UBC Theses and Dissertations

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UBC Theses and Dissertations

Radiation from tensile fractures. Mansinha, Lalatendu 1962-12-31

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RADIATION FROM TENSILE FRACTURES by LALATENDU MANSINHA B . S c , Indian I n s t i t u t e of Technology, 1957 M.Tech., Indian I n s t i t u t e of Technology, 1959  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE, DEGREE OF DOCTOR qF 'PHILOSOPHY i n 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 t h e s i s i n p a r t i a l f u l f i l m e n t of  the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. f o r extensive  I f u r t h e r agree that permission  copying of t h i s t h e s i s f o r s c h o l a r l y purposes may  granted by the Head of my Department or by h i s  be  representatives.  I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission.  f HVS  Department of  1C  ^>  The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. Date  2 q ^  O O u W  \Q  6?  The U n i v e r s i t y 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 I n s t i t u t e of Technology, 1957 M.Tech., Indian I n 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. J.A. D.L. J.C.  Chinnery Jacobs Livesey Savage  W.F. Slawson R.W. Stewart E. Teghtsoonian W.H. White  E x t e r n a l Examiner: Jean-Claude De Bremaecker Rice U n i v e r s i t y  RADIATION FROM TENSILE FRACTURES  GRADUATE STUDIES F i e l d of Study:  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 l i m i t e d extent, S waves) has been studied by seismologists to provide information on the f o c a l mechanism of earthquakes. In this, thesis we investigate the.inverse problem: knowing the type and form of displacement at the focus at the f o c a l i n s t a n t , 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 t e n s i l e f r a c t u r e i n a glass p l a t e . 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 i n d i c a t e d . The experiments reported i n t h i s thesis c o n s t i t u t e a p a r t i a l test of a recent theory by Knopoff and G i l b e r t (1960) on f i r s t motions from seismic sources. The type of fracture studied 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 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 r a t i o s P / P and S /P d i f f e r s i g n i f i c a n t l y i n magnitude from the theory i n many azimuths. I t i s suggested that the discrepancies are possibly due to the neglect i n the theory of non-linear e l a s t i c e f f e c t s near the t i p of the f r a c t u r e . e  e  Q  qo  Seismology  Introduction to Quantum Mechanics Theory of Measurements Nuclear Physics Advanced Geophysics Radioactive and I s o t o p i c Processes i n Geophysics Modern Aspects of Geophysics  J . Grindlay J.R. Prescott J.B. Warren J.A. Jacobs R.D. R u s s e l l J.A. Jacobs  Related Studies: Applied Electromagnetic Theory Mineral Deposits  G.B. W.H.  Walker White  - i i -  ABSTRACT  The first  m o t i o n o f P waves  waves) has  t h e s i s we  we P  on  the  to a v e r y  form of displacement the a z i m u t h a l  of e l a s t i c  Initial  Two  inverse at  the  provide  problem; knowing the  type  focus  thermally  types  at  of  the  the  technique.  induced  tensile  of a recent  The  in this  theory  Gilbert  The  type  results  show s i g n i f i c a n t  discrepancies with  tures.  The  f r o m the the  theory,  ratios P  theory  Q  of non-linear  fracture.  f o r both I n i t i a l 0  and  S@/PQ  i n many a z i m u t h s .  d i s c r e p a n c i e s are  theory  /Pg  p o s s i b l y due elastic  distriThe  partial  (1960)  on  of f r a c t u r e Gilbert.  the  and  differ  to  The  that  Extended  Frac-  i n magnitude  I t i s suggested to the n e g l e c t  e f f e c t s n e a r the  Our  theory.  S motion i s opposite  s e n s e o f the m e a s u r e d f i r s t the  Extended  azimuthal  c o r r e s p o n d s t o C a s e 3 o f K n o p o f f and  by  studied:  and  studied  predicted  fracture in a  thesis constitute a  sources.  source  i s indicated*  by K n o p o f f and  f i r s t motions from s e i s m i c  first  The  o f f r a c t u r e s have b e e n  S wave a m p l i t u d e s  experiments reported  focal instant,  sense o f  ( B i l a t e r a l l y propagating) Fractures  b u t i o n o f t h e P and  S  extent,  this  ( U n i l a t e r a l l y propagating) Fractures.  test  to  s e n s e o f the  In  distribution  energy i s a  glass plate.  limited  seismologists  S motion, u s i n g model seismic  and  o f the  f o c a l mechanism o f e a r t h q u a k e s .  i n v e s t i g a t e the  study  distribution  (and  been s t u d i e d by  information  and  geographical  that in  the  t i p of  the  TABLE OF CONTENTS Page ABSTRACT  i i  ACKNOWLEDGMENT  i l l  L I S T OP FIGURES  i v  L I S T OF SYMBOLS  v i  L I S T OF TABLES CHAPTER I  -  v i i INTRODUCTION  1.1  General  1.2  Previous  Works  2  1.3  Method o f I n v e s t i g a t i o n  7  CHAPTER I I  -  1  FRACTURE PHENOMENON  2.1  Definition  10  2.2  Velocity Consideration  11  2.3  Criterion  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  2.5  S t r e s s e s Around a P r o p a g a t i n g  2.6  Radiation  f o r Crack I n s t a b i l i t y  from a Propagating  Crack  Fracture Fracture  ..  15 19 23  CHAPTER I I I - INSTRUMENTATION AND EXPERIMENTAL TECHNIQUE 3.1  Scaling Considerations  30  3.2  Method o f I n d u c i n g  31  3.3  E l e c t r o n i c D i s p l a y and R e c o r d i n g 3.3.1  Ceramic  3.3.2  Filter  Fracture  Transducer System  System  .... 36 37 39  Page  3.4  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 R e c o r d i n g  41  3.3.5  T r i g g e r i n g System  41  M e c h a n i c a l Arrangement  CHAPTER I V  -  42  PROCEDURE  4.1  C a l i b r a t i o n of Detectors  44  4.2  V e l o c i t y Measurements  46  4.3  Identification  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  Determination  o f the F a r - F i e l d  50  4.6  Symmetry o f I n i t i a l  o f P and S waves  Region  Fractures  52  CHAPTER V - RESULTS 5.1  General  54  5.2  Initial  5.3  Extended F r a c t u r e s  Fractures  57 58  CHAPTER V I - DISCUSSION 6.1  General  60  6.2  High Frequency  61  6.3  Diffraction Effects  62  6.4  Correspondence o f F r a c t u r e s D i s l o c a t i o n Models  to Other 63  6.5  V e l o c i t y of Fracture Propagation  64  6.6  Non-Linear E f f e c t s  65  6.7  A c t u a l Correspondence o f 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 T h r e e D i m e n s i o n s .... 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  - iii  -  ACKNOWLEDGEMENT  It  i s a p l e a s u r e to a c k n o w l e d g e  the h e l p I  have r e c e i v e d f r o m D r . James C. S a v a g e . suggested and  seismology  after time  that I investigate  D r . Savage  problems of e l a s t i c i t y  w i t h model s e i s m i c t e c h n i q u e , and  t h a t devoted  a c o n s i d e r a b l e amount  i n d i s c u s s i n g i d e a s and p e r f o r m i n g  o f h i s own experiments.  I have l e a r n e d a l o t f r o m w o r k i n g w i t h htm, and I take him.  this  o p p o r t u n i t y - t o e x p r e s s my i n d e b t e d n e s s t o  I also  t h a n k D r . R. D. R u s s e l l , who a c t e d a s  my s u p e r v i s o r d u r i n g D r . S a v a g e ' s a b s e n c e , a n d 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 e x p e r i m e n t a l work. T h i s work was f i n a n c i a l l y from  the American Petroleum  supported  Institute.  by g r a n t s  -  iv -  L I S T OF FIGURES Page Fig.  1.  Method  Fig.  2.  I n f i n i t e l y long fracture d i m e n s i o n a l body  Fig.  Fig. Fig.  3.  4. 5.  of applying  gas flame t o g l a s s  plate.  72  i n three72  I n f i n i t e s i m a l part 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 m o t i o n a t any p o i n t on t h e X - Z p l a n e  72  C o o r d i n a t e s y s t e m and d i r e c t i o n o f p o s i t i v e unit vectors ,  73  D i s t r i b u t i o n o f 6£ 1951)  with ©  (after Joffe,  :  74  Fig.  6.  Figure  shows volume V bounded by s u r f a c e s  Fig.  7.  Fig.  8.  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 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 o f the temperature d i s t r i b u t i o n shown i n f i g u r e 7  75 76  Fig.  9.  Method o f i n d u c i n g  Fig.  10.  Method o f e x t e n d i n g s h o r t  Fig.  11.  Method o f e x t e n d i n g l o n g  Fig.  12.  S c h e m a t i c c i r c u i t f o r m e a s u r i n g 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  Schematic c i r c u i t f o r studying from f r a c t u r e s i n g l a s s p l a t e s  79  Fig. Fig. Fig. Fig,  13. 14. 15. 16.  I n i t i a l Fractures  S'. 74  fractures fractures  77 77 77  radiation  T r a n s d u c e r 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 o f P Q /PgQ r a t i o  80  Transducer and f r a c t u r e p o s i t i o n s f o r determination o f Sg / P ^ r a t i o  80  T r a n s d u c e r 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# 1 7 . Fig. Fig.  18. 19.  Attenuation fractures  o f P waves a t 9 0 ° f r o m  Attenuation fractures  o f P waves a t 0 ° f r o m  Fig. Fig.  Fig.  20. 21. 22.  23.  81 Initial 82  T y p i c a l record f o r the determination o f $ / 90 I n i t i a l Fractures  p  Fig.  Initial  p  r  a  t  i  o  f  r  o  m  T y p i c a l r e c o r d f o r SQ /P@ I n i t i a l Fracture  ratio  from 85  Typical record f o r P Q / P Q Q ratio Extended F r a c t u r e  from 86  Record showing 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 f r o m E x t e n d e d F r a c t u r e s i n t h e r e a r quadrant  86  P l o t o f measured P Q / P ratios I n i t i a l Fractures .V  from 87  q n  Fig.  24.  P l o t o f measured S 0 /P© I n i t i a l Fractures  ratios  from  Fig.  25.  P l o t o f measured P e / P q Q Extended F r a c t u r e s  ratios  from  P l o t o f measured S @ / P g Extended F r a c t u r e s  ratios  Fig.  26.  85  88 87  from  Fig.  27.  F i g u r e s h o w i n g change o f Q w i t h propagation o f the f r a c t u r e  Fig.  28.  F i g u r e shows c o n t r i b u t i n g segment o f infinite fracture front  88 89 90  - v i -  L I S T OF SYMBOLS  For  convenience  f r e q u e n t l y used  a partial  i n the t e x t  are d e f i n e d wherever  |3>  Transverse  7^  Direction  ^  Velocity Plane  Other  wave  velocity.  velocity.  cosine. of Propagation  of fracture,  polar coordinates. at ( r , 6 ) with  Pg  (A)  P wave r e c o r d e d  PQ  (r)  P wave a m p l i t u d e s  a t angle  t r a n s d u c e r A.  & , fordifferent r .  E P§  P  (R, 9  ) from Extended F r a c t u r e s .  I P  P  (R, 6  j from  Q  P@ / P  g o  SQ/TQ  (B)  (R,  (R, 9  a t (R, 6  ) to P  amplitude  a t (R, 6  ) to P  amplitude  90°).  R a t i o o f S amplitude at  SQ  I n i t i a l Fractures.  Ratio o f P amplitude at  symbols  they a r e i n t r o d u c e d .  L o n g i t u d i n a l wave  )  o f symbols  i s given here.  oC  (r, 0  list  ) .  S wave r e c o r d e d  at ( r , 9 ) with  transducer  B.  - v i i -  L I S T OF  I.  Table  II.  P9/P  III.  S@/P@  IV.  PQ / P  V.  s  showing G  e/ 0 p  Ratio  O  Ratios 9  0  Ratio Ratios  y  TABLES  , d and f f o r d i f f e r e n t  for Initial forInitial  0 .  Fractures  Page 91 92  Fractures  93  f o r Extended F r a c t u r e s  94  f o r Extended F r a c t u r e s  VI.  Calculated S Q / P  VII.  Calculated S Q / P  g o  9  q  95  Ratios  for Initial  Fractures  Ratios  f o r Extended F r a c t u r e s  96 96  - 1  -  CHAPTER  I  INTRODUCTION  1.1  General. A r e c e n t p a p e r by K n o p o f f  predicted  Gilbert  of various types.  model o f a f r a c t u r e of displacement then used  or s t r a i n ;  to p r e d i c t  here  This theory  a dislocation  elasticity  the e a r l i e s t  consists  o f an  waves f r o m  fractures  The radiation  test  t o the  of p a r t i c u l a r  d i r e c t i o n and adequately  nature  a fracture  i n geophysics  of f r a c t u r e  the r e s u l t  of  work  this elastic  the  i s to apply  Other  This  where  problem  the i s not  ( e . g . , a r e deep  questions  o f energy  in strain of  between  the  the p r o p e r t i e s  i t i s p o s s i b l e that f r a c t u r e  of a dislocation  the p a r t i t i o n i n g  of  i n earthquakes  i n some e a r t h q u a k e s  i n displacement?).  concern  i n the s t u d y  the r a d i a t i o n p a t t e r n .  known; i n d e e d ,  play a role  earthquakes than  interest  importance  The  study o f  i n v e r s e problem o f i n f e r r i n g from  trans-  types.  o f e l a s t i c waves f r o m  o f the f r a c t u r e  not  of c e r t a i n  principal  theory i s  the d i s l o c a t i o n . experimental  a  component  l o n g i t u d i n a l and  t h e o r y , as w e l l as a g e n e r a l e x p e r i m e n t a l  is  has  takes as  in a certain  linear  verse motions a s s o c i a t e d with  results  (1960)  t h e r a d i a t i o n p a t t e r n o f e l a s t i c waves a s s o c i a t e d  with fractures  reported  and  does  focus  rather  interest longitudinal  - 2 -  and  t r a n s v e r s e waves a n d t h e q u e s t i o n o f w h e t h e r f r a c t u r e  tends  to propagate  fracture)  i n ,a s i n g l e  or simultaneously  ("bilateral f r a c t u r e ) .  Previous The  sive.  literature  on s e i s m i c s o u r c e s  I n the f i r s t  (72 x 33 x 13 c m s ) .  different  as a source  of detector.  on a b l o c k o f  agar-agar  a variable capacitor  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 He a l s o v e r i f i e d  a result of  t h a t a p u r e l y downward m o t i o n a t a p o i n t on t h e i s a c c o m p a n i e d b y upward m o t i o n i n t h e i m m e d i a t e  surrounding In  simulates with  the t r a n s d u c i n g  Surface motions a t various d i s t a n c e s  waves were i d e n t i f i e d .  area  exten-  experimental  p a p e r , he u s e d  f r o m t h e s o u r c e were s t u d i e d w i t h  surface  i s not  s t u d i e s on t h e g e n e r a t i o n o f  waves, u s i n g w i d e l y  Lapwood,  have  (1952 t o 1955) p u b l i s h e d a s e r i e s o f  p a r t s o f a loudspeaker  type  directions  questions.  p a p e r s on h i s e x p e r i m e n t a l  techniques.  i n two o p p o s i t e  Works.  Kasahara  elastic  (unilateral  The e x p e r i m e n t s r e p o r t e d h e r e  a b e a r i n g on a l l o f t h e s e  1,2  direction  the p o i n t .  the f i f t h  paper o f t h i s  an e x p l o s i v e s o u r c e .  t h i n rubber  f i l m was  c a v i t y were 0.6  cms  Kasahara  A cylindrical  embedded  90 x 30 x 30 cms i n d i m e n s i o n s .  series,  cavity  i n a block of  lined  agar-agar  The d i m e n s i o n s o f t h e  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 in  by an i m p u l s i v e i n c r e a s e  t h e c a v i t y w i t h a n a i r pump.  and  a t v a r i o u s p o i n t s away f r o m  Motion  of pressure  a t the epicenter,  i t , were s t u d i e d .  l o n g i t u d i n a l a n d s u r f a c e waves were i d e n t i f i e d .  Only The f o r m ,  as w e l l a s t h e a m p l i t u d e , o f t h e d i s p l a c e m e n t changed a s one moved away f r o m Kasahara  the e p i c e n t e r .  a n d a number o f o t h e r  (Howes e t a l (1953) , T a t e l Oliver Levin Kato of  (1954), Evans e t a l (1954),  e t a l (1954), P r e s s e t a l (1954), K n o p o f f e t a l (1955), O ' B r i e n  (1955), K n o p o f f  e t a l ( 1 9 5 6 ) ) , have p e r f e c t e d  seismic modelling.  to  to use.  propose  Oliver,  (1956),  the theory and technique  a r e cumbersome a n d e x p e n -  P r e s s and Ewing  (1954) were t h e f i r s t  and u s e two d i m e n s i o n a l m o d e l s t o s t u d y s e i s m i c  phenomena.  They e s t a b l i s h e d  certain plate The symmetric  (1955),  M o s t o f t h e e a r l y w o r k s were on  thtfee d i m e n s i o n a l m o d e l s , w h i c h sive  investigators  the correspondence  between  v i b r a t i o n modes a n d body waves. group  velocities  o f symmetric  p l a t e waves have a p r o n o u n c e d  and a n t i -  dependance on k h ,  where k i s t h e wave number a n d h t h e t h i c k n e s s o f t h e plate with of  (see appendix the p l a t e  the f i r s t  established the  II).  F o r wavelengths  l o n g i n comparison  t h i c k n e s s , k h i s s m a l l and the d i s p e r s i o n  symmetric  mode i s s m a l l .  (appendix I I ) that  I t has been  the f i r s t  two d i m e n s i o n a l m o d e l c o r r e s p o n d s  symmetric  mode i n  to l o n g i t u d i n a l  - 4 -  body waves i n t h r e e d i m e n s i o n s . o f the asymmetrical symmetric  F o r s m a l l k'h t h e v e l o c i t y  ( f l e x u r a l ) mode i s much l e s s  o r s h e a r modes a n d t h u s does n o t i n t e r f e r e  t h e s e modes i n s e i s m i c m o d e l l i n g . displacement  i n the plane  thickness.  o f t h e two d i m e n s i o n a l m o d e l  o f 1/16 i n c h a n d a f r e q u e n c y  radiation  first  attempt  K a t o and T a k a g i pulse applied  a plate  of plate thick-  o f the order o f I O k c / s . 2  t o study  from non-propagating  m o d e l s was b y P r e s s  produces  independent  O l i v e r , P r e s s and E w i n g u s e d  The  with  S h e a r waves w i t h  show no d i s p e r s i o n and have v e l o c i t y  ness  than  the e l a s t i c  faults  in  wave  two-dimensional  (1956, 1957) and i n d e p e n d e n t l y b y  (1957).  A high voltage, short duration  to a p i e z o - e l e c t r i c  bimorph  a short duration mechanical  transducer  displacement.  This  simulates a singlet  source.  combination  bimorph t r a n s d u c e r s can s i m u l a t e any  of such  force  or displacement  focal  Instant.  combinations  systems a c t i n g a t the f o c u s a t the  A t the c e n t e r o f a t h i n  o f benders  d o u b l e t and q u a d r u p l e t benders,  A properly oriented  served to simulate sources.  sheet  singlet,  Two o p p o s i t e l y p h a s e d  s e p a r a t e d by a s h o r t d i s t a n c e , form a d o u b l e t  source; f o u r form a quadruplet slit  circular  source.  between t h e d o u b l e t b e n d e r s ,  late a fault.  With s i l i c o n e  to study a l u b r i c a t e d  fault.  By i n t e r p o s i n g a  P r e s s was a b l e t o simu-  as a l u b r i c a n t  he was a b l e  The arms o f t h e d o u b l e t  -  pointed fault  i n the  0 ° and  180°  o r i e n t e d i n the  both with  changed slit,  S amplitude 90°.  is  25°  shifted  dipole. the  The  But  along  termination  the  Press  the  slit  ( 1 9 5 1 ) , who  face  and  fault  plane  solution.  The  the  G waves.  ( 1 9 6 1 ) , and of  the  Rayleigh converted  i s supported system of be  found  methods a r e  p e r i o d m a n t l e R a y l e i g h waves and  oscillations  zone.  earthquake  of  Smith  i n contrast  the  which  been d e s c r i b e d  Ben-Menahem  crack.  by  (1961).  The  of  e a r t h f o r h i g h mode numbers  on long  theoretical  Benioff, Free  a  focus  based  e a r t h and  by  surface  (1961) r e p o r t e d  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  b a s i s f o r t h e method h a s  null  simple  to  being  This  Toksoz  parameters of  a  S wave  c o n d i t i o n s o f a moving G r i f f i t h  two  no  (1959) s u g g e s t s  shows t h a t a  Ben-Menahem and  method t o r e c o v e r  and  the  i s due  i n t o s h e a r waves.  boundary  Press,  the  With  to 90°  ( n o t n e c e s s a r i l y R a y l e i g h waves) c a n  satisfy  slit.  i n c r e a s i n g to  m o t i o n i n the 30° H e a l y and  and  S wave p a t t e r n  s i d e from that of a  the anomalous s h e a r d i s t r i b u t i o n  waves  the  sense o f shear motion i s c l o c k w i s e ,  the work o f J o f f e  from  slits.  to the  counterclockwise  the  P wave  the p r e s e n c e o f a s l i t ,  to 30°  a  amplitudes at 0°  i s a t a minimum a t 0 ° , In  waves t r a v e l l i n g at  The  i n the p r e s e n c e o f a  L a t e r work by that  vanishing  without  significantly  maximum a t  to  and  directions, representing  same d i r e c t i o n .  d i s t r i b u t i o n p a t t e r n had 90°,  5 -  Press  spheroidal are  - 6 -  equivalent  t o 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 p r o p a g a t i n g i n o p p o s i t e d i r e c t i o n s . phase s h i f t  between v e r t i c a l and  horizontal  f o r p r o p a g a t i n g waves i s 9 0 ° , and For a p o i n t source equally  are produced. direction, radiation  and  I f the energy  be  i n opposite d i r e c t i o n  Toksoz used  test  this  the ceramic  A  rupture  v a r i e d u n t i l resonance  source.  The  occurred.  superimposed  s o u r c e was  translation composite moving  time  kept  intervals  after  T h i s i s the  time  i n radius  The  The  of 2°.  o f the s o u r c e  to  frequency  was by  theorem  I n s t e a d o f a moving  The  the r e c e i v e r  was  s o l u t i o n s were  by a n amount e q u a l t o  f o r a 2° i n t e r v a l .  record i s approximately  and  used  reciprocity  fixed while  shifting  and  Nodes were c o u n t e d  o f a moving r e c e i v e r  moved i n d i s c r e t e  phase  velocity.  12"  transducer.  r e c e i v e r a t the r i m o f the p l a t e . the u s e  the  s i n u s o i d a l f u n c t i o n was  source  180°  F o r c a s e s where  i s unequal,  aluminum p l a t e  excited  one  t h e o r y , P r e s s , Ben-Menahem  a circular  i n thickness.  permits  90°.  180°.  of 0° or  i n only  assume some i n t e r m e d i a t e v a l u e .  To  excite  shifts  i s radiated  case f o r f a u l t i n g w i t h a f i n i t e  1/16"  f o r s t a n d i n g waves  phase  the phase s h i f t w i l l  shift will  displacement  e x c i t a t i o n , R a y l e i g h waves a r e  i n a l l directions,  The  e q u i v a l e n t to a  the  Such a finite  source. A more p r e c i s e method i s m e n t i o n e d b y P r e s s ,  Ben-  - 7 -  Menahem and T o k s o z , b u t t o d a t e published. static  A moving source  ceramic  source  no r e s u l t s  i s simulated  transducers pulsed  suitable  time  ducers.  The n e t i m p r e s s i o n  have  been  by a s e r i e s o f i n sequence,  d e l a y s between p u l s i n g o f a d j a c e n t  with  trans-  i s a source moving i n d i s c r e t e  steps w i t h an e q u i v a l e n t v e l o c i t y  t o a c o n t i n u o u s l y moving  source.  1.3  Method o f I n v e s t i g a t i o n . T h e r m a l s t r e s s e s a r e 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 a n open f l a m e  t o the s i d e o f the p l a t e a t a p o i n t  a b o u t l-ir cm f r o m , and a l o n g linear  scratch previously inscribed  When s u f f i c i e n t a fracture  tensile  initiates  its  prolongation.  the  fracture  after  t h e e x t e n s i o n o f , a 1/2  termed  b u i l d s up a c r o s s  (see f i g . l ) . the s c r a t c h ,  a t the s c r a t c h and p r o p a g a t e s  The s u d d e n d i s p l a c e m e n t  gives rise P',  stress  on t h e p l a t e  along  associated with  t o s y m m e t r i c p l a t e waves  to d i s t i n g u i s h  cm  (here-  f r o m P waves i n t h r e e  dimensions) and t r a n s v e r s e S waves. It is  should  be e m p h a s i z e d h e r e  by no means a d i r e c t  initiated body.  analogue o f the problem o f a n  fracture propagating  To e s t a b l i s h  that the experiment  i n a three  the connection  between the two we  have t o c o n s i d e r t h e f o l l o w i n g a n a l o g i e s . that a  dimensional  First,  t h e waves i n t h e t h i n p l a t e a r e r e l a t e d  three dimensional  we  will note  t o waves i n  body s u b j e c t t o t h e c o n s t r a i n t o f  - 8 -  plane the  strain.(i.e.,  v a n i s h i n g components o f s t r a i n  s i d e s o f the p l a t e ) .  We  note  that i n either  s i g n i f i c a n t motions a r e i n the plane unconstrained to  plate,  motions permit  i n t h e symmetrlo  a variation  a  o f volume p e r u n i t a r e a  o f the  o f volume p e r u n i t  Thus t h e u n c o n s t r a i n e d  constrained plate having  bility.  This analogy  Oliver, Press  perpendioular These  c o n s t r a i n e d p l a t e come a b o u t  matter.  through p l a t e may  of as  compressi-  h a s b e e n f o r m a l l y e s t a b l i s h e d by  and E w i n g  strain.  of  be t h o u g h t  an u n u s u a l l y h i g h  (1954).  The s e c o n d s t a g e  the c o n s t r a i n e d p l a t e s t o form a s o l i d  body i n p l a n e  area o f  compression  argument i n v o l v e s t h e J u x t a p o s i t i o n o f a v e r y of  I n the  p l a t e wave.  p l a t e , whereas a l l v a r i a t i o n s the  case the  o f the p l a t e .  however, s m a l l m o t i o n s  the p l a t e do o c o u r  across  In order  l a r g e number  three  t o be d e f i n i t e  o f the  dimensional l e t us  choose a s e t o f c a r t e s i a n axes o r i e n t e d so t h a t a l l m o t i o n is  parallel  propagates  t o t h e X-Z p l a n e . i n the X d i r e c t i o n .  analogue o f a f r a c t u r e Initiates  fig.  Gilbert  The t h r e e  that the f r a c t u r e dimensional  i n a p l a t e i s then a f r a c t u r e  i n the X d i r e c t i o n  t o t h e X-Z p l a n e  2).  imagine  whloh  s i m u l t a n e o u s l y a t e v e r y p o i n t on t h e Y a x i s a n d  propagates parallel  We  so t h a t every  I s a t a l l times  On t h e o t h e r hand  seotlon  i d e n t i c a l (see  the t h e o r y o f K n o p o f f and  (1960) c o n s i d e r s a f r a c t u r e w h i c h h a s o n l y a n  infinitesimal  length along  t h e Y a x i s and p r o p a g a t e s  along  -  the X a x i s  ( f i g . 3).  w i s h to model. i n our  This  argument.  We  I t i s the l e a d s us will  t i o n s i n the X-Z  plane  f i r s t motions of  the  We  see  9 -  to the  restrict  f r a c t u r e centered  the  f i r s t m o t i o n s o b s e r v e d on  from the  remainder of  having  had  upon t h e  be,  attenuation  o f t h e wave i n t h e  p r o p o r t i o n a l to r  tional  to r "  c o n c e r n us conclude should for  sure,  the  the  1  i n the  here.)  that  give  the  the  On  8  experimental  on  the  point  two  cases,  of the  the  waves.  i n f i g u r e 2. theory  to  disturbance fracture  the  geometric  amplitude propor-  T h i s d i f f e r e n c e need these  of  observation.  e x p e r i m e n t s and  basis of  the  length  infinite of  step  only  contribute  plane,  we  observa-  consider  not  a r g u m e n t s we i n this  correct radiation pattern  test  to  transverse  experiments described  fracture depicted  final  d i f f e r e n c e i n the  i n our  theory. the  a  and  infinitesimal  the X-Z  (There w i l l  being  and  o r i g i n may  the p o i n t s  time to r e a c h t o be  third  l o n g i t u d i n a l and  from f i g u r e 2 that only an  problem which  ourselves  ( s e e f i g . 3)  the  not  latter  may  thesis  i n the  X-Z  plane  I t i s therefore  o f K n o p o f f and  Gilbert.  an  - 10 -  CHAPTER I I  FRACTURE PHENOMENON  2.1  Definition. We  surface  define  i s created  under the a c t i o n distribution. velocity;  general  volume.  a s a p r o c e s s by w h i c h a  i n the i n t e r i o r  o f an i n t e r n a l or e x t e r n a l  The f r e e  surface  free  medium  stress  may expand a t a  finite  than the  o r t r a n s v e r s e wave v e l o c i t i e s i n t h e medium.  the f r e e  surface  will  e n c l o s e a penny  shaped  A s t h e two p l a n e b o u n d i n g s u r f a c e s o f t h e p e n n y  s h a p e d volume a p p r o a c h e a c h o t h e r , z e r o , and i t i s p o s s i b l e assume t h a t  there  plane before  the passage o f the f r a c t u r e  difference  fracture  plane.  radiation  plane.  We  o f the f r a c t u r e  front.  correspond  i n the d i f f e r e n c e plane.  suddenly  o r s t r a i n across the time  to a propagating  dependstep  o f displacement and/or s t r a i n I t follows  o f the type d e s c r i b e d i s due t o a c t i o n  The  then corresponds to a  I f we assume a s t e p f u n c t i o n  the f r a c t u r e  fracture  front  i n displacement  ence, a f r a c t u r e would function  t o speak o f a f r a c t u r e  t o , and on b o t h s i d e s  passage o f the f r a c t u r e applied  t h e volume t e n d s t o  i s no r e l a t i v e m o t i o n o f p a r t i c l e s  immediately adjacent  across  o f an e l a s t i c  t h i s v e l o c i t y i s presumably l e s s  longitudinal In  fracture  that  i n an i d e a l  above, a l l e l a s t i c  a t the f r a c t u r e  front.  wave  - 11 -  In  l i t e r a t u r e t h e t e r m s c r a c k and f r a c t u r e  been used  as interchangeable  existence  of free  we u s e t h e t e r m and  'propagating  referring original  2.2  terms t o d e s c r i b e t h e  s u r f a c e i n t h e medium.  'fracture' fracture'  thesis case  f o r t h e dynamic  case.  When  i n the f i e l d  their  h a s been  maintained.  Velocity Considerations. loffe  t i o n around determined  (1951) h a s shown t h a t  by c o n s i d e r i n g  varies with  the s t r e s s d i s t r i b u -  t h e head o f a p r o p a g a t i n g  velocity.  fracture  the f r a c t u r e  moving s u r f a c e d i s t u r b a n c e s .  w  In this  and 'crack' f o r s t a t i c  to works o f other authors terminology  have  as a system o f  The s t r e s s  For a stationary  c a n be  distribution fracture  ( s e e f i g . 4 f o r c o o r d i n a t e s y s t e m ) i s a maximum some Ir-R  d i s t a n c e ahead o f the c r a c k i n the d i r e c t i o n o f propagation  (0 - 0 °  that  the f r a c t u r e  increasing 8  that  I t i s due t o t h i s maxima o f t e n s i l e propagates  ahead i n a s t r a i g h t  until  <5~Q  a t a b o u t 0.6  direction  o f p r o p a g a t i o n and w i l l  a division of available  energy  With  o f t h e t r a n s v e r s e wave  i s independent  a t t h i s v e l o c i t y the f r a c t u r e  line.  stress  6~e Co©)  v e l o c i t y the v a r i a t i o n o f  decreases  velocity  ).  of 0  will form  .  I t i s expected  have no branches,  of fracture  f o r m a t i o n o f branches the v e l o c i t y should  preferred leading  formation. continue  to  With  t o be  -  maintained  a t less  12  -  0 . 6 (3 .  than  T h e r e f o r e we c a n c o n s i d e r  0 . 6 p a s t h e l i m i t i n g o r maximum v e l o c i t y of  a tensile  fracture.  T h e r e h a s b e e n no o t h e r types  of propagating  intuitive fracture  o f propagation  reasoning  t h e o r e t i c a l work o n o t h e r  discontinuities. one c a n n o t  However,  s e e a method b y w h i c h a  can propagate a t v e l o c i t i e s higher  t u d i n a l wave v e l o c i t y .  from  than  the l o n g i -  P o i n t s away f r o m t h e f r a c t u r e  c a n n o t know o f t h e e x i s t e n c e o f t h e f r a c t u r e  before a  equal  wave e l a p s e s .  t o the t r a v e l  time  C a s e s have been o b s e r v e d at  apparent  velocity. that  apparent  i n which f r a c t u r e s  v e l o c i t i e s i n excess  a t more t h a n  stress  glass  of  that  (1937)  t h e maximum t e n s i l e  to  i na high  (Schardin, 1959).  have  experimentally  fracture  velocity i n  o f t h e t r a n s v e r s e wave  T h i s maximum v e l o c i t y  stresses  i s practically stress  Glass plates  independent  a t t h e moment o f with high  inter-  gave i d e n t i c a l maximum v e l o c i t y a s t h o s e  which stresses velocity  cases i s  the fraoture  resulting  wave  velocity.  temperature and o f the o v e r a l l  fracture nal  has caused  i s 1500 m/sec, a b o u t 0 . 5  velocity.  travelled  o f the l o n g i t u d i n a l  one n u c l e u s ,  Schardin and S t r u t h verified  have  Presumably t h e e x p l a n a t i o n i n such  the high t e n s i l e  originate  o f the l o n g i t u d i n a l  time  have been r e m o v e d .  i nplates  from  Measurement o f f r a c t u r e  under h i g h compressive  loads a l s o  gave  - 13 -  no  velocity  variation with  We  load.  can thus e s t a b l i s h an e m p i r i c a l  between  , t h e maximum f r a c t u r e  wave v e l o c i t y .  v e l o c i t y and t h e s h e a r  This i s  1  n  --  0-6/J  o-6  =  From t h e f o r e g o i n g i t would a p p e a r  experiments  described i n this  to see f r a c t u r e 1 mm/sec.  fractures  no  thesis  velocity,  velocity  view  fracture  (1 mm, be  During the  than  t h i s may o n l y  b e i n g made u p o f i n t e r m i t t e n t  o f 0.6 j3 .  only a short distance but The m a j o r e v i d e n c e  has been observed  t h e maximum v a l u e .  supportthat  velocity  A f r a c t u r e may s t o p o f 0.6 (3>  o f r e s o l u t i o n o f measuring  1 ^ sec) a t r a n s i t i o n  (1959)  t o move a t a l o w e r  suddenly a f t e r moving a t a v e l o c i t y the l i m i t s  less  i s the observation o f Schardin  once i t a t t a i n s  may  i t was n o t uncommon  creep forward a t v e l o c i t i e s  each o f w h i c h advanced  the f u l l this  that a fracture  However, t h e r e i s e v i d e n c e t h a t  be a n a p p a r e n t  ing  JjjT  a t a n y v e l o c i t y b e t w e e n 0 and 0.6 p> •  propagate  at  relation  , but w i t h i n  instruments  to a lower v e l o c i t y  could not  observed. It  i s o f some i m p o r t a n c e  to note  that  the f r a c t u r e  v e l o c i t y d o e s n o t depend on t h e t h i c k n e s s o f t h e p l a t e being used. between  $^  This i s also and |2> .  seen from  the e m p i r i c a l  The t r a n s v e r s e wave v e l o c i t y  relation |3  i s  - 14 -  Independent o f p l a t e t h i c k n e s s .  2.3  Criterion To  f o r Crack  Instability.  e x p l a i n t h e d i s c r e p a n c y between t h e t h e o r e t i c a l  estimate  of strength of solids  Griffith  (1920, 1922) s u g g e s t e d  be  due t o t h e p r e s e n c e  solids. such at  The c r a c k s  calculate in  could concentrate  e n e r g y due t o t h e p r e s e n c e  The r e l i e f  of s t r a i n  Griffith  that the crack would spread  Inglis  the case  (very f l a t  hole  of a  crack  I t was p r o p o s e d b y o n l y i f the  the increase i n s u r f a c e  (1913) c a l c u l a t e d  about an e l l i p t i c a l that  two new s u r f a c e s .  i s g r e a t e r than  exceeded  i s o p p o s e d by t h e e n e r g y  to create  strain  locally  I t i s possible to  required  in  the s t r e s s  s t r e n g t h m i g h t be l o c a l l y  l o w mean s t r e s s .  the s t r a i n  a plate.  t h a t the d i f f e r e n c e might  o f l a r g e number o f s m a l l c r a c k s i n  that the t h e o r e t i c a l  a comparatively  and a c t u a l " measurements  the s t r e s s  decrease energy.  distribution  i n a stressed plate.  He  suggested  i n which the e c c e n t r i c i t y approaches u n i t y  ellipse)  model f o r a c r a c k .  the e l l i p t i c a l  hole  The change i n s t r a i n  t h i n p l a t e due t o t h e p r e s e n c e  should  be a g o o d  energy i n a l a r g e  o f an e l l i p t i c a l  crack i s  g i v e n by  W =-  (3-P)  * c V  where T  =  tension (2-1)  2c = l e n g t h o f c r a c k ^  = Poisson's  ^if*- s Lame's  ratio .  constants  - 15  -  I n the a b o v e 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 n a r r o w ; i . e . the a n g l e crack i s 0°. created.  As  the  Potential  per u n i t  between the  crack propagates energy of  the  sides of  the  s u r f a c e energy i s  s u r f a c e o f the  crack,  t h i c k n e s s o f the p l a t e i s R = 4cS  The  t o t a l decrease  due  t o the presence  W  —  Griffith's  R  where S = s u r f a c e t e n s i o n . o f the p o t e n t i a l energy o f the o f the  crack i s given  =  -  c o n d i t i o n t h a t the  ( 3 - ! = > ) A C T  or  2.4  two  r  treatment  o f the  static  a r o u n d a s t a t i o n a r y c r a c k was  by W i l l i a m s  i s given  by  Crack.  stress distribution  done by W e s t e r g a a r d  u s i n g a complex s t r e s s f u n c t i o n . exploited  extend  (2-2)  16/AS  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 A  by  4 C 5  c r a c k may  2  system  This technique  (1939) was  later  (1956) i n a more c o m p l e t e a n a l y s i s  - 16  of  the problem.  We  g i v e below a b r i e f account  method o f W i l l i a m s .  4  will  C  3  above s t r e s s  s o l u t i o n o f the  function  s\n(>-0>  o f the  +  C«.  function w i l l  chosen  as the p o s i t i v e r o o t s o f  edges a t ^  -.0  S i n ( a o(. )  -  and  •+  equation takes the form  sin  cos  ^  7\  o( = 27\  of a crack w i t h f l a n k angle  >  equations  type  ( * - 0 > ]  satisfy  free  t h e c a s e where  the  (2-4)  stress  For  of  - O  g i v e the s t r e s s  -I-  The  The  F  V "  -  =  (2-5)  conditions of  oi  i f the  A  are  Siri oC  , c o r r e s p o n d i n g t o the <p  -  A  = n/^ , where n = 1,2,3,  = 0, and  2.7!; thus  case  = 0 t h e above requiring  giving  the  stress  function  n/ + i 2  |T ( r , >  , ry ) 2  r  [ C , Sin (n/ -r a  M-  c^co^Cya + O"^ (2-6)  - 17  Solving tions  the e q u a t i o n s  f o r the s t r e s s  07  Williams  obtained  the  solu-  as  1  00 e)  -  a, j-5cos I  4Ta  ^ cos ^  ]  where 0  +  b,  ( - S S ^ |  + ^  ^  = ^  - K  j  (2-7)  0^ Cr ; B)  =  1  a.  4ri  4  _  D,  I  3 cos  I  _  2  SAW ©  +  2  3 sun 3 §  1  4. 4 a  33  cos  O C  R  A  V  J  +  .  .  .  .  (2_e)  - 18 -  4-r a  ^  cos &  - 2a ^ 2 8 2  For convenience two  istic  c a s e when a ^  square  root  =0.  by W i l l i a m s .  a x i s o f the crack 0~  e  are  I n both  singularity  Some i n t e r e s t i n g  the r e s u l t s  Firstly, stress  stresses.  and t h e a n t i -  cases a c h a r a c t e r -  i s obtained.  observed  9  c a n be t r e a t e d i n  The s y m m e t r i c  i n p l a t e s under  observations  the shear  the p r i n c i p a l  2  c a s e when bj_ = 0  c r a c k i s more common, b e i n g tension.  + o Cr*")* . . . ( . )  the s o l u t i o n  p a r t s , the symmetric  symmetric  -t 3 cos  c a n be made  at Q  =0,  from  along the  v a n i s h e s , making  The v a l u e s  of  <r  0> and  a n d CTQ  r  g i v e n by  <r r (0')  =  cr^ ( o ° )  = - cxtr  Hi  «^  »  *  T h i s means t h a t a s t a t e o f two d i 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 n e a r t h e end o f a c r a c k , a l o n g  i t s axis  *  *  - 19  2.5  S t r e s s e s Around a P r o p a g a t i n g The  t i o n around  work o f J o f f e  a brief  account  in this  (1951) on  chapter.  o f h e r method and  Consider plate.  be  We  distributions  the problem reduces  propose  speed. plane  propagates  We  in a  thin  i s the p l a n e  the Y d i r e c t i o n .  The  (T^ =  to propagate.  By  keeping  the  coordinate  direction.  uniform  passes  The  the  at  on y = 0,  a  that  crack  tension T acts  boundary c o n d i t i o n s are  =0  be  constant  system such  o f t h e p l a t e , and  large  crack w i l l  crack of  (the crack)  transverse For  to that of a p l a t e under  i n the p o s i t i v e X  here  infinite  large  the ends o f t h e  d e f i n e our  been  to give  at  tension across which a disturbance constant  distribu-  results.  expected  independent o f each o t h e r .  in  stress  f r a c t u r e has  a c r a c k o f l e n g t h 2c  t h e c r a c k may  the s t r e s s  t h e X-Y  the  I f the p l a t e i s under a s u f f i c i e n t l y  t e n s i o n T,  length  Fracture.  the head o f a p r o p a g a t i n g  mentioned before  2c  -  -c  <x  <c  - 20 -  Considering with  s e p a r a t e l y the m o v i n g p a r t o f t h e s y s t e m ,  T removed  we  have  on y = 0 ,  symmetry p e r m i t s  plane.  Defining u x  us to consider only and  x  Uy.  = x - ^ t  half  as the displacements,  and  where ^  i s the  fracture  t h e b o u n d a r y c o n d i t i o n s f o r a dynamic  propagation  elastic  system are  -T  at  y = 0  -c <^x<^c  =  0  at  y = 0  a l l x'  =  0  at  y = 0  These b o u n d a r y c o n d i t i o n s have by s u r f a c e d i s t u r b a n c e s displacement  U-v*. =  velocity  =  ^  c a n be r e p r e s e n t e d  <^o  the u p p e r  of  without  <^x  0  CT^=  The  -o  propagating  lx'\ t o be  Such  c  satisfied  at a velocity  i n the Z d i r e c t i o n .  y ^  disturbances  by  A T exp ( - T s ^ ) S  cos sx'  (2-10)  - 21 -  =  fe  Here  Displacements  we  I  -  T  -  o f t h e above  i n c r e a s e o f y, i f  =  exp(-^s^)  £  s  \  i  _  +  type  _io)  2  vanish r a p i d l y with  T* ) 1  =  Using  0  °  n y  the  the c o n d i t i o n  =  °  o b t a i n the r e l a t i o n  The g e n e r a l f o r m o f t h e d i s p l a c e m e n t ing  ( 2  1 - •  i s positive.  P ( aft*"  cos sx'  the  expressions  over a l l p o s i t i v e  values of S •  i s g i v e n by  integrat-  -  22 -  Thus  s x ' OiS  and OO  cos-sx  Substitution (T  will The  %  o f t h e above e x p r e s s i o n  ( * + 2/M ^  =  -r  into  e+c.  *  give expressions f o r the various s t r e s s unknown Ag  i s determined  from  components.  the boundary c o n d i t i o n s .  From t h e r e l a t i o n  o~~  sin  v  one  c a n determine  different (r, 0  8  .  ^  For  variation until  t h e hoop s t r e s s  and  a t constant  t  n  e  i n the d i r e c t i o n  of  0~ w i t h e  ©  r  and  the p o l a r coordinate  y = 0.  F o r low ^  a t a b o u t 0.6^3 j  <^0.6|3>  2  In figure  (1951) t h e v a r i a t i o n •  shows a maximum  ~_ 1 (T^y S \ n 9 COS0  az c o s " e  The o r i g i n s o f  Joffe  for various  ^  4  ©  ) are at x = c  duce f r o m  The  2  n  els  °op  of  5 we  system repro6  <7© w i t h  stress  distribution  o f the c r a c k a t Q  decreases  with  = 0°.  increase of  (TQ d o e s n o t change w i t h © .  t h e r e f o r e the f r a c t u r e  propagates  ina  - 23  straight  line.  But  fracture  has  preferred  nearly  equal stresses  ahead o f  2.6  no  when  the  (1956) and  K n o p o f f and  Gilbert  radiation pattern  as  surface  plane.  The  system. the +  (1958) has  from v a r i o u s  In  inner  the  normal c o i n c i d e s w i t h I t should  be  limiting  i t goes t o  surface,  -  fault the  displacement  tensors  terms between the  two  to  plane,  Z axis  and  the  p o s i t i v e and  outer the  fault  in a  so  cartesian surfaces  s'  positive direction.  The  that  the  dis-  i s oriented  f o r the  s u b s c r i p t a f t e r the  represent  motion  surfaces  case the  infinity  i t reduces  noted here  s u p e r s c r i p t and  first  by  types of p r o p a g a t i n g  the  i s shrunk u n t i l  given  been used  (1960) t o d e t e r m i n e the  o u t w a r d l y drawn n o r m a l i s the and  angles  e l a s t i c wave e q u a t i o n  de Hoop  i s expanded u n t i l  the  as  Fracture.  C o n s i d e r a volume V bounded by  inner  that  the  shown i n f i g u r e 6.  surface  the  d i r e c t i o n of propagation  from a P r o p a g a t i n g  by K n o p o f f  S'  a p p r o a c h e s 0.6/3  e x i s t over a wide range o f  i n t e g r a l of  locations.  ^  fracture.  Radiation An  -  differences  negative  stress in  Z sides  of  and  these the  fault. We inside to  the  assume t h a t , volume V;  2)  1)  there  that  the p r o b l e m i s i n t e r i o r  the  to the  are only  no  body  surface  infinite  forces pertinent  elastic  medium  - 24 -  and  3) we a r e i n t e r e s t e d o n l y  (the h i g h tion in  frequency  o f the s u r f a c e s ' b y suddenly  .  We  t ^ j  f o l l o w here  Knopoff and G i l b e r t For  excita-  discontinuities  o r the displacement  tensor  and d e f i n i t i o n s o f  (1960).  o l a r i t y we d e f i n e h e r e c e r t a i n q u a n t i t i e s :  = t ^ j  tensor  Displacement  applied  the terminology  Displacement Stress  mptions  s o l u t i o n ) from the impulsive  the s t r e s s tensor  |\j  i n the f i r s t  tensor  = (°C- 2p )  - ^ ( a ^ j  v  4  r\j  Retardation with P  velocity  -  [utco] = u. Ct - %0  Retardation with  velocity  -  < u . W >  Coordinates  S  o f the p o i n t o f observation  Coordinates  o f a p o i n t on the  fault  Distance  r  Direction  cosines  T;  plane  =  V i a )  "-Ji  - 25 -  In  the 2-Z p l a n e  the d i r e c t i o n  =  cosines o f r The d i f f e r e n t i a l geneous  isotropic  ct grad where f / p  T  z  -  (5L - ^^/^  = COS 0  -  (J- -  -  *'Vr  s v / n  e q u a t i o n o f m o t i o n o f a homo-  e l a s t i c medium i s  d i vU -  |3 c u r l  curl U -  i s t h e body f o r c e d e n s i t y .  = f/p  The s o l u t i o n  of this  e q u a t i o n by K n o p o f f and de Hoop i s  (2-12)  where  t h e o p e r a t o r Gj_j  i s d e f i n e d by  (2-13) In  t h e a b s e n c e o f body f o r c e s ,  n o r m a l v e c t o r on S  f  and n o t i n g t h a t t h e u n i t  p o i n t i n g i n t o V i s n e g a t i v e we  - j^M^J- ' - J ds  have  f < i C* ' ~ Gt  x  s (2  14)  6  - 26 -  The  operator  G-.y  ing  terms i n v o l v i n g 1/r  Slj and  jp^j  i s a p p r o x i m a t e d f o r l a r g e r by n e g l e c t and 1 / r  Oj^Ti  + ^ < * > ( ^ l "UTS) r  (2-15)  is  The m o t i o n a t a remote  point  c a n be w r i t t e n a s t h e  sum o f P wave m o t i o n a n d t h e S wave m o t i o n .  Thus  (2-16)  (2-17)  For  first  motions a t large  r  we may  J as  originating along  the  (2-18)  "J  the second term i n the d i f f e r e n t i a t i o n The d i s t u r b a n c e  in  Jtt  write  a very  small  even  For f i r s t  S  .  as  m o t i o n s we a r e i n t e r e s t e d  range o f time, and w i t h i n  over the surface  2  though the p o i n t p r o p a g a t e s  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  uniform  o f f as r  a t l a r g e r may be c o n s i d e r e d  from a p o i n t ,  the f a u l t p l a n e .  falls  this  interval  t o be s u b s t a n t i a l l y  , e l i m i n a t i n g the need f o r  27  integration  over  -  ds . f  I t must be n o t e d h e r e ment t e n s o r s u s e d of  here  these q u a n t i t i e s  density.  We  by  differ  that from  the s t r e s s the s t a n d a r d  c e r t a i n dimensions  change t h e d i m e n s i o n s  include  the e l e m e n t a l a r e a d s ' .  factors  they w i l l  and  displace-  definitions  of velocity  o f t ^ j and p±j  Since these are  not a f f e c t the c a l c u l a t e d  to  and now  constant  radiation  pattern. The  expressions f o r u^  and  u^  s  t h e n become  (2-20) In polar  c o o r d i n a t e s t h e above e q u a t i o n s become  4 ^oc^CL p -  n 21  r  (2-21)  where  - 28  The  r  vector  and  0j  to  17  i s a unit and  pointing unit  is a unit  lies  i n a plane  i n the p o s i t i v e  between  v e c t o r i n the r a d i a l  direction  vector i n a d i r e c t i o n at r i g h t  v e c t o r i n the  angle  -  containing  direotion  j direction  ej  and  r^  is  is  e~j  from  and  e^  e^.  The  given  by  angles  r,  to  ,  r,  and .  sine of  The  the  a. i  - Tj  83 = (1 An  actual fault  o f a l l o r some o f and  Gilbert  p a t t e r n hy  the  )«  o r f r a c t u r e may  terms i n the  have d e t e r m i n e d  the  c o n s i d e r i n g each o f  be  a  e x p r e s s i o n Wy  f i r s t motion the  combination Knopoff  radiation  t e r m s o f w.  i n turn,  and  J assuming  the  other  time dependance  seven  we  can  t o be  zero.  For a step function  write  and  The  fault  propagates  HCCp) i s the u n i t For a this is  i n the x d i r e c t i o n w i t h  tensile  f r a c t u r e o f the except  u  a  type are  considered zero.  Then  to  W  C**- £) 2  i  :  ,  step f u n c t i o n .  t h e s i s a l l t e r m s i n w^  equal  velocity ^  ^±  , (* '2f?)£±± x  ,  **^»  in w^  - 29  Noting  that  on  X-Z  L  H(t  - r/oL  -  ) « *&- 6 ( t - v/di  plane  LoC  f  r  oOf  I  d i r e c t i o n of first  (T^j  some f u r t h e r the  This  T  j u s t below t h a t  previous  ments c a u s e d by  a  (2)  sign  of  and  the  There appear  to  be  completely,  error.  the  displace-  Our  equations f o r  of  s__  term o f  i s c l e a r l y only  the a  ( i n Knopoff  and  from t h e i r equation  s i g n o f a l l terms c o n t a i n i n g  latter  of  not  the  This  sign  i n s i g n w h i c h p a r t l y , but  type 3 d i s l o c a t i o n  first  the  c h a n g e s the  equation.  G i l b e r t ' s nomenclature) d i f f e r  i n the  (2-23)  G i l b e r t a p p e a r t o have o v e r l o o k e d  normal to S .  errors  (S-S2)  P*s  i n t e g r a l i n t h e i r equation  given  cancel  the  J  o(»  (i  K n o p o f f and  the  obtain,  r  ox the  ) , we  and  i n the  second of  typographical  the  (12)  In  inclusion two  error.  equations.  - 30 -  CHAPTER  III  •INSTRUMENTATION AND EXPERIMENTAL TECHNIQUE  3.1  Scaling  Considerations.  Knopoff  (1955) shows t h a t  i n seismic  models the  relationship g = t should hold.  Here  g  and  i s the g e o m e t r i c a l  the P o i s s o n ' s r a t i o s c a l e and  v  primary aim i s t o study  scaling plate the  I n t h i s thesis our r a d i a t i o n o f a moving  so that  only  sufficiently  the f a r f i e l d  and t h e f a r f i e l d condition  record  f a r from  i s detected.  plate  means t h a t  theory requires plate  that  other.  o b s e r v a t i o n s be made a  dimensions t h i s c o n d i t i o n  wavelengths.  The  r e q u i r e m e n t s oppose e a c h  number o f w a v e l e n g t h s away f r o m t h e s o u r c e .  with small  the  factor,  We m u s t , however, s a t i s f y t h e t h i n  r e s t r i c t i o n s and a l s o  farfield  finite  factor.  factor, p  the time s c a l e  the f a r f i e l d  considerations.  fracture  large  t  scale  T h e r e f o r e we do n o t "have t o c o n f o r m t o t h e r i g i d  thin plate The  factor,  the v e l o c i t y scale  source.  p = v = 1  For  c a n be a c h i e v e d  On t h e o t h e r h a n d , t h i n  the w a v e l e n g t h be l o n g  only  plate  compared  with  thickness. In  the s e r i e s o f experiments described  have u s e d a p l a t e  thickness  o f 3 mm,  h e r e we  w i t h measured P  f  wave  - 31  velocity At  o f 5500 m/sec and  a f r e q u e n c y o f 100  ^  t=  5 5  near  the wavelengths  and  ^(a  o f 3700 m/sec.  are  =  c m s  a r e much g r e a t e r t h a n  the  »  plate  M o s t o f the o b s e r v a t i o n s have b e e n made a t  t a n c e s g r e a t e r t h a n 25 verify  an S wave v e l o c i t y  cms  e  Thus t h e w a v e l e n g t h s thickness.  kc/s  -  that  field,  cms  from  the s o u r c e .  the m e a s u r e m e n t s were n o t an a t t e m p t  was  In order to  influenced  made t o measure t h e  a t t e n u a t i o n o f the e l a s t i c waves f r o m  dis-  by  the  geometric  the f r a c t u r e .  For  i two  d i m e n s i o n a l waves the a m p l i t u d e  the f a r f i e l d , fall  i n the near  o f f more r a p i d l y .  given  i n s e c t i o n 4.5.  o u t t o be that  but  - 0.53  ±  field  Details  0.08  o f the  experiment  (Standard E r r o r ) .  the e x p e r i m e n t a l r e s u l t  verifies  d i s t a n c e s g r e a t e r t h a n 20 cms  to  any  -  8  in  should are  r  came  It is felt  t h a t measurements  do n o t  involve near  field  appreciable extent.  Methods o f I n d u c i n g F r a c t u r e . An  in  the a m p l i t u d e  The m e a s u r e d e x p o n e n t o f  at  3.2  s h o u l d v a r y as r  inhomogeneous t h e r m a l s t r e s s f i e l d  the g l a s s p l a t e  region there  by  (see f i g . 1 ) . is differential  stresses  are developed  the a p p l i c a t i o n o f h e a t  i s s e t up to a s m a l l  B e c a u s e o f the t h e r m a l g r a d i e n t , expansion,  and  i n the p l a t e .  r a d i a l and As  hoop  i s w e l l known,  - 32  fracture and  i n glass  heated will in  i s r e a d i l y i n d u c e d by  the hoop s t r e s s e s  initiate  fractures.  region  be  affected.  F o r the p u r p o s e  sufficiently  T e m p e r a t u r e m e a s u r e m e n t s on flame  f o r 60  radius  low  a r e p e t i t i v e source was  filtered  an a c t u a l from No  fracture.  to a pick-up  detectable therefore  applied  and  was  the e l a s t i c  heated  arrival  devised.  as  the  i n the two  t o the p l a t e .  properties  An from  signal  signals  from  2 cm  j o i n i n g s o u r c e and  observed  times  plate  The  then a p p l i e d  change i n t h e t i m e o f a r r i v a l . that  glass.  zone w o u l d  transducer.  f l a m e was  t h e s o u r c e a l o n g the l i n e  t h e f l a m e was  of  after heating  along a glass  t h e same way  change i n t h e s i g n a l was  that  plate  of  C.  transmitted  The  experi-  any  properties  whether t h i s heated  i n exactly  of t h i s  gave a c e n t r a l  o f a n u l t r a s o n i c wave, an e x p e r i m e n t s i g n a l was  glass  waves  the temperature  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  ultrasonic  of the  to prevent  the g l a s s  seconds  a t 200°  To d e t e r m i n e suffice  that  a l t e r a t i o n o f the e l a s t i c  zone o f 5 mm  central  In addition, p l a s t i c  i t i s important  t h e h e a t s o u r c e be  w i t h a gas  i n the  to  the p r o p a g a t i o n o f e l a s t i c  take p l a c e .  ment, t h e r e f o r e ,  stress,  suited  becomes t o o h i g h , the p r o p e r t i e s  t h e p l a t e may  significant  tensile  are thus admirably I f the temperature  be a l t e r e d , and  y i e l d i n g may  -  away  receiver.  minutes  T h e r e was  also  no  I t i s concluded  o f the g l a s s  were  - 33 -  n o t 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 t o d i s t o r t the r a d i a t i o n p a t t e r n  o r the a r r i v a l  i s further  time o f the e l a s t i c  waves.  This  s u p p o r t e d by the f a c t  elastic  c o n s t a n t s o f t h e t y p e o f g l a s s u s e d do n o t change  more t h a n 3 p e r c e n t o v e r a t e m p e r a t u r e  that the  range  from 0 ° t o  2 0 0 ° C. I n Appendix computing zone. 45  I c a l c u l a t i o n s a r e shown f o r  r a d i a l a n d hoop s t r e s s e s  The a p p r o x i m a t e  temperature  sec o f the a p p l i c a t i o n  The  radial  temperature that  the c e n t e r o f the heated  8.  I t i s seen  increase  rapidly  zone, w h i l e t h e t e n s i l e  hoop s t r e s s r e a c h e s a maximum v a l u e a t a b o u t  1.5 cm f r o m  the c e n t e r o f t h e h e a t e d zone.  We have o b s e r v e d  fracture  a t about  point  does i n f a c t  of application To  initiate  originate  that the  1.5 cm f r o m the  of heat. a fracture a very shallow  scratch  a b o u t OS cm l o n g  i s made on t h e s u r f a c e  Heat  1.5 cm f r o m t h e end o f t h e s c r a t c h b y a  gas 30  Is applied flame.  After  t o 60 s e c o n d s )  7.  s e t up by t h e  i s shown i n f i g u r e stresses  after  i s shown i n f i g u r e  distribution  the compressive r a d i a l  towards  distribution  o f the flame  and hoop s t r e s s gradient  s e t up by s u c h a h e a t e d  a period  of build-up of stress  a fracture  gates i n the d i r e c t i o n  o f the g l a s s  forms  sheet.  (about  spontaneously and propa-  o f the s c r a t c h .  The s c r a t c h a c t s a s  a f l q w i n t h e g l a s s a n d t h u s d e t e r m i n e s where t h e f r a c t u r e  - 34 -  b e g i n s and a l s o propagates. insure  influences  the d i r e c t i o n i n which i t  Occasionally,  longer  fracture originating i n a previously  medium i s , f o r c o n v e n i e n c e , Fracture.  When a p r e v i o u s l y  fractures  e x i s t i n g f r a c t u r e i s extended  before,  however, t h a t  ment was p e r f o r m e d . distances  the I n i t i a l  I n the  fractures  To r e s o l v e  T h e r e was a p o s s i -  the doubt, an e x p e r i -  The g a s f l a m e was a p p l i e d  At a distance  propagate  f r a c t u r e s m i g h t be  was no f r a c t u r e f o r m a t i o n  5 minutes.  at different  At a distance of even a f t e r  heating  o f 5 cms, a f r a c t u r e was  t o f o r m a f t e r 4.5 m i n u t e s .  The f r a c t u r e  remained  f o r a s h o r t w h i l e a n d t h e n p r o p a g a t e d b o t h ways.  end m o v i n g t o w a r d s  reaching for  that I n i t i a l  f r o m t h e edge o f t h e s c r a t c h .  cms t h e r e  observed  scratch.  as i s observed v i s u a l l y .  propagating b i l a t e r a l l y .  The  The  f r a c t u r e s i s , as stated  0.5 cm l o n g  i t was assumed  unilaterally,  static  them.  b y t h e a p p l i c a t i o n o f a g a s f l a m e a b o u t 1.5 cms  beginning  for  t o produce  f o r producing I n i t i a l  away f r o m t h e end o f a  10  These two t y p e s o f  a r e d i f f e r e n t i n many a s p e c t s a n d d i f f e r e n t  t e c h n i q u e s have t o be employed procedure  unbroken  termed h e r e a s a n I n i t i a l  i s termed a s a n E x t e n d e d F r a c t u r e .  bility  were made t o  a straight fracture. A  it  scratches  t h e f l a m e came t o a s t o p  the c e n t r a l heated  fracture formation  zone.  The h e a t i n g  after time  decreased w i t h the distance  required o f the  - 35  point  of a p p l i c a t i o n of flame  scratch.  end  or  f o r m a t i o n was o f the  usually  proceeds  the  To  the g l a s s p l a t e in  addition  coming to a s t o p . visual  1.5  required for cm  and  causes  the  heated  i n the  zone i s a l s o u s e d  Therefore existing  of  to  at a  preserve (fig.9)  actually  the I n i t i a l  fracture,  hoop  stress  The  central  energy.  i s formed, a l a r g e c o n c e n t r a t i o n o f  fracture.  Our  time  fracture  hoop s t r e s s  to extend  s e t up  t h e n have t o overcome t h e  a  pre-  i s t o l o c k i n enough  f r a c t u r e which w i l l  extends.  stress  singularity).  energy  problem here  at  will  to  stop.  certain point.  i t requires very l i t t l e  The  which  i s heated  to lock i n s u f f i c i e n t  around the e x i s t i n g  s p o t A.  end  cms  Extended f r a c t u r e i s  t i p ( t h e w e l l known r " ^  the  10  to form, w h i l e h e a t i n g spot A  case  energy the  flame  heating a t spot B  f o r producing  to stop f r a c t u r e  o c c u r s a t the  the  unilateral  a spot A  o f the f r a c t u r e  technique As  The  fracture  o t h e r end  Once a f r a c t u r e  sees.  It i s this  c e n t r a l zone, w i t h z e r o or compressive  i s used  30  (about  impression of  f o r f u r t h e r use,  the I n i t i a l  the  time  the  s t a n d a r d i z e o b s e r v a t i o n s , and  causes  more i n v o l v e d .  edge o f  t h a t moves away f r o m  t o s p o t B.  The  t o be  the  a considerable distance  erroneous  propagation.  found  fracture  more) b e f o r e  gives  from  The minimum d i s t a n c e and  fracture The  -  be r e l e a s e d  T h i s i s done by  by h e a t i n g s p o t B compressive  heating (fig.10)  stress at  A.  - 36  In  addition  fracture  a s p o t G has  from It  -  t o be h e a t e d  from  i s also  possible  to extend  t o be h e a t e d  (figure  11),  the f r a c t u r e  I n t h i s case  to p r e v e n t  e x t e n d i n g i n t h e wrong d i r e c t i o n .  D has  t o be h e a t e d  away, and  to prevent  the s p o t B has  the  t o be h e a t e d  from  that  of extending short  Extended fractures. of  The  compression  after  the  fractures  passage  from  to lock  crossing  'go w i t h a b a n g ' .  preceded  by a number o f  audible  'tInks'  are not as  proceeds  which u s u a l l y  to s e t o f f the t r i g g e r .  The  3.3  Electronic Two  circuits used  is differ-  does are  These a r e  shocks'  zone  spurts.  do n o t have enough fore  Initial  through the  fractures  shocks'.  t o some p r o c e s s a l o n g t h e f r a c t u r e t h e t i p o f the  spot  energy.  sudden a s  i n short  Extended  'fore  the  running  in  o f the zone o f c o m p r e s s i o n  fracture  spot  fractures.  o f the f r a c t u r e  i s s l o w and  the  Of c o u r s e  the f r a c t u r e  in  fracture  Thus t h e t e c h n i q u e f o r e x t e n d i n g l o n g f r a c t u r e s ent  the  extending i n d e f i n i t e l y .  t h e d i r e c t i o n B D by h e a t i n g a t E . C has  to prevent  Only  the always discrete  energy  are probably  interface,  and n o t  due at  fracture.  Display  and  Recording  System.  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  have been u s e d .  The  first  f o r v e l o c i t y measurements and  system  electronic  (fig.12)  c a l i b r a t i o n of  is detectors.  - 37  The  second  radiation first  system i s used f o r s t u d y i n g from propagating f r a c t u r e s  system  i s essentially  O l i v e r , P r e s s and modification of  Ewing  elastic  (fig.  13).  (1954), w h i l e the second the s p e c i a l  source of e l a s t i c  energy  s q u a r e wave g e n e r a t o r .  the  c o n s i s t s o f ceramic band p a s s f i l t e r  addition a trigger  sweep.  The  second  of  preamplifier  system  circuit  because  e l a s t i c waves f r o m f r a c t u r e .  comes f r o m a d e t e c t o r n e a r fracture. discussed  3.3.1  The  various  in detail  Ceramic Barium  (10 -  The  20  detecting part  of  preamplifier  - camera.  amplifies  a  triggers  the  i s similar  i s used  The p u l s e  transducers -  t h e p u l s e g e n e r a t o r and  oscilloscope  The  - oscilloscope  take-off  circuit  by a h i g h v o l t a g e  by a l o w f r e q u e n c y  cycles/sec)  t h a t no  i s the  f o r the f i r s t  transducer activated  generator i s triggered  s i g n a l from  by  requirements  short d u r a t i o n p u l s e from a p u l s e g e n e r a t o r .  - variable  The  nature of i n d i v i d u a l o b s e r v a t i o n s .  a ceramic d i s c  circuit  wave  t h e same a s t h a t u s e d  o f the same t o f i t  nonrepetitive The  is  -  In  triggering  to the above  502  except  o f the h i g h e r energy triggering  the e x p e c t e d o r i g i n  signal of the  components o f b o t h t h e s y s t e m s a r e  i n the f o l l o w i n g  sections.  1  Transducers. Titanate  e l a s t i c waves a n d  t r a n s d u c e r s were u s e d  c o n v e r t them t o e l e c t r i c a l  to p i c k  signals.  up Two  - 38 -  t y p e s o f t r a n s d u c e r s were u s e d : The  cylinders  diameter, axis.  and a r e m a i n l y  sensitive  0.05 x 0.16 x 1.27  the v o l t a g e g e n e r a t e d the s e n s i t i v i t y .  cm.  to motion  cm i n  along the  strips  of  by t h e two  strips  The b e n d e r s  a r e most  differential  of  a d h e s i o n , and a r e v e r y d i r e c t i o n  sensitive.  a l o n g t h e edge o f a p l a t e  o s c i l l a t i o n s , w h i l e benders  c a n be u s e d  such  add up a n d sensitive  oriented with f l a t  to detect P  I f the bender  s i d e a l o n g the d i r e c t i o n  g a t i o n , m a i n l y S waves a r e p i c k e d u p . by 9 0 ° , s o t h a t  flat  side  Cylinders  on t h e s u r f a c e o f  t o d e t e c t e i t h e r P o r S waves.  orientation  of  displacement perpendicular to the plane  c a n o n l y be u s e d  the plate  barium  They a r e p o l a r i z e d  to  Is  benders.  fused together to form a p a r a l l e l e p i p e d  dimensions  increase  and  cm i n t h i c k n e s s and 0.64  The b e n d e r c o n s i s t s o f two t h i n  titanate  that  a r e 0,25  cylinders  of propa-  A change i n i s perpendicular to  t h e d i r e c t i o n o f p r o p a g a t i o n , means t h a t m a i n l y P waves are  detected. The  fibre  rods  d i s c s and b e n d e r s  to avoid r e f l e c t i o n s  were mounted i n l o w v e l o c i t y f r o m end o f t h e r o d s .  f u r t h e r prevent r i n g i n g both benders w i t h foam r u b b e r . ducers  a n d d i s c s were  The f r e q u e n c y r e s p o n s e  i s n o t known-, and c a n be e x p e c t e d  over the range difficulty,  o f f r e q u e n c i e s used.  To  clamped  o f the t r a n s t o be  To a v o i d  non-uniform this  most o b s e r v a t i o n s were made i n a v e r y n a r r o w  - 39  frequency filter  range.  B o t h h i g h and  were s e t a t t h e  attenuation point cut-off  then  (10  occurs  and  voltage output  4 volts at  at  h i g h e r f r e q u e n c i e s and  necessary. radiated  on  c p s ) ; the 1.30  the  study  the  3db  times  the  ranges  range, but  up  is  lower is  e n e r g y o f waves to give a An  transducer  amplification  to a l l o w the  oscilloscope  s i g n a l t o he  of properly  screen.  System.  filter  electromagnetic  system i s necessary  and  acoustic noise  amplified.  Major sources  fluorescent  lamps,  noise  transducers  frequency  order of m i l l i v o l t s .  Filter A  switches,  i s caused  to e l i m i n a t e e x t e r n a l  t h a t i s p i c k e d up  of electromagnetic and  n o r m a l 60  noise  cycle  and  are hum.  by movement i n the l a b o r a t o r y ,  m a c h i n e r y i n the b u i l d i n g ,  passing  t r u c k s , and o c c a s i o n a l l y  conversation. As  rather  discussed e a r l i e r ,  critical;  receiver On  the  some amount o f a m p l i f i c a t i o n  I n the p r e s e n t  of the  Aooustic  o f the  from a crack i s s u f f i c i e n t  displayed  loud  a t 0.77  t h e e x t r e m e low  i s usually sufficient  3.3.2  cut-off d i a l s of  same f r e q u e n c y  to  100  low  frequency. The  output  -  i t must be  to source  the o t h e r hand  t h e c h o i c e o f wave l e n g t h i s  s m a l l enough so  distance i s at least t h e w a v e l e n g t h must be  that  the  s e v e r a l wavelengths. large i n  - 40 -  comparison w i t h  the plate thickness.  In order  to f u l f i l  5 these both  conditions a frequency h i g h e r and l o w e r A  310 of  The f i l t e r  combinations.  The g a i n  i s zero  i n the mid-frequency  i s unavoidable. range,  i n c r e a s i n g a s one moves away on b o t h  the mid-frequency.  between  I n any v a r i a b l e band  some amount o f p h a s e d i s t o r t i o n  distortion  (Kron-Hite  consists of several sections  and i n p u t t e r m i n a l s i s u n i t y .  gradually  set  frequencies being undesirable.  resistance capacitance  pass f i l t e r , The  cps i s r e q u i r e d ,  c o m m e r c i a l v a r i a b l e band p a s s f i l t e r  AB) was u s e d .  output  c l o s e t o 10  sides of  When t h e h i g h a n d l o w c u t - o f f a r e  a t t h e same f r e q u e n c y ,  t h e phase d i s t o r t i o n i s  negligible. A s e t t i n g o f the low-cut-off d i a l  at 2 kc/s  effectively  eliminates ordinary electromagnetic  and acous-  tic  (This i s s t r i c t l y  amplifica-  noise.  t i o n o f 100; t h e s e t t i n g  true f o r a signal  i s liable  to vary with  higher  amplification),  3.3.3  A m p l i f i e r System, The  high  i n t e r n a l g a i n o f t h e T e k t r o n i x 502  o s c i l l o s c o p e makes a n e x t e r n a l a m p l i f i e r u n n e c e s s a r y . calibration necessary source  For  and f o r v e l o c i t y measurements i t i s sometimes  to detect s i g n a l s  of elastic  energy.  from a pulsed  transducer as a  The e n e r g y o f waves i s l o w a n d  - 41  amplification a  case  o f the  -  o r d e r o f 10  a c o m m e r c i a l b a t t e r y powered a m p l i f i e r  or a l a b o r a t o r y designed  instrument  have s u i t a b l e a m p l i f i c a t i o n and  3.3.4  i s essential.  D i s p l a y and The  i s used.  low  noise  In  such  (Tektronix) Both  types  characteristics.  Recording.  filtered  and  a m p l i f i e d s i g n a l f r o m the  trans-  d u c e r i s d i s p l a y e d on  the  screen of a Tektronix  or  535  osoilloscope.  i s used  502  A DuMont camera w i t h a P o l a r o i d back  to photograph the  display.  The  oamera b a c k  Is  m o v a b l e , p e r m i t t i n g s e v e r a l e x p o s u r e s t o be made on same  print.  3.3.5  T r i g g e r i n g System. To  special  record  the  disturbances  i n s t r u m e n t a t i o n was  f r o m the f r a c t u r e s  developed.  two-dimensional model s e i s m i c  experiment  I n the  usual  (Oliver, Press  E w l n g , 1954)  a short duration, high voltage  pulse  applied  transducer  of  to a  energy. so  This pulse  be  observed.  of e l a s t i c able  also triggers  In  i s plaoed  o f the f r a c t u r e . signal  the  present  To p r o v i d e  very near The  the  a  source  and  is  elastic  t h e o s c i l l o s c o p e sweep p o i n t s i n t h e medium  i n v e s t i g a t i o n s the  energy i s a propagating  time o f o r i g i n .  deteotor  The  to p r o v i d e  that f i r s t motions at d i f f e r e n t  can  the  a  fracture with triggering  expected  source  unpredict-  signal,  p o i n t of  d e t e c t o r i s a Barium T i t a n a t e  from the bender i s a m p l i f i e d , c r u d e l y  a  origin bender.  filtered  - 42 -  of  l o w f r e q u e n c y hums, and f e d i n t o  circuit  the s i n g l e - s w e e p  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 .  from t h i s  oscilloscope  i s used  T e k t r o n i x 502 o s c i l l o s c o p e  A gating  signal  t o t r i g g e r t h e sweep o f a  on w h i c h t h e P o l a r o i d  camera  i s mounted. Because o f the h i g h energy o f t h e e l a s t i c generated glass This  by f r a c t u r e s , a c o u s t i c  plate i s long  oscilloscope  compared w i t h screen.  the r e c o r d i n g  t h e beam sweep t i m e a c r o s s t h e  To p r e v e n t  c o n f u s i o n o f the f i r s t  succeeding signals  oscilloscope  the  T e k t r o n i x 535 g e n e r a t e s one g a t i n g  the  s c r e e n once.  the f i r s t  i t i s imperative  s t o p a f t e r t h e beam h a s moved  the  receiving  p e r s i s t s i n the  f o r t i m e s r a n g i n g up t o t e n s o f m i l l i s e c o n d s .  recorded motions with that  'noise'  of a train  The s i n g l e  signal  the s i n g l e  of  after  of triggering signals  Hence, t h e 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 t h e s c r e e n once, u n t i l  across  sweep c i r c u i t  t r i g g e r a m p l i f i e r , and then s h u t s o f f u n t i l  across  waves  from  reset.  sweeps  sweep c i r c u i t  Is  reset.  3.4  M e c h a n i c a l Arrangement. Glass plates  cm were u s e d  o f d i m e n s i o n s 45 x 45 cms o r 60 x 60  i n the experiments.  c o m m e r c i a l l y a s window g l a s s . plates  The p l a t e s  The c o s t  i s a p p r o x i m a t e l y 60 c e n t s .  No  are used  f o r the l a r g e r  special  treatment  - 43  was  -  g i v e n t o the g l a s s p l a t e s .  mately  3 mm,  The  approxi-  a l t h o u g h p l a t e s o f o t h e r 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 t i m e was wave t o be r e f l e c t e d - a up f r e e  t h i c k n e s s was  too s h o r t f o r an  sufficient  o s c i l l a t i o n o f the p l a t e .  elastic  number o f t i m e s  to s e t  Therefore elaborate  m o u n t i n g a n d s u p p o r t i n g a r r a n g e m e n t s f o r t h e p l a t e were unnecessary.  The p l a t e was  wooden b a r s 1.2  cm w i d e .  edges o f t h e g l a s s The  s u p p o r t e d h o r i z o n t a l l y by  T h e s e were p l a c e d a l o n g t h e  plate.  ceramic  t r a n s d u c e r s were mounted i n f i b r e  r o d s and clamped w i t h r u b b e r r o d s a r e clamped onto movable The  c o u p l i n g between  l o o s e , making remote.  two  o r foam p l a s t i c .  The  fibre  b e n c h e s made o f s t e e l  the benches and f i b r e  the p o s s i b i l i t y o f the whole  rods  rods.  was  system  ringing  - 44 -  CHAPTER  IV  PROCEDURE  4.1  C a l i b r a t i o n of Detectors. The  characteristics  s i g n i f i c a n t l y with vidual for  the type  ceramic p i e c e s .  choosing  up  only  i n t h e 100 k c / s r e g i o n ,  There should  longitudinal oscillations, Without a standard the frequency  2) e x h i b i t no  a  simple.  and v i c e  i s oriented f o r  versa.  and convenient  method o f  response o f detectors  L e t us d e f i n e as n o i s e  d i r e c t P a n d S wave f r o m a p o i n t  pulsed  transducer)  on a n e l a s t i c  S v i b r a t i o n s a r e termed  source  transducer  s i g n a l a considerable i s improperly  observed a f t e r  damped  The p r i n c i p l e than  (such as a The d i r e c t  I f we u s e a  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 to the t r u e  one c a n u s e -  any s i g n a l o t h e r  plate.  signals.  directional  oscillations  o n l y an e m p i r i c a l system o f c r o s s c h e c k i n g . is  which  be a minimum o f p i c k  o f t r a n s v e r s e waves when t h e d e t e c t o r  obtaining  those  and 3) show d e f i n i t e  t o l o n g i t u d i n a l and t r a n s v e r s e plate.  indi-  A p r o c e d u r e has been worked o u t  r i n g i n g a t that frequency  of the e l a s t i c  varies  o f mounting, and w i t h  from a group o f d e t e c t o r s  1) show a r e s p o n s e  sensitivity  o f each transducer  P and  detector  observe i n a d d i t i o n  amount o f n o i s e .  then a r i n g i n g w i l l  the P o r S s i g n a l i s r e c e i v e d .  I f the be  As the  - 45 -  e x a c t wave f o r m o f t h e e l a s t i c unknown, one c a n compare If  they a r e n o t s i m i l a r  the s i g n a l s from  transducers w i l l  reasonably  two d e t e c t o r s .  i n form and phase,  the d e t e c t o r s i s u n s u i t a b l e . of  s i g n a l a t 100 k c / s i s  t h e n one o f  Testing of a large  p r o v i d e a group  number  whose r e s p o n s e s a r e  identical.  The e l a s t i c  i n preliminary  testing  is  a p u l s e d transducer which g i v e s a mechanical  impulse  to  the p l a t e  generator. from  on r e c e i v i n g a 1 k v s p i k e p u l s e f r o m a p u l s e The a m p l i t u d e  such a source  o f the e l a s t i c  i ssmall,  than those r e c e i v e d is  source used  b e i n g about  from f r a c t u r e s .  done t o d e t e r m i n e  signals  received  100 t i m e s s m a l l e r  Hence a n o t h e r  i f the d e t e c t o r s m a i n t a i n  response  at signals  driven.  Two d e t e c t o r s a r e p l a c e d a t t h e same  test  their  o f h i g h a m p l i t u d e s , and a r e n o t over-  w i t h respect to an I n i t i a l  fracture.  position  The r e s u l t i n g  signals  s h o u l d be i d e n t i c a l a t l e a s t u p t o 200 m i c r o  seconds  after  the r e c e i v i n g o f the f i r s t  Instrument cut-off after  d i a l s were  filtering  oscilloscope 20 ^Ls/cm.  Settings  - The f i l t e r  s e t a t 100 k c / s .  i s around  signal. h i g h and low  The s i g n a l  1 t o 10 m i l l i v o l t s .  amplitude The  s e t t i n g s were f r o m 1 mv/cm t o 10 mv/cm, a n d  - 46  4.2  -  V e l o c i t y Measurements. M e a s u r e m e n t o f P and  p l a t e s was The  made u s i n g a p u l s e  circuit  transducer  (1 kv)  placed  s p i k e was  a t the at  calculated  slope  from the  time of a r r i v a l An P and  elastic  10  cm  intervals.  o f the  a g a i n s t the  r e c o r d i n g P and  However, o n l y  the  first  4.3  i s used  m o t i o n can  P  be  Unfortunately,  the  phase  the  estimate  and  i s the  i n the m i d s t S detector.  i s not  plot  of  cms  As  the  or  be u s e d than  S  f o r each the  first  Moreover,  the e r r o r .  S Waves. first  the  case  of spurious In order  was  higher  desired P  d e t e c t o r s can  an  S  velocity  d e t e c t o r s a b o u t 40  arrival  and  p i c k e d up w i t h a good d e a l o f this  o f the  the  o n l y f o r r o u g h measurements.  method p e r m i t s  The  by  line  Thus, the method I s l e s s a c c u r a t e  I d e n t i f i c a t i o n of P  arrives  velocity  f r o m a f r a c t u r e i s o f much  signals.  and  diso  P and  S waves f r o m a f r a c t u r e .  two  12.  distance.  i t i s e a s i e r t o p i c k out  one  The  straight  amplitude  fracture.  i n figure  a l t e r n a t i v e method o f m e a s u r i n g  disturbance  source.  a p p l i e d to a ceramic  S waves i s by p l a c i n g two  a p a r t and  for a  i n glass  edge o f a g l a s s p l a t e .  waves were r e c o r d e d  of  generator  f o r t h i s measurement i s g i v e n  A high voltage  of  S wave v e l o c i t i e s  f o r the  p i c k up  to s o r t out  of the  its first  confidence. S  phase w h i c h  the P first  phase motion  - 47  of  the S wave i t i s n e c e s s a r y  and  t o employ s u c h a i d s a s  for  S waves.  fied to  -  i t was  first  noticed  that  S motion could of  the P and  Determination The  controlled  be  S s i g n a l s had  been  identi-  quite  similar  t h e S w a v e f o r m was frequency  obtained  range i n v e s t i g a t e d . identification  source.  two  recording  ation of  o f the R a d i a t i o n  experimental  and  Pattern. of each  procedure.  However, f o r t h i s  the  the  S waves.  have a b a t t e r y o f i d e n t i c a l  the  of  by m a t c h i n g p e a k s  the r a d i a t i o n p a t t e r n f r o m a s o u r c e should  records time  n o n - r e p e t i t i v e nature  the  several  calculated arrival  a d d i t i o n a l check upon the  troughs  4.4  the  After sufficient  the P w a v e f o r m i n t h e  T h u s an  to study  observation  I d e a l l y , to  observe  like a fracture, instruments  surrounding  s e r i e s of experiments  c h a n n e l s were a v a i l a b l e .  Moreover,  coupling of i n d i v i d u a l detectors  makes i t seem u n l i k e l y t h a t more t h a n two  one  the  to the  channels  only variglass  could  be u s e d s u c c e s s f u l l y . The variable.  So  energy r e l e a s e d f o r the  direct  by  plotting  radiation pattern a monitoring monitor i s placed with  respect  placed  at a fixed  to the  Q  o f e i t h e r the P o r  detector i s necessary.  distance  fracture.  at various angles  each f r a c t u r e i s extremely  The  with  and  other respect  at  a fixed  detector  S  This angle  is  to the f r a c t u r e .  - 48  -  The m e a s u r e d a m p l i t u d e s a r e n o r m a l i z e d w i t h r e s p e c t monitor.  The  as f o l l o w s : direction; plane  coordinate the f r a c t u r e  system i s a l s o  axis.  wise  i s Z.  The  the o r i g i n  to the f r a c t u r e , origin  o f the  i n the  cartesian Q  i n an a n t i c l o c k w i s e manner f r o m t h e p o s i t i v e  X  outward  o f the p o l a r system.  s y s t e m we  define  (following Knopoff  r a d i a l motion as p o s i t i v e ,  t a n g e n t i a l motion as p o s i t i v e ,  and  o f the c o o r d i n a t e  direction  vectors.  of positive unit  a m p l i t u d e o f P waves a t 0  The  s y s t e m and  of P at 9 0 ° .  F o r S waves t h e S @ / P Q  ratio  been d e t e r m i n e d a t v a r i o u s a z i m u t h s . a t the p a r t i c u l a r  determine S@ / P  g o  f o r any 0  Without large was  any  compensation  scatter  Knowing  azimuth i t i s p o s s i b l e  the two m e a s u r i n g  transducers.  Q  / P Q Q and  SQ / P  A method has b e e n d e v i s e d w h i c h  B.  First  to  contact  f o r such unequal coupling,  of unequal coupling.  d e s i g n a t e d A and  the  .  i n the v a l u e s o f P  observed.  the e f f e c t  and  to  amplitude  A m a j o r p r o b l e m has been the v a r i a b l e between t h e g l a s s p l a t e  the  azimuth i s  amplitude  ratio  and  anticlock-  e x p r e s s e d as a n o n - d i m e n s i o n a l r a t i o w i t h r e s p e c t  has  is  on t h e X - Z p l a n e .  F i g u r e 4 shows t h e d e t a i l s  P @ /Pgo  X  Angle  I n the p o l a r  Gilbert)  before, i s  p r o p a g a t e s i n the p o s i t i v e  the d i r e c t i o n normal  o f the p l a t e ,  measured  system, as d e f i n e d  to the  L e t two  q  a ratios  eliminates  t r a n s d u c e r s be  A i s used f o r r e c o r d i n g P Q ,  - 49 -  and  B f o rrecording P  without Fg  g o  ,  from a f r a c t u r e F^.  d i s t u r b i n g A o r B, a d i f f e r e n t  i s u s e d s o t h a t A now r e c o r d s P  g o  P CA)/ P 6  The  qo  (60  a n d B r e c o r d s Pg> .  P Ceo/ p (A) 6  qo  mean P  now i n d e p e n d e n t  + P U»/P,.CA)] = P / P ,  C B )  e  s  ratio,  therefore, represents  f r o m two d i f f e r e n t figure  record  only  of course the  14.  fractures.  Transducers  the cos y  E a c h measured  t h e mean o f two o b s e r v a t i o n s The a r r a n g e m e n t i s shown  A and B a r e o r i e n t e d so a s t o  component o f P r a d i a t i o n .  i s independent o f y  .  The a n g l e s  The r a t i o  0 , y  s e p a r a t i o n s r , f , a n d d a r e shown i n t h e f i g u r e .  d i s t a n c e r between t h e f r a c t u r e is  origin  30 cms. From geometry /FjCD  =  ^_CF-jA +  or  Y 2 - T  -  ®  or  1^  z  */  f  =  %  therefore  0  o f unequal c o u p l i n g a t the contacts o f  t r a n s d u c e r s A and B and the g l a s s p l a t e .  in  location  The r a t i o s a r e  and  4r[PeW/ ^ is  fracture  >  This constitutes a s e t o f readings.  Next,  1  y  +  2  ICAF  -  9  - ®/2  and The  and t h e t r a n s d u c e r s  -  = r  and  j  and  d =  50  -  sin y  r «>s7  The  quantities ^  , d and f a r e c a l c u l a t e d  and  different  and a r e shown i n T a b l e  Q  the S amplitude  As arrangement  cannot  the o r i e n t a t i o n joining  i s s m a l l a t 9 0 ° , t h e same  o f the transducers a t an angle  study  0  ratio  i s again possible.  then  , from  Designating  t h e f r a c t u r e F-^.  B, a d i f f e r e n t records S Q P  The  C A )  .  g  and B t o r e c o r d disturbing A or  i s used  We  so t h a t A  have t h e r a t i o s  S CA)/p (B)  and  mean g i v e s S Q / P Q  coupling.  location F  e  e  , which i s independent  The a r r a n g e m e n t f o r t h i s  of trans-  the t r a n s d u c e r s a s  Next without  and B r e c o r d s P Q  VBV e The  fracture  However, i f  the interchange  A and B, a s b e f o r e , we u s e A t o r e c o r d P © SQ  origin  amount o f p i c k up o f P waves  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 .  the S © / P  Besides  to the l i n e  the p o i n t o f o b s e r v a t i o n t o the f r a c t u r e  which w i l l  ducers  1.  f o r S wave m e a s u r e m e n t s .  be u s e d  would cause a s i g n i f i c a n t  we  f o r r = 3 0 cms  of  unequal  i s shown i n f i g u r e  15.  s e p a r a t i o n r i s e q u a l t o 2 5 cms.  4.5  Determination The  valid  first  o f the F a r - F i e l d  Region.  motion theory o f Knopoff  o n l y i n the f a r f i e l d  region, i . e . , a  and G i l b e r t i s sufficiently  - 51 -  l a r g e number o f w a v e l e n g t h s away f r o m t h e s o u r c e . considered distances within  necessary  therefore  o f 5,5 cm.  the s o u r c e  amplitude  r"^  i n the f a r f i e l d  wavelengths  spreading  o f energy  o f t w o - d i m e n s i o n a l waves i s p r o p o r t i o n a l t o region.  the p o i n t o f o b s e r v a t i o n region  a r e indeed  r e g i o n f o r P waves w i t h  Because o f the g e o m e t r i c  the  field  to determine whether the  o f 25 and 30 cms f r o m  the f a r f i e l d  I t was  the value  Here r i s the d i s t a n c e o f  from the source.  o f the a t t e n u a t i o n  I n the n e a r constant i s  higher. As of  before,  transducers  coupling.  the arrangement i n v o l v e d  A and B t o e l i m i n a t e  the e f f e c t  interchange of unequal  Transducer A i s f i r s t used t o r e c o r d P  G  O  20 cms,and B i s u s e d t o r e c o r d P Q a t r cms d i s t a n c e Q  fracture  F-j_.  is  i s used  a t r cms and B r e c o r d s shown i n f i g u r e 16.  proportional  from  Next, without d i s t u r b i n g e i t h e r A o r B , a  new f r a c t u r e l o c a t i o n F g PQQ  at  to r  3 1  .  s u c h t h a t A now  P Q Q a t 20 cms.  The a r r a n g e m e n t  L e t u s assume t h e a m p l i t u d e i s 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 t h e r a t i o s  P,o W Cr)  Pa C=0)CA) 0  records  D  CO (A)  r 20  - 52  or  In  A least  n*c  square  a n a l y s i s gave t h e v a l u e  p o i n t s are p l o t t e d  value  of n  20  a<0  - 0.53 The  -  indicates  ±  0.08  on  log log scale  t h a t the  ( standard  made f o r P Q .  The  study, w i t h value  - 0.45  This confirms plotted  4.6  on l o g l o g s c a l e  The  is  i d e n t i c a l arrangement, case  ( standard  i n figure  came t o  was  be  error)  The  data i s  18.  Symmetry o f I n i t i a l F r a c t u r e s .  that I n i t i a l fracture one  stop we  cms  17.  region.  for n in this 0.09  be  i n figure  the p r e v i o u s m e a s u r e m e n t s .  As mentioned  at  i  to  error)  r e g i o n r ^ 20  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 A similar  of n  f r a c t u r e s are b i l a t e r a l  is initiated  end  (figure  the f r a c t u r e  had  i n s e c t i o n 3.2,  reason  by  9).  The  stress  i s heated  too f a r .  that there might  r a d i a t i o n between the A and symmetry by  o t h e r end  determined  i n nature.  inducing thermal  from p r o p a g a t i n g  to s u s p e c t  a test  the B s i d e s .  mostly only  to  Therefore  be a n We  The  unequal  tested for  simultaneously measuring amplitudes  at  45°  - 53 -  and  135°  S waves  measurements 0.96 we  1 0.08  find  from  o f the r a t i o  of  fractures. ^^5/^4.5  (standard d e v i a t i o n  no e v i d e n c e  fractures  Initial  induced  of b i l a t e r a l  by o u r m e t h o d .  The  rough  came t o be  o f t h e mean). asymmetry  of  Therefore Initial  - 54 -  CHAPTER  V  RESULTS  5.1  General. Energy from I n i t i a l  ranges from the audio eigenfrequency  fractures  i n t o t h e 200 k c / s r e g i o n .  The  of the glass p l a t e s d i d not permit  observe and r e c o r d fracture  and Extended  the audio  i s signalled  frequencies.  by a c l e a r l y  audible  c a t i n g measurable energy i n the audio  us to  But each 'ping',  range.  indi-  There i s  some r e l a t i o n s h i p between t h e s h a r p n e s s o f t h e ' p i n g ' sound a n d e n e r g y c o n t e n t region. usually  Fractures with show h i g h e r  i n the high 'pings'  amplitudes  o f high audio  frequency  amplitude  noise  100 k c / s frequencies  i n t h e 100 k c / s r e g i o n .  Almost a l l f r a c t u r e s that produced audio  frequency  very  loud, but low  ('bangs') showed no a p p r e c i a b l e  i n t h e 100 k c / s r e g i o n . The  P  'waveform' o b s e r v e d  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 e x c e p t a n a r r o w f r e q u e n c y around to  t h e 100 k c / s r e g i o n .  observe  I t i s remarkable  the r e p r o d u c i b i l i t y  a by  trough.  A l l first  a) m e a s u r i n g  suring  extrema,  The  two p e a k s a n d  m o t i o n m e a s u r e m e n t s have b e e n made  the height  the v e r t i c a l  therefore  o f t h e 'waveform'.  u s u a l P waveform c o n s i s t s o f t h r e e  band  o f the f i r s t  separation  extremum, o r b) mea-  between t h e f i r s t  and t h e  - 55  second  extrema.  when t h e  first  waves) the  procedure  arrival  i s higher  I t i s not  too  the  i s experienced  S motion.  The  of  spurious P  appears  t h a t the  waveform a t  this  beginning  three  extrema c h a r a c t e r i s t i c  S motion,  then,  such i s indeed three as  the  19  identified three  and by  20.  kc/s  the k i n k s  the  identification  extrema.  extrema,  be  altered  up  P of  the The  first  I f the  having  then and  but  first  of looking f o r  three  been  It  resembles  to i d e n t i f y  the  a  S small  first  measure. on  as  can  be  S If  the  i n magnitude as  i s unlikely,  S three  Definite  and  by p i c k  our measurements, based  Such a p o s s i b i l i t y  f r o m some t y p i c a l P and figures  of  three  difficult case,  of  w a v e f o r m o f P wave.  extrema waveform, w i l l  sign.  usual  a question  the  to  S wave,  of S i s obscured  f r o m t h e P w a v e f o r m by  motion preceding  motion i s extremely  the  identification  Then t h e  i s the f i r s t  waveform i s d i f f e r e n t first  to f i n d  of v a r i a b l e amplitude.  frequency.  S motion i s simply  signal  S m o t i o n have n o t  S w a v e f o r m a t 100  first  (as f o r S  (b) i s e m p l o y e d ,  first  i n the  oscillations  The  only  less.  difficult  difficulty  exact  s m a l l o r when  i f procedure  e r r o r s o f measurement a r e  easy.  been used  is indistinct.  Measurements o f so  (b) h a s  extremum i s v e r y  exact  noise r a t i o the  The  -  well  seen  e x t r e m a w a v e f o r m shown i n  a r r i v a l o f S waves c a n  t h a t a p p e a r i n some t r a c e s .  extrema waveform f o l l o w s the k i n k s .  be The  However,  - 56 -  all  S arrivals  a r e n o t so d e f i n i t e  rise  to the ambiguity  here  because  opposite  of f i r s t  the sense  to that The  motion.  first  Q  0  Q f o r both I n i t i a l  various  The  S r a d i a t i o n p a t t e r n c a n be d e t e r m i n e d  The P  e  of  errors  P  =  P  and E x t e n d e d  9 / 90 P  x  S  © /  fractures.  from  P  0  b e i n g compounded a r e much h i g h e r t h a n f o r e i t h e r  / P Q o r SQ / P Q  0  .  However, we c o u l d f i n d  no d i r e c t  means  d e t e r m i n i n g S e/PgQ r a t i o w h i c h w o u l d a l s o p e r m i t u s t o  interchange coupling. of  0/ 9O  S motion i s  have b e e n m e a s u r e d  for  S  this  by K n o p o f f a n d G i l b e r t .  / P Q and S / P ^  0  giving  We m e n t i o n  o f the observed  predicted  ratios P  and c l e a r ,  transducers to eliminate Scatter  the r a t i o s  ation  a t t h e same 0  .  of unequal  i n s u c c e s s i v e measurements We u s e t h e s t a n d a r d  devi-  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 s t a n d a r d  deviation ing  i s observed  the e f f e c t  o f t h e mean a s m e a s u r e o f s c a t t e r .  definitions  o f t h e s e q u a n t i t i e s have b e e n  The f o l l o w used  X  Mean  Standard  Deviation  Standard  D e v i a t i o n o f t h e mean  (f  07.-  xl n -1  £1 in  -  t h e n v a l u e s o f P Q /Pgo  where Xj_ r e p r e s e n t s the 10  same 0  •  57 -  I n g e n e r a l the standard  t o 20 p e r c e n t  o f t h e mean.  t h e mean i s o f c o u r s e much The Figures given  5.2  lines  lower. i n T a b l e s I I t o V I I and  Theoretical  v a l u e s a t each azimuth a r e  measured P ^ / P  i n figures  O  and S Q / P Q  deviation  o f t h e mean i s a l s o p l o t t e d  calculated  good f o r 3 0 ° ^ 0 P0/P Q Q  from  <90°.  i s almost  = 45°.  smaller.  The s e n s e  theory.  the 0  Q  / P  ratios will <^45°  a r e shown  For P Q / P Q Q  agreement w i t h & ^  theory  range.  0  measure-  i n magnitude  4 5 ° t h e measured magnitudes a r e  r a t i o we  differ  and G i l b e r t  the observed  The S Q / P  of S motion i s opposite  0  The t h e o r e t i c a l  the agreement i s  I n the 0 < 3 0 ° range i n value.  The  as v e r t i c a l  o f Knopoff  C o n s i d e r i n g the disagreement  and t h e S  SQ/PQQ  For  .  the formulae  constant  ments show r e a s o n a b l e to 0  ratios  23 a n d 24 a n d t a b l e s I I a n d I I I .  shown f o r c o m p a r i s o n .  ratio  G  as a measure o f u n c e r t a i n t y .  curves,  in  deviation of  I n i t i a l Fractures.  standard  by  ranged  f o r comparison.  plotted  up  S@ / P Q a t  r  deviation  The s t a n d a r d  i s presented  23 t o 2 6 .  The  are  data  o  from  that predicted  i n both P © / P Q  can conclude  9  that the  the t h e o r e t i c a l  values  -  5.3  Extended We  dislocations  58 -  Fractures.  believe  that Extended  fractures  o f the t y p e d e s c r i b e d  are  unilateral  by K n o p o f f and  Gilbert  as t h e i r model I I I .  I f so, then the measured r a t i o s  of  PQ/PQQ  d i f f e r e d markedly  There  and S Q / P Q  of d i r e c t i o n of motion at P Q  is  no r e v e r s a l  to  PQQ* a s p r e d i c t e d  PQ/PQQ  by t h e o r y .  are p o s i t i v e ,  source.  By  theory,' t h e m o t i o n  A l l measured r a t i o s  s h o u l d be  t h e Q <^ 2 5 ° r e g i o n .  in  the f o r w a r d q u a d r a n t i n s t e a d except f o r Q  with respect  i n d i c a t i n g m o t i o n away f r o m  in  fore,  from t h e o r y .  towards  The m e a s u r e d P Q / P Q of around  Q  = 90°, the n o r m a l i z i n g p o i n t ,  26,  to f i g u r e  shows l a c k o f a g r e e m e n t not  go  there the of  we  with theory.  i s no s i n g u l a r i t y  f i r s t quadrant.  As  i s opposite  i n the I n i t i a l that  ments f r o m  the Extended  waves was  from I n i t i a l  2 - 5  scatter,  ratio.  The  ratio  also does  quadrant,  value of S fracture,  0  / P  the  Q  in  sense  g i v e n by t h e o r y . were e n c o u n t e r e d i n m e a s u r e -  fractures.  Firstly  the amplitudes  t i m e s l o w e r t h a n t h e a m p l i t u d e o f waves  fractures.  greater  the  As P a m p l i t u d e  i n the m e a s u r e d  Several d i f f i c u l t i e s  of  the S Q / P ^  see t h a t  falls  values.  t o z e r o o r become n e g a t i v e i n t h e f i r s t  S motion  the s o u r c e  - 100°. There-  m e a s u r e d v a l u e s do n o t a g r e e w i t h t h e o r e t i c a l Referring  the  maxima  Q  of  due  T h e r e f o r e measurements  to the d e c r e a s i n g  second d i f f i c u l t y  was  had  s i g n a l to noise  that Extended  fractures  -  tended  to creep a short d i s t a n c e before going w i t h  'bang', as i t w e r e . waves were quadrant in  59 -  that  e m e r g e n t , and n o t s h a r p .  t h e f r e q u e n c y was  the f o r w a r d  bandwidth  The r e s u l t was  quadrant,  (figure  22,) .  about h a l f  i n spite  t h e r e f o r e are of dubious  true  of S Q / P Q  d a t a f o r SQ/TQ  measurements.  of the  measurements  S  frequency  o f the n a r r o w  value. We  b o t h P and  I n t h e 90°<C© <^180°  The m e a s u r e m e n t s  rant  a  filter  i n t h e b a c k quad-  This i s p a r t i c u l a r l y  have n o t p r e s e n t e d  any  i n the 9 O ° < ^ 0 < ^ 1 8 O ° r e g i o n .  - 60  -  CHAPTER V I  DISCUSSION  6.1  General. Prom t h e r e s u l t s  it  can  be r e a l i z e d  c o n s i d e r a b l y from disagreement  presented  In the  those  c o u l d be  of Knopoff  due  contemplated  Model 3 d i s l o c a t i o n , satisfy  and  Gilbert.  to various f a c t o r s ,  dent  and  w h e t h e r we  have  i n the  The  the  t o the  major  type  Gilbert  in  taken  care  theory.  the a t t e n u a t i o n f a c t o r ,  we  of  their to  From  c a n be  the  confi-  t h a t o u r measurements have been made i n t h e f a r  field  r e g i o n , as  Fracture Gilbert should  state  envisaged  i n the t h e o r y .  d e f i n i t e l y corresponds  correspond  with  way,  dislocation.  i n the d i f f e r e n c e  direction  from  t h a t the  t h e r e m i g h t be However, f r o m  t h a t m o s t o f P and  Initial  The  Extended  S energy  of u  z  .  fracture  B u t we  and  Fracture  t o the u n i l a t e r a l p r o p a g a t i o n o f  certainty  while  The  to Model 3 of Knopoff  f o r b i l a t e r a l propagation.  discontinuity  one  and  by K n o p o f f  the v a r i o u s a s s u m p t i o n s  determination of  chapter  t h a t our measured v a l u e s d i f f e r  ones b e i n g w h e t h e r o u r m o d e l c o r r e s p o n d s dislocation  last  a  can  propagates  only  only  a b i l a t e r a l propagation  of  o u r measurements i t a p p e a r s i s r a d i a t e d i n the  Extended F r a c t u r e s .  makes t h e p o s s i b i l i t y o f a b i l a t e r a l  The  forward  extreme  dislocation  asymmetry  - 61 -  p r o p a g a t i o n from a u n i l a t e r a l  6.2  High  unlikely.  Frequency. The  for  fracture  first  theory o f Knopoff  motions,  and G i l b e r t  i . e . the h i g h frequency  i s only  s o l u t i o n s . But  t h e y have n o t g i v e n a n y c r i t e r i o n f o r d e t e r m i n i n g a  certain  frequency  are f o r c e d  A most e m p i r i c a l  i s that  case  criterion,  the f r e q u e n c y  i s much s m a l l e r t h a n present  dimension,  by the f a c t  the a c t u a l f r a c t u r e  is  %  Let X  a t the source.  be t h e t i m e  I n the case o f  probably returns to  Our c r i t e r i o n  f o rhigh  interval  frequency  over which f i r s t  extrema.  motion  I n our usage  between a r r i v a l  (P o r S wave) and t h e s e c o n d  of observation.  i s further  t h e t h e o r y assumes a n i d e a l  00*B i s t h e d i r e c t i o n o f f r a c t u r e point  motion  m e a s u r e m e n t s a r e made on a s i g n a l .  w o u l d mean t h e time  signal  I n the  c r i t e r i o n h a s no  the displacement  a short time.  as f o l l o w s .  amplitude  for static  c a n be shown t o be l o n g e r  and t h i s  that  step-function displacement  zero a f t e r  dimensions.  The i n t e r p r e t a t i o n o f f i r s t  complicated  valid  the t h e o r y assumes a p r o p a g a t i n g p o i n t s o u r c e .  the source  meaning.  T h e r e f o r e we  i s h i g h i f the wavelength  the source  Hence a n y a r b i t r a r y w a v e l e n g t h than  whether  t o u s e o u r 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. sources,  i s s u f f i c i e n t l y high.  valid  o f the  In figure  27  p r o p a g a t i o n and A i s t h e  I n the i n t e r v a l  the t i p o f f r a c t u r e  - 62  to 0 ' ,  moves f r o m 0 9  and  In  where 00'  t h e a n g l e AO'B  sufficiently  -  i s 0+  .  60  .  the p r e s e n t  i s an  <  t i o n o f the  get  6.3  the  total 0  .  Diffraction  theory  time  P@  /Pgo  through  due  elastic  over  determine at  the  at  0  range  frequency  as  the  The  frequency,  kc/s.  with  ^30°  range.  to  spots  The  T h i s c o u l d be due or d i f f r a c t i o n a t  effect  could not  be  type  temperature  o f hot The  measured procedure  similar  to that described  squares  determination  the  s p o t has  refraction  significant  as  been effects  the  o f g l a s s change o n l y  range o f 0°  i f t h e d i s a g r e e m e n t i s due  =0°.  100  a  shows marked d i s a g r e e m e n t  the h o t  spot  e d g e s , we  frac-  i s considered  p e r i o d o f the h i g h  p r o p e r t i e s of t h i s  per cent  time  s m a l l i n s e c t i o n 3.3.  t o the hot  small.  ratio  edges o f the f r a c t u r e s . shown t o be  i s negligibly  Effects.  o n l y i n the ©  refraction  t h a t X> i s  constitutes a small  I f the  corresponding  The  contend  is  30  o u t e r l i m i t and  rough measure o f the we  a n g l e AOB  case  rT h i s o0  We £9  s m a l l i f the angle  The  to 200°C.  to the  experimental  i n s e c t i o n 4.5.  o f the  To  diffraction  the a t t e n u a t i o n c o n s t a n t and  3  s e t up The  e x p o n e n t o f r gave  for P is  least the  value  -  0.45 This  i s equal  at 0  -  of P © / P  edges  the  We  can  Q  conclude  SQ / P Q  -  (standard  attenuation  the  limits  of  error.  would d i f f e r that  of F r a c t u r e s  have so  f a r assumed  Gilbert.  the  anomalous  significantly  of P g / P  at  the 1/2.  from no and  g o  to O t h e r D i s l o c a t i o n Models that  correspond  instead  and  o r more o f  the  one  t h e r e must be  strain  to a  other  would  t o K n o p o f f and  precludes ment.  The  asymmetry  same a p p l i e s  g l a s s and  scatter  on  w o u l d be  other  to s t r a i n  e x i s t s i s due one  of  to Model 3  the  theory  both sides  in of  the the  difference of  fracture  plane  components o f  components.  i n the m e a s u r e d r a t i o s .  displace-  Any  o f a d i s c o n t i n u i t y i n the  r e a s o n s f o r the We  fail  t o see  u  The  t o random i n h o m o g e n e i t y the  3  states  difference  of  of  fractures  G i l b e r t ' s M o d e l 3.  both sides  a d i f f e r e n c e i n the  that  The  d i s c o n t i n u i t y i n the  conditions  Induced  combination of Model  components on  A  symmetry o f  that  models.  fracture plane.  fractures  correspond  a suddenly applied  or displacement  correspond  the  It is possible  studied  bility  Had  d i f f r a c t i o n plays  thermal s t r e s s i n glass p l a t e s  that  f o r P measured  to d i f f r a c t i o n  r o l e i n the m e a s u r e d v a l u e  Correspondence  K n o p o f f and  the  constant  f o r © < ^ 3 0 ° b e e n due  therefore,  error).  ratios.  We by  q  0.09  attenuation  significant  6.4  t o the  90°, w i t h i n  value  ±  63  observed  the  d i f f e r e n c e o f any  in  possiother  -  strain  6.5  or displacement  Velocity  the  that  fact  We must  this  that  oc .  is  preponderant. 90°.  c o u l d be t h e a p p a r e n t propagates  the value of ^  F o r ~$ —> 0 , t h e t e r m  to  The r a t i o P  Obviously this  expression f o r u  g  Q  contrary  b u t we  velocity  i n short  due t o  spurts.  o f our assumption can range involving  /PgQ—  > 0  0  f  i s n o t the r i g h t  anywhere 3-  ^  o  being  r  a  1  1  from  N  ®  except  solution.  The  i s  Here a g a i n the term i n v o l v i n g The r a t i o  Thus S Q / P @  Fractures  t o move a t s l o w e r v e l o c i t i e s ,  the consequences  i n that  0  —> 0 ° .  the v e l o c i t y o f  i s a constant w i t h a value of  the f r a c t u r e  examine  incorrect  Q—>  that  T h i s i s t h e maximum m e a s u r e d v a l u e .  have b e e n o b s e r v e d contend  component.  of Fracture Propagation.  propagation o f fracture .  -  s e c t i o n 2 . 2 we c o n t e n d  In  0.5  64  SQ/PQ  ratio will  f will  i s preponderant f o r then vary as t a n 9  have a s i n g u l a r i t y a r o u n d  t o our measurements.  0-?9O°,  -  65 -  There I s a p o s s i b i l i t y propagation  that the v e l o c i t y o f  o f f r a c t u r e may m o m e n t a r i l y r e a c h  o f t h e l o n g i t u d i n a l wave v e l o c i t y . curve ^  for j  5  = cC w i l l  = _^/S and  curve  still  ^ = oo .  Non-linear The  linear  to doubt r~s  I t c a n be s e e n  would It  and G i l b e r t  assumes t h a t  There i s evidence  the e x i s t e n c e o f the well-known  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  Secondly  the p o s s i b i l i t y  experimental  of a plastic  photographs presented i n  (1959) show a s e t o f p r o p a g a t i n g a t the t i p .  are absent. tip.  = oC.  to the f r a c t u r e .  Firstly  also a set of static  the  g o  i n t h e immediate n e i g h b o u r h o o d o f t h e t i p o f  the c r a c k .  'coronas'  5  even w i t h  of a crack s t r o n g l y suggests  Schardin  that the P ^ / P  theory i s a p p l i c a b l e i n the r e g i o n  adjacent  this.  zone  between t h e c u r v e s f o r  theory o f Knopoff  singularity  yield  0  Effects.  elasticity  immediately  I f so, the P ^ / P g  i s a v e r y 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 ,  6.6  fall  the value  I n t h e same p h o t o g r a p h s  fractures,  The c o r o n a  but i n these  i s indicative  The r a d i a t i o n p a t t e r n f r o m  t h e r e f o r e be c o m p l i c a t e d  with  there i s  the coronas  of a p l a s t i c a propagating  zone a t fracture  by t h e zone o f p l a s t i c i t y .  i s c o n c e i v a b l e t h a t the i n i t i a l  wave w h i c h i s c o n v e r t e d  fractures  disturbance i s a  plastic  i n t o a n e l a s t i c wave some d i s t a n c e  -  from the are  fracture.  i n the  elastic extent  far field  theory  explained  effects  by  plane  the  28  the  that  the  theory  have no  o f K n o p o f f and  we  mentioned  infinitesimal,  fault  extending  symmetry, we The  plane  plane  of  the  vation  i s A,  and  OA  and  are  measure-  Gilbert  in Plate  our  could  to  Fracture  f i r s t m o t i o n a t A,  we  makes a n g l e  only  s h a l l now  along  on  - Z. 0  with  the p o s i t i v e Y  obser-  the X a x i s .  AC  t h e X and  e l e m e n t 00'  case  point of  Z  axes  t i m e o f measurement  the  X  direction.  (which i n our The  -  Figure  i n the  the Y  of  the  discuss  our measurements.  o f measurement is X  X - Z  length of  s h a l l only discuss  the  the  i n f i n i t e s i m a l part the  to i n f i n i t y  plate)  During  t h a t on  f r o n t propagating  p r o j e c t i o n s o f AO  respectively.  and  l e n g t h on  infinite  the  AD  our  the  the f r a c t u r e .  I n a c t u a l case  of f i n i t e  direction.  idea of  d i s a g r e e m e n t between  f i r s t m o t i o n comes f r o m an  shows the  Because o f  linear  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  s e c t i o n 1.3  i s not  effect  where  Dimensions.  d i r e c t i o n and  is  But we  measurements  a n - e l a s t i c zone a f f e c t s  fracture front.  section  our  source,  A c t u a l Correspondence of F r a c t u r e  In  the  the  i n the n e i g h b o u r h o o d o f  i n Three  the  c e r t a i n that  region of  It i s possible  m e a s u r e m e n t s and  6.7  are  -  is applicable.  t o w h i c h the  ments.  be  We  66  ( X  ) of  contributes to  the  - 67  f i r s t m o t i o n a t A.  -  Hence f o r P waves,  A O >/o<  Now  (AO'  In our  - AO)  cLX  Therefore, 00'  =  -  (AO'  2.75  - AO  .2 )'v.*  5500  =  AO  By  = 30  =  (33  signals  from a p a r t o f the  fault  as  i f a l l o u r measurements a r e  ,  because have new T  z  infinitesimal.  to formula affect and  the  r.  we  coordinate  s y s t e m a t 0'.  have, from  equivalent  to  At  s i g h t i t appears  first  see of P the  Defining 28  long,  which  T h i s i s not  t h a t the  f o r AO',  the f i g u r e  measurements,  cm  neglect  and  our  cms,  f r o n t 28  a t t e n u a t i o n d e p e n d s on  t o d e t e r m i n e Tx  we  (2 - 22)  can  = 13.7  s  invalid.  the a m p l i t u d e s We  )  t h i c k , are  c e r t a i n l y not  could  cms.  - 30  is  Referring  m/s  i n s e c t i o n 1.3  t h o u g h made on a p l a t e 3 mm  in  ^OC  cms.  the a r g u m e n t p r e s e n t e d  that  =  X  case OC  and  =  ( o r S)  so.  only f a c t o r s are  changes  change i n l e n g t h  r,  , which i s small. with  respect  these  as  to  Tx.  We  a and  -  68  -  OC  OC  AO  AO'  AO  AO'  OD  has a d i r e c t i o n cosine  the  signals  fault  £  motion.  o f the f a u l t  from that  b u t i t must  o f AO b y  be remembered  f r o m the end o f t h e c o n t r i b u t i n g  segment  o f the  end o f t h e t i m e o f m e a s u r e -  Therefore  i n the a c t u a l  case  the e r r o r  r a t i o f o r 0 ° from Extended  i s much Fractures  t h e o r e t i c a l v a l u e o f 0 . 0 1 , when t h e d i r e c t i o n  gives  the f o r m u l a  differs effect  o f summation  from  by 9 p e r c e n t .  o f the c o n t r i b u t i o n  The v a l u e  earthquake  It still  fort  o f the whole  seismology,  o f the o r d e r o f 10  The segment  f o r S wave measurements i s c a n be done. the time p e r i o d  waves i s o f the o r d e r o f 1 s e c o n d .  to our p e r i o d  cosines  the measured r a t i o o f 0 . 5 6 .  , and s i m i l a r c a l c u l a t i o n s In  elastic  (2 - 22) a r e r e d u c e d  considerably  be s m a l l .  10 / * 5  that  a n d have a r e l a t i v e l y m i n o r e f f e c t on t h e f i r s t  The P Q / P Q Q  lower.  will  i s large:  segment  differing  f r o n t a r r i v e a t the t a i l  ment  in  This  ^  x  Thus t h e end o f t h e c o n t r i b u t i n g  9 per cent.  1 £  - OD AO_ _ ^ 2>p_ AO AO' * 3?>  AO  front  T  s.  This  of  corresponds  The s c a l i n g  factor  5 is  thus 1 0 .  Therefore,  1 km i n t h e f i e l d .  1 cm on t h e m o d e l c o r r e s p o n d s t o  Our r e s u l t s , then, a r e 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 a t 30 km d i s t a n c e f r o m a 26  km d e e p .  the  fault  too c l o s e  During  t h e time  b u t a s we have r e c o r d e d be  valid  i n the f a r f i e l d  considering  time a t i n f i n i t e v e l o c i t y .  source, i t i s p o s s i b l e  of fracture  and in  I n s t e a d of  i n the p l a t e ,  i s widening  propagation  of our r e s u l t s  as t h e moving  with  t h e Y-Z p l a n e  expressions f o r u  X,  The  An  direction,  The f r a c t u r e Our r e s u l t s  a s m e a s u r e m e n t s o f the f i r s t  m o t i o n on  i n u , propagating at  i n the X d i r e c t i o n . and u  •  of e l o n g a -  i n the Y d i r e c t i o n .  from a d i s c o n t i n u i t y  velocity  simply  Referring to  segment 0'00'' a s t h e X  t h e p l a n e o f t h e p l a t e a s the Y-Z p l a n e .  c a n be r e i n t e r p r e t e d  source.  segment.  2 and 28, we now c o n s i d e r t h e d i r e c t i o n  t h e p l a t e now p r o p a g a t e s  as the  i s s m a l l compared w i t h t h e  o f c o o r d i n a t e s i s now n e c e s s a r y .  o f the c o n t r i b u t i n g  infinite  then,  increasing  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  interchange  tion  should  i n t h e X-Z p l a n e  segment i n t h e Y % p l a n e  means t h a t t h e s o u r c e  figures  seismology:  to c o n s i d e r t h e t i p o f t h e  propagation of the f r a c t u r e  velocity  30 km i s  segment 00', ( F i g . 28)  the t i p of the f r a c t u r e  contributing  rate  that  our r e s u l t s  interpretation  The c o n t r i b u t i n g  elongates with  The  realize  motion  o v e r much l o n g e r d i s t a n c e s , a t l e a s t u p t o 100 km.  possible.  moving  We  s p e c i a l l y i n earthquake  Another i n t e r e s t i n g is  of r e c o r d i n g of f i r s t  a d v a n c e s a b o u t 3 km. to the s o u r c e ,  fault  a r e unchanged  z  The  theoretical  i f we  take  ^  =  6 0  - 69a -  and m e a s u r e  6  and  a r e the same a s shown i n f i g u r e s  S  Q  except  /P  &  that  from the Y a x i s .  The c u r v e s f o r P  /Pg  0  23 t o 26,  t h e y now r e p r e s e n t v a l u e s on t h e Y-Z p l a n e .  The e f f e c t o f w i d e n i n g  source  fracture  i n the p l a t e ,  now i n the Y-Z p l a n e ) w i l l  However,  even w i t h  still  d  (due t o t h e p r o p a g a t i o n o f  this interpretation,  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  be  small.  o u r measurements the t h e o r y .  - 70 -  CONCLUSIONS  We have s t u d i e d e l a s t i c wave r a d i a t i o n propagating Initial in  tensile  the l o n g i t u d i n a l  i n t h e 100 k c / s r a n g e  fractures  (Bilateral) Fractures  a direction  normal  the P r a d i a t i o n  t o the f r a c t u r e ;  amplitude  i s r e a s o n a b l y c o n s t a n t a t about  Fractures,  9  For i s a maximum  t h e P wave  drops s t e a d i l y u n t i l  P amplitude a t 90°.  from  i n glass plates.  amplitude  the  and t r a n s v e r s e  = 5 0 ° , a f t e r which the  F o r Extended  35 p e r c e n t o f  (Unilateral)  t h e P a m p l i t u d e maxima i s i n t h e f o r w a r d  quadrant;  t h e P wave a m p l i t u d e f o r  direction  of propagation) i s high,  P amplitude a t 90°.  0  = 0° ( i n the  b e i n g about h a l f the  I n a l l cases the f i r s t  P motion i s  away f r o m t h e s o u r c e .  F o r S waves, the f i r s t  away f r o m t h e f r a c t u r e  p l a n e , towards  the normal  The e x p e r i m e n t e s s e n t i a l l y  constitutes  fracture. partial  test  Gilbert  ( 1 9 6 0 ) , f o r t h e i r Case 3.  the  measured  theoretical For  to the a  o f t h e F i r s t M o t i o n T h e o r y o f K n o p o f f and  first  For I n i t i a l  P motion amplitude d i f f e r s  v a l u e i n magnitude o n l y f o r  Extended F r a c t u r e s  from the p r e d i c t e d For  motion i s  9  from the  <^ 3 0 ° r a n g e ,  the measured magnitudes  values a t a l l points,  except  S waves, t h e most s i g n i f i c a n t d i f f e r e n c e  sense o f the f i r s t  Fractures  motion.  The m e a s u r e d  differ 6  = 90° „  i s i n the  first  S motion  - 71  lias a s e n s e o p p o s i t e suspect and  -  t h a t p r e d i c t e d by  t h a t the d i s c r e p a n c y  t h e o r y may  elastic  be  effects  due  theory.  between the  t o the n e g l e c t o f  a t the  source  We  experiment the  non-linear  o f 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 ) . It of  i s impossible  e x p e r i m e n t and  theory  described  as f i r s t  defined.  Nevertheless,  t o make a p r e c i s e  s i n c e the d u r a t i o n o f what i s  motion i n the  theory  i s not  i t seems l i k e l y  that  d i s c r e p a n c i e s between the e x p e r i m e n t and above a r e call  first  theory  eight  to  we  the  theory  I f this  i s s o , we  conclude  adequate to d e s c r i b e a l l the  have t e s t e d o n l y one  described  particular  feel  the d i s c r e p a n c i e s a r e  c a d t doubt upon the u s e f u l n e s s o f  the  that  the  observations. case  independent d i s l o c a t i o n models proposed  G i l b e r t , we  cases.  precisely  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 w o u l d  motions.  i s not  Although  and  real  comparison  so  by  out  Knopoff  serious  theory  of  as  i n other  -  72 -  GAS FLAME  HORIZONTAL JLASS PLATE SCRATCH  Fig.  1.  Method o f a p p l y i n g gas flame t o g l a s s 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 long fracture i n three-dimensional 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 t o t h e Y a x i s and p r o p a g a t e s i n the X d i r e c t i o n . The d i s p l a c e m e n t i s i n t h e Z direction.  plate,  F i g . 3. I n f i n i t e s i m a l part of f r a c t u r e that contributes t o f i r s t m o t i o n a t any p o i n t on t h e X - Z p l a n e .  - 73  -  Z  X DIRECTION F R A C T U R E  F i g . 4.  O F PROPAGATION  C o o r d i n a t e s y s t e m and d i r e c t i o n positive unit vectors.  of  - 74 -  1.5  k*0.9/3 5=0.8/9 -  /  A  /  /  = 0.5/3  0.5  0  1  30  —> Fig.  5.  Distribution  of  1  60  90°  e <5Q  with  6  (after  Joffe,1951).  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 n o r m a l n t o S' i s o u t w a r d . I n the l i m i t i n g c a s e t h e o u t e r S' goes t o i n f i n i t y and t h e 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  o O  -  L_  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 o f 0.5 cm r a d i u s i s s u d d e n l y r a i s e d t o a t e m p e r a t u r e o f 2 0 0 ° C a t t i m e t = 0. The f i g u r e shows t e m p e r a t u r e d i s t r i b u t i o n a f t e r 45 s e c o n d s ,  -  76  -  E •Si  (SiDq F i g . 8. Stress the t e m p e r a t u r e  distribution distribution  )  i n the g l a s s p l a t e b e c a u s e shown i n f i g u r e 7.  of  - 77 -  S C R A T C H  Fig.  9.  Method o f 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 a v e r a g e 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.  EXISTING  B  F R A C T U R E  Fig.  10.  Method  of extending  C ,  E i i.  EXISTING  Fig.  11.  c  short fractures,  B a  D *  F R A C T U R E  Method o f e x t e n d i n g  long f r a c t u r e s .  -  LOW  78  TRIGGER P U L S E TRIGGER G E N E R A T O R TAKE-OFF INPUT HIGH VOLTAGE PULSE TAKE-OFF  FREQ.  OSCILLATOR  SOURCE T R A N S D U C E R  V  P R E AMPLIFIER  RECEIVING TRANSDUCER  TRIGGER INPUT  FILTER  AMPLIFIER  SIGNAL INPUT  O S C I L L O S C O P E A N D C A M E R A  Fig*  IS.  S c h e m a t i c c i r c u i t f o r m e a s u r i n g P a n d S wave v e l o c i t i e s i n glass plates.  - 79 -  TRIGGER D E T E C T O R  M O V E A B L E FRACTURE  D E T E C T O R  AMPLIFIER  POINT OFORIGIN F I L T E R  GLASS PLATE  FILTER  MONITORING \DETECTOR  TRIGGER, INPUT  SINGLE S W E E P  SIGNAL INPUTS OSCILLOSCOPE AND C A M E R A  Fig.  13.  Schematic c i r c u i t f o r studying fractures i n glass plates.  radiation  from  - 80  -  F R A C T U R E  Fig.  14.  T r a n s d u c e r 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 o f P 9 /P90 r a t i o . D i s p l a c e m e n t i n the d i r e c t i o n o f a r r o w s a t A and B c a u s e s upward swing o f 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  Fig.  15.  T r a n s d u c e r 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 tion of S e / P Q r a t i o . The a n g l e 6 may be replaced with 2 7^ - Q f o r s m a l l 6 .  - 81  -  5  j i_  6 7 8 91  /20  r  Fig.  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 Fractures. The s l o p e i s n. The d a s h e d l i n e s a r e n •+ t h e s t a n d a r d e r r o r .  F R A C T U R E S  |e  Fig.  16.  r  ^  T r a n s d u c e r 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 .  - 82  Fig.  18.  Attenuation Fractures.  -  o f P waves a t 0 °  from  Initial  -  83  -  DESCRIPTION OF FIGURES 19, 20, 21 and 22.  Traces of a c t u a l Each f i g u r e  shows f o u r  records.  t r a c e s on w h i c h  m e a s u r e m e n t s a r e n e c e s s a r y f o r a s i n g l e P@ / P q ratio  measurement.  trace.  or  S /P Q  t r a n s d u c e r used  to record  the p a r t i -  The t o p two t r a c e s a r e r e c o r d s f r o m a  f r a c t u r e F.. . The l o w e r two t r a c e s a r e f r o m a f r a c t u r e with  e  The l e t t e r A o r B i n t h e p a r e n t h e s e s  shows t h e p a r t i c u l a r cular  0  amplitude  the t r a n s d u c e r f u n c t i o n s  F  i n t e r c h a n g e d (see f i g u r e s  14  and 1 5 ) . In  figures  19, 20 and 21 t h e t y p i c a l  waveform o f P i s seen. a kink signified by a t h r e e third the  the a r r i v a l  extrema waveform.  t r a c e , and o n l y  trace  extrema  of figure  o f the S wave, w h i c h  20,  i s followed  The k i n k i s a b s e n t i n t h e  two o f t h e t h r e e e x t r e m a  a r e seen as  t r a c e went o f f s c a l e . Referring  first 21.  I n the second  three  P motion The f i r s t  towards  i s always S motion  the normal Figure  to figures  14 and 15, we  outwards,  i n figures  19, 20 a n d  i s a l w a y s away f r o m the f r a c t u r e ,  to i t .  22 shows t h e d i f f i c u l t y  waves f r o m E x t e n d e d  see t h a t the  fracture  i n the r e a r  f r e q u e n c y i s low, t h e e x a c t a r r i v a l s  of identifying quadrant.  are i n d i s t i n c t ,  S  The and  - 84 -  a m p l i t u d e measurements a r e d i f f i c u l t There 22 b e c a u s e amplified tures .  i s a higher noise  the s i g n a l s 2,5  t o make.  level  i n figures  21 a n d  from Extended f r a c t u r e s a r e  t i m e s more t h a n s i g n a l s  from I n i t i a l  frac-  - 85  (See  description  on  -  page  83)  _ 86 -  p 3  o  ( A  »  I 5 8  P  (  A  )  I58 »  S  P  90  >  (A  ©  p  3  0  S  .58 <> A  20 AO ^  p  < > B  .e 5  R  Extended Fig.  ( B  3Q  90  21. " (See d e s c r i p t i o n r  ») Extended  F i g . 22. o n pages 83-84)  - 87  20  40  60  80 9  Fig.  23.  100  120  [40  \60  180  •  P l o t o f measured P Q /Pgn r a t i o s from I n i t i a l Fractures, E a c h p o i n t i s t h e 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 s t a n d a r d d e v i a t i o n o f t h e mean. The t h e o r e t i c a l c u r v e i s shown f o r c o m p a r i s o n .  -0.5 Fig.  25. P l o t o f m e a s u r e d ? Q/^QQ Fractures.  ratios  from Extended  6  Jig.  26,  - 88  -  P l o t of measured S g , / P fractures,  e  r a t i o s from  Extended  89.  Pig.  27.  F i g u r e s h o w i n g change o f 6 propagation of the f r a c t u r e .  with  90.  Fig.  28.  F i g u r e shows c o n t r i b u t i n g infinite fracture front.  segment  of  - 91 -  TABLE  I  (see F i g u r e  14)  0  y  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  d  f  -  92 -  TABLE  P  0  /Pgo  Ratio  II  forInitial  Fractures  0  n  P /P Q  Mean  Standard ' Deviation  0  5  0.33  0.07  0.03  0.09  10  5  0.33  0.06  0.03  0.13  22.5  5  0.40  0.05  0.02  0.23  30  5  0.35  0.04  0.02  0.31  45  5  0.60  0.07  0.03  0.55  60  5  0.30 0.31 0.32 0.26 0.45 0.26 0,34 0.42 0.29 0.36 0.43 0.47 0.36 0.42 0.34 0.37 0.30 0.34 0.41 0.32 0.55 0.66 0.65 0.50 0.62 0.98: 0.92 0.83 0.90 0.80 1.00  0.89  0.07  0.03  0.77  90  e  g  Normalizing  Standard Deviation o f the Mean  Point  Theory  1.00  -  93 -  TABLE  s  9 /P@  0  n  22.5  8  30  9  45  5  90  4  Ratios  „ , S /P 6  6  2.40 2.82 1.83 1.82 2.28 3.85 5.48 4.13 2.67 2.62 2.00 2.18 3.27 4.02 2.19 3.74 1.85 2.79 1.94 2.05 3.30 2 75 - o!56 0.12 0.15 0.00  III  forInitial  Mean  Fractures  Standard D  e  v  i  a  t  i  o  n  Standard Deviation o f the Mean  m 1  Theory  3.05  1.30  0.46  -  4.59  2.73  0.78  0.26  -  4.16  2.57  0.57  0.25  -  2.72  - 0.07  0.33  0.16  0.00  - 94 -  TABLE P@ / P g  9  n  U  0  Ratio  Pe/Pqn 9  9  IV  f o r Extended  M  e  a  n  0  Fractures  Q. , , ^anaara Deviation  Standard Deviation o  f  t  h  Theory  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  0.04  0.02  1.04  0.16  0.08  0.07  1.12  0.16  0.09-  0.35  Normalizing Point 0.48 0.05  0.03  1.00 0.75  0.25  0.07  0.04  0.56  0.08  0.07  0.04  0.38  -  0.19  - 95 -  TABLE S Q /T?Q  0  n  22.5  7  30  5  45  4  90  4  135 ) 150 ) 158 )  5.00 3.14 4.83 2,70 4.10 4.63 3,03 3.36 3.50 , 3.09 4.03 3.34 2.60 1.80 1.80 2.60 - 0.36 0.10 0.31 - 0.50  Ratios  -  V  f o r Extended  Fractures  Standard Deviation of the Mean  Mean  Standard Deviation  3.92  0.95  0.36  43.52  3.46  0.35  0.16  - 24.33  2.20  0.46  0.23  -  5.55  0.38  0.19  -  0.62  0.11  No  acceptable records.  Theory  -  96  TABLE  VI  Calculated S Q / P Q Q Ratios  0  S g /P 9  22.5 30 45 90  0  e  S  e  /P  Q 0  6 / 90 p  Ratios  f o r Extended  Standard . _. Deviation  S  0.51 0.62 0.38  following relations  e/R  Se  - 1.05 - 1.29 - 1.46 0.00  Fractures  Standard Deviation  J  3,60 2.46 0*11 The  0.48 0,28 0.28 0,16  Theory  VII  cr  30 45 90  Fractures  Standard Deviation o f the Mean  1.35 0.82 0.64 0.33  TABLE S  forInitial  Standard Deviation  1.22 0.96 1.54 - 0.07  Calculated  -  o  f  t  h  Theory  e  Mean 0.24 0.32 0.19  - 1.61 - 1.93 - 0.62  have be/en u s e d :  - 97 -  APPENDIX  STRESS F I E L D  I  OF AN A X I A L L Y SYMMETRICAL  TEMPERATURE  GRADIENT I N A THIN PLATE  It  i s o f some i n t e r e s t  distribution  i n a thin  small c i r c u l a r area. will  the  plate  linear  the ambient coefficient  strain relation which,  subject  which  Clearly  be a x i a l l y s y m m e t r i c .  a t u r e above  to calculate  the  stress  i s heated over a !  the temperature  We u s e T t o d e n o t e  field t h e temper-  room t e m p e r a t u r e and o f thermal expansion.  t o denote The  stress-  i s g i v e n by t h e Duhamel-Neumann l a w , t o the assumption  o f plane stress  ( G\  = 0)  may be w r i t t e n a s  1-1  1-2  Two o f t h e e q u i l i b r i u m by v i r t u e  equations are s a t i s f i e d  o f the f o r e g o i n g assumptions;  is  dr  r  identically  the other equation  -  This  equation  i s satisfied  r  =  The c o m p a t i b i l i t y  r Substituting  dr  1  98 -  by t h e s t r e s s  and  r  cr~  function  -  d CD. d r  9  e q u a t i o n f o r r o t a t i o n a l symmetry i s  e  +  dr  £  I - l , I - E , and 1-4  r dr  "  r  r  ^  1-5  O  i n t o e q u a t i o n 1-5  1  - "IE  we  Replacing  t h e a b o v e , we  1-6  cr r u  obtain  r (J) i n 1-4  -- J J->L  1-7  r w i t h e x p r e s s i o n 1-7  Tr  get  dr  dr  T  i f  1-4  dT  Integrating  (p  dr -v J1L  we  have  +  JS*.  1-8  2  £t 1-9  -  For  finite  99 -  s t r e s s e s a t t h e c e n t e r Kg must  normal  s t r e s s must  normal  temperature d i s t r i b u t i o n  The f i n a l  vanish as r - > »  expressions f o r  r  CT  Tr 1  (i.e.  and  R  .  be z e r o .  The  Thus f o r any-  T — ? 0, a s r — ?  0  0  )  0~Q a r e  4r  i _ i i  Jo  1  T r dr To e v a l u a t e t h e i n t e g r a l s , we need a function of r . Jaeger given of  the s o l u t i o n s  time t ^ O r  and g i v e s  a glass plate 45 s e c o n d s  determine T as  (1959).  Jaeger has  f o r the case i n which an i n n e r  'a' i s h e l d  f o rvarious  1-12  F o r t h i s we u s e t h e c o m p u t a t i o n s o f  (1955) a n d C a r l s l a w a n d J a e g e r  radius  j  region  a t a c o n s t a n t temperature f o r t a b l e s f o r the v a r i a t i o n  time t .  of T against  E x p e r i m e n t a l measurements made u p o n  h e a t e d by a g a s f l a m e f o r a p p r o x i m a t e l y  indicated  'a  T  t o be a b o u t  0.5 cms a n d T a b o u t  2 0 0 ° C. Initial seconds a f t e r We have u s e d calculations. are  fractures  start  the a p p l i c a t i o n  between 30 s e c o n d s and 60  o f a gas flame  t o the p l a t e .  t h e mean v a l u e o f 45 s e c o n d s f o r o u r The p h y s i c a l p r o p e r t i e s  o f the;'glass  used  - 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 Coefficient  85 x  10  -  0.20  cal/gm  Density  -  2.5  gm/  Diffusivity  -  5.6  x IO"  Specific  = 0.5  t = 45  expansion -  heat  We r  of L i n e a r  cm.  assume a u n i f o r m t e m p e r a t u r e The  CTQ  the  main  w i t h r a t t = 45 text.  cm 3  cm  2  /sec  from r = o to  v a r i a t i o n of T(r) versus r f o r  sees i s g i v e n i n f i g u r e  and  °C  sees  7;  the v a r i a t i o n  of  i s given i n figure  <T  R  8,  in  - 101 -  APPENDIX  II  PLATE WAVES  O l i v e r , P r e s s and Ewing that  i t w o u l d be p o s s i b l e  plates The  (1954) f i r s t  t o u s e e l a s t i c waves i n t h i n  t o s t u d y p r o b l e m s o f s e i s m i c wave p r o p a g a t i o n .  principal result  symmetric  of their  s t u d y was t h a t t h e  p l a t e wave i n a t h i n p l a t e was  o f t h e d i l a t a t i o n a l wave i n a s o l i d . a stringent  requirement  sufficiently  that  the p l a t e  the a n a l o g  There  i s , however,  t h i c k n e s s be  s m a l l compared w i t h t h e w a v e l e n g t h  Extensive a  investigations  Ewing  (1917),  T o l s t o y and U s d i n  (1954),  We r e p r o d u c e the problem The u,v,w  (1953),  (1889),  O l i v e r , P r e s s and  and E w i n g , J a r d e t z k y and P r e s s here  studied.  o f the v i b r a t i o n s o f  t h i n p l a t e have b e e n p u b l i s h e d b y R a y l e i g h  Lamb  noted  (1957).  those r e s u l t s which a r e important i n  studied here. f o l l o w i n g n o t a t i o n h a s been displacement  used:  i n t h e X, Y, Z  directions  respectively. ^lc>£  etc.  stress  acting  i n the X d i r e c t i o n on a  plane normal t o the Y ®  dilatation  axis.  - 102 -  (p  and I f p  - displacement  -  ^ i f*'  potentials  density  - Lame's  constants  p p Let us consider planes of  z = ± h, w i t h  the p l a t e .  The  plate  a thin plate  the X and Y axes i n the median  The t h i c k n e s s  c a n be c o n s i d e r e d  bounding s u r f a c e s  bounded by t h e  free  o f the plate Infinite  plane  i s , then, 2h.  i n e x t e n t , w i t h the  o f n o r m a l and t a n g e n t i a l  Following  Rayleigh  functions  i n the s o l u t i o n s o f the e q u a t i o n s o f m o t i o n ikrr  involve  x only  This  implies  tion  i s finite,  (1889) we assume t h a t  stresses.  through e  that while —  We w i l l  II.  v i s finite  can  U  - —  i n the y d i r e c -  of solutions:  and v v a n i s h e s .  and u and w  vanish.  Two d i m e n s i o n a l s y m m e t r i c  vibrations  be w r i t t e n  the displacement  have two s e t s  u and w a r e f i n i t e  symmetric  and y does n o t appear a t a l l ,  - 0.  I,  Case I .  a l l periodic  of a thin plate.  and a n t i -  The d i s p l a c e m e n t s  i n the form  -  ii-L  , w  - -T-L  .+ _ r _ L  i i - i  -  where wave  t h e p o t e n t i a l s Cp  103 -  and y  a r e s o l u t i o n s o f the  equations  ^ The must v a n i s h  at  boundary c o n d i t i o n s across  the f r e e  A G  22  +  are that  surface.  2^  II-2  be  r  1  ^  Thus  =  duo-  c>u,\  d*.  c)2  by  c>2  the s t r e s s e s  0-  =  0  >  at 2 = ±V» n _  We assume a s o l u t i o n o f t h e f o r m  (p r (/\  where  &>oK  \ ) \ k  x  Da-  -  +  ,  g>  cosK  0*)  K "- k * 2  ,  i  e  (yit  = t£  - kx.)  y  - co  3  - 104  On we  substituting  o b t a i n the  -  equation  II-4  i n I I - l and  II-3,  equations  4  tosr\V\ tank  \>h 4  c o e f f i c i e n t s A and  Now  we  plane  II-6  Oh  The  can  II-5  D can  ^  '  be  separated  f r o m B and  c o n s i d e r a m o t i o n symmetric w i t h  z *> 0 w h i c h i s g i v e n  respect  C. to  by  ^ (cot -*x)  II-7  F o r waves l o n g compared w i t h the p r o d u c t s equation  vh,  kh,  II-6 reduces  (  x k  )  Oh may  be  the  thickness  considered  small.  2h, The  to  - 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  of plate  longitudinal  waves a s  Therefore  The  Cu  anti-symmetric  (p  'Y  Antisymmetric involves  -  D  motion  bending  II-9  m o t i o n i s g i v e n by t h e f u n c t i o n s  A  -  4 '  r  , sunh  , \)2:  cosh  e  \) 2.  t ( t o t - k*0  e  ( o t h e r w i s e known a s f l e x u r a l waves)  o f the p l a t e .  F o r waves l o n g  w i t h 2h t h e e q u a t i o n I I - 6 r e d u c e s  compared  to  11-11  These waves a r e d i s p e r s i v e ,  the phase v e l o c i t y  decreasing  to zero f o r long wavelengths. Case I I . A n o t h e r s o l u t i o n in  the plane  o f the p l a t e .  conditions II-3 apply.  Thus w  Consider  =  i n v o l v e s only motion 0.  The same  boundary  the equation o f motion  - 106 -  As  —  ^©  -  x  velocity  ,  M  ^  11-12  that the displacement  t h e wave e q u a t i o n .  at  +  —  i s z e r o , we f i n d  satisfies  The  .  1>  f  v  itself  The e q u a t i o n i s  P  1  i s g i v e n by \t±  w h i c h i s t h e s h e a r wave  4 f>  velocity  in infinite  To  ( P  satisfy  i n which  si/nn ^2  i s of the form  •+ Q cosh u z j e ^  11-13 = o  either  PsO,  case  -  Q - ^ c  11-14  os  whence V  =.  P  O l i v e r , Press the  The s o l u t i o n  the boundary c o n d i t i o n s o f  V  o r Q = 0,  solids.  equivalence  11-15  \)V  and E w i n g  (1954)  have e s t a b l i s h e d  between t h e l o n g wave a p p r o x i m a t i o n  of  -  the  first  plate.  plate  107 -  mode a n d d i l a t a t i o n a l waves i n a  T h u s , i t h a s b e e n shown t h a t  t a t i o n a l waves e x i s t  i n thin plates.  shear and  thin  dila-  The v e l o c i t y o f  s h e a r waves i n t h i n p l a t e s  i s t h e same a s f o r i n f i n i t e  medium, w h i l e  o f the p l a t e  the v e l o c i t y  wave i s m o d i f i e d . of  F o r long wavelengths, the v e l o c i t y  f l e x u r a l waves i s s m a l l , and t h i s  motion a r r i v e s plate  dilatational  so l a t e  that  type o f p l a t e  i t does n o t i n t e r f e r e  wave with  d i l a t a t i o n a l a n d s h e a r waves i n two d i m e n s i o n a l  modelling.  - 108  -  BIBLIOGRAPHY  B e n i o f f , H., F . 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