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Friction-induced oscillations in rock systems Wong, Kwong Ching 1974

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FRICTION-INDUCED  OSCILLATIONS  IN  ROCK  SYSTEMS  by KVONG B . S c ,  A  THESIS THE  CHING  N a t i o n a l  T a i p e i ,  Taiwan  IN  OF  FOR  THE  APPLIED  the  1971  PARTIAL  REQUIREMENTS  i n  U n i v e r s i t y  Taiwan,  SUBMITTED  MASTER  WONG  FULFILMENT DEGREE  OF  OF  SCIENCE  Department o f  M e c h a n i c a l  We  accept  r e q u i r e d  THE  t h i s  t h e s i s  E n g i n e e r i n g  as  c o n f o r m i n g  to  s t a n d a r d  UNIVERSITY  OF  BRTTiSii  A p r i l ,  197^  COLUMBIA  the  In  p r e s e n t i n g  r e q u i r e m e n t s of  B r i t i s h  i t  f r e e l y  agree f o r  t h i s f o r  an  a v a i l a b l e  t h a t  I  o r  h i s  that  p u b l i c a t i o n ,  t h i s  t h e s i s  f o r  w r i t t e n  i n  p a r t i a l degree  f u l f i l m e n t at  t h e  The  Date  U n i v e r s i t y  L i b r a r y  s h a l l  r e f e r e n c e  and  s t u d y .  I  e x t e n s i v e  may  be  c o p y i n g  g r a n t e d  b y  o f  t h e  I t  i s  p a r t  o r  t h e  f i n a n c i a l  o r  i n  g a i n  w h o l e , s h a l l  n o t  be  t h i s  Head  r e p r e s e n t a t i v e s .  o f  8,  /L'f^/  M e c h a n i c a l  o f  B r i t i s h  E n g i n e e r i n g  C o l u m b i a  Canada  f , f<?~?4.  make  f u r t h e r t h e s i s  o f  my  u n d e r s t o o d c o p y i n g  a l l o w e d  p e r m i s s i o n .  U n i v e r s i t y  Vancouver  the  the  KWONG  Department  o f  that  f o r  p u r p o s e s b y  agree  f o r  p e r m i s s i o n  s c h o l a r l y  i n  advanced  C o l u m b i a ,  Department  my  t h e s i s  CHING  WONG  o f  w i t h o u t  iii ABSTRACT  F r i c t i o n - i n d u c e d v i b r a t i o n s and the r a t e dependencies  of f r i c t i o n were s t u d i e d f o r s t e e l ,  and f o u r types o f r o c k . Cameron was  a n a l y s i s of  m o d i f i e d to i n c l u d e a f o r c e r a t e parameter  used by Johannes. r e l a t i n g the s t a t i c mentioned  The mathematical  glass  A new  e q u a t i o n was  as  used i n the a n a l y s i s  c o e f f i c i e n t of f r i c t i o n w i t h above  f o r c e r a t e parameter.  T h i s e q u a t i o n was  found  to be v a l i d f o r each of the m a t e r i a l s used which p o s s e s s e d stick-slip vibrations.  Reasonable  c o r r e l a t i o n s were  o b t a i n e d between the a n a l y t i c a l curves and the e x p e r i m e n t a l data.  S t i c k - s l i p v i b r a t i o n t h e o r y was  mechanics o f s h a l l o w earthquakes.  used to e x p l a i n  the  A rough c a l c u l a t i o n of  the permanent d i s p l a c e m e n t of the 19C-6 San F r a n c i s c o earthquake was  o b t a i n e d and found to be v e r y c l o s e to  the r e p o r t e d v a l u e .  A f o r m u l a f o r e s t i m a t i n g the p e r i o d  of the f a u l t d i s p l a c e m e n t was i n t r o d u c e d .  i v TABLE  OF  CONTENTS  C h a p t e r  Page  I  II  1-1  I n t r o d u c t i o n  1-2  H i s t o r i c a l  . ' .  .  .  1  .  .  1-2-1  E n g i n e e r i n g  1-2-2  G e o l o g i c a l  .  .  B a c k g r o u n d B a c k g r o u n d  .  .  .  .  .  .  .  .  .  .  .  .  .  .  1  .  1 .  5  T h e o r y  .  9  III 3-1  E x p e r i m e n t a l  3-2  I n s t r u m e n t a t i o n  3- 3  Specimens S l i d i n g  IV  4- 2  V  and  .  26  .  33  P r e p a r a t i o n  of  S u r f a c e s  E x p e r i m e n t a l 4- 1  A p p a r a t u s  36  R e s u l t s  P r o c e d u r e  39  R e s u l t s S t e e l  .  4-2-1  1020  A- 2 2  GX  4-2-3  P y r i t e  k-2-k  B l a c k  M a r b l e  4-2-5  Green  S c h i s t  4-2-6  G r a n o d i o r i t e  -  3.  s s  •  ••• .  .  .  « .  . •  .  .  .  • .  • .  .  .  .  • .  .  .  . •  41 ©  .  ^  "1 42  42 .  .  .  42 43  D i s c u s s i o n 5- 1  L a b o r a t o r y  5-2  A p p l i c a t i o n  R e s u l t s t o  E a r t h q u a k e  M e c h a n i c s  »  .  70  .  .  72  V  Table  of contents  cont.  Chapter  Page  V I  6-1  Conclusions  81  6-2  Recommendations  83  Appendix I  References  8k 88  vi L I S T OF  FIGURES  Figure  Page  1-1a  Stick-Slip Vibrations  3  1 - 1b  Quasi-Harmonic  3  2- 1  Model  2-2  p-V C h a r a c t e r i s t i c Slope Form  Curve  of a  2-3  p-V  Curve  of a Negative  Vibrations  Characteristic  Slope  10  of the S l i d i n g System  a t Low  Velocity  2-4  Humped F o r m  2-5  Possible  2-6  Linearized  2-7  Phase  2-8  Phase P l a n e Diagrams Velocities  2-9  Phase  2 - 10  p-9  Curve  3 - 1a  The  Arrangement  o f a p-V  Form  o f p-V  p-V  as  15  Characteristic  Curve  .  15  Characteristic  Curve  .  16 17  Curve  of a S t i c k - S l i p Cycle  Diagram  13  Region  Relationship  Diagram  Positive  at Different  i n Equation  Driven = 0  of S t i c k - S l i p at V  19  . . . .  20  . . .  22  (2-18)  of Apparatus  24  and 27  Instrumentation 3-1b  Close-Up View  3-2  Schematic Diagram  3-3  Pressure  3-4  Configuration o f the P l a t f o r m and the Suspension Device Wheatstone Bridge C i r c u i t of the S t r a i n  32  Ring  34  3-5  of Apparatus  Source  28  of the Apparatus  29  . . . .  System  31  ..  . .  3- 6  Specimens  38  4_1  r j ^ V e r s u s Log^ &  4- 2  a  Q  Versus Log  i r )  8  Steel Steel  with with  Lubricant Lubricant  .  kk  .  45  v i i L i s t  of  f i g u r e s  c o n t ,  F i g u r e  Page Versus  4-3  p  4-4  a  4-5  y  4-6  a  4-7  p  4-8  a  4-9  p  4-10  a  4-11  p  4-12  a  4-13  p  4-14  cx V e r s u s  4-15  p  4-16  a  4-17  p  4-18  a  g  Versus  L o g  Versus g  Log  L o g  Versus Versus  Log  Versus g  g  Log  Log  Versus  Versus s  Versus Versus Versus  4-19  Versus  9  1  1  1  1  1  Q  1  Q  0  Q  9  1  Q  Q  9  Log^^Q L o g ^ d L o g  Q  46  .  .  .  .  47  49 .  .  .  .  .  .  50 51  G l a s s  52  Glass  G l a s s  53  w i t h  G l a s s  w i t h  Dry  P y r i t e  L u b r i c a n t L u b r i c a n t  .  54  .  55 56  P y r i t e  Wet  57  P y r i t e  58  P y r i t e  P y r i t e P y r i t e  9  .  Wet  w i t h w i t h  B l a c k  .  .  .  .  59  L u b r i c a n t  60  L u b r i c a n t M a r b l e  61  . . .  62  . . .  63  p  4-22  a  4-23  Phase  Plane  Diagram  of  S t e e l  67  4-24  Phase  Plane  Diagram  of  G l a s s  67  4-25  Phase  Plane  Diagram  of  P y r i t e  68  Versus Versus  9  Log^^Q Log^Q  Q  M a r b l e  .  4-2 1  Q  B l a c k  .  a  1  Wet  .  4-20  s  L o g  1  .  48  G l a s s  Wet  Wet  9  .  G l a s s  Dry  9  Q  S t e e l  .  S t e e l  D r y  Wet  9  S t e e l  S t e e l  Dry  Q9  Log L o g  9  9  Q  Log  Log  Q  on  Dry  Dry  ©  Q  Log  Versus  s  Q  L o g  Versus Versus  1  ©  Q  1  on  Water  1  L o g  Versus  g  1  Water  0  Q  L o g ^ e  Versus  s  1  Wet  G r a n o d i o r i t e  . . .  64  Wet  G r a n o d i o r i t e  . . .  65  v i i i L i s t  of  f i g u r e s  c o n t .  F i g u r e  Page  4-26  Phase  Plane  Diagram  of  B l a c k  M a r b l e  .  .  .  .  68  4 - 27  Phase  Plane  Diagram  of  G r a n o d i o r i t e  .  .  .  .  69  5-  S t r i k e - S l i p  Mechanism  73  Index Map of S h o t p o i n t s and S e i s m i c P r o f i l e s i n the G a b i l a n and D i a b l o Ranges, C e n t r a l C a l i f o r n i a  75  5-3  A  76  5-4  M o d e l  5-2  1  Pure  Shear of  an  F a u l t  M o d e l  of  E a r t h q u a k e  E l a s t i c  Zone  V i b r a t i o n  System  .  .  77  1. CHAPTER 1-1.  INTRODUCTION: In  that  r e c e n t y e a r s , new t e c t o n i c  relative  movements  of adjoining  ^.ithosphere  are the results  convergence  and divergence  usually  accompanied  that  faulting  slip  vibrations  shallow  conditions. so-called  blocks i n the  be used  i n zones  relative  mechanism to explain  of  movements a r e  Geologists  vibrations  suggest  and that  stick-  earthquakes at  Some o f t h e s e  time  dependent have  been  friction  the  amplitude  periodicity  under  of static  concerned  of rock  of the rate  with  of stick-slip  of earthquakes  from  studied  triaxial  stress  with the  friction, but the stress  rate  friction.  sensitivity  induced vibrations  been  are concerned  present research i s concerned  investigation the  studies  mechanism  characteristics The  o f r o c k have  by other investigators  previous studies  dependent  an  o f the f r i c t i o n and  of rock which  with  systems,  especially  the i n t e n s i t y and  c a n be e s t i m a t e d .  HISTORICAL: 1-2-1. When  to  These  suggests  depth.  experimentally  1-2.  .  by earthquakes.  might  theory  of the actions  i s an earthquake  Stick-slip  no  I  ENGINEERING  BACKGROUND:  two s u r f a c e s a r e l o a d e d  a shear force,  movement b e t w e e n  the force these  that  together and subjected  resists  the relative  two s u r f a c e s i s d e f i n e d  as the  2. f r i c t i o n  f o r c e .  c r i t i c a l  v a l u e ,  take  the  f o r m  when  i t  v i b r a t i o n  may  f r i c t i o n whereas  i s  a  curve  o c c u r s  was  not  t h a t  to  In  f r i c t i o n  at  v e l o c i t y  and they  i n  low  a  speed  system  curve  to  Leben  produce (3)  o b s e r v e d  c o n s t a n t Bowden, p r o b a b l y  was  and Moore due  and to  he  the i f  the the  1-1b,  i s  m o t i o n (k)  w e l d i n g  i n  measure the  the  s u g g e s t e d the  d i d i n  t h a t  s u r f a c e  t w e n t i e t h k i n e t i c  F o u r that  1939»  In  f o r c e  r e s e a r c h  the  were  between  p r o c e e d e d  phenomenon  there v e r s u s  rubbed when  j e r k s .  t o -  Bowden  m e t a l l i c not  of  y e a r s  f r i c t i o n  s u r f a c e s  m o t i o n .  form  d i s p l a c e m e n t  s u g g e s t e d of  c u r v e ;  the  the  p e r i o d i c a l l y .  f r i c t i o n  of  i s  common  e a r l y  that  two  form  However,  f o u n d  (2)  the  c u r v e .  a  to  f r i c t i o n a l  T a b o r  the  the  i s  s a w - t o o t h  F i g .  s l o p e  i n t e r m i t t e n t on  1 - 1 a ,  t r i e d  K h a i k i n  n e g a t i v e  that that  W e l l s  where  a  u n t i l  f l u c t u a t e d  worked  has  system  f r i c t i o n a l  s i n u s o i d a l  a r e a  s e l f - e x c i t e d  v i b r a t i n g  the  s i t u a t i o n s .  ( l ) ,  and  r e g i o n  of  o r  f r i c t i o n - i n d u c e d on  may  phenomenon,  of  v i b r a t i o n  t h i s  when  K a i d a n o v s k i i  e x i s t  g e t h e r  v a r i e t y  1929»  m e a s u r i n g  l a t e r , must  a  a  w h i c h  the  F i g .  v i b r a t i o n ,  a  of  v i b r a t i o n ,  s i m i l a r  commenced  c e n t u r y .  h i s  i n  forms  to  commences,  f l u c t u a t i n g  d e p e n d i n g  F r i c t i o n - i n d u c e d and  The  two  v i b r a t i o n  q u a s i - h a r m o n i c  i n c r e a s e d  f r i c t i o n - i n d u c e d  o b s e r v e d  S t i c k - s l i p  i s  m o t i o n  d i s p l a c e m e n t  t i m e ,  be  f o r c e  g r o s s  c a l l e d  the  i n d u c e d  has  shear  v i b r a t i o n .  v e r s u s  c o n d i t i o n s .  w h i c h  of  When  p l o t t e d  the  r e l a t i v e  o c c u r s ,  v i b r a t i o n . i s  As  s u r f a c e s ,  r e m a i n In  19^3»  f r i c t i o n  was  at  p o i n t s  l o c a l  TIME FIG.  1-1 a .  STICK-SLIP  VIBRATIONS  TIME FIG.  1-1b.  QUASI-HARMONIC  VIBRATIONS  4. of  c o n t a c t  w h i c h In  out  i n  r e q u i r e d  the  t h a t  any  p l o t t e d  u s i n g i n  1949»  c y c l e  of  the  on  a  phase  p r e d i c t e d  w i t h  i n c r e a s i n g  s i z e  i n c r e a s e d  the  w h i c h  w i t h  s t a t i c  showed d r i v i n g  a m p l i t u d e  appeared  reached  s u f f i c i e n t l y D e r j a g i n ,  a n a l y s i s , s t a t i c the  S i n g h  and  In of  v i b r a t i o n  c o u l d 1962,  T o l s t o i  s t i c k - s l i p  a  P o t t e r  d e c r e a s e d  that  be  v e l o c i t y . i n c r e a s e d  j u n c t i o n s  time  of  s t i c k  i n c r e a s e d  to  e x p e r i m e n t a l  of  the  v i b r a t i o n  and  d r i v i n g  might  (12)  (10)  the v e l o c i t y  or  p r o v i d e d  v i b r a t i o n s .  r e d u c i n g  s i m i l a r  w i t h  h i s  so  c o u l d  t h a t  the  D u d l e y  (6),  t e c h n i q u e  f r i c t i o n  the  s t i c k .  v e r s u s  i n c r e a s e d ,  once  v i b r a t i o n  g i v e  as  p o i n t e d  v a l u e .  and  f r i c t i o n  f r i c t i o n - i n d u c e d  c o e f f i c i e n t  h i g h  of  was  a l s o  v i b r a t i o n  a m p l i t u d e  out  f o u n d  (11)  k i n e t i c  the  d i e  Push  a n a l y s i s  t h a t  n e g a t i v e  method  v i b r a t i o n  p u b l i s h e d  v e l o c i t y  to  d u r i n g  s u g g e s t e d  of  of He  g r a p h i c a l  of  (8)  and  he  that  as  t h e o r e t i c a l  time  (9)»  r e g i o n  d i s p l a c e m e n t  c o e f f i c i e n t  d e c r e a s e d  a  of  R a b i n o w i c z  1957  a  s t i c k - s l i p  c u r v e .  f r i c t i o n - i n d u c e d  v e l o c i t y .  i n  a  g r a p h i c a l  d r i v i n g  i n c r e a s e d  r e s u l t s  s  a m p l i t u d e  1951»  In  1  that  o b s e r v e d  the  In  maximum.  was  d e v e l o p e d  p l a n e  t h a t  of  v e l o c i t y  L i e n a r d  (7)»  They  a  e x i s t e n c e  v e r s u s  of  d e f o r m e d .  c o n f i r m e d  (5)  m i c r o - d i s p l a c e m e n t  S w i f t  t h a t  the  f r i c t i o n  By and  p l a s t i c a l l y  B r i s t o w  1945»  o s c i l l a t i o n s l o p e  were  the  be  a  u s e f u l  an  i n c r e a s e  By  a u s i n g  d i f f e r e n c e way of  to  h i s of  suppress  damping  r e s u l t .  o b s e r v e d  i n c r e a s i n g  that  the  v e l o c i t y ,  a m p l i t u d e w h i c h  was  q u i t e  d i f f e r e n t  p r e d i c t e d .  He  d i e d  a  One  out  at  y e a r the  (the  v e l o c i t y  f r i c t i o n  but  the  by  f o r c e by  9,  Green  (15)>  l e n g t h a r e a  of  of  s h e a r  time  and  t h a t  f o r  m e t a l s .  there  B.C.,  and  was  to  an  the  v i b r a t i o n  by  of  the  H i s  was  be  time  o f  s t i c k - s l i p i n  t h a t  s h e a r  was  p r o v e d  v i b r a t i o n .  when  no  of  t h a t of  f o r  out  s t i c k ,  the  m a t t e r  s u g g e s t e d  time  s u g g e s t e d  s t a t i c  e l e c t r i c a l  z e r o  mechanism  of  t r a n s v e r s e  f i n d i n g  asymptote  p r e d i c t e d .  p o i n t e d  o b s e r v e d  the  a n a l y s i s  v e l o c i t y  c o n c l u s i o n  to  c u r v e .  mechanics  governed  on  what t h a t  the  c o n t a c t . p l a s t i c  c o n t a c t the  r a t e  g r o w t h , curves  p ~© g  BACKGROUND:  earthquake to  of  was  r e c o r d e d  s u p e r n a t u r a l  c l a s s i f i e d  n a t u r e  the  (14)  T h i s  (16)  upper  a t t r i b u t e d (17)  about  r e d u c e d  g o v e r n i n g  f i r s t  A r i s t o t l e  a c c o r d i n g  M a r i o n  c o u l d  i n c r e a s e  depend  GEOLOGICAL  The ago  not  the was  was  c r i t i c a l  Johannes  s u r f a c e s  i n v o l v e d .  d i d  was  1 - 2 - 2 .  the  of  t h e o r e t i c a l  the  s t u d y i n g  o b v i o u s  a  out)  l o a d .  when  f o r c e  1972,  In d e f o r m a t i o n  no  was  c o n t a c t  n o r m a l  1971 »  t h a t  not  (7)  S w i f t  a m p l i t u d e  i n  and  d i e d  and  f r i c t i o n - v e l o c i t y  showed  concept  was  D u d l e y  the  the  a p p l i c a t i o n  the  a c r o s s  a p p l y i n g  (13)  new  of  i n  r e p o r t e d  of  p r o p o s e d .  r a t e  c o n d u c t i v i t y of  was  by  t h a t  v i b r a t i o n a  t h a t  v i b r a t i o n  v i b r a t i o n  d i v i d e d  Green  of  1969»  of  p o i n t  Cameron  when  In  r e s u l t  i n d i c a t e d  c e r t a i n  a m p l i t u d e  a m p l i t u d e  the  a l s o  l a t e r ,  that  s t a t i c  from  the  f o r c e s .  e a r t h q u a k e s e a r t h  about In  i n t o  movement  kOOO about  s i x  y e a r s 350  types  o b s e r v e d .  In  6. 132  A.D.,  the  direction  Chang Heng  of the f i r s t  In Mallet  (17)  chain.  Up  suggeated  ( 1 7 ) made a n i n s t r u m e n t  the middle suggested  impulse  t o t h e end o f t h a t  wide  century,  systematic  century,  of earthquakes the formation  collapse  of mountains,  explosions, upheaval striking  under  an e l a s t i c  led  (19)»  i n 1911, t o i n t r o d u c e  Reid  theory of  after  April  strain.  of mountains, and  one  snapping  observatory  (18) i n c l u d i n g  of volcanoes,  and t h e most  Robert  several ideas  outbreak  of mountains,  indicated  earthquake.  of the nineteenth  a world  as t h e cause  o f an  which  were the  the  downthrow  the sudden  This  latter  proposal  his elastic  he h a d s t u d i e d t h e San F r a n c i s c o  rebound  earthquake  18,1906. In  internal  the early  twentieth century  s t r e s s l e a d i n g up  to fracture  be  the mechanism o f earthquakes.  to  the shallow  earthquake  This  the growth  ( 2 0 ) was concept  mechanism accepted  of  believed to i s very  by  close  most  contemporary g e o l o g i s t s . In the  1910, Taylor  similarity  coasts drift  i n shapes  and i n t r o d u c e d which  formerly  u n i t e d and  movements crust. currents  suggested  would  be  ( 2 1 ) and Wegener and geology  the s t r i k i n g that  that  those  i n the mantle  ocean r i d g e s , and t h a t  of several  hypothesis  similar  of continental  adjacent  and Hess rose  fracture  and seperated  sea floor  spreading  coasts apart.  of the  ( 2 3 ) suggested  out  adjacent  t h e c o n t i n e n t h a d moved  associated with  Bernal, Dietz  (22) pointed  that  were The  earth's convection  under  the mid-  was  associated  7. w i t h  t h i s  c o n c e p t .  c o n t i n e n t a l  d r i f t ,  s p r e a d i n g , and  c o u p l e d  was  the  T h i s  t h e o r y  the  In  a d d i t i o n ,  i s l a n d  to  depths  arcs  was  r a t h e r  h i s  et  a l  be  w i t h  s i n g l e  i960,  phenomena  the the  i d e a  e a r t h ' s  1959»  of  i n  e x p e r i m e n t s f o u n d of  the at a  f a u l t  t e c t o n i c  t h e o r y  w h i c h  i n  {2k)  the  1968.  e n t i r e  oceans  mechanism  of  the  t r a n s f o r m  E a r t h .  f a u l t s  and  that  s i l i c a t e s h e a r i n g h i g h  few  deep  showed  f o c u s  and  e v i d e n c e  so  o f t e n  h i g h  he of  over f o u n d  t h a t  v i b r a t i o n G r i g g s  o c c u r r e d  Bridgman of  and  t e m p e r a t u r e  r o c k s .  m i n e r a l s .  f o c a l  e a r t h q u a k e s .  s t i c k - s l i p  s t i c k - s l i p  at  m e l t i n g  and  were  p o r p h y r y  phenomena  o c c u r  shear  s t r e s s  c r u s t  p r e s s u r e  e a r t h q u a k e s  o n l y  that  o b s e r v e d  w i t h  that  c o u l d  that  (26),  compounds  t r a n s f o r m  a r c s ,  at  of  o c c u r r e d .  on  f l o o r  Sykes  and  f r a c t u r e  c r y s t a l s  sea  i s l a n d  s u g g e s t e d  (25)  mechanism  on i n  i960,  worked  i n o r g a n i c shear  i n  at  and  p r o d u c e d  k i l o m e t e r s ,  based  than  of  of  of  e x p l a i n e d .  the  depths  t h e o r y  of  c o n t i n e n t s  h y p o t h e s i s  the  and  f r i c t i o n - s l i d i n g  ten  l a b o r a t o r y  (27),  O l i v e r  f o r c e s be  g l o b a l  e x p l a n a t i o n  the  or  J a e g e r i n  an  Hess  s u g g e s t i o n  new  Orowan  than  i n c r e a s e d  f l o w  of  1959,  s h o u l d  a  Wegener  l i t h o s p h e r e  I s a c k s ,  c o u l d  l e s s  t h e o r y  the  and  and  the  f o r  shear  f r a c t u r e  f a u l t i n g H i s  by  f o r m a t i o n  In due  of  p r o v i d e d  of  i n  w i t h  b a s i s  i n t r o d u c e d  T a y l o r  the ,\Dietz  u n d e r t h r u s t i n g  p r o v i d e d  at  The  (28),  200 t h a t  the  s t i c k - s l i p  b e h a v i o u r . In  1966,  B r a c e  and  B y e r l e e  (29)  p o i n t e d  out  t h a t  8. at  depths  be  the  be  c o n s i d e r e d  the  l e s s  mechanism to  m a j o r i t y  J a p a n  of  e l a s t i c  earthquakes,  and  be  l a r g e - s c a l e  s t i c k - s l i p .  t h i s  t h e i r was  i n  the In  1970  was  at  h i g h  s t a b l e  at  low  e x p e r i m e n t s  to  They  t h a t  showed  and  and  i n v e s t i g a t e  f r i c t i o n  l o a d ,  and  t h a t  the  time  (33)»  d e p e n d e n t ,  r e s u l t s  to  work  r e q u i r e d .  The  s p e c i a l shear  he  d i d  and  on  f r i c t i o n to  d i v i d e d  not  of  the  i n d u c e d the  by  1971  »  made of  l i m i t e d  upon  a  to  and  any  and  of  of  n o r m a l  the  s t i c k - s l i p .  f r i c t i o n  p r o v i d e  s e r i e s  q u a r t z - r i c h  t e m p e r a t u r e  t h a t  i n  r o c k s  was  s t i c k - s l i p  o b s e r v a t i o n . of  p r e v i o u s phenomena  p r e s e n t  work  v i b r a t i o n  parameter  normal  s l i d i n g  l i m e s t o n e .  of  f r i c t i o n a l  purpose  r e f e r e n c e  f o r c e  the  the  t e m p e r a t u r e .  i n i t i a t i o n  r e v i e w  and  are  t e m p e r a t u r e ,  h i s t o r y  t h i s  a m p l i t u d e  s t i f f n e s s  that  f r i c t i o n a l  not  and  c o n c e r n i n g  out  low  t h a t  they  f i n d i n g s  knowledge  s t i c k - s l i p  r e p o r t e d  f o r e g o i n g  more  f r i c t i o n  the  s u b s t a n t i a t e The  that  i n  r a t e ,  These  i n  depended  1972,  but  the  at  ) ,  was  p r e v i o u s  i n  2  the  s t i c k - s l i p  c r i t i c a l  t h a t  h i g h  ( 3  C a l i f o r n i a  r e s u l t s  p o i n t e d  c o u l d  n o t e d  (30)»  s t r a i n  and  at  Handy  that  D i e t e r i c h  they  i n  They  1968  c u r r e n t  p r e s s u r e  and  i s  of  (31)»  r o c k ,  specimen  o f  might  e a r t h q u a k e s  In  s u r r o u n d i n g s .  p r e s s u r e  Drennon  d e p t h .  e x p e r i m e n t a l  l i g h t  s t i c k - s l i p . u n s t a b l e  s h a l l o w  of  that  e a r t h q u a k e s  i n d e p e n d e n t  p r o p e r t i e s  s u r p r i s i n g  f r i c t i o n - s l i d i n g  of  at  f r o m  s t i c k - s l i p  k i l o m e t e r s ,  d e s t r u c t i v e  o c c u r r e d  c o n c l u d e d o f  t h a n . 2 5  l o a d .  ©,  of  the  r e s e a r c h  s u g g e s t e d  of  was  r o c k s  was r o c k r a t e  to  s t u d y  systems o f  the w i t h  a p p l y i n g  CHAPTER  II  THEORY  The of  the  present  t h e o r e t i c a l a n a l y s i s i s an  work  of  Cameron.  coefficient  of  friction  whereas, Johannes suggested  that  rather  theory  produced by  is  static  than  time the  a m a s s M,  load  rate  coefficient  of  stick.  Johannes  of  The  time  of  static stick, 9,  parameter, friction  of  and  depended  modification  concept  a  a vibrational  spring with  damping  coefficient  mass  the  the  not  of  slider  with  lower M,  respect  and to  a f f e c t the  i s zero, well  static  the  equilibrium between M  that  platform  the  datum.  x,  the  That  of  slider on  the  the  of  the  friction  a  consists  dash  pot  than  moves w i t h  i s , any  with  Assume  much h i g h e r  a  motion  of  that  the  that  of  velocity of  M  platform.  spring f  the If  platform  and  the  dash pot  between M  slider  the  V  will  i s in i t s equilibrium position  force, F , of  and  s h o w n i n F i g . 2-1. i s very  position.  and  as  K  platform  shear  displacement,  system which  stiffness  movement  forces  as  C  the  When t h e  The  a  of  the  ANALYSIS:  Consider  as  dependent  that  considered.  THEORETICAL  V  was  introduced  the  upon ©  Cameron assumed  extension  the  i s measured  coefficient  i s not  and  zero,  of  the  are  zero  platform.  from  this  friction, mass  and  JJ,  FIG.  2-1.  MODEL  OF  THE  SLIDING  SYSTEM  11.  M will with  stick  to the platform  velocity  resultant  V.  force  shear  force.  gross  relative  to  This  situation  of the spring  Then,  the platform will  last  i s moving  until  the  and the dash pot equals  i f the platform  motion between  moves  t h e mass  slightly  the  further,  and t h e p l a t f o r m  star  occur. During  is  when  almost  zero, Kx  where W  the s t i c k  period,  the motion  + C i + Mx  i s normal  =  load  i n which  of the s l i d e r  relative  c a n be  motion  expressed  Wp  (2-1)  s  and i s equal  t o Mg  i n the  present  system. Assume  that  the v e l o c i t y  x  = V =  of the platform  i s constant, i . e .  Constant  Therefore, dV _ Equation  _  =  (2-1) Kx 9  force  dx =  x  =  becomes:  + C x = WjJ i s defined  (2-2) as t h e r a t e  d i v i d e d by the normal  Assume  that  W i s time F dt  From  0  v  W  = Kx =  KV  acting  on t h e  independent, F  ;  equation (2-2): F = ^ ( K x + = K x + Cx f  load  of applying  " ¥ Cx)  for x =  0  the  shear  surface.  as  12.  Therefore,  4  = f  At that  is  at  contact  ( 2 - 3 )  the  the  end  surface Kx  instant of  is  stick  equal  + Cx = ¥ U  Consider whereby  the  term  when  the  slider  period,  to  the  the  starts shear  static  to  slip,  force  friction  at  the  force. ( 2 - 4 )  g  case  Cx i n  the  where  x  and C are  very  ( 2 - 4 ) becomes  equation  small  negligible,  then, Kx  =  Wp  s  Ps = ¥  < -5) 2  During of  the  slider  Consider  the  x  and U i s system  the  is  that  x  Consider as  shown  of  the  achieve region.  in  Fig.  curve. steady  the  in  this  the  motion  platform.  situation: (2-6)  of  a non-linear condition exist  a  friction function  that  of of  the  velocity.  frictional  region  sliding  vibrations  of  negative  slope  with  positive  slope  in  curve.  a jj-velocity  2 - 2 , with  sliding  Friction  as  cycle  0  must  System  the  WTJ  ^  necessary  friction-velocity  of  same  coefficient  usually  there  the  equation  + Mx =  kinetic  is  portion  longer  ^ V,  The occur  no  + Cx  the  and  is  slip  governing  Kx where  the  p  g  having motion  induced  curve  o c c u r r i n g at a p-V  curve  the  lowest  similar  throughout  the  vibration will  not  to  whole  point  this  will-  velocity  commence  13.  .  2-2  p - V CHARACTERISTIC SLOPE FORM  CURVE  O F A  POSITIVE  14.  build-up  because the  applied A  Fig.  of  force second  region  found  velocity  often  shown i n F i g . 2 - 6 . expressed  P  behaviour  occurs  i n the low  i s increased.  quasi-harmonic  vibration  shown  i n Fig.  quasi-harmonic  stick-slip oscillations descriptive  i n the low  of the  general  i s shown b y F i g .  f r i c t i o n - v e l o c i t y curve i n F i g . 2 - 3 which  This  2-5.  f o rstick-slip  i s linearized  p - V relationship  linearized  that  as can  as: p  =  m  Equation Fig. 2-6.  i.e.:  the f r i c t i o n  possessing  of materials  i s shown  i n the low  I t i s believed  assumed t h e form  A p - V curve  typical  oscillations  of  show  behaviour A  to  to  slope  as v e l o c i t y  out that  materials  region.  frictional  of negative  and d i m i n i s h e s  that  oscillations  i n opposition  similar  Stick-slip vibration  when t h e p - V c u r v e  existed  be  a region  (34) pointed  Ko  was  of U - V curve,  i n Fig. 2-3 describes  most m a t e r i a l s .  velocity  It  form  i s shown i n F i g . 2 - 3 .  region  p - V curve  resistance  i s impossible.  2 - 2 , but baving  velocity the  of f r i c t i o n a l  +  |*(v - i ) ( 2 - 6 ) c a n be  Put equation  Kx  + Cx  + Mx  Kx  +  + fiv)k  (C  (2-7)  =  linearized into  (2-7)  W ( p  +  m  + MX  =  Define:  ^ = £(C + (W) (KM)"  2  with  equation  the  help  ( 2 - 6 ) :  pV - p i ) w  ( p  m  + jiv)  (2-8)  2  VELOCITY FIG.  2-3  fJ-V C H A R A C T E R I S T I C CURVE OF A NEGATIVE S L O P E A T LOW VELOCITY REGION  VELOCITY FIG.  2-4  HUMPED CURVE  FORM  OF  A  U-V  CHARACTERISTIC  16.  FIG.  2-5  POSSIBLE  FORM  CHARACTERISTIC  OF  U-V  CURVE  VELOCITY  FIG.  2-6  LINEARIZED  p-V  RELATIONSHIP  CURVE  18. A standard  solution = |Ky  x The  first  to equation + pv) + e  m  derivative - A U  x  (2-8)  i s :  -ACJ,t  -  IAcoscO t  + Bsin^tJ  d  (2-9)  of x gives: t  d  r  = (0 e  (A + AB)sin<^t I  j^B - A A ) c o s t J t -  d  d  (2-10) The  second d e r i v a t i v e = -o  x  of x gives: l[ ^  e  d  A  solution  on  a phase-plane  In  equation (2-8) Kx  let  +  i  ( 2 - 1 1)  2  of equation  as shown  ) + 27iBjcosw t  - A ) - 2AA] s i n c j j t j  [B(1 A  - A  1  (2-8) c a n a l s o be  i n F i g . 2-7  + ( C + p>¥)x + MM  = Wiy^  presented  (lk,3k).  + /3V)  x = y, ..  d y  x = y-fdx  or  3  Equation  (2-8) becomes: Kx  + ( C + /3W)y + M y g  = W(U  + (IV)  m  (2-12)  dy Settxng  = 0, t h i s X  =  For changes. zero at a  gives  K Fm P ) (  +  V  various  " S  isocline  three  curve.  the an can  higher  trajectory equilibrium only  exist  than  ceases  P  t o meet  i*'^) the trajectory each w i t h a limit  the cycle at c r i t i c a l V, a s shown  x = V line  Obviously,  at a velocity  lower  curve.  ) y  trajectories  critical  situation.  isocline  F i g . 2-8a showa  l o w V a n d F i g . 2-8b showa velocity  +  ( C  slope  driven velocities,  F i g . 2-8 s h o w s  slope  the zero  than  cycle V.  i n f i g .  and s p i r a l s  stick-slip o r equal  a  At 2-8c, into  vibrations tothe  0  1  J  3  2  V  FIG.  2-7  P H A S E D I A G R A M OF A S T I C K - S L I P  CYCLE  21 . c r i t i c a l of  the  v a l u e .  v i b r a t i o n  A p p e n d i x  when  the  At  =  p o i n t From  n e c e s s a r y  c r i t i c a l  v e l o c i t y 0,  c o i n c i d e s  f o r  c o n d i t i o n  the  are  e x t i n c t i o n  c o n s i d e r e d  i n  =  1  so i s  2 - 7 ,  F i g .  w i t h  the  x  approaches  2  p o i n t  c o n s i d e r  the  case  z e r o ,  that  i s  c o i n c i d e s  w i t h  p o i n t  as  shown  i n  as  0  —  6 1,  .  and  2 - 9 .  F i g .  KFso  ( 2 - 9 ) ,  E q u a t i o n  i n  t r e a t m e n t  ( 2 - 5 ) ,  e q u a t i o n  U r  the  a p p r o x i m a t e  p o i n t  X  Where  an  d r i v e n  0 , 3  at  c o n d i t i o n s  I. As  V  The  s t a t i c w i t h  = —y 1  t  Q  c o e f f i c i e n t = 0 ,  =  of  f r i c t i o n  at  V  =  0 .  y i e l d s  + A  Kr m  T h e r e f o r e ,  = K ^ F s o " Pm>  A  At  2  p o i n t  i n  ( -^) 2  2 - 1 0 ,  F i g .  t  =  •• ^  , d  t h e r e f o r e ,  * - lV - ^ [ I ( 2  Let  OC  be  the  m  a m p l i t u d e  of  P s o  th©  - j)  (-'5)  V  v i b r a t i o n  • i P . o - SP. * • [I<P.O-M  To  c a l c u l a t e  o t h e r  than  the  z e r o , x  o  =  t 2. In the  f a c t ,  the  d r i v e n  a m p l i t u d e  of  the  g i v e n  by  vJ\  v i b r a t i o n  at  a  v e l o c i t y  assume:  x . 1  *  CO, d  v e l o c i t y  v e l o c i t y  d u r i n g  that  the  s l i p  i s  d r i v e n  so  much  v e l o c i t y  h i g h e r can  be  than  F I G .  2-9  PHASE  DIAGRAM  OF  STICK-SLIP  AT  V  =  23. neglected  when p l o t t e d  on a d i s p l a c e m e n t - v e l o c i t y  The  arising  this  error  from  (2-16),  Equation calculate  the amplitude  U -6  as shown  in  results  1 - u  u  r  'so Equation  (2-17)  —  y r  TTd  1  +  b  amplitude  The  a curves  as: +  Pm  < - ) 2  m  i  —  - y so 'm  —  1  +  ^  b  ><  1  +  e  of vibration  ~ >  ( -  of the vibrating  system  M  2  V —»~ 0 ,  c a n be c a l c u l a t e d  (2-19)  i . e . 9 —*- 0, e q u a t i o n a  =  K  (  1 y - y iso m  ^ d +  b  from  X  1  +  becomes:  )  e  0  +  a" *) 7  '  v  which  i s t h e s a m e as  equation  (2-16)  W h e n V — — oo , 9 —v oo , a n d c l e a r l y t*  and  velocity.  a t V —*• 0 .  from  equation  (2-19)  0  the amplitude  9  equation  r  *(u - U )(1 K^rso ' nr  i  >  e  (2-19). If  1 8  e  Knowing a l l the parameters the  work,  becomes:  = ¥  a  of previous  i n Fig. 2-10 i s introduced.  F i g . 2 - 1 0 c a n be e x p r e s s e d  Ps =  small.  of the vibration.  the experimental  curve  i s very  t h e r e f o r e , c a n be a p p l i e d t o  Concerning g  assumption  diagram.  approaches  zero  at a very  high  driven  25. i f  F ~  P  so  =  ra  a  °»  K a i d a n o v s k i i ,  i s  a  c u r v e .  l a t t e r K h a i k i n  c o n d i t i o n  r e g i o n  c l e a r l y  from  e q u a t i o n  (2-19)  = 0  T h i s  n e c e s s a r y  then  of  r e s u l t and  f o r  n e g a t i v e  agrees  S i n c l a i r  the  w i t h and  e x i s t e n c e  s l o p e  i n  the  the  s u g g e s t i o n  o t h e r s , of  t h a t  o f  the  i n t e r m i t t e n t  f r i c t i o n - v e l o c i t y  motii  26. CHAPTER 3-1.  EXPERIMENTAL The  the  San  APPARATUS:  r a t e  Andreas  III  of  d i s p l a c e m e n t  f a u l t  has  been  at  the  f a u l t  (35)  r e p o r t e d  zones  to  be  i n  about  —8 i n c h e s  2  per  y e a r ,  A c c o r d i n g l y , at  v e r y  the  low  10  X  f r i c t i o n a l  d r i v e n  i n c h e s  mechanisms.  i n c h e s  second  A  can  i s  s e c o n d .  m e a n i n g f u l  d r i v e n  be  per  c h a r a c t e r i s t i c s  v e l o c i t i e s  e a r t h q u a k e per  6  or  v e l o c i t y  a c h i e v e d  i n  an  of  r o c k  i n  the  as  low  s u r f a c e s s t u d y 10  as  h y d r a u l i c a l l y  of  ^ d r i v e n  a p p a r a t u s . The  h y d r a u l i c  p r e s e n t  r e s e a r c h  drove  s l i d i n g  i n  a  3-2.  F i g .  type f s u p p l y a c t e d The  a  maximum  p s i g  w i t h  d r i v e  the  l e n g t h  of  a  c a p a c i t y  p i s t o n  u s e d  Between n e e d l e These p i s t o n  to the  the  and  of  used  a  the  enough f o r t h  e n t r a n c e  c o n t r o l  the  c y l i n d e r  v a l v e s n e e d l e  back  as  i n  the  the  about  p i s t o n h y d r a u l i c  system  w h i c h  r e s e r v o i r .  a c c u m u l a t o r that  s c h e m a t i c a l l y  a x i s ,  i n  the  w h i c h  and  p r e s s u r e  so  f o r  3-1  t i l t e d  used  i n  the  2000  was  p r e s s u r e  f i v e  times  c o u l d  over  a  i n c h e s .  At was  l a r g e  was  used  c y l i n d e r  F i g .  was  and  p r e s s u r e  i n  d r i v e n ,  pump  e l i m i n a t o r  a p p a r a t u s  h y d r a u l i c  shown  a c c u m u l a t o r  w o r k i n g  15  as  the  a  e l e c t r i c a l l y  An  p u l s e  of  of  d i s p l a c e m e n t  s y s t e m . as  c o n s i s t e d p l a t f o r m  An  c o n s t a n t  u n i t  to  the  d i r e c t i o n  and  way  v a l v e ,  end  of  at  each  v a l v e s  were  used  to  the  n e e d l e  g i v e  f o u r  p i s t o n  f o u r  p l a c e d  When  of  a  the  were  movement.  c y l i n d e r  the  p r e c i s e  v a l v e s  were  way  v a l v e  movement. 0.25  i n c h  c y l i n d e r . c o n t r o l f u l l y  of  the  opened  ARRANGEMENT  OF  APPARATUS  AND  INSTRUMENTATION  FIG.  3-1b  CLOSE-UP  VIEW  OF  APPARATUS  ro oo  c  FIG.  3-2  S C H E M A T I C D I A G R A M OF T H E APPARATUS A. HYDRAULIC CYLINDER, B. NEEDLE VALVE C. LEAD WEIGHTS, D. PISTON ROD E. SLIDERS, F. STRAIN RING  30. a  speed  the  two  n e e d l e  10  ^  per  g a l l o n  1.5  o i l  on  i n c h  a  b l o c k s .  p a n e l  by  to  system  Rubber  wheels  by  e l e c t r i c  the  used the  between c a r t  The 0.5  i n c h  clamped a  the  p l a t f o r m .  s o c k e t F i g .  A  j o i n t  was  a l l o w e d  and  r a i l s o  to  the  to  v a l v e  moved  the  10  l a t h e  o i l  to  i s  were  shown  i n  p i l l o w  s o l v i n g  a t t a c h e d p a r t s  were  20  l o n g .  v a l v e s  2  b l o c k s These  s u s p e n s i o n  d e v i c e  s i m i l a r  to  b a l l  a f f e c t i n g  p l a t f o r m , r o d  of  degree  the  w h i c h  of  the  t i l t i n g  weighed  h y d r a u l i c  a  p l a t f o r m ,  c o n t a c t  24  and  t h a the  of  of  as  the  by  to  a  were suppor  and shown  the  pounds,  c y l i n d e r  wide  b l o c k s  used  s m a l l  so  i n c h e s  were  a  were  3-3•  s l i d e r s  the  s y s t e m .  hoses  Three  between  o r d e r  produced  b e d .  l o c a t e d  i n  F l e x i b l e  F i g .  specimens  a  the  c a r t  n e e d l e  the  on  of  c o n f i g u r a t i o n  i n  i n c h e s  mounted  v i b r a t i o n  the  The  was  v i b r a t i o n  pump.  and  s t r o k e ,  r o d  f o u r - w h e e l  the  to  i n c h  b e a r i n g  the  as  r e t u r n e d  24  p i s t o n  o t h e r  f r o m  f r e e l y .  system  and  a  low  was  a  a s s i s t e d  i s o l a t e  way  w i t h o u t  p i s t o n  s o u r c e  f o u r  t h a t  The  on  the  was  so  3-k,  mounted  when  v a l v e .  had  b a l l  A l l  as  O i l  way  v a l v e s  c y l i n d e r .  s t a t i o n a r y  t h i c k to  n e e d l e  o b t a i n e d ;  speed  w h i c h  l o c a t e d  and  be  f o u r I  arrangement  used  a  d i a m e t e r  motor  the  of  the  i n c h  p r e s s u r e  were  c o u l d  arrangement  1  be  a c h i e v e d .  c y l i n d e r  The  were  the  be  c e n t r a l l y  the  c o u l d  c l o s e d ,  t h r o u g h  m o u n t i n g  s e c u r e d  s e p e r a t e  c o u l d  a  p r o b l e m .  h y d r a u l i c to  bed  and  second  a l m o s t  h y d r a u l i c  T h i s  a l i g n m e n t  were  tank  bore  l a t h e  p e r  second  The a  i n c h e s  v a l v e s  i n c h  15  a  of  i n  p l a t f o r m s l i d e r s  was  connect^  s t r a i n - r i n g  FIG.  3-3  PRESSURE  SOURCE  SYSTEM  33. h a v i n g  a  s t i f f n e s s  p l a t e n  w e i g h i n g  p l a t f o r m . p o s t  It  w i t h  b o t h  so  s i d e s  that  a r i s i n g  formed  by  on  the  l o a d  IN  3^2.  X  d i r e c t l y f o r c e  120  ohm  was  so  used  that  d i s c  and  an  the  u p r i g h t  the  the on  f r i c t i o n a l  e l i m i n a t e d  the  to  c i r c u l a r  l e a d  w e i g h t s  moment,  d r i v e  pounds  380  of  e q u a l l y  tend  of  the  c e n t e r  d i s t r i b u t e d o f  of  would  w e i g h t  per  i n c h .  on  the  b r i d g e any  o s c i l l o g r a p h  d i s p l a y w i t h  b e t t e r  i n s t r u m e n t  BL-932  D.C.  The  the w i t h  c o u l d  form  ram  a  and  on  be  p l a c e d  o f  f o r c e .  r e c o r d i n g  permanent matched  to  b r i d g e to  so  that  F o u r ,  as  B r u s h  b r i d g e gauges  o s c i l l o s c o p e  found A  shown  the  However, was  The  the  of  s t o r a g e  r e c o r d s . a  mounted  s t r a i n - r i n g .  r e s i s t a n c e a  of  measured.  the  f r i c t i o n  was  be  s u r f a c e s  p l a t f o r m  c o n n e c t e d  the  A m p l i f i e r  the  was  was  work,  g i v e  s t i f f n e s s  Wheatstone  p r e v i o u s  to  a  the  In  c h a r t  r u b b i n g  had  c o u l d  a m p l i f i e r  paper  the  s t r a i n - r i n g  mounted  v a r i a t i o n  m o n i t o r e d . to  to  on  w h i c h  p l a t f o r m  were  connected  that  a c t i n g  h y d r a u l i c  gauges  A  f o r c e  s t r a i n - r i n g  the  a c t i n g  3-5»  be  a  pounds  were  c i r c u i t c o u l d  by  s t r a i n  F i g .  was  T h i s  N i n e  t o t a l  f r i c t i o n  between  the  i n  r a i l .  f o r c e s ,  a  top  STRUMENTATION:  10^  gauges  l o a d  the  t h r o u g h  w e i g h t  A  p l a t e n .  measured  1.94  c i r c u l a r  i n c h .  to  e q u a l i z a t i o n  s i d e .  had  The was  each  u n b a l a n c e d  h o l e s  a  per  screwed  p a s s i n g  g i v i n g  on  was  normal  from  toward  c e n t r a l  l i n e  the  t h e r e b y  of  pounds  1cA  X  pounds  23.5  c e n t e r  f o r c e s  p l a t f o r m  1.94  c o n s i s t e d  i t s  p l a t f o r m  of  an to  B r u s h  d o u b l e  be  a  Model t r a c e  35. u l t r a l i n e a r b r i d g e the  a m p l i f i e r  pen  of  f o r c e s When the  o s c i l l o g r a p h  the  to  an  a c c u r a c y w i t h  of  the  The by  a  was  Sanborn d e s i g n  such  a  l o n g  a  enough  s i n c e  the  n o m i n a l  of  magnet  to  the  of  to  cover  a m p l i t u d e  l o a d  moving  w i t h  of  the  than  a  was  one  s t i c k - s l i p  X  k  i n c h e s  10  MM a  v e r y  e q u a t i o n  x  f o r  x  + Kx  was  f o u r  m o n i t o r e d  6-LV-4,  an  w h i c h  i n d u c t i o n  magnet.  The  i n c h e s .  thousand  c o i l  l i n e a r l y  It  was  s t i c k - s l i p s  v i b r a t i o n  was  of  the  o r d e r  - x  + - x  = p  and  shows can x  k  X  i n  (2-5)  i n c h e s  10  c h a p t e r  f o r  g l a s s .  two: (3-1)  system  C  i s  n e g l i g i b l e  and  the  Wp  =  y  and  wy  =  damped  ( l o ,  v a l u e s  of  the  p-V  curves  r e c o r d e d  (3-2) t h a t ,  be  P, m  a c c e p t a b l e .  to  + Kx  x ,  p y r i t e  e q u a t i o n  Mx  (3-2)  and  s i g n a l s  Cx  l i g h t l y  E q u a t i o n f o r  +  reduces  o r  be  - 3  C o n s i d e r  In  pounds,  magnitude  -k of  to  moving  d r i v e  r e c o r d  50  p l a t f o r m  t h r o u g h  the  d i v i s i o n .  of  model  of  to  to  c h a r t  c o n s i d e r e d  t r a n s d u c e r  of  per  s l i d i n g  i n d u c e d  more  p o s s i b l e  t r a n s d u c e r  v e l o c i t y  t h i s  was  was  v o l t a g e  s u f f i c i e n t l y  normal  the  b a r  was  o u t p u t  pounds  0.13  v e l o c i t y  v o l t a g e  s t r o k e  It  measurement  LVsyn  p r o p o r t i o n a l u s a b l e  the  the  a m p l i f i e d  of  v e l o c i t y  as  that  be  that  o s c i l l o g r a p h .  compared e r r o r  c o u l d  so  w i t h  o b t a i n e d 3k)  •  p r e s e n t d u r i n g  The  a by  s u i t a b l e the  v e c t o r  c o e f f i c i e n t  experiment s l i p .  c h o i c e  were  of  sum  of of  the  k i n e t i c  o b t a i n e d  s c a l e s  i n  f r i c t i o n , the  36.  A Kistler of  0.1  volt  servo  per unit  gravity  with  the displacement  from  which  t h e p-V  oscilloscope.  3-3.  S P E C I M E N S AND  were  prepared  polishing exists  o f r o c k were  equipment.  SLIDING used  w i t h a diamond  Pyrite,  o f o c t a h e d r a l and  properties 10^ bars In  of green  s p e c i m e n was at M i l e  on  green  has  a homogeneous  and  (36).  In addition  very  t h e Hope  easily.  slide  Highway.  I t s shear  of 0 . 7  X  i s composed i n a  form  of 0 . 3 1 5  of the The  crystals.  third  i n British  I t has been  X  rock  Columbia classified  a hard metamorphic limestone,  a coarse  i t sconstitution  iron  mechanical  w i t h a shear modulus  Granodiorite,  r o c k , has  The  and  cubic  and g a r n e t ,  the o r i e n t a t i o n  splits  structure  schist,  crystals.  Black marble,  quartz-diorite.  (36).  by  saw  modulus  w i t h a shear modulus  the Hope-Princeton  a black marble.  igneous  schist  obtained from  as  10^ bars  schist,  ( 3 6 ) , are affected  addition,  a chlorite  as m a g n e t i t e  These  a mineral of  (36).  fabric-like  displayed  i n the tests.  10^ bars  of non-homogeneous  transducer  SURFACES:  i n metamorphic r o c k s , w i t h a shear  of v a r i o u s m i n e r a l s , such  combination  c o u l d be  crystals  Green s c h i s t ,  sensitivity  the v e l o c i t y  and f i n i s h e d  i n the form  i n the  at s l i p  P R E P A R A T I O N OF  types  disulphide,  used  t r a n s d u c e r and  Four  geological  was  relationship  on an  specimens  accelerometer with a  granular  between  modulus  that  i s 0.25 X  to the rock m a t e r i a l s ,  steel  of  0.229  X  intrusive of  granite  10^  bars  and g l a s s  were  37. a l s o  t e s t e d .  The  e x p e r i m e n t a l hand,  i s  a  r e s u l t s  i n  marble of  The  l e n g t h of  than Epoxy The  and  of  be  and  to  c a r e f u l l y  below, i n  and  o r d e r  to  t i e  w i t h  G l a s s ,  r o c k s  on  and  were  s l i d e r s  e a r l i e r  the  o t h e r  p o s s e s s e s  a  glue  washed  a l l  s l i d e r s shown  the  to  However,  an  i n  t h o r o u g h l y  g r i n d i n g  i n  a  to  BLACK  MARBLE  60  GREEN  SCHIST  220  GRANODIORITE  53  i n c h  a cut  squares  a v a i l a b l e  r u n n i n g  f o r  g l a s s  and  s l i d e r  F i g .  3-6. the  f i n i s h e s  d i s t a n c e  l e n g t h s  r o c k as  d i s t i l l e d  c o n t a m i n a n t .  90  were  the  and  w i t h  w i t h  r e s p e c t i v e l y .  SURFACE F I N I S H C. L. (micro i n c h e s ) PYRITE  one  r o c k s  of  and  i n c h .  onto  s u r f a c e s average  squares  0.4  i n c h e s  o b t a i n e d  the are  cut  P y r i t e  s l i d e r s  of  a c c o r d i n g  f o r  e a s i l y  s c h i s t  0.25  t e s t .  i n c h  1.25  were  v a r i e d  c u t t i n g  remove  and  o b t a i n e d  each  t h i c k n e s s  c u t t i n g .  ground  to  i n  green  a  0.375  s l i d e r s  A f t e r were  w i t h  r a i l s  were  used  r a i l s  of  b e f o r e  c o u l d  was  some  used  cut  The  g l a s s  the  i n c h e s  12  were  squares  r o c k  i n c h e s  a  w o r k e r s .  to  were  i n c h .  t h i c k n e s s e s  s i z e k  s l i d e r s  0.5  G r a n o d i o r i t e w i t h  s i m i l a r  s l i d e r s  i n c h  0.75  o t h e r  p r o v i d e d  s t r u c t u r e .  Three  t h i c k n e s s  r e s u l t s  of  m a t e r i a l  homogeneous  b l a c k  s t e e l  A.  of  of more  s t e e l .  a d a p t e r s .  specimens shown w a t e r  FIG.  3-6  SPECIMENS A.  D.  GREEN  GLASS,  SCHIST, E.  B.  BLACK  1020 STEEL,  MARBLE,  F.  C.  PYRITE,  GRANODIORITE.  39. CHAPTER  IV  EXPERIMENTAL  k-1.  RESULTS  PROCEDURE: The  general preparation of the test  been d e s c r i b e d i n chapter received  additional  three.  treatment  However,  immediately  specimens  has  a l l specimens before  test  as  follows: (a)  Steel  trichloro-ethylene same a s was newly  used  washed  sliders  and r a i l s  and a l c o h o l .  i n the test,  A  was  were washed  layer  by  of lubricant,  immediately  surfaces i n order to limit  coated  the  on t h e  atmospheric  contamination. (b) water. that  Tap w a t e r  would  results. for  one  from  Rock specimens  adhere The  specimens  and h a l f  t h e y were (c)  order were  not used  hours  were  they were  put to  a t room  were  spread  sliders  frictional  i n an oven a t was  evaporated  c o o l e d t o room  100°C away  temperature  cleaned with trichloro-ethylene  the grease  on t h e s u r f a c e s .  sliders  were  c l a m p e d on a l a t h e on t h e r a i l s  over  then  the  and d i s t i l l e d  The  i n  specimens  water,  and  condition.  Three which  to avoid minerals  heated  moisture  distilled  test.  then washed w i t h a l c o h o l  dried  then  so t h a t  G l a s s was  t o remove  i n order  t o the s u r f a c e s and a f f e c t  the specimens,  before  was  was  were washed w i t h  the r a i l s .  p l a c e d on two p a r a l l e l bed.  a n d was The  Lubricant,  coated  rails  i f needed,  evenly by moving the  p l a t f o r m was  p l a c e d on t h e  sliders  4o. and of  connected  to a strain-ring  w h i c h was f i x e d  the piston r o d of the hydraulic  cylinder.  were  p u t on t h e w e i g h t - c a r r i e r o v e r  load  could  be v a r i e d from  increments  o f about With  to  More  the valves  first  on r o c k s , used  two h u n d r e d  s e t of data  ceased  to flow,  reduced  s t i l l  running  about  at high  on s t e e l  twenty  was r a i s e d  the valves  was d r i v e n  w e r e made  l o w sjiieed;  were  speed.  before  the  were  made  strokes  a n d no r u n - i n  the driving  gradually until  thereby  t o two i n c h e s In  procedure  was  the vibrational  whereby  The inch  system, C were  both  ranging  from  the stiffness,  constant;  0 and normal  load.  K  the only These  to vary  K and C  without  the system. diameter  or the length One i n c h  would be accomplished  over  speeds  fluid  r e l a t e d to the structure of the apparatus  to 1.25 inches.  cycles.  existed  were  i t was n o t p o s s i b l e  disassembling  sliders  were  the hydraulic  high  per second.  considered  restrictions  s p e e d was s e t a t a  giving driving  t h e damping c o e f f i c i e n t ,  variables  slip  t o 4 3 0 pounds i n  the o i l pressure  platform  was t a k e n ;  the tests,  then  0.75  Normal  f o rglass.  value,  and  closed  strokes  at a fairly  In  zero  weights  the platform.  5 0 pounds  t h e pump  on and t h e s l i d e r  than  Lead  40 p o u n d s .  1 0 0 0 p s iand w i t h  turned  a t t h e end  A reasonable such  a short  of the sliders of displacement  i n about  consistency distance.  two h u n d r e d  were of the stick-  of the surface  condition  41. RESULTS:  4-2.  F o r e a r t h q u a k e s , and  the  c o e f f i c i e n t  a m p l i t u d e  i n t e n s i v e l y  purposes  o f  was  u s e d  as  o i l  d a t a  water  u s i n g  1020  ( L . P . ) and  r e s u l t s  of  d i f f e r e n c e are p  the  and  same  ©  curves  ©  c u r v e s  were  f o r  a l l  p y r i t e  t e s t s  were  specimens  and  g l a s s  c o n d u c t e d .  a l t h o u g h  a f t e r  s u f f i c i e n t  s t e e l from  f o r on  and  r e s u l t s  t h a t  d r y  d i f f e r e n t  l u b r i c a n t ,  d r y  on  the  w i t h  ii.P.  l i g h t  the  F i g s .  no  4-1, The  s i g n i f i c a n t  F i g s .  F i g s .  and  p e t r o l a t u  l u b r i c a n t .  gave  II.P.  s t e e l ,  s t e e l ,  c o n d i t i o n s ;  c o n d i t i o n s .  p e t r o l a t u m  s t e e l  water  as  w i t h  l i g h t of  f o u r  (ll.P.)  water  w i t h  of  under  4-3  and  4-4  and  4-6  are  almost  i n  the  t e s t s  f o r  d r y  g l a s s ,  4-5  r e s u l t s  are  the the  s t e e l .  GLASS:  F i g s .  F i g s .  F i g s .  l o g ^  l o g ^  o f  o b t a i n e d .  t e s t e d  the  Three  and  on  been  l u b r i c a n t ,  water  4 - 2 - 2 .  g l a s s .  mechanism  v e r s u s  u n l u b r i c a t e d  p e t r o l a t u m  curves  as  u s e d  was  r e s u l t s  (X  and  had  d i s p l a y  4-2  v e r s u s  the  STEEL:  heavy as  f r i c t i o n  l u b r i c a n t  was  S t e e l namely,  e x p l a i n i n g  c o n s i d e r e d .  p e t r o l a t u m  4 - 2 - 1 .  of  v i b r a t i o n  L u b r i c a t e d Water  of  4-11  4-9 and  c o n d i t i o n s  4-7  and  and  4-10  4-12  4-8  are  were are  show the  c o n s i d e r e d  the  the  r e s u l t s  r e s u l t s  r e s u l t s  of  f o r g l a s s  wet w i t h  g l a s s . L.P.  of  42.  Water  gave  no  increased mineral and  U  obvious  the amplitude by o i l gave  an  an o b v i o u s  order of f i v e .  drop  o f IJ  amplitude  4-13  through  results  conditions. and  4-2-4.  The  t o 4-18  of pyrite  addition  are the s t a t i c  u n d e r d r y , wet  of water  lowered  BLACK  Figs.  4-19  water  was  values  conditions 4-20  and used  investigated  as  no  GREEN  that  was  vibration  were  The  friction  scattered  tests  induced  were  and  0 . 3 2 to  pyrite.  material.  amplitude  Relatively  high  results  were  curves  friction  of the d r y marble  that  conducted  with  v i b r a t i o n was  the difference  could  was  obtained.  between p  n o t be measured. green  The  green  observed. g  the amplitude  o f d r y and wet  respectively.  from max  examined f o r t h i s  the l u b r i c a n t .  so s m a l l  friction  lubricated  SCHIST:  friction  believed  friction  and  TJ  of dry  show t h e f r i c t i o n  but very  Several  of  0.4,  MARBLE:  were f o u n d .  4 - 2 - 5 .  to 5 0 ^ of that  the amplitude  Two  schist  of  0 . 9 to  , from  f  but  layer  PYRITE:  Figs.  0.22  A  but  mm  4-2-3*  and  i n t h e maximum f r i c t i o n  0 . 6 5 to 0 . 1 5 .  . , from  i  change  and of  I t of  was  green  stick-slip  average  schist  schist,  were  coefficients 0 . 6 6 and 0 . 5 8  4 . 3  4-2-6.  GRANODIORITE: Granodiorite  the  experiments.  was v e r y  similar  l o g ^  to that  The  of f r i c t i o n  but gave  of about  versus  a r e shown i n F i g s .  rock used i n  behaviour  of black marble.  vibration,  0 and amplitude  diorite  the only igneous  The f r i c t i o n a l  gave no s t i c k - s l i p coefficient  was  of Dry  a fairly  0.72.  maximum  coefficient  of f r i c t i o n  coefficient  of friction  was  0.55-  granodiorite constant  The f r i c t i o n  l o g ^ 0 curves  4-2 1 a n d 4-22  granodiorite  o f wet  versus grano-  respectively.  was O.65  and the k i n e t i c  FIG. k-k  a VERSUS L O G  1rt  ©  WATER ON  STEEL  4  2  1 F I G .  4-7  p  g  VERSUS  L 0 G  1  Q  0 9  1 DRY  G L A S S  2  L 0 G  1  Q  9  P  51.  o o  1.0  .6  F I G .  k-9  p  V E R S U S  LOG  9  J  I  I  T  o  1  WET  G L A S S  LOG 9 1 q  ro  53.  •CD  O T— O  CO CO  < O  H  C D  O hJ CO CO  a w  > |f0  o H  I  C5 H fa  J •UT  01  X  o CO  o  o CM  55.  LOG F I G .  4-13  U  g  VERSUS  LOG,„©  '10'  DRY  PYRITE  1 0  9  a  ON  TABLE  PARAMETERS  2  3 , 4  225.©  a ^, /o 7.  c  /'/ . IZ  b  d  A ('A  ;./3o  22525  c  i-'9-  6  /2 2  6 6-7  2 3 2 2 • .272< 2 . 3 / 5  0.  42-26  2233 4222  2.3/5  3.307  2434  5 765 .cava 7  3.62/  4.34  42.^  2552  29 4  225  / 37 8  7-699  7 8 /. 9 4 /a/. 2 2 9 4  ANDT H E THEORY  263.6  U94-  /3  SYSTEMS  9.09  224.-5  5 , 6 224,5  OF EXPERIMENTAL  a  W (lb)  Fig.  1,  I:  2.c<02  G/9  0.30  /  .  £_  £. 4 6  3. ^  0 55  0,33  2. 6  A 42  Oft 5  0.7 Z  7  7J53  24 2  0.?2  465  0,674-  ^87  3. A2  6.38  6). 15  // . 2 2  s7>  Q  / 3 , /4  /. 9 4  22.5  24-32. / . 2 8 /  4 4 4  6,326  0. 3/015. / 4  /5 . /6  194-  tzs.o  3 375,  4,44  0*22.6  66.18  C.2C6 4 62  2' 180 4 57  6A 6 5  <2. S 5  /7.  /S  3/0.0  79.20 224.2 24.22  769.9  • f.94- 38-4-5 A 9 4  76 J 7  294  7 6)-60  NOTE:  P.  34.4  38.  =  a  O.BZQ 3,  233/2  +  be  7.J4-6  1.S32.  2. 3  a  (  m  IS  !• 75 /  1  Fm =  « = o Fs - F ) A  4S45  /?5  + be  c(iog  1  0  9)  +  F  m  /?. /a 2422  67.  S c a l e :  FIG.  S c a l e :  FIG.  F o r c e D i s p l a c e m e n t V e l o c i t y 4-23  PHASE  PLANE  PHASE  D i v  =9.7  1  D i v  =  DIAGRAM  F  vs  X  X  vs  X  F o r c e D i s p l a c e m e n t V e l o c i t y 4-24  1  l b s  1 D i v = 0.0005 i n  PLANE  1 D i v D i v 1 D i v 1  = = =  DIAGRAM  0.104  i n / s e c  OF  STEEL  9.7  l b s  0.0005 O.O635 OF  i n i n / s e c  GLASS  68,  S c a l e :  FIG.  F o r c e  1 D i v  =  7.76  V e l o c i t y  1  =  .0373  Displacement  4-25  PHASE  PLANE  1 D i v D i v  DIAGRAM F  S c a l e :  FIG.  F o r c e D i s p l a c e m e n t V e l o c i t y  4-26  PHASE  PLANE  = 0.0004  1 1 1  D i v D i v D i v  OF  vs  l b s  i n  i n / s e c  PYRITE  X  = 15.12 l b s = 0 . 0 0 0 7 8 i n = 0.084 i n / s e c  DIAGRAM  OF  BLACK  MARBLE  69.  FIG.  4-27  PHASE  PLANE  DIAGRAM  OF  GRANODIORITE  70 CHAPTER  V  DISCUSSION  5-1.  LABORATORY The  the  purpose  stick-slip  objective  system. slip and  analysis  analysis  w o r k was t o  to earthquake  to relate  and other  showed  arising  K  =  The presented equation  from  that  pr s o - u rm  results  (2-19)  result  were  suggested their  analysis  (1  stick-  and U »  g  m  t o be  e-")  experimental  T h e o r e t i c a l curves  (2-18)  i n order  work a r e  based  on  t o compare  indicated  two  d e s c r i b i n g t h e p-©  even  frictional  metals  agreement  and  non-metals.  though rocks behaviour  the amplitude  was v a l i d  of  approximate  was f o u n d +  of  e  f o r both  Second,  between U The  The comparison  that,  theory with reasonable present  b  also plotted  equation  of metal.  analysis  four.  was v a l i d  be b r i t t l e ,  +  the amplitude  the amplitude  of the present  experiment.  First,  relationship  this  mechanics.  properties of the  increased.  i —Tld >  (  i n chapter  theory with facts.  as v e l o c i t y  study  systems w i t h the  was p r o p o r t i o n a l t o t h e d i f f e r e n c e  A  that  was made  to the f r i c t i o n a l This  equation  to  of rock  of applying the results  diminished  This  of the present  vibrations  A theoretical vibration  RESULTS:  are considered  i s similar to  results  indicating  f i t the  that the  f o rthe prediction  of the  71.  amplitude  of stick-slip The  the  experimental results  coefficient  slip and  vibration.  vibration  of f r i c t i o n  i s a ratio  material  t o t h e b u l k volume  mechanical internal  o f t h e volume  properties  spaces  rock  i s reduced  bulk  strength  frictional  i n the structure  values might  value  t o 6C*jb o f i t s o r i g i n a l  of  friction  within the  i n water.  then  value,  Mineral  o f a l l specimens  The  are influenced  by  I f the  some i n f l u e n c e The a d d i t i o n  and black-marble lowered  and glass.  of a  and t h e s t r e n g t h o f  be e x p e c t e d .  on p y r i t e  steel  of pores  when i t i s i m m e r s e d by water  stick-  The p o r o s i t y  of the material  i s altered  that  by the porosity  of the material.  water  with  be a f f e c t e d  of the material.  material  suggested  and t h e amplitude o f  of rocks would  the d u c t i l i t y  also  on  of  the f r i c t i o n a l  b u t gave no  change  o i l lowered the c o e f f i c i e n t  t o about  60f> o f  original  values. The  effect  of  rock material  to  possess  the of the  or  from  stable  rocks would  high porosity into  sliding,  vibration  be i n c r e a s e d  rocks would the rubbing  granodiorite but reduced  results  o f igneous  and that  be d e c r e a s e d system.  amplitude  When t h e p r o p e r t i e s  c o n s i d e r e d , t h e above  amplitude of s t i c k - s l i p  introduced  I t caused  amplitude of pyrite.  t h e r o c k s were  porosity  on t h e s t i c k - s l i p  was n o t u n i q u e .  stick-slip  stick-slip  of water  i m p l i e d th; or low  of sedimentary i fwater  was  72. Ductility to  deform  i s d e f i n e d as  permanently  without  ductilities  of p y r i t e  and  hydrostatic  p r e s s u r e , were and  higher  values.  indicated  these  that  pyrite,  dry  and  steel  dry  the  suggested be 5-2.  The  that  the of  will  be  shear  generated  between U )so steel  0.06,  of  the  and  steel  with  was  much  TJ of r m  wet  petrolatum 0.2  and  increase of ductility  stick-slip  over  results  and  0.1  low  the of  material  vibration  might  ductility. EARTHQUAKE MECHANICS:  mechanics  of  earthquake  Wilson  (37)  at opposite sides  by  1020  U w i t h the fm  amplitude  considered.  stress  and  0.5^  t o be of  fairly  experimental  coincidence of  A P P L I C A T I O N TO The  The  black-marble,  between U 1 so  a function  ductility  were 0.008, 0.02,  respectively. difference  the  material  The  under  (36)  found  of  (36).  fracture  difference  pyrite,  ability  black-marble  20$> r e s p e c t i v e l y , than  the  fault  i n 19^5 of a  displacements  stated  that  strike-slip  the  fault  was  tectonic  d i v e r g e n t r i d g e s as  shown i n F i g .  A number o f  shallow earthquakes  have  5-1.  strike-slip total slip  fault  l e n g t h of fault  earthquake  zones.  zone.  In  (39»4o)  and  the  fault  the  earthquake.  600  about  was  The  San  miles  the A p r i l ,  the  Andreas  (38), 1906  displacement  active  over  a  fault,  has San was  a  having  typical  at a  strike-  Francisco entirely  length of  Permanent displacement  occurred  as  270  horizontal  miles  large  as  during 21  73.  A  Zone  o f  d i v e r g e n c e  A V  S t r i k e - s l i p Zone  f a u l t  o f  d i v e r g e n c e  FIG.  5-1  STRIKE-SLIP  FAULT  MECHANISM  7k.  feet  were  observed  diminished  as  along  the  This  decrease  distances  from  the  each  width  side  of  this  kilometers the  of  fault  the  each  highly  is  the  and  series  basic rock  this  granitic  granitic the  San  of  the  ground  (35)•  length  of  the  inactive  i n the  The  lays  fault, the  elastic (40).  zone by This  then  of-freedom  was  elastic  after  year  elastic  as  existed  of  The  wide  The  fault  coastal In  about  two  occur  the  because  the  zone would  per  i n  the zone  whole  build rate  year  inactive  the  equal  rock  slip-rate  inches  would  to  active  the  mostly  the  structure  system  throughout  strain  of  Granitic  surface  (39).  are  of  sediments  rocks. the  10  as  epicenter,  the  reported that  uniform  zone  up of  the  i n the slip-  zone.  fault,  the  system  the  earthquakes  o u t s i d e the  the  was  increased  consisted mainly  underneath  fault  slip-rate  active  t o be  granitic  i t was  at  became  displacement.  metamorphosed  Nevada  fault  elastic  (4o).  fault  geology  i n Sierra  zone y e a r  length of  the  foundation joins  fault  the  geologic structure  When t h e w i d t h the  displacement  deformation  thought  adjacent  that  along  I f the  the  t h a t an to  the  slightly  and  Andreas  displacement rate  the  Generally, large  zones.  the  of  (4l).  structure  of  prior  z o n e was  side  d i s t u r b e d and  Franciscan  implied  fault  t r a c e and  s h o w n i n F i g . 5-2  and  i n permanent  fault  elastic  on  fault  transverse d i s t a n c e from  greater.  on  the  pure  of  zone  i s compared  i t i s possible  to  shear  shown i n F i g .  c o u l d be  vibrational  elastic  model  as  simplified  system  as  as  approximate  with the 5-3  a single-degree-  shown i n F i g .  5-k.  75.  Fig.  5-2  I n d e x Map o f S h o t p o i n t s a n d P r o f i l e s i n t h e G a b i l a n and Ranges, C e n t r a l C a l i f o r n i a .  Seismic Diablo  F I G .  5-3  A  PURE  SHEAR MODEL  OF  ELASTIC  ZONE  77.  w  K  AAAA  FIG.  5-4  MODEL SYSTEM  OF  AN  1  EARTHQUAKE  VIBRATION  78. The system  a m p l i t u d e  s i m i l a r  to  F i g .  y  r so  F o r  9  <<  and  1  the  s m a l l  e q u i v a l e n t  the  s t i c k - s l i p  5-^  vas  g i v e n  - y  «-&P.oand  of  +  by  v i b r a t i o n  e q u a t i o n  of  a  ( 2 - 1 9 ) :  b e  rm  damping  the  e q u a t i o n  reduces  to  F >( > 2  (5-=)  M  s t i f f n e s s ; o f  the  system  i s  g i v e n  by  G K = 5D  (5.3)  T h e r e f o r e ,  where  G  i s  the  shear  Brune, upper be  l i m i t  of  Assume  g r a n o d i o r i t e normal  l o a d  Henyey  w h i c h  s t r e s s  that  =  D  =  m i l e s  13  _  O.65  3 8 5  b a r s  The  r e s u l t s  of  =  r a t e  5  f o r  14.5  ~ X  10~  1  0  10  that  the  cm/year  5  the  the  f a u l t  would was  e q u i v a l e n t  x  385 s e c "  g r a n o d i o r i t e  X  x  365  e q u a t i o n  .  =  k  (36) ,,  '  k  p  s  . / . / l  i  -1 x  24  x  3600  s  1  i n  (5-4)  n  2  c h a p t e r  65  i n t o  b a r s  g i v e s ,  (35) 4.4  6  p s i  6  i n / y e a r  2  X  10 X 14.5 X 63360 i n  Fso = °' Fm =  S u b s t i t u t i o n  of  of  then  0.25  KV w  =  X 13  =  l o a d - r a t e •  s u g g e s t e d  g e o l o g y  g o  G  (40), 0.25  G  D a  a  m a t e r i a l .  becomes:  v  U s i n g  g e o l o g i c a l  (42)  at  the  P  has  the  Roy  „ .2^|p . U s i n g  o f  and  i n i t i a l  b a r s .  250  modulus  y i e l d s  f o u r  gave:  e  c  79.  = TT<F - v )  A  80  o r  a  =  254  i n c h e s  =  21.2  f e e t  I f  we  be  g r a n o d i o r i t e ,  of  about  the  c o n s i d e r  t h a t  X  0.2  G D KV  v  *  C h o o s i n g  the  _  U ' s o  T h i s  o f  analogous  t h a t  to  I f  are  the  a d d i t i o n ,  5  5  .  )  s t r u c t u r e  a  n  h o r i z o n t a l  the  i s  c  h  at  e  i  the  t o  t h e  same  as  d i s t u r b a n c e .  1906  t o  be  g r a n i t e .  G  f o r  (36). e  3  s e c "  1 0  f a u l t  d i s p l a c e m e n t  n e a r l y  6  5  2  p  s  . /. / i  i  s  e  n  -1  c  1  d i s p l a c e m e n t  12  X  b a r s  as  =  f e e t ,  21  g i v e s  3.25 _  X  _  .  ,  "  u  * ' u  y  e  0.65  b a r s  385  s u g g e s t s  that  w o u l d  c o n c l u s i o n ,  p o s s i b l e .  I n  >  = 0.65 - 0.079 = 0.57  be i f  s t i c k - s l i p  r e a s o n a b l e  known,  Q  10 X 14.5 p s i _ _ 63360 i n = ' 3.52 X 2 X 385 X 3 6 5 x 24 x 3600  =  g r a n i t e  In  _  5  X  250  ¥  c a l c u l a t i o n  f r i c t i o n  i s  f  = —  1 m  6  s t r u c t u r e  - S K _ 21  F  U  <  Fm - 2W " 2 x 3 8 5 x 1 4 . 5  T T  t h e r e f o r e ,  X  13  10"  X  f o r  r o c k  ~ 14.5  xj  Tso and  0.2  0  g i v e  b a r s  =  (  d i s p l a c e m e n t  h o r i z o n t a l  TT  •t  the  10^  ¥  = 4  w o u l d  r e p o r t e d  C o n s i d e r i s  ^ . 5  g e o l o g i c  T h i s  d i s p l a c e m e n t  g r a n i t e  the  i t  f e e t .  21  m  38?X 4.4  X  e s t i m a t e s  the  a m p l i t u d e  the  p e r i o d  the  s h a l l o w  v i b r a t i o n , f a u l t  and o f  o f  k i n e t i c  c o e f f i c i e n t  o f  0.57.  o f  g e o l o g y  the  the  the the  e a r t h q u a k e  then  i t  w o u l d  d i s p l a c e m e n t s l i p - r a t e  e a r t h q u a k e f a u l t  o f can  movement appear  a r e t h e be  d i s p l a c e m e n t s  f a u l t p r e d i c t e d . c a n  be  80. estimated  as:  a  WD. £G  <Pso- Pm>< >  " W(P.owhere T rate  i s the  applied  period  along  the  2  P.) of  disturbance  fault.  (5-5) and  e i s the  slip-  81. CHAPTER CONCLUSIONS:  6-1.  The r o c k  have  used  f o r  o f  VI  f r i c t i o n  been the  r o c k s  that  To r e s u l t s  i n v e s t i g a t e d .  purpose  to  have  been  G l a s s  and  the  of  f o u r  s t e e l  types  were  f r i c t i o n a l  o f  a l s o b e h a v i o u r  m a t e r i a l s .  the  p r e s e n t  work  the  f o l l o w i n g  o b t a i n e d . h y d r a u l i c a l l y  r e s u l t s  range  v i b r a t i o n  comparing  o t h e r  The  f r i c t i o n  v e l o c i t y  of  of  summarize  (1) good  i n d u c e d  from  i n  the  ^  10  d r i v e n  v e r y  i n / s e c  low  to  2  a p p a r a t u s  v e l o c i t y i n / s e c  p r o v i d e d  r a n g e .  A  was  e a s i l y  the  f o l l o w i n g  a c h i e v e d . A  (2)  P -9  e q u a t i o n  g  was  f o u n d  i n  f o r m ,  p  r w h i c h  was  m e t a l s .  found The  s u g g e s t e d and  s t i c k - s l i p  e  be  there  a p p l i c a b l e w h i c h  c o e f f i c i e n t s  of  Under were  g r a n o d i o r i t e . but  a  f a i r l y  and  0.55  when  wet.  not  show  o s c i l l a t i o n  On  and  e q u a t i o n  c o n d i t i o n s  f o r  a l l  non-  a l s o  between  s t a t i c  o t h e r d r y  but  JJ  r e l a x a t i o n  specimens  s c h i s t  c o n s t a n t  when  t h i s  m e t a l s  f r i c t i o n .  Green  the  b o t h  d i s c o n t i n u i t y  s e v e r a l  a c h i e v e d  f o r  f i t t e d  no  o s c i l l a t i o n s and  b  was  (3)  s c h i s t  to  +  rm  so  r e s u l t s  that  k i n e t i c  - p  of  hand  gave about  e x c e p t no 0.66  o b v i o u s when  g r a n o d i o r i t e  produce  g r e e n  v i b r a t i o n s  d r y  d i d when  82. wet. The  (4) i n  c h a p t e r  f o u r ,  c u r v e s .  The  appeared  to  s u g g e s t e d of  the  gave  be  v e r y Water  t h a t  the  m e t a l  but  the  e f f e c t  the  and  non-metal  a l s o  of  f o r  m i n e r a l was  appeared  to was  the  g o v e r n i n g An  earthquake  v i b r a t i o n  t h a t  f r i c t i o n a l  mechanism  of (9)  the  e q u a t i o n  i n  w i t h  as  p l o t t e d  t h e o r e t i c a l  m e t a l s  and  non-metals  l u b r i c a n t r o c k s  f o r  might  be  some a  r o c k s  f u n c t i o n  was  r o c k s on  a  good  l u b r i c a n t  not  and  g l a s s .  T h i s  the  s u r f a c e  c o n d i t i o n s  o n l y  s u g g e s t e d of  t h a t m e t a l  same.  the  an  measured  c a l c u l a t i o n  o i l  of  v i b r a t i o n  w i t h t h a t  s i z e  a p p r o x i m a t e  f o u n d  T h i s  of  r e s u l t s  m a t e r i a l .  b e l i e v e d  c o n d i t i o n s  d i s p l a c e m e n t  of  c o r r e l a t e  It  (8)  good  a m p l i t u d e s  m a t e r i a l s . f a c t o r s  a  o i l  the  The  (7)  b e h a v i o u r  was  M i n e r a l  f o r  agreement  f r i c t i o n  of  a m p l i t u d e  s i m i l a r .  (5)  p o r o s i t y  and  good  f r i c t i o n a l  (6)  t e s t e d  f r i c t i o n  the  of  a l o n g  s u p p o r t e d s t i c k - s l i p  the B r a c e  zone  to  the  and  an  one  of  known  permanent  f a u l t  might  the  a m p l i t u d e  B y e r l e e ' s  o s c i l l a t i o n s  be  on  maximum  earthquake  the  a m p l i t u d e .  based  gave  s u r f a c e s  of  might  v i b r a t i o n  c a l c u l a t i o n  c l o s e  the  d u c t i l i t y  d u c t i l i t y  earthquake  v e r y  of  zone. s u g g e s t i o n  be  the  e a r t h q u a k e s . The  a m p l i t u d e  of  v i b r a t i o n  was  f o u n d  from  of  83.  Fso" V] An  estimate  of the period eG  6-2.  v  Fso  + b© o f earthquakes  i s given  by  t V  RECOMMENDATIONS: (1)  F r i c t i o n measurements a t h i g h d r i v e n v e l o c i t y  were l i m i t e d by the p r e s e n t  apparatus.  r e s u l t s a t h i g h ©, an apparatus  To o b t a i n  friction  w i t h a wider v e l o c i t y range  s h o u l d be used. (2)  From the m e c h a n i c a l  p o i n t o f view,  t e s t s under u n i a x i a l s t r e s s were used.  friction  However, f o r s t u d i e s  i n the f i e l d o f rock mechanics, h y d r o s t a t i c p r e s s u r e be c o n s i d e r e d .  should  The h y d r o s t a t i c p r e s s u r e i n the t r i a x i a l  s t r e s s s t a t e i s commonly o b t a i n e d by s u r r o u n d i n g  the t e s t  specimens w i t h p r e s s u r i z e d f l u i d .  I n o r d e r to p r o v i d e a  triaxial  apparatus  s t r e s s s t a t e , the p r e s e n t  would r e q u i r e  extensive redesign. (3) temperature,  The p r e s e n t experiments were made a t room whereas at depths o f about  10 k i l o m e t e r s the  temperature would be c o n s i d e r a b l y h i g h e r .  The e f f e c t o f  v a r y i n g temperature s h o u l d be c o n s i d e r e d when r o c k is investigated further.  friction  84. APPENDIX DETERMINATION  OF  i s  at  V  c  *  There x  where  U [ sc  i s  From  =  V  x  3  at  c r i t i c a l ,  f o l l o w i n g  that  c o n d i t i o n s :  c o e f f i c i e n t  =  of  f r i c t i o n  at  V  V  .  c v  1  ' (1-3)  " I c V  = 0  3  (l-2)  c  (1-5)  t  =  0' a t  and  t  =  t^  0  p o i n t  at  i n  1-1,  F i g .  3»  p o i n t  (2-9)J  equation x  i s  c  C h o o s i n g  From  the  V  (1-1)  s t a t i c  = SP-  3  *  when  e x i s t  (2-13):  e q u a t i o n  X  case  must  the  o  the  KFSC  o  x  9 :  CRITICAL  C o n s i d e r  I  = i ( p  o  m  K Fm (  3  +  p )  +  /  y  +  (-)  A  J  c  i  c  V  )  +  6  J  [AcosCO t d  6  + Bsinu> t J  3  d  3  (1-7) From  (2-10):  e q u a t i o n *o  ^ d  =  (  "  B  A  A  =  I  1 r  —A  x  ( - >  )  6J e  1_(B  d  -  AA)coscJ t d  -  8  + A B j s i n o ^ t 3- ]  (A  (1-9) From  (2-1  e q u a t i o n x'  =  3  -w  1):  2 d  e  "^HlS I I[A(1 a  J  [B(1 S u b s t i t u t e  e q u a t i o n V  B  =  __9_ d  +  (l-2)  T\A  -  A )  i n t o  2  2 A )  -  T  + 2ABJ c o s o t d  27\A] s i n c ^ t  e q u a t i o n  3  j  3  +  (l-10)  (l-8):  (1-11)  85. S u b s t i t u t e and  e q u a t i o n s  s i m p l i f y : -V  (C  and  (1-3)  + W(i)  A  u>  K  t  d 3  (l-1l)  i n t o  r  _  [Acoscu t  +  d  e q u a t i o n  (•—  (l-7)  s i n ^ A A ) s i  +  d  t  ]  3  ( l - 7 a ) S u b s t i t u t e and  (l-4)  e q u a t i o n s  (1-11)  and  i n t o  e q u a t i o n  (l-9)  s i m p l i f y : _ £  d  e  3  =  d  W  ( ^ ) c o s c o  t  d  -  3  |A(1  + A  )  +  j ^ J s i n ^ t 'd " 3 ( l - 9 a )  S u b s t i t u t e and  (1-5)  e q u a t i o n s  (1-11)  and  i n t o  e q u a t i o n  (l-10)  s i m p l i f y :  V  0  =  JA(1 + A ) + 2X^jcosW t  +  2  d  3  v [TJT( and  -  1  A  2  )  "AA(1  oJ,t„ d 3  =  ) ] s i n ^  2  t  d  (1-IOa)  H  = — , d  s i m p l i f y Acos6J t^  (H  W (C  +  d  or  E q u a t i o n  , +  d  t  3  ( l - 1 2 )  -  A  )  AA(1  -  ( l - 7 a )  HcoscJ t d  and  the  PH)cosw t d  . ^  CO.  w i t h  A  )  e q u a t i o n  ( l - 9 a ) :  =  -  3  +  +  [A(1  A )  +  2  n ] s i n w  d  t  3  W|i)  K (A  + 27\  )  + AAjsincJ^t  K =  A  (1-12)  e q u a t i o n  +  d  P  +  tan  •(1  Let  A  a l s o : A(1  Let  +  , =  tan and  3  =  above  e q u a t i o n  |P[A(1  +  A  (H  +  A A ) j s i n w  -1J j _ . P[A( 1 e q u a t i o n  2  +  A — +  A  )  ( l - 1 3 )  +  )  becomes  +  H]  d  t  3  PH H]  must  -  (H be  + A A) e q u a l .  (1-13)  |  86. T h e r e f o r e ,  A(  2  - A ) 2  H(1 The  + A  1  above  e q u a t i o n  e q u a t i o n  i n A  +  )  2 A H  A  + A )  AA(1  2  c o n be  + A ) +  PJA(1  s i m p l i f i e d  + PH H ] -  2  t o  g i v e  a  [  + A A]  H  q u a d r a t i c  A: 2  +  P ( 1  H  2  A  )  2  [ p ( A  A H ( 3 P A  2  +  2  + 2 A-  +  - 2 ) ( 1  P  1 ) -  2 A ] =  +  A  2  )  +  0  (1-14)  T h e r e f o r e ,  A  = ( 2 -  A  i s  I n e q u a l i t y  2  A  -  2  2  S u b s t i t u t e  o f  R  2 P ( P +  = (Fm W  o  e q u a t i o n  r e a l , ( 5 P  +  ( l - 1 5 ) +  K  P  V C  i n t o  -  +  that  4P  (2-18)  >  0  2  ( 1 - 1 6 )  t h e  i n  1  +  c h a p t e r  ( l - 6 ) , -R  7\  (1-17)  ) "  t w o :  •d Fm b©  (1-18)  +  +  becomes  K  •d  -—•  Fso- F  (I-16)  must  + be  s a t i s f y  +  ( 1 - 1 9 )  fm  t h e boundary  ( I - 1 9 ) :  KFm  4)  r e s t r i c t s  >  W  e q u a t i o n  i s :  e q u a t i o n  2P(  Fso- Fm  o E q u a t i o n  2  that  4P + 4)]  -  2  p a r a m e t e r s .  1  ( i - l )  be  2)A  1  V.  - 2 P ( P + 2)A + (5P  c o n d i t i o n  system  e q u a t i o n  x  a  ( 1 - 1 5 )  2  must  i s  (1-16)  v a r i a b i l i t y  E q u a t i o n  J5P A  t h e r e f o r e 5 P  C o n s i d e r  37\P)i  P -  r e a l  + A )&•>  1  2P(  where:  R  -R  =  +  [KP  R 2P(1  +  V  7\  c  =  c o n d i t i o n  i n  87. w  1  K  , Ad  (1-20)  fin  +  m From  ( 2 - 3 ) :  e q u a t i o n V  =  c  K  c  t h e r e f o r e ,  KP  Let  Z  2P(1  +  + A )cJ rc 2  d  =  ,-1  =  2P(  1  +  — r  J  U r so  U l m  1 -  , • +  b  9  - F c  Tm  (1-21)  7\ ) 2  t h e r e f o r e , •d+1 b0  +  c  I f known i s ,  c r i t i c a l  e q u a t i o n  o c c u r s  the  o r  0  -0  F s o Fm  parameters can  ( l - 2 2 )  n o t .  C  be can  -  of  found be  Z  =  the from  used  to  0  (1-22)  v i b r a t i n g e q u a t i o n p r e d i c t  system ( l - 2 2 ) .  whether  are That v i b r a t i o n  REFERENCES  88.  1.  W e l l s , J . H . , "Kinetic Boundary F r i c t i o n " , ( L o n d o n ) , V o l . 1 ^ 7 , p . k^k, 1 9 2 9 .  2.  K a i d a n o v s k i i , N . L . and K h a i k i n , B . E . , J o u r n a l of Technical Physics, U . S . S . R . , V o l . 3 , p. 9 1 , 1 9 3 3 .  3.  Bowden, F . P . and L e b e n , L . , "The N a t u r e o f Sliding and the A n a l y s i s o f F r i c t i o n " , P r o c e e d i n g s of the Royal Society, Series A, V o l . 1 6 9 , p. 3 7 1 , 1 9 3 9 .  k.  Bowden, Applied  5.  Bristow, J . R . , " K i n e t i c Boundary F r i c t i o n " , Proceedings o f the R o y a l S o c i e t y , V o l . 1 8 9 , p . 8 8 , 1 9 ^ 5 .  6.  L i e n a r d , A . , "Etude des O s c i l l a t i o n s Entretenues", Rev. Gen. E l e c t . , V o l . 2 3 , p. 9 0 1 , 1 9 2 8 .  7.  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