E G E N E R A T O R HIGH PULSE S O U R C E T R A N S D U C E R R E C E I V I N G T R A N S D U C E R TRIGGER TAKE-OFF VOLTAGE TAKE-OFF V F I L T E R SIGNAL INPUT TRIGGER INPUT A M P L I F I E R O S C I L L O S C O P E A N D C A M E R A F i g * IS. Schematic c i r c u i t f o r measuring P and S wave v e l o c i t i e s i n g l a s s p l a t e s . - 79 - M O V E A B L E D E T E C T O R F I L T E R F I L T E R T R I G G E R D E T E C T O R F R A C T U R E POINT OFORIGIN A M P L I F I E R GLASS PLATE M O N I T O R I N G \ D E T E C T O R SIGNAL INPUTS TRIGGER, INPUT S I N G L E S W E E P O S C I L L O S C O P E A N D C A M E R A F i g . 13. Schematic c i r c u i t f o r s t u d y i n g r a d i a t i o n from f r a c t u r e s i n g l a s s p l a t e s . - 80 - F R A C T U R E F i g . 14. Transducer and f r a c t u r e p o s i t i o n s f o r determina- t i o n of P 9 /P90 r a t i o . Displacement i n the d i r e c t i o n of arrows a t A and B causes upward swing of o s c i l l o s c o p e t r a c e . F R A C T U R E S F i g . 15. Transducer and f r a c t u r e p o s i t i o n s f o r determina- t i o n of S e / P Q r a t i o . The angle 6 may be r e p l a c e d w i t h 2 7̂  - Q f o r s m a l l 6 . - 81 - j i_ 5 6 7 8 9 1 r/20 F i g . 17. A t t e n u a t i o n o f P waves a t 90 from I n i t i a l F r a c t u r e s . The slope i s n. The dashed l i n e s are n •+ the standard e r r o r . F R A C T U R E S |e r ^ F i g . 16. Transducer and f r a c t u r e p o s i t i o n s f o r the d e t e r m i n a t i o n of the f a r - f i e l d r e g i o n . - 82 - F i g . 18. A t t e n u a t i o n o f P waves a t 0° from I n i t i a l F r a c t u r e s . - 83 - DESCRIPTION OF FIGURES 19, 20, 21 and 22. Traces of a c t u a l r e c o r d s . Each f i g u r e shows f o u r t r a c e s on which amplitude measurements are necessary f o r a s i n g l e P@ /Pq 0 or S Q / P e r a t i o measurement. The l e t t e r A or B i n the parentheses shows the p a r t i c u l a r transducer used to r e c o r d the p a r t i - c u l a r t r a c e . The top two t r a c e s are r e c o r d s from a f r a c t u r e F.. . The lower two t r a c e s are from a f r a c t u r e F w i t h the transducer f u n c t i o n s interchanged (see f i g u r e s 14 and 15). In f i g u r e s 19, 20 and 21 the t y p i c a l three extrema waveform o f P i s seen. I n the second t r a c e o f f i g u r e 20, a k i n k s i g n i f i e d the a r r i v a l o f the S wave, which i s followed by a three extrema waveform. The kink i s absent i n the t h i r d t r a c e , and o n l y two of the three extrema are seen as the t r a c e went o f f s c a l e . R e f e r r i n g to f i g u r e s 14 and 15, we see t h a t the f i r s t P motion i s always outwards, i n f i g u r e s 19, 20 and 21. The f i r s t S motion i s always away from the f r a c t u r e , towards the normal to i t . F i g u r e 22 shows the d i f f i c u l t y o f i d e n t i f y i n g S waves from Extended f r a c t u r e i n the r e a r quadrant. The frequency i s low, the exact a r r i v a l s are i n d i s t i n c t , and - 84 - amplitude measurements are d i f f i c u l t to make. There i s a h i g h e r n o i s e l e v e l i n f i g u r e s 21 and 22 because the s i g n a l s from Extended f r a c t u r e s are a m p l i f i e d 2 ,5 times more than s i g n a l s from I n i t i a l f r a c - t u r e s . - 85 - (See d e s c r i p t i o n on page 83) _ 86 - p 3 o ( A » P I 5 8 ( A ) SI58 ( B » P90 (A> © 20 AO ^ p 3 0 < B > Extended R 3 Q 90 S.58 <A> p . 5 e » ) Extended F i g . 21. r " F i g . 22. (See d e s c r i p t i o n on pages 83-84) - 87 20 40 60 80 100 120 [40 \60 180 9 • F i g . 23. P l o t of measured P Q /Pgn r a t i o s from I n i t i a l F r a c t u r e s , Each p o i n t i s the mean o f s e v e r a l measurements. The v e r t i c a l l i n e r e p r e s e n t s standard d e v i a t i o n o f the mean. The t h e o r e t i c a l curve i s shown f o r comparison. -0.5 F i g . 25. P l o t of measured ? Q/^QQ r a t i o s from Extended F r a c t u r e s . 6 - 88 - J i g . 26, P l o t of measured Sg,/P e r a t i o s from Extended f r a c t u r e s , 89. P i g . 27. F i g u r e showing change of 6 with p r o p a g a t i o n of the f r a c t u r e . 90. F i g . 28. F i g u r e shows c o n t r i b u t i n g segment of i n f i n i t e f r a c t u r e f r o n t . - 91 - TABLE I (see F i g u r e 14) 0 y d f 0° 45° 21.1 21.1 10° 40° 23 19.3 22.5° 33.8 25 16.7 30° 30 26 15 45° 22.5 27.75 11.5 60° 15 28.9 7.8 90° 0 30 0 - 92 - TABLE I I P 0 /Pgo R a t i o f o r I n i t i a l F r a c t u r e s Standard Standard 0 n P e / P g Q Mean ' D e v i a t i o n Theory D e v i a t i o n o f the Mean 0 5 0.30 0.33 0.07 0.03 0.09 0.31 0.32 0.26 0.45 10 5 0.26 0.33 0.06 0.03 0.13 0,34 0.42 0.29 0.36 22.5 5 0.43 0.40 0.05 0.02 0.23 0.47 0.36 0.42 0.34 30 5 0.37 0.35 0.04 0.02 0.31 0.30 0.34 0.41 0.32 45 5 0.55 0.60 0.07 0.03 0.55 0.66 0.65 0.50 0.62 60 5 0.98: 0.89 0.07 0.03 0.77 0.92 0.83 0.90 0.80 90 1.00 N o r m a l i z i n g P o i n t 1.00 - 9 3 - TABLE I I I s 9 /P@ R a t i o s f o r I n i t i a l F r a c t u r e s Standard „ , Standard D e v i a t i o n m 1 0 n S 6/P 6 Mean D e v i a t i o n of the Theory Mean 22.5 8 2.40 3.05 1.30 0.46 - 4.59 2.82 1.83 1.82 2.28 3.85 5.48 4.13 30 9 2.67 2.73 0.78 0.26 - 4.16 2.62 2.00 2.18 3.27 4.02 2.19 3.74 1.85 45 5 2.79 2.57 0.57 0.25 - 2.72 1.94 2.05 3.30 2 75 90 4 - o!56 - 0.07 0.33 0.16 0.00 0.12 0.15 0.00 - 94 - TABLE IV P@ / P g 0 R a t i o f o r Extended F r a c t u r e s Q. , , Standard 9 n Pe/Pqn M e a n ^ a n a a r a D e v i a t i o n Theory U 9 9 0 D e v i a t i o n o f t h e Mean 0 30 45 90 135 150 180 3 4 0.55 0.52 0.60 0.84 0.97 1.21 1.12 1.03 1.31 1.02 1.00 0.45 0.54 0.46 0.23 0.16 0.29 0.32 0.15 0.00 0.08 0.56 1.04 1.12 0.04 0.16 0.16 0.25 0.08 0.07 0.07 0.02 0.08 0.09- N o r m a l i z i n g P o i n t 0.48 0.05 0.03 0.04 0.04 - 0.19 0.07 0.35 1.00 0.75 0.56 0.38 - 95 - TABLE V S Q /T?Q Ratios f o r Extended F r a c t u r e s Standard Standard 0 n Mean D e v i a t i o n D e v i a t i o n Theory o f the Mean 22.5 7 5.00 3.14 4.83 2,70 4.10 4.63 3,03 3.92 0.95 0.36 43.52 30 5 3.36 3.50 , 3.09 4.03 3.34 3.46 0.35 0.16 - 24.33 45 4 2.60 1.80 1.80 2.60 2.20 0.46 0.23 - 5.55 90 4 - 0.36 0.10 0.31 - 0.50 - 0.11 0.38 0.19 - 0.62 135 ) 150 ) 158 ) No a c c e p t a b l e r e c o r d s . - 96 - TABLE VI C a l c u l a t e d S Q / P Q Q R a t i o s f o r I n i t i a l F r a c t u r e s Standard 0 Sg /P Standard D e v i a t i o n Theory 9 0 D e v i a t i o n o f the Mean 22.5 1.22 1.35 0.48 - 1.05 30 0.96 0.82 0,28 - 1.29 45 1.54 0.64 0.28 - 1.46 90 - 0.07 0.33 0,16 0.00 TABLE V I I C a l c u l a t e d S e / P Q 0 R a t i o s f o r Extended F r a c t u r e s e S 6 / p90 Standard Standard . J_. D e v i a t i o n D e v i a t i o n o f t h e cr Mean Theory 30 45 90 3,60 2.46 0*11 0.51 0.62 0.38 0.24 0.32 0.19 - 1.61 - 1.93 - 0.62 The f o l l o w i n g r e l a t i o n s have be/en used: S e/R Se - 97 - APPENDIX I STRESS FIELD OF AN AXIALLY SYMMETRICAL TEMPERATURE GRADIENT IN A THIN PLATE I t i s o f some i n t e r e s t to c a l c u l a t e the s t r e s s d i s t r i b u t i o n i n a t h i n p l a t e which i s heated over !a s m a l l c i r c u l a r a r e a . C l e a r l y the temperature f i e l d w i l l be a x i a l l y symmetric. We use T to denote the temper- at u r e above the ambient room temperature and to denote the l i n e a r c o e f f i c i e n t o f thermal expansion. The s t r e s s - s t r a i n r e l a t i o n i s g i v e n by the Duhamel-Neumann law, which, s u b j e c t to the assumption of plane s t r e s s ( G\ = 0) may be w r i t t e n as 1-1 1-2 Two of the e q u i l i b r i u m equations are s a t i s f i e d i d e n t i c a l l y by v i r t u e of the f o r e g o i n g assumptions; the other e q u a t i o n i s dr r - 98 - Th i s equation i s s a t i s f i e d by the s t r e s s f u n c t i o n (p i f r = r a n d cr~ - d CD. 9 d r 1-4 The c o m p a t i b i l i t y equation f o r r o t a t i o n a l symmetry i s r e d r + £ r ^ O 1-5 S u b s t i t u t i n g I - l , I-E, and 1-4 i n t o equation 1-5 we get d r 1 r dr " r 1 - " I E d r d T dr 1-6 I n t e g r a t i n g the above, we o b t a i n T r r 1-7 R e p l a c i n g (J) i n 1-4 w i t h e x p r e s s i o n 1-7 we have T r dr -v J1L + J S * . cr - J J L ur - -> 1-8 2 £t 1-9 - 99 - For f i n i t e s t r e s s e s a t the ce n t e r Kg must be z e r o . The normal s t r e s s must v a n i s h as r - > » . Thus f o r any- normal temperature d i s t r i b u t i o n ( i . e . T — ? 0, as r — ? 0 0 ) The f i n a l e x p r e s s i o n s f o r CTR and 0~Q are Tr 4r i _ i i r 1 Jo 1 Tr dr j 1-12 To evaluate the i n t e g r a l s , we need determine T as a f u n c t i o n o f r . F o r t h i s we use the computations of Jaeger (1955) and C a r l s l a w and Jaeger (1959). Jaeger has gi v e n the s o l u t i o n s f o r the case i n which an i n n e r r e g i o n of r a d i u s 'a' i s h e l d a t a constant temperature f o r time t ^ O and g i v e s t a b l e s f o r the v a r i a t i o n of T a g a i n s t r f o r v a r i o u s time t . Ex p e r i m e n t a l measurements made upon a g l a s s p l a t e heated by a gas flame f o r approximately 45 seconds i n d i c a t e d 'a T to be about 0.5 cms and T about 200° C. I n i t i a l f r a c t u r e s s t a r t between 30 seconds and 60 seconds a f t e r the a p p l i c a t i o n of a gas flame to the p l a t e . We have used the mean value of 45 seconds f o r our c a l c u l a t i o n s . The p h y s i c a l p r o p e r t i e s of the;'glass used are - 100 - Thermal c o n d u c t i v i t y - 0.0028 c a l / s e c cm C -7 C o e f f i c i e n t of L i n e a r expansion - 85 x 10 S p e c i f i c heat - 0.20 cal/gm °C D e n s i t y - 2.5 gm/ cm D i f f u s i v i t y - 5.6 x IO" 3 cm 2 /sec We assume a u n i f o r m temperature from r = o to r = 0.5 cm. The v a r i a t i o n o f T ( r ) versus r f o r t = 45 sees i s given i n f i g u r e 7; the v a r i a t i o n o f <TR and CTQ w i t h r a t t = 45 sees i s g i v e n i n f i g u r e 8, i n the main t e x t . - 101 - APPENDIX I I PLATE WAVES O l i v e r , P r e s s and Ewing (1954) f i r s t noted that i t would be p o s s i b l e to use e l a s t i c waves i n t h i n p l a t e s to study problems of s e i s m i c wave p r o p a g a t i o n . The p r i n c i p a l r e s u l t o f t h e i r study was t h a t the symmetric p l a t e wave i n a t h i n p l a t e was the analog of the d i l a t a t i o n a l wave i n a s o l i d . There i s , however, a s t r i n g e n t requirement that the p l a t e t h i c k n e s s be s u f f i c i e n t l y s m a l l compared w i t h the wavelength s t u d i e d . E x t e n s i v e i n v e s t i g a t i o n s o f the v i b r a t i o n s o f a t h i n p l a t e have been p u b l i s h e d by R a y l e i g h (1889), Lamb (1917), T o l s t o y and U s d i n (1953), O l i v e r , P r e s s and Ewing (1954), and Ewing, J a r d e t z k y and Press (1957). We reproduce here those r e s u l t s which are important i n the problem s t u d i e d here. The f o l l o w i n g n o t a t i o n has been used: u,v,w displacement i n the X, Y, Z d i r e c t i o n s r e s p e c t i v e l y . ^lc>£ e t c . s t r e s s a c t i n g i n the X d i r e c t i o n on a plane normal to the Y a x i s . ® d i l a t a t i o n - 102 - (p and I f - displacement p o t e n t i a l s p - d e n s i t y ^ i f*' - Lame's constants p p L e t us c o n s i d e r a t h i n p l a t e bounded by the planes z = ± h, w i t h the X and Y axes i n the median plane of the p l a t e . The t h i c k n e s s o f the p l a t e i s , then, 2h. The p l a t e can be considered I n f i n i t e i n exten t , w i t h the bounding s u r f a c e s f r e e of normal and t a n g e n t i a l s t r e s s e s . F o l l o w i n g R a y l e i g h (1889) we assume t h a t a l l p e r i o d i c f u n c t i o n s i n the s o l u t i o n s o f the equations of motion ikrr i n v o l v e x o n l y through e and y does not appear a t a l l , T h i s i m p l i e s t h a t w h i l e the displacement i n the y d i r e c - t i o n i s f i n i t e , — - 0. We w i l l have two s e t s o f s o l u t i o n s : I , u and w are f i n i t e and v v a n i s h e s . I I . v i s f i n i t e and u and w v a n i s h . Case I . Two dimensional symmetric and a n t i - symmetric v i b r a t i o n s o f a t h i n p l a t e . The displacements can be w r i t t e n i n the form U - — - i i-L , w - -T-L .+ _r_L i i - i - 103 - where the p o t e n t i a l s Cp and y are s o l u t i o n s o f the wave equations ^ a t 1 r be II-2 The boundary c o n d i t i o n s are that the s t r e s s e s must v a n i s h a c r o s s the f r e e s u r f a c e . Thus 2 2 A G + 2 ^ ^ = 0- duo- c>u,\ d*. c)2 by c>2 = 0 > at 2 = ±V» n _ 3 We assume a s o l u t i o n o f the form (p r (/\ &>oK Da- + g> cosK 0*) e i (yit - kx.) where \ ) \ k x - , K2"- k* , = t£ y - co - 104 - On s u b s t i t u t i n g equation I I - 4 i n I I - l and I I - 3 , we o b t a i n the equations 4 I I - 5 tosr\V\ \>h t a n k Oh 4 ^ ' I I - 6 The c o e f f i c i e n t s A and D can be separated from B and C. Now we can c o n s i d e r a motion symmetric w i t h r e s p e c t to plane z *> 0 which i s g i v e n by ^ (cot -*x) II-7 F o r waves l o n g compared w i t h the t h i c k n e s s 2h, the products kh, v h , Oh may be c o n s i d e r e d s m a l l . The equation I I - 6 reduces to ( k x ) - 4- K V = 0 n_ 8 - 105 - From t h i s we o b t a i n the v e l o c i t y o f p l a t e l o n g i t u d i n a l waves as Therefore Cu r 4 ' I I - 9 The anti-symmetric motion i s g i v e n by the f u n c t i o n s , , t ( t o t - k*0 (p - A sunh \)2: e ' Y - D cosh \) 2. e Antisymmetric motion (otherwise known as f l e x u r a l waves) i n v o l v e s bending of the p l a t e . For waves l o n g compared w i t h 2h the equation II-6 reduces t o 11-11 These waves are d i s p e r s i v e , the phase v e l o c i t y d e c r e a s i n g to zero f o r long wavelengths. Case I I . Another s o l u t i o n i n v o l v e s o n l y motion i n the plane of the p l a t e . Thus w = 0. The same boundary c o n d i t i o n s I I -3 a p p l y . Consider the equation o f motion - 106 - 1 > . f — x - — + , M ^ 11-12 As ^ © i s zero, we f i n d t h a t the displacement v i t s e l f s a t i s f i e s the wave equ a t i o n . The equation i s at 1 P The v e l o c i t y i s g i v e n by \t± which i s the shear wave 4 f> v e l o c i t y i n i n f i n i t e s o l i d s . The s o l u t i o n i s of the form - ( P si/nn ^2 •+ Q cosh u z j e 11-13 To s a t i s f y the boundary c o n d i t i o n s o f ^ = o e i t h e r P s O , i n which case V - Q c - o s ^ 11-14 or Q = 0, whence V =. P \ ) V 11-15 O l i v e r , Press and Ewing (1954) have e s t a b l i s h e d the equivalence between the l o n g wave approximation of - 107 - the f i r s t p l a t e mode and d i l a t a t i o n a l waves i n a t h i n p l a t e . Thus, i t has been shown that shear and d i l a - t a t i o n a l waves e x i s t i n t h i n p l a t e s . 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