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Numerical analysis of rock failure and laboratory study of the related acoustic emission Zou, Daihu 1988

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NUMERICAL  ANALYSIS  OF ROCK FAILURE  AND LABORATORY  OF T H E R E L A T E D ACOUSTIC  EMISSION  by DAIHUA ZOU B.Sc, A  China Mining Institute,  THESIS S U B M I T T E D  IN P A R T I A L  T H E REQUIREMENTS DOCTOR  1982  FULFILMENT OF  FOR T H E DEGREE OF  OF  PHILOSOPHY  in Mining Engineering THE F A C U L T Y OF GRADUATE  STUDIES  Department of Mining and Mineral Process Engineering  We  accept this thesis as  conforming  to the required standard  T H E UNIVERSITY  O F BRITISH  January  1988  © Daihua Zou,  1988  COLUMBIA  STUDY  In  presenting  degree freely  at  this  the  available  copying  of  department publication  of  in  partial  fulfilment  University  of  British  Columbia,  for  this or  thesis  reference  thesis by  this  for  his thesis  and  scholarly  or for  her  of  The University of British C o l u m b i a 1956 Main Mall Vancouver, Canada V6T 1Y3 Date  DE-6(3/81)  3AMU A Ay  lp{>  I  I further  purposes  gain  the  shall  requirements  agree  that  agree  may  representatives.  financial  permission.  Department  study.  of  be  It not  is be  that  the  for  Library  an shall  permission for  granted  by  understood allowed  advanced  the that  without  make  it  extensive  head  of  copying my  my or  written  ABSTRACT  Sudden rock failure in the in  underground  mines.  The  basic  unresolved. This thesis describes the  doctoral  Process  candidate  form of rockbursting has mechanism  of  long been  this  a problem  phenomenon  is  still  the research work on this problem conducted by  Daihua Zou in  the  Department  of  Mining  and  Engineering at The University of British Columbia, under the  Mineral  supervision  of Professor Hamish D.S. Miller.  This research project was violent rock failure and was  undertaken in order to investigate the process of  achieved  by examining  various aspects of the rock  failure mechanism.  The assumption that acoustic emission can be used as a reliable means of predicting  rock  failure  was  investigated,  as  well  as  the  possibility  that  violent  rock failure could occur in any mine rock.  As process  part  of  the  analogous  stage.  The  under  a variety  to  stick-slip of  research,  shearing  is  phenomenon conditions.  a  rock  failure  postulated has  to  been  be  which  using could  violent  rock failure were determined.  tested  from rock  results  obtained show a correlation with field measurements  specimens  order to verify the  testing  under  results  different  same time, loading  a  rise  to  model possible  emissions were  The  made  A  post-failure  numerical  give  conditions.  postulated. the  acoustic  from limited experiments,  ii  was  important at  analyzed  The conditions A t the  mechanism  experimental  in a mine. In  a numerical acoustic  model  was  stick-slip their  developed,  process  associated  acoustic  not  which on any  acoustic  emissions  were  is  unique  acoustic  emission simulated  to  in  theory. be  that  it  This  realistically  under  various  is  model  entirely allows  simulated.  loading  rock  With  conditions  kinds of rocks. The case of a hard or a soft intercalation was  iii  based  on  the  tests and this  for  model, different  also modelled.  TABLE  OF CONTENTS  Abstract  ii  Table of contents  iv  List of Tables  viii  List of Figures  ix  Acknowledgement  •  xiii  Chapter 1. Introduction 1.1. Introduction  1 1  Chapter 2. Basic Concepts of Rockbursting and its Control 2.1. History 2.2. Characteristics of Rocks 2.3. Field Investigations of Rockbursts 2.3.1. Mining Activity 2.3.2. Mining Depth 2.3.3. Geological Conditions 2.3.4. Properties of a Rock Mass 2.3.5. Geometry of Openings 2.4. Development of Rockburst theory 2.5. Warning Methods 2.5.1. Closure Measurement 2.5.2. Stress Measurement 2.5.3. Microseismic Monitoring 2.6. Rockburst Control 2.6.1. Optimization of Mining. Layout 2.6.2. Destressing 2.6.3. Rock Support 2.7. Summary  .  5 5 6 9 10 10 11 11 12 12 14 14 15 15 18 18 19 20 21  Chapter 3. Failure of a Massive Rock 3.1. General Concepts 3.2. Fracturing Process 3.3. Detection of Fracturing 3.4. Failure Development and the Shearing Process 3.5. Determination of a Failure Plane 3.6. Summary  23 23 24 27 29 34 36  Chapter 4. Failure by a Process of Shearing 4.1. General 4.2. The Law of Friction 4.3. Shear Strength 4.4. Effects of Environment 4.4.1. Normal Pressure 4.4.2. Temperature  37 37 38 43 47 47 50  iv  4.4.3. Pore Pressure AAA. Time Dependencj' 4.5. Stick-slip Phenomenon 4.6. Summary Chapter 5. Theoretical Shear Model: Constant Friction 5.1. Mathematical Model 5.2. Solutions to the Differential Equation 5.3. Model Results 5.3.1. Slip Time 5.3.2. Slip Distance 5.3.3. Stick Time 5.3.4. Comparison with Laboratory Results 5.4. Discussions 5.5. Summary  51 51 53 55 57 57 60 62 62 64 64 66 69 73  Chapter 6. Slip Behavior under Various Conditions 74 6.1. Summary of Rock Properties 74 6.1.1. Frictional Coefficient 74 6.1.2. Cohesion 75 6.1.3. Elastic Modulus 76 6.1.4. Uniaxial Compressive Strength 76 6.2. Seismic Effect 77 6.2.1. Formulation of Seismic Radiation 77 6.2.2. Characteristics of Seismic Radiation Coefficient .... 83 6.3. Mathematical Model 83 6.4. Energy 85 6.5. Numerical Solution 87 6.5.1. Introduction to Runge-Kuta Method 88 6.5.1.1. First Order Differential Equation . 88 6.5.1.2. Simultaneous Differential Equations .... 88 6.5.2. Application to the Numerical Model 89 6.6. Programming 90 6.7. Numerical Results 94 6.7.1. Effects of Major Factors 94 6.7.1.1. Effect of Cohesion 94 6.7.1.2. Effect of Frictional Coefficient 95 6.7.1.3. Effect of Elastic Modulus 96 6.7.1.4. Effect of Normal Load 99 6.7.1.5. Effect of Loading Speed 100 6.7.2. The Variation of Slip Behavior 102 6.7.2.1. Maximum Slip Distance 104 6.7.2.2. Stick Time 104 6.7.2.3. Force Drop 106 6.7.2.4. Energy Release 106 6.7.2.5. Average Energy Release Rate and Energy Release Ratio 107 6.8. Summary 108 v  Chapter 7. Transition Conditions and Violent Failure 7.1. General 7.2. Transition Conditions 7.3. Slip Behavior in Shear Test 7.4. Occurrence of Violent Failure 7.5. Summary  Ill Ill Ill 116 119 122  Chapter 8. Effect of 8.1. 8.2. 8.3. 8.4. 8.5.  124 124 125 128 129 132  Sudden Loading General The Effect of Excessive Load Occurrence of Sudden Loading Occurrence of Violent Failure in Compressive Test Summary  Chapter 9. The Nature of Rockbursting 9.1. General 9.2. Violent Rock Failure along a Natural Fault 9.3. RockBursting in a Massive Rock Mass 9.4. Influence of Other Geological Structures 9.5. Influence of Mining Conditions 9.5.1. The Shape and Size of a Pillar 9.5.2. Mining Rate 9.6. Estimation of Possible Violent Failure 9.7. Prevention of Violent Failure 9.7.1. Mining Design 9.7.2. Destressing 9.7.3. Support 9.8. Summary  134 134 135 141 142 145 145 147 148 150 151 155 156 158  Chapter 10. Laboratory Study of Acoustic Emission at Rock Failure 10.1. Introduction 10.2. Test Program 10.2.1. Specimen Preparation 10.2.2. Equipment 10.2.3. Test Procedure 10.3. Test Results 10.3.1. Acoustic Emission from Compressive Tests .. 10.3.2. Acoustic Emission from Direct Shear Tests . 10.4. Discussions 10.5. Summary  160 160 161 162 162 165 167 167 173 181 188  Chapter 11. Precursory Signals in Comparison with Field Measurements 11.1. General 11.2. Precursory Signals in the Laboratory Tests 11.3. Precursory Signals in Field Monitoring 11.3.1. Precursory Signals prior to Rockbursting 11.3.2. Typical Examples 11.4. Comparison 11.5. Summary  191 191 191 192 192 193 194 199  vi  Chapter 12. Numerical Simulation of Acoustic Activity at Rock Failure 12.1. Mathematical Model 12.2. Energy Estimation 12.3. Count of Event 12.4. Limits to the Model 12.4.1. The Logical Position 12.4.2. The Physical Condition 12.4.3. Conditions for Stick-slip 12.5. Numerical Solution by Runge-Kuta Method12.6. Programming 12.7. Modelling Results 12.7.1. Resemblance to the Testing Results 12.7.2. The Total Energy Released versus the Energy 12.7.3. After Shocks 12.8. Summary  202 202 207 209 211 211 211 213 214 216 219 219 Seismic 223 224 225  Chapter 13. Acoustic Activity under Different Conditions 13.1. Acoustic Emission as Normal Pressure Varies 13.2. Acoustic Emission as Loading Speed Varies 13.3. Acoustic Emission as Elasticity Varies 13.4. Acoustic Emission under Multiple Elasticity 13.4.1. A Hard Intercalation 13.4.2. A Soft Intercalation 13.5. Summary  227 227 232 236 241 242 244 246  Chapter 14.  248 248 251  Conclusions 14.1. Conclusions 14.2. Recommendations for Further Research  Bibliography  253  Appendix I. List of F O R T R A N Program M O D E L 1 and Sample Appendix II. List of F O R T R A N Program M O D E L 2 Appendix III. List of F O R T R A N Program M O D E L 3 Appendix IV. List of B A S I C Program M O D E L 4  vii  and Sample  Results Results  and Sample Results  and Sample  Results  257 262 267 271  LIST  OF  TABLES  4.1 Regression analysis of velocity-dependent  coefficient  of friction  42  4.2  Constants for empirical formula of slip-velocitj' dependent friction  44  6.1  Summary of rock properties  75  6.2  Effect of cohesion on slip behavior  95  6.3 Effect of friction coefficient  on slip behavior  99  6.4  Effect of elastic modulus on slip behavior  6.5  Effect of normal load on slip behavior  103  6.6  Effect of loading speed on slip behavior  103  8.1  Effect of sudden loading on slip behavior  127  8.2  Stress estimation on failure surface of rock specimen in compression  132  10.1  Identification and mechanical properties of compressive  10.2  Mechanical properties of shear specimens  viii  99  specimens  167 179  LIST  OF  FIGURES  2.1  Complete stress-strain curve for unconfined rock specimen  2.2  Complete stress-strain curve for unconfined and confined Tennessee  7 marble  3.1 Mechanism of brittle fracture of rock in multiaxial compression 3.2  8 25  Front, top and side views of the central section of sample showing locations of events that occurring in the dynamic cracking region  32  3.3  Unconfined Charcoal Gray Granite I in advanced stage of failure  33  3.4  Schematic showing shear failure plane  35  4.1  Simple model for shearing  39  4.2  Velocity dependent friction. A , B and C refer to different experiments  41  4.3  Friction strength of sawcut and fault surfaces of variety of rock types under different conditions of temperature, rate and amount of water  44  4.4  a) Postulated bilinear shear strength; b) the effect of slip velocity  48  4.5  Sliding characteristics of stick-slip and stable sliding on sawcut surfaces  ..  4.6 The effect of temperature on the friction strength of dry gabbro  49 50  4.7 Transition from stable sliding to stick-slip as a function of normal stress, stiffness and surface finish  55  5.1  Simple shear model  58  5.2  a) Load-displacement curve for a typical shearing test, b) the oscillation of load-displacement curves on a magnified scale a) One cycle of the oscillation of figure 5.2b) on an enlarged scale;  67  b) the same  68  5.3  showing displacement against time  5.4  Model results showing changes  of each parameter with time  5.5  Model results: a) force-displacement curve; b) displacement-time curve  70  6.1  Simulating the effect of seismic radiation  78  6.2  A n element of the semi-infinite string  79  6.3  Shearing resistance  as a function of slip velocity and seismic radiation ix  69  ...  82  6.4  Flow chart for program M O D E L l : numerical shearing model  92  6.5  Flow chart for program M O D E L 2 :  93  6.6  Change of slip behavior parameters with cohesion  96  6.7  Change of slip behavior parameters with friction coefficient  97  6.8  Change of slip behavior parameters with elasticity  98  6.9  Change of slip behavior parameters with normal load  6.10  sensitivity  analysis  101  Change of slip behavior parameters with loading speed  102  7.1  Flow chart of program M O D E L 3 : transition analysis  113  7.2  Transition conditions for stick-slip and stable sliding  115  7.3  Transition conditions showed  117  8.1  Variation of slip parameters with the ratio of initial shear force over  as loading speed against elasticity  the shear strength  126  9.1  Stress components  on a natural fault in the rock mass  136  9.2  Stress redistribution after excavation of an opening in the rock mass ....  137  9.3  Streamline of stress change due to mining activity  138  9.4  Possible sliding of highly stressed blocks  139  9.5  Stress change due to an opening around a fault  140  9.6  The loading and the failure path of rock pillar  143  9.7 The loading and the possible failure path of a working face  144  9.8  146  Stress redistribution due to mining around a hard intrusive  9.9 The intersection at two roadways should be made round as shown by the dot line in order to reduce stress concentration  152  9.10  Adjusting mining sequence to achieve better  153  9.11  When mining across a fault, it is better to approach it from the upper panel in order to reduce unnecessary  stress condition  high stress  9.12  Proper support in advance can reduce the incidence of violent failure  10.1  x Loading diagram for acoustic emission test  154 ..  157 166  10.2  Acoustic emission from uniaxial compressive test for specimen #1  169  10.3  Acoustic emission from uniaxial compressive test for specimen #2  170  10.4 Acoustic emission vs axial load for specimen #1  171  10.5 Acoustic emission vs axial load for specimen #2  172  10.6  174  Shear strength of sawcut and breakage surfaces  10.7 Acoustic emission from breakage specimen #5  under direct shear test ..  175  10.8  under direct shear test ..  176  Acoustic emission from breakage specimen #7  10.9 Acoustic emission from sawcut specimen #4  under direct shear test  177  10.10  Acoustic emission vs shear displacement for specimen #5  178  10.11  Acoustic emission vs shear displacement for specimen #7  178  10.12  Acoustic emission vs shear displacement for specimen #4  179  10.13  Effect of normal pressure on event rate from spcimen #4  182  10.14  Effect of normal pressure on energy release from spcimmen #4  183  10.15  10.16  10.17  10.18  Acoustic emission vs shear displacement at various normal pressure on specimen #4  184  Acoustic emission from sawcut specimen at sudden shear loading, by releasing normal pressure at 1, 2.5 and 4.5 ksi level, respectively ......  186  Acoustic emission from breakage specimen at sudden shear loading, by releasing normal pressure at 1, 2.5 and 4.5 ksi level, respectively  187  Effect of rock type on acoustic emission  188  11.1 Microseismic event rate and relative energy plotted for one week before and three days after the May 15 event  195  11.2 Event rate, corner frequency and event energy measured over a period of 25 days, covering two rockbursts  196  11.3  Schematic seismic spectrum, clarifying: low-frequency amplitude level, cornor frequency, and high-frequency amplitude decay  198  11.4 The relationship between size and number of seismic events  200  12.1 Diagram of acoustic activity model  204 xi  12.2a) Flow chart for program M O D E L 4 :  acoustic simulation  217  12.2b) Flow chart of the computation part in program M O D E L 4  218  12.3  220  Computer results fro the numerical acoustic model  12.4a) Complete pattern of acoustic activity prior to failure, showing after shocks  221  12.4b) Complete pattern of acoustic activity prior to failure, showing similarity between  total and seismic energy  222  13.1a) Numerical acoustic emission at normal pressure 500  Pa  229  13.1b) Numerical acoustic emission at normal pressure 1 K P a  230  13.1c) Numerical acoustic emission at normal pressure 10 K P a  231  13.2a) Numerical acoustic emission at loading speed 0.01  233  13.2b) Numerical acoustic emission at loading speed 0.1 13.2c) Numerical acoustic emission at loading speed  m/s m/s  234  1.0 m/s  235  13.3a) Numerical acoustic emission at elastic modulus 100 M P a  237  13.3b) Numerical acoustic emission at elastic modulus 1 M P a  238  13.3c) Numerical acoustic emission at elastic modulus 100 K P a  239  13.3d) Numerical acoustic emission at elastic modulus 30 K P a  240  13.4 Numerical acoustic emission with a hard intercalation  243  13.5 Numerical acoustic emission with a soft intercalation  245  xii  ACKNOWLEDGEMENT The author would like to thank: The Chinese Government for funding this project during the first two Dr.  Hamish D.  S.  Miller  for his  continuous  supervision,  great  years.  help during  this research and funding the rest of this project. Professor  C. 0.  Brawner,  Dr.  Ross  Hammett  and  Professor  their helpful advice and comments  throughout this program.  Dr.  members  A.  Hall  Department  and of  other  Mining  Faculty  Engineering  for  and  Graduate  their  valuable  A . Reed  students  in  discussion  for  the and  encouragement. Professor acoustic  J . S. Nadeau in the Department of Metallurgy for lending us emission  equipment  and  technians  M r . Frank  R. Gutenberg for their help during the laboratory tests. Ms. Sylvia Paulin for proof reading the thesis.  xiii  Schmidiger  the  and M r .  To MY  xiv  PARENTS  1.  CHAPTER  1.1.  INTRODUCTION  Rock  failure  unexpected on  INTRODUCTION  and  aspects  referred  can  of  to  place  severe  rock  rockbursts  mines,  gradually  problems  sudden  as  underground  take  dating  can  failure.  suddenly.  result.  This  Large  and  these  back  to  depth has continued to increase  or  the  research  sudden  have  When  long  rock  of  occurs  therefore  failures  been  beginning  it  a  concentrates  in  a  serious  this  is released  century.  the  violent  failure  of  a  characterized by the  that results.  rock mass  under  opening  by expulsion  [1,2,3].  a  high  non-violent  rock  failure  stress stored  of rock in varying quantities  Therefore, by  this  type  its  of  way  Rockbursting is  sudden release of a large amount of strain energy characterized  As  are in  mining  in recent years, the problem is becoming critical.  usually  and by the damage  mine  problem  More and more mines with no previous history of bursting are being  Sudden rock failure is  suddenly,  failure  suddenness,  the  is  in which energ}'  generally  field,  defined  as  accompanied  by  in the from the  rock mass and surface  distinguished  absence  affected.  of  from  warning  of an normal  and  the  intensity of the resulting damage.  Once violent rock failure occurs, it can give rise to various problems in a mine, from  depending the  mining  on  focus  induced  of  the  energy  released  the  event.  If  seismic  event,  with  Richter scale [1],  the  a  and  large a  the  amount  magnitude  distance of  of  energy  possibly  the  mine  is  released  reaching  5.5  opening by on  a the  effect could be similar to that of a small earthquake. The  1  Introduction / 2 result  can  structures rock  be,  and  frequently  and facilities.  failure.  more. In fact, of rock failure  production[4]. were  failure  and  damage  aspect of violent  rock failure  is  African  mine  this year  killed nine  threat  people and injured  has  a  major  cause  of  fatalities,  alone,  more  than  680  cases  of  and  the  been  1975  [5].  causing  73  fatalities  to  one rockburst  earliest days of gold mining in South Africa,  in these mines,  48,000 man-shifts  mine  since the  During  reported  to  its  and casualties are often a direct result. For example,  that occurred in a South  kind  catastrophic  Millions of dollars are lost annually due to this kind of  The most dangerous  miners' lives,  many  is,  damage violent loss of  this  and  loss of  rock  failure  more  than  A rockburst occurred in a mine in Ontario two years  ago  claimed the lives of four miners.  Although progress,  the  research  initiated  in  the  last  few  results have not been satisfactory,  decades  and the  has  achieved  some  problem of violent rock  failure in mines is still unresolved. This is because first of all, the mechanism of violent  rock failure  is  not well  understood  and as  a result  the  conditions  which  cause this kind of rock failure are unknown.  Because violent to  rock failure  predict  dollars  or  has  to  is  virtually  never  any  in underground openings,  give  any  warning  to  such  been spent on field research  the  progress  for  rockburst  South  there  is  African  very  physical it is an  visual  evidence  extremely event.  difficult  Each  year,  Chamber  in of  1986/87 Mines  but which  to  practice  millions  of  of rockburst prediction and control, but  slow. The Government of Ontario spent 4.2  research  in  prior  little is  progress the  has  earliest  been and  million dollars reported.  still  the  The  leading  Introduction / 3 rockburst  research  group  in  the  world,  million dollars since its establishment measure groups  of  success  throughout  in  has  in 1964  predicting violent  the  world  spent  have  than  an  estimated  50  and only recently has it had some  rock  faced  more  failure  the  in  same  a  mine.  difficulty  All research in  predicting  rockbursts that arise as a result of not having reliable precursors.  Because expensive  field research of violent rock failure in operating mines is a very  and difficult exercise,  this  research  attempts  to study  the  problem by  applying numerical analysis and laboratory experiments in an attempt to derive a method or to provide a guideline for subsequent field work. The major  objectives  are: I.  to  investigate  the  conditions  which  may  give  rise  to  violent  rock  failure,  discussed in chapters 3 through 9, and II. to find precursive signals for such an event, given in chapters  In  order to  find  the  conditions  causing  violence,  the  a  fault  qualitative fracturing researchers explain  or joint assessment  is  plane. made  intrinsic to to  violent  discontinuities,  be  the  rock  Failure  in  of  common  the  basic failures  any  both  cases  factors.  failure of massive mechanism as  a  of  result  should  such as faults or bedding planes.  be  of rock  rock as well examined  and  as a  mining, stress induced  rock and is considered by some  violent of  In  13.  mechanism  failure will be studied first. Violent failure can occur in massive on  10 to  rock  sudden  failure  [2].  slips  along  Others  [6]  geological  Whether violent failure occurring  in these two conditions is independent or related needs studying.  Introduction / 4 The would failure.  emission  appear It  is  to  of  have  for  this  acoustic the  noise  greatest  reason  from  material  potential  therefore  for  that  specimens will be monitored in laboratory conditions techniques.  undergoing  giving acoustic  stress  warning  of  emissions  and modelled  loading  impending from  rock  using numerical  CHAPTER In  order  2. B A S I C  to  CONCEPTS  provide  some  OF ROCKBURSTING  background for  study  of  violent  from previous research on rockbursts are investigated ninety  major problems existing  was  in this chapter.  results  More than  are summarized in the following and  HISTORY  20th century. The earliest in  1908  generally rock  failure,  in practice are also listed.  Rockbursting in underground mines the  rock  published papers have been reviewed but only the more relevant ones are  referred to here. The results of this survey  2.1.  A N D ITS C O N T R O L  [5]  not  and  in  Ontario  mines  in  1929  because the  [1],  in South  [7].  Rockbursting  is  not very high unless high tectonic stresses exist. However,  the  as mining depth increases,  or  a  vein  of  dyke  gravitational  Africa  the  exist  mines  was  in 1898  at the beginning of  load on  problem becomes greater faults  reported as early as  report in India was  a problem in shallow  structure is  natural  was  particularly in a mine  material  or  competent  where  orebodj'  is  attempt  to  intercalated in a moderate to hard rock matrix.  Much understand results  research  precursory  rockbursting has  and to prevent  achieved,  improvement  into  of  we  in  initially called  techniques,  seismicity  such as  were  carried out  the  before  monitored  violent  rock  5  an  From  the  this problem. With  the  rock  mass  failure  avoiding high stress concentration,  incidence of rockbursting can be reduced.  in  "earth tremors".  getting better in understanding  monitoring  signals  control methods,  are  what  been  shows  occurs.  destressing,  By etc.,  some using the  Basic Concepts of Rockbursting and its Control / 6 2.2.  CHARACTERISTICS  Various  properties  factors  in  of  violent  have  little  Most  rocks  exhibit  behave  rocks  failure.  rock,  potash  OF ROCKS  tensile  are It  considered  is  well  strength  brittle  plastically,  because  known  but  have  characteristics particularly  that  they  geological  relatively  under when  are  high  inherent  materials,  such  compressive  compression,  under  important  high  although  as  strength. some  like  and  low  confinement  loading speed.  Generally,  a rock will behave  strength  as  capacity  of rock to  rock  in  figure  curve  place.  The elastic  varies  widely  with  failure  kinds  of  behavior  Beyond  rock  when  varies  types  loaded  greatly.  the  modulus,  different  when the  the  peak  in  Even  load is  It  with  can be  the  failure level,  seen from this  confining  stage, where  rock the  pressure, will  "flow"  or the  slope  point  for  that  the  when  the  the  strength the  same  of the  its  A,  there is no energy accumulation.  or  the  does.  confining  O A part  of  the  similar  the  post  will  be  as shown in figure  modulus  pressure  does not  reaches  load. In the is  is  behavior  Furthermore, at  strain energy  complete  However,  rock, this  elastic  at a constant  accumulated  constant  compression.  case of plastic behavior, to the extent that energy  process,  strength,  of rock. Pre-failure behavior uniaxial  deformation continues  behavior, upon rupture the the  figure  but  less than  held  either brittle or plastic when under different confinements [8].  stress is  load will decrease dramatically. Eventually,  even when  takes  all  2.1.  support external  will deform continuously  failure  for  illustrated  elastically  change  the a  2.2  post  certain  case of brittle  fully released, is dissipated  while  in the  in  flow  Basic Concepts of Rockbursting and its Control / 7  Fig. 2.1  Complete stress-strain  curve for unconfined rock specimen et al, [56])  (from Starfield  Basic Concepts of Rockbursting and its Control / 8  Fig.  2.2  Complete stress-strain curves for unconfined and confined Tennessee marble (after Wawersik et al, [ 8 ] )  Basic Concepts of Rockbursting and its Control / 9 Therefore,  a  probably violently  rock  mass  may  fail  under low confinement  gradually  under  if the energy  high  confinement  is released  suddenly.  may suggest that rockbursting or violent rock failure will never take deep confined zone,  but possibly  at or near the  surface  and This  place in a  of an opening or where  relaxation has taken place.  Laboratory dependent  work  upon the  conventional  or  stress-strain  curve  implies that the  has  testing  "soft" being  machine  obtained  when  rock failure process  a rock specimen  [8] because of the  behavior  of  rock  mass  [9,10] that  the  rock failure  process  is  will  fail  tested  in  gradually, a  stiff  with  testing  a  complete  machine.  This  depends not only on the rock properties but  However, for some kinds of rock, the violent failure  cannot be completely  machine  a  shown  machine. Rocks which fail abruptly when tested in a  testing  also on the loading system. of  also  inherent is  controlled only by stiffening  characteristics  related  to  other  of the  the  testing  rock. In addition, the  environmental  variables,  such  as  temperature, time and pore pressure.  2.3. F I E L D I N V E S T I G A T I O N S It  is  observed  from  field  OF ROCKBURSTS  investigations  stress zones or in areas near geological with  mining  activities.  Many  other  conditions, rock properties, geometry well.  that  rockbursts  usually  occur  structures and are also closely  factors,  such  of openings,  as  mining  depth,  in  high  associated geological  etc. contribute to rockbursting as  Basic Concepts of Rockbursting and its Control /  10  2.3.1. M i n i n g Activity More  rockbursts occur during and immediately  non-extraction and results  periods.  Mining  disturbs  the  after  stress  excavation  are created than  equilibrium in  the  rock  mass  in a redistribution of stresses. The rapidity of stress change is very  important to rock failure. A sudden change  of stress brought about, for example,  by blasting may be the immediate cause of violent failure [6]. Therefore the high speed  of  stress  change  induced by  blasting  may  have  higher  risk  of  causing  rockbursts than that by relatively low speed, continuous excavation.  2.3.2. M i n i n g Depth Rockbursts meters  are  but  occurred  usually  can  within  excavations  occur a  and  components  experienced at  depth  quarries.  of the  at  shallower of  less  This  depth.  than  can  existing tectonic  depth  be  starting In  300  at  some  meters,  accounted  for  around  cases,  600  -  1000  rockbursts  have  as  well  as  by  the  high  stress field. The general tendency  severity  and the frequency of bursts  because  of the increase of the gravity stress. As mining goes deeper,  increases  and  differently stress  field  stress  on  in  the the  might a  rock  away  post-failure become  failure  are expected  from  stage  as  hydrostatic  surface.  the  However,  mine  shown  at  great as  to increase  the  in  openings figure  depth,  excavation  surface  horizontal is that  the  with mining depth  may 2.2.  thus  in  In  confinement  behave addition,  reducing the  the  shear  disturbs  the  in-situ stress field and relaxation occurs in and around the mining openings,  the  potential for rockbursting will be enchanced. This is because differentials created with an increase in depth.  process  quite  of the greater stress  Basic Concepts of Rockbursting and its Control /  11  2.3.3. Geological Conditions Rockbursts are usually associated hard  intrusion.  In-situ  stress  with geological  fields  gravitational stress field, tectonic vicinity  of  these  geological  distributions  the  being  mass  concentrations  more  caused  highly  will certainly increase  by  rapid  stress  from  The presence  within the  amount of strain energy present. fault  arise  stress field and the  structures.  introduce uneven rock  can  structures, such as a fault, or a three  stress concentrations of  geological  stressed  than  others.  properties  rockbursts  are  more  related  tend to occur more often  sedimentary  rocks. This,  the  compressive  commonly  may  stress  field,  to  cause  strong  as  and a  will  These  local  stress  violent  failure. In  fact,  this  is  [11].  and  in rock failure. For, while brittle  does not imply that  inversely  measure  rocks  than  to  soft  in igneous and metamorphic rocks than in  rocks. Strain energy,  considered  The more energy same  however,  strength  structures  Mass  rocks, they  in soft, sedimentary  in the  In the case of a fault, a sudden slip along the  of a rock mass are important factors usually  the  risk of violent failure because of larger  considered to be the cause of many shallow earthquakes  The  sources:  stress field, resulting in some parts of  the  change  2.3.4. Properties of a Rock  different  of  rockbursts will not occur  which is proportional to the proportional  the  tendency  to  the  of  a  elastic  square of  modulus,  rock mass  to  is  burst.  stored, the higher the risk of bursting exists. Therefore, in the the  rock  mass  with  higher  compressive  higher capacity of energy storage is more likely to burst.  strength  and  hence  Basic Concepts of Rockbursting and its Control /  12  2.3.5. Geometry of Openings The  geometries  of  This  is  say  burst,  not but  openings  to the  underground openings that  relative  a  smaller  positions  can be significant,  are  or a of  also  bigger  openings  closely  opening will be and  the  pillar  the  field,  irregularities  all and  openings hence  or a  another  opening or with  the  opening  and  stope should be  should be  stopes  abnormal  opening  to  rockbursts.  more likefy shapes  to  between  and the irregularities of mining structures are usually  more burst prone because of uneven stress concentrations. in  related  any  stress  such  that  geological  parallel to  the  must  be  According to  carefully  concentrations.  weakness,  such  direction of the  planned  The  it will not make as  experience to  orientation  an  acute  a fault.  avoid of  angle  an with  The axis of  major principal  stress in  order to minimize the stress concentration.  2.4.  DEVELOPMENT  OF ROCKBURST  THEORY  Since the earliest stage in rockburst research, vavious theories  have been used to  interpret the  in South Africa  committee  phenomenon  was  formed to  of domes,  of rockbursting [1]. investigate the  Early  in  1915,  problem. Committee  members  a  suggested  the  concept  zones of fractured rock around stopes and concluded  the  domes supported load and also transferred load to pillars. The removal and  failure of pillars may cause a dome to fail, giving bursts. During the late by  the  theory  excavations  due  of to  elasticity, stress  the  concept  concentration  was  of  fracture  used  and  development the  sufficiently  that  1920, around violent  fracturing could result in bursts.  Prior  to  1930,  all hypotheses were based  primarily  on observation. Little  Basic Concepts of Rockbursting and its Control / 13 effort was made to understand the mechanism causing a burst. During this time, the number and severity of bursting increased. By the end of causes  of  rockbursts were  accepted:  (1)  the  pressure-dome  1930s, two main  theory  using  stress  concentration around mine openings to account for rockbursts in mines where the veins  dipped  steeply,  and  (2)  the  cantilever  veins were mostly flat-lying. Both theories measured the  behavior of  stope  walls  In  1938,  experimental  the  results  first was  used  in  mines  where  the  were based primarily on observed and  and suited  application of various control methods  continued and became more severe  theory  to  a  particular  geometry.  Despite  as a result of these theories, bursting  as mines went deeper.  mathematic proposed  accepted by mine operators because  to  model explain  at that  based  on  elastic  rockbursting but  time,  it was  could not be used to predict mine behavior. During the  felt  theory it  and  was  that  never  mathematics  1950s, mathematics  was  paid more attention and the theory of elasticity was used to a greater extent.  In  1963,  be analyzed by  Cook  an energy  excavation  must  Later  further  he  [12]  be  in  proposed that the  mechanics  approach. To control bursting, the  small  suggested  stability problem in the same  amounts [13]  that  that  it  is  loading system initiated,  stiffness,  causing  might  way as a specimen behaves  instantaneously violence  could be  rockbursts  the specimen is stiffer than the loading system, the  of rockbursts could best  [10].  loads  the  In  other  energy  dissipated be  release  at  nonviolently.  considered  as  a  in laboratory tests. If  excessive strain energy stored in  rock  structure further when failure  words,  depending  on  the  relative  the specimen will fail violently or nonviolently if energy can or can not  Basic Concepts of Rockbursting and its Control / be extracted rockbursts  from the  are  loading system  controlled  by  the  at failure. It was  rate  at  which  concluded in 1966  energy  is  released  14 that  as  an  excavation is made.  The massive and  stiffness  approach  rock. However,  this  there  approach can not  this case, the  is  certainly  has  been  explaining  rockbursts  in  no further work published on this  a  topic  because in  failure takes place as shearing. In addition, this approach can not  summary,  mechanics  in  explain rockbursting along natural faults,  correlate the violence to the acoustic  In  valid  of  rockbursts  rockbursting  are  activity preceding the violent failure.  have  been  adequately  still  unclear  described.  because  little  Yet  the  basic  research  has  been  directed towards how a burst occurs.  2.5.  WARNING  During the warnings  past  METHODS study  of rockbursting, major efforts  of impending rockbursts. Methods such  measurement pre-failure  and  microseismic  behavior  of  the  monitoring  rock  mass,  as  have  with  have  been made to provide  closure been  the  measurement,  used  last  one  to  stress  monitor  having  the  the most  potential.  2.5.1. Closure Measurement This is the such  a primitive method used  possible as  locations  closure  of  for a  to pinpoint the  rockbursts.  tunnel  or  a  It  is  stope,  found  areas that  between  of large deformation and large  roof  ground  and  floor,  movements, sometimes  Basic Concepts of Rockbursting and its Control / 15 precede  a  burst [14],  occurring over warning  of  in the  order of  a long period.  impending  While  failure,  10  the  this  times  as  abnormal  method  rapid  rate  cannot  as  normal  movement  of displacement  reliably  predict  gives  and  a  locate  rockbursts.  2.5.2. Stress Measurement Stress  measurement  at various points in a mine made  over  a long period will  show the change of the stress field as mining proceeds. The areas of high stress concentration which usually precede the potential burst zones can then be located. While  this  should be  burst,  because  nature of the  of  possible  the  wide  analytically since  variation of  high  geological  stress  is  conditions  necessary and  the  for  a  changing  stress field at different regions throughout the mine, the accuracy  of this method is not sufficient either.  2.5.3. Microseismic Monitoring Microseismic  monitoring is  the  use  of a geophysical technique  which has  had a  long history in oil and mineral exploration fields, but its use in mining is fairly recent.  Experience has  proven the  microseismic technique  to  be  quite  successful  and encouraging, especially since the introduction of the electronic computer, which makes possible online data processing. This technique promises to provide warning of impending violent rock failure, and is therefore described in more detail.  The  principle of this method is based on the  redistribution induced by mining, self-adjustment fracturing  which  is  accompanied  by  acoustic  fact that during the  stress  takes place in the rock mass by emission,  or  rock  noise  which  is  Basic Concepts of Rockbursting and its Control / 16 audible  or  subaudible  and will  be  later. By recording the acoustic can be detected here  is  the  and the  synonymous  acoustic  discussed  signals  energy  activity  and  the  estimated.  emission  final  detail  in  the  next  chapter and  with a transducer, the microseismic event  released  with acoustic  in  The term microseismic event  of rock. Then  failure  may  a relationship between  possibly  be  established  from  continuousl}' monitored data.  The  recording system  amplifiers,  cable  used  for  microseismic  and a central processor.  monitoring include  The signal of a microseismic event is  recorded and transformed into an electrical signal by a geophone, an  amplifier, the  amplified signal  processor, which analyses rate,  seismic  found  in  failure  is  by  energy  laboratory usually  then being  transmitted by  the signal and gives final  rate, energy  ratio or whatever  studies  in  and  preceded by  monitoring acoustic  a  emission  field  the  results  rock  to  a central  in the form of event  that  in a  [15].  the  in acoustic mass  passed through  cable  is needed  monitoring  sharp increase from  geophones,  It has  been  impending  rock  emission. Therefore, successive monitoring  period, it is theoretically possible to predict a coming failure and to give warning in advance if an abnormal pattern of acoustic emission occurs.  If the velocity with which the shock wave known, it is possible to locate the seismic shock  wave  co-ordinates differences detecting  from of  of  the  the the  points  in  source  detecting  first the  arrival rock  to  the  stations of mass  the are  propagates in the rock mass is  event, provided the travel time of the detecting are  shock  point  known wave,  measured  is  measured  [1.6].  Usually,  usually  P-wave,  by  setting  up  an  and the at  the time  several  array  of  Basic Concepts of Rockbursting and its Control / sensors  at  different  locations.  The  wave  velocity  can  be  determined  by  17 a  calibration test with a man-made signal as a source event.  There One  is  are  the  three  single  channel  sensor and a simple and  major  types  system,  processor.  of monitoring systems  in  which  consists  is  portable  It is effective  gives warning signals within its  and  use  in  when  an unstable  field.  a  single  of  over a radius of about  coverage  the  20  meters  condition  occurs  and a failure is pending. It cannot give the  exact location of the  unstable  and  single  an  improvement  systems, than  which  the  system,  first  can  on this is monitor  system.  which  has  a  Finally,  an  array  the  system  consisting  of several  wider  region  and give  similar but  the of  most commonly  from  7  to  up  used  to  is  32  in the rock mass to be monitored. It has  signal  system consisting  and hard copy printer, etc. events within  channel  better source  geophones  different locations processing  the  area,  results location  installed  a more  at  sophisticated  of a minicomputer, a recorder, visual monitor  This system is able  to accurately  locate the  ± 1 0 feet or even better and pinpoint any unstable  area  seismic  whenever  it occurs.  The  major problem of the  microseismic  technique  is  its  low  reliability in  rockburst prediction. Few rockbursts have been successfully  predicted in the  nor  fact,  has  a  potentially Africa  just mine  key  successful  [17].  becoming  reliable  prediction  Nevertheless,  wide  spread.  precursor  this  of  yet  been  rockburst  method  Moreover, the  found. is  only  still  has  a  final  goal  of  In  the  evidence  on  from  reported  bright future a  past,  monitoring  and  its  system  of  South use is  is not  to predict a burst, but more importantly, to locate seismic "hot spot" in a and so provide an early warning so that measures can be taken to avoid  Basic Concepts of Rockbursting and its Control /  18  the coming problem.  2.6.  ROCKBURST  Rockbursting reaches  seems  great  objective  CONTROL to  depths,  be and  inevitable everj'  of rockburst control is and  measures  in use today are these:  the  consequently  optimization  to  of  effort  to  incidence  in  cases  should  eliminate  minimize  mining  some  the  be  or at  taken  to  from  to  prevent  to  avoid the  when  control  least to  damage  layouts  particularly  it.  reduce  the  mining  the  burst.  unnecessary  So  the  bursting  The major  high  stress  concentrations, the  destressing  of an area concerned  burst or to reduce  the  incidence of bursting when high stress builds up, and the  introduction  of  rock  support  system  that  can  handle  the  results  of  rockbursts.  Usually these three methods are used in combination so as to get better  results.  2.6.1. Optimization of M i n i n g Layout The  optimized  mining  layout  offers  the  most  effective  measure  of  control, and at the stage of designing the mining system,  unnecessary  concentrations  rule  design  for  properties  it in  concentrations 1.  should varies a  be  avoided.  with  the  There geological  particular mine,  as much as possible.  and  the  is  no  general  conditions, general  mining  principle  is  for  high stress  the  method to  rockburst  optimum and  reduce  rock stress  For instance:  In pillar operation, ore should be recovered as much as possible.  If sprags,  Basic Concepts of Rockbursting and its Control / 19 pillar remnants,  or complete  pillars have  to be left in the  mined-out  areas,  they should be evenly distributed for best stress distribution. 2.  Pillars should be approximately the same size and shape,  and large enough  to support the overburden. 3.  Roof spans possible  projecting over the mined-out areas  or else  provided with  support that  should be kept as  ensures  that  the  short  as  roof beds do  not fracture. 4.  The axes of the  workings should be parallel to the  direction of the  major  principal stress in order to minimize the stress concentration. 5.  Sequential  extraction  from  strata  or  from  stages  and  horizons  should  be  adopted for multi-layer mining.  2.6.2. A  Destressing  high  stress  necessary stress  condition  giving for  concentrations  incidence  the  depth,  thus  source  further  bursting  rise  to  large  rockbursting to  can  be  avoided,  stress  occur  or  if  fractured zone ahead  with  reducing away a  excavation  the from  deeper of  stress the  zone  of the  a  working  areas  broken  rock.  openings—the  rock  a  high  stress  The purpose  or  and  massive  working face  concentration,  of  differences  in  of rockbursting will decrease greatly.  extend  before  field  at  and  can  be  preconditioning  is  a  Therefore  if  lowered,  of destressing  moving  can  [18],  the is  to  normal fracturing  cushioning  Destressing  stress build-up, or at the stress concentration zone [19]  rock.  over the least  gradients  the be  to  the  trouble  effects used  prevent  of  either high  to reduce the high stress  Basic Concepts of Rockbursting and its Control / 20 or to shift it further into the rock mass.  The destressing mass in the these and  area to be destressed,  holes—the blasting  method  is  infusion  method  them—the  to  fracturing  process usually consists of drilling deep holes into the rock  "soften"  the  rock  [19,20],  blasting the  mass,  or loading these  method  rock  thus  then either injecting high pressure water into  [18,19].  mass  within  decreasing  the  The  the  holes  basic  area  to  with  explosives  principle be  stress gradients  of  destressed  and therefore  this by its  capacity of storing energy and reducing the potential for rockbursting.  It  should  always  such that  the  rock  otherwise,  unexpected  be  mass  kept  will  results  in mind  not  lose its  and damage  that  the  extent of rock  abilit3' to will  sustain  occur due  to  the  fracturing is external load,  over-deformation of  the rock mass.  2.6.3. Rock Support Suitable rock supports which can handle the results of rockbursting are important in reducing the damage to mining openings. Because rockbursting generates strong shock  waves,  rock surface, the  source.  sametime, If  the  tolerate  as  the  compressive  wave  a reflection tensile wave  reaches  in tension  rockbursting is a rapid action and the  the  supporting system  can  rapid deformation, the  interface  between  air and  is induced which propagates backwards to  As such, the rock mass will fail  rock  the  reduce damage  the  at the  surface.  deformation rate is effect  of  the  tensile  A t the  very high. wave  and  can be reduced to minimum. Usually  the rapid yielding hydraulic prop is used in stopes and the grouted steel cable is  Basic Concepts of Rockbursting and its Control / 21 used in tunnels  2.7.  [58].  SUMMARY  Despite  extensive  rockbursting  is  research  not  yet  over  many  years,  properly understood  and  the  actual  therefore  the  mechanism conditions  of  which  give rise to violent failure are not clear. The latest theory of rockbursting is the energy has  and stiffness  been  massive  approach proposed  reported.  This  approach  in  1965  [13]  to  explain  seems  but since  then  little work  rockbursting  well  in  a  rock, but it has difficulties in:  explaining the rockbursts occurring along natural faults, determining the  stiffnesses  around an underground opening  and the  loading  system of a mine, correlating the rockburst with the acoustic activity that preceds the bursting. Therefore this  theory needs improving or another hypothesis  should be  postulated  to explain rockbursting.  While  the  use  of  microseismic  monitoring has  improved the  technique  of  locating potential rockburst sites, the reliability of predicting the precise time of a rockburst pre-failure  is  still  acoustic  low.  Sometimes  emission,  of predicting rockbursts is after  many  years  failures  but often faced  research.  this  worldwide  This  makes  occur  with  pattern is and little it  doubtful  a  recognizable  absent [21]. progress that  as  has  pattern  of  The difficulty been reported  used  at  present  microseismic activity or acoustic emission can serve as a reliable precursive signal for violent rock failure.  Basic Concepts of Rockbursting and its Control / 22 In serious  summary,  problem as  rockbursting has the  had  a  mining depth continues  long  history  to increase.  and  has  become  It is usually  a  related  to rock properties, mining conditions, geological environment and rapidity of stress change.  While  progress  has  been  achieved  problem is still far from being solved.  as  a  result  of  past  research,  the  CHAPTER  3.1. G E N E R A L To  3. F A I L U R E  OF A MASSIVE  ROCK  CONCEPTS  study the mechanism of violent rock failure, it is important to understand the  failure  of  a  mssive  failure  by  its  rock.  suddenness  Violent and  the  rock  failure  severity  of  is  different  damage.  In  from mining  normal  rock  excavations,  rock usually fails in the form of spalling, breaking, roof sag, collapse of a pillar, or  closure of an opening, etc.  term can  action be  and usually  have  controlled and the  installing proper supports failure,  These normal failures have  a relatively slow long  some visual evidence  to  damages at  the  they  cause  right time.  prior  final  failure.  They  can be reduced to minimum by  However,  rockbursting Violent rock  as described before is an instant action, accompanied by the release of a  tremendous  amount  of  strain  energy.  There  is  usually  no  visual  evidence  in  advance.  It is violence.  therefore  The  rock  important to  mass  is  an  understand the anistropic,  conditions  which give  nonhomogeneous  geological  rise  material.  Because it contains many weaknesses, such as joints, beddings, foliations, etc., mechanical properties are not solely these  weaknesses.  especially  on  strength.  The  a  Most short  kinds term  development  of  of base,  to  its  dependent on the material itself but also on rocks  are  for  fractures  they in  characterized have  intact  little  rock  is  by  brittle  plasticity an  and  tensile  important  process  that should be taken into account when considering violent rock failure.  23  behavior,  Failure of a Massive Rock / 3.2.  FRACTURING  PROCESS  The  development  rock  the  generally  of  accepted  Bieniawaski  (1967)  experimental  theory  [22]  results,  fractures  and  has  been  studied  by  many  of brittle fracture of rock is is  Bieniawaski  rock in multiaxial compression,  used  in  this  postulated  figure  research.  the  five  researchers,  and  the  one  developed  From  his  research  stages of brittle  24  by and  fracture of  3.1:  1. closing cracks, O-I 2. linear elastic deformation,  I-II  3. stable fracture propagation, II-III 4. unstable  fracture propagation, III-IV  5. forking and coalescence of cracks, IV-V.  The  behavior of rock fracturing is mainly described by the  curve of linear  stress versus linear axial strain. These stages of brittle fracture of rock apply and  for  tension.  In  tension,  processes of stable and  duration plane  due  to  compared  the  fact  with  in  however, unstable  that,  in  compression  crack  fracture  closure  will,  propagation  of course, will be  tension,  a  crack  will  where  a  crack  does  be  absent  of very  propagate not  generally  in  small  its  propagate  own in  its  own plane but in the weak direction.  By of  fracture  this theory, development.  load, the pre-existing corresponding deformation  before  to  a  takes place,  compressive  stress  the is  whole process is induced  in the  a  matter  rock under  small cracks or Griffith cracks close first up to stress level  point  under  As  failure  I  further  in  figure loading.  3.1.  Then  After  the  stress  rock has  shows reached  a  perfect  point  II  elastic where  Failure of a Massive Rock /  "3- O  3reQ  Maximum strain  Fig.3.1 Mechanism of brittle fracture of rock in multiaxial compression Bieniawaski, [22])  (from  25  Failure of a Massive Rock / 26 fracture initiation begins or the preexisting cracks begin to extend, propagates  forward  in the  material. The fracture propagation continues  strength failure at point IV. However, between process  is  somewhat  first stage, between fracturing energy  can be  released  stopped by crack  development  and the  the  stage,  second  becomes  different  the  load  decreases  value  point III,  the  other  the  crack  constant,  hand,  extension  between  propagation after  be  divided into  the  length.  stopping loading because  fracturing  self-maintained,  maitaining  and can  is  two  stages.  not  at this  sufficient  to  III-IV,  means  fracture  the the  with the  crack velocit.y which quickly energy  elastic  energy  released  Therefore  at  the  energy  is  second  a fracture will continue to extend.  required  lowered  from  which means  the  stage,  at  crack  stage,  the  is  to  during  crack  level. On  increases  the  by  the terminal  stress  extension  and  stopped  maintain  reaches  if  fracture  unstable  be  some  even  elastic  However  cannot  Because  required  the  propagation  constant.  this  During  stress.  fracturing  the  fracturing  maintain the  directly controlled by  points  which  is  until  points II and IV, the  points 11-111, fracture propagation is stable,  by  microfracturing  load  Any increase of load will  is  with held  accelerate  the fracture propagation.  Obviously, during unstable fracture propagation, the elastic from crack extension create  new  crack  energy  released  can not be completely consumed in maintaining fracturing to  surfaces.  This  released  energy  can  also  be  possibly  converted  into several other forms of energy losses in addition to the crack surface energy: kinetic energy, plastic energy, energy  dissipated  on  the  breakdown  of  atomic  bonds  at  the  tips  of  Failure of a Massive Rock / 27 extending cracks. energy  changes  due  to  mining such  as  caused by  artificial  rock breaking,  heat removed due to ventilation, etc.  From Bieniawaski's study [22], the  present  movement also  discussion,  of the  faces  to  approach  found  approaches its dissipate hence  the  its  except  the  of the a  all other energy losses can be neglected in  kinetic  extending  constant  energy, crack.  value  which  is  associated  However, this  once  the  with  kinetic energy  crack  velocity  the is  quickly  terminal velocit3' during unstable fracture propagation. In order to additional energy,  surface  energy  by  the  crack  forking  tends  in  the  to increase weak  its  direction  surface to  form  area and additional  cracks.  The fracture material, forking  onset of forking represents  propagation. This  point IV in figure 3.1. will  lead  to  macrofractures.  These  fractured  to  The  zone,  form  Fracturing is process  coincides  of  many  macrofractures a  new  with  the  failure  strength  Once this transition has taken place,  coalescence  proof of this suggestion  3.3. D E T E C T I O N  the  transition  a transition within the process of unstable  will  surface  microfractures, eventually  on which the  together  final  failure  the  successive  consequently  join  of  forming  within takes  the  place.  will be provided later.  OF FRACTURING  an important characteristic mechanism within of a rock mass. But of  fracturing  is  not  visible  and  most  of  the  acoustic  accompanying these microfractures are not audible to human ears because  emission of the  Failure of a Massive Rock / 28 tiny  amount of energy released or their high frequencies  noted  previously,  part  of  the  elastic  energy  [15,23,24]. However, as  released  from  crack  extension *  accompanying associated  the  microfracturing  with the  is  converted  movement of the  crack  into  surfaces.  kinetic  energy  which  This portion of the  is  energy  will propagate spherically outwards through the movement or vibration of particles of  rock  weak,  until  it  is  completely  it can be detected  dissipated.  Although  the  vibration  by suitable instrumentation and after  is  extremely  amplification can  be converted into audible sound.  By  detecting  development  of  the  material. In fact, structural  the  fracture  frequencies  prior  whereas  lower frequencies.  acoustic  process  energy,  it  hence  the  and  is  possible  potential  to  study  failure  of  the the  acoustic emission testing has been widely used in material and  engineering.  micro-fractures  released  Results to  large  from  failure events  previous  result are  in  studies small  preceded  In laboratory tests, acoustic  by  [23,24]  events,  have which  showed have  macro-fractures,  that  higher  which  have  activity generally increases sharply  prior to the failure of a rock specimen.  One  question  emission  from  rock  generated  from  a  emerges:  what  specimens  in  rockburst ' or a  is  the  relationship  laboratory  natural  tests  between  and  the  the  seismic  earthquake? Theoretically, the  emission should be similar for these two cases because  acoustic events acoustic  the fracture process  itself  should be similar if the materials and loading conditions are the same. The only difference [25]  will  compared  be his  a matter  of scale.  laboratory  results  Many of  seismologists  microfracturing  agree  with  behavior  of  this.  Mogi  rock  with  Failure of a Massive Rock / 29 earthquakes similar  and concluded  to  observed  fracture  experiments  observed  that  sequences  analogous  that  to  occurrence  be  earthquakes, a  scale  behavior  of microfractures  and  suggested  model  he  of crustal  the  specimen  failure  may  or  in  can their  Therefore,  rockbursts and the microseismic  and by  be  regarded  damage. a  as  From  similar the  correspond  rockburst are  comparison,  above observations  to  the  of  extremely  acoustic  earthquakes  FAILURE discussed  in a manner  either  of  a  similar  in  terms  be  used  emission  can  in  of to  actually  based  before,  monitor  failure can be considered similar emissions should  to  the the  AND T H E SHEARING  follow  PROCESS  failure process of a rock mass is failure  strength.  However,  the  a matter  previous  of fracture  concentrated  on the fracture itself. On a macro scale, fracturing seems to  randomly in the rock mass at first. As loading continues,  stress,  seismic  upon this principle. Then  up  shear  a  for these two cases.  DEVELOPMENT  the  their  seismologist,  development  in  shock.  should apply for rockbursting as well. The  monitoring of rockbursts is  similar patterns  develop  also  foreshock  main  view  to that prior to a rockburst. In other words, the acoustic  As  Mogi  similar to to  very  laboratory  deformation.  failure is  point  the microfracturing process prior to the specimen  3.4.  that  is  earthquakes.  earthquake  emission.  might  of  statistical  also found that the microfractures radiate elastic energy  Rockbursting  natural  behavior  the  the buildup of microfracturing before  and  Scholz [23]  that  direction  which  usually  coincides  with  discussion  was  initiate  these fractures tend to  the  planes  of  gradually forming a zone of fracturing. This zone usually  maximum has  the  Failure of a Massive Rock / 30 highest the  stress concentration and is where  strength  additional develop rock  point,  energy  as  cracks  is  start  available  a result of the  mass  to  loading will  sustain  speed  fork  from  failure occurs. As loading reaches  in  crack  the  weak  extension.  load  decreases  failure process.  This  direction  The  available internal energy.  external  up the  to  final  forking  Because  after  the  forking  when  enough  process  will  the  ability of the  strength  point, further  process  quickly joins  the  existing fractures, forming a macro-fracture surface within the fracturing zone.  From  this  moment,  the  failure is  similar to a shearing process.  In other  words, the shear stress and shear strength control the stability. At this moment, if  the  external  externa]  load  load is  is  removed,  lowered  and  the  failure  remains  may  in balance  not  develop  with the  further.  If  the  supporting ability of  the rock mass, or if the shear stress and shear strength are in equilibrium, the failure or  will develop gradually. If the  increases  further, the  external load remains at the  failure will  develop  quickly  and  even  strength level  violently  if  the  resultant shear stress is too high.  Take the failure of a rock specimen in compression as known  that  compressive machine. from  the  adjusted  the test  same may  fail  rock  which  gradually  failed when  violently  tested  on  during a  the  servo-controlled machine receives  deformation of  the  rock  to  prevent  excessive  deformation.  and  the  When  a  conventional  servo-controlled  This is because  specimen  an example. It is  testing  a feed-back signal  load  on  the  the  failure  specimen  is  strength  is  approached, or when a failure surface is initiated, the failure process becomes a shearing process.  A t this stage, the  supporting ability of the  specimen decreases  Failure of a Massive Rock / 31 rapidly  to  the  shear  viewed  as  the  residual  load  is  reduced  strength  on  strength  the  failure  of that  quickly enough  to  rock  meet the  surface. at  the  This  post  decreasing  ability  failure  speed  is  usually  stage.  of the  If  supporting  ability of the rock specimen,  the failure occurs gradually and non-violently  complete  can  stress-strain  conventional  testing  prevent the  specimen  curve machine  usually  has  no  be  obtained.  ability  to  failure. Therefore, after  the  On the  lower  the  and a  hand,  the  can  not  and  point, the  decrease of  supporting ability of the specimen together with the release onto the  specimen of  the  strain  energy  stored  in  the  extremely  rapidly. Usually violence  of  energy.  strain  A  testing  strength  other  load  the  machine  make  the  is observed because of the  typical  example  of  failure  of  this  will  be  failure  happen  high speed  given  in  the  release  chapter  on  sudden loading.  The  formation  demonstrated  by  experimental  study  emission. to  the  the  experiments. and  A  surface  few  traced  the  throughout  the  of  the  specimen.  3.2.  fracture  However,  events  This means  the  ago,  fracturing zone  Scholz  process  [23] by  can  conducted  locating  be an  acoustic  some stress level which may correspond  unstable  on a plane which corresponds closely in figure  years  fracturing  He observed that events below beginning  within  propagation, above  with the  that  stress  observed  that the fracturing process  appear  to  level  be  group  failure surface,  will eventually  scattered tightly such  as  lead to  the  occurring  in  formation of a failure surface.  This  failure  surface  underground structures  can  also  be  observed  from  damage  and rock failures. Underground investigations  of rockbursts  Failure of a Massive Rock /  32  Fig. 3.2 Front, top and side views of the central section of the sample showing locations of events occurring in the dynamic cracking region (from Scholz, [23])  Failure of a Massive Rock / indicate  that  failure is  usually  particular  example  the  plane has  a conical shape  compression. Figure 3.3 of  failure,  failure of  where  the  occurred along  maximum  measurements  place  along  failure  planes  case of the failure of a rock pillar,  macro-failure  surface  this  which made  fractures  or  surfaces.  where  the  A  failure  and is very similar to the failure of rock specimen in  shows an unconfined rock specimen  compression. of  takes  33  surface, In  field  induced  has  study  in the  of  stope  been an  well  acute  in an advanced stage developed. angle  rockbursts,  to  The  the  final  direction  observations  roof during excavation  indicate  that fractures dipping outwards from the face are likely to cause burst [2].  Fig.3.3 Unconfined Charcoal Gray granite I in advanced stage of failure Wawersik et at, [ 8 ])  and  (after  Failure of a Massive Rock / 34 3.5. D E T E R M I N A T I O N As  can be  the  formation of a  by  seen  from  shearing. This  OF A FAILURE previous discussion,  failure surface surface  will be a fracturing  PLANE the  fracture development  on which the  may or may  not be  failure is  will  eventually  lead to  completed  a plane. For an intact rock, it  surface, which is not necessarily  the plane where maximum  shear stress exists and can be determined as following.  In  underground mines,  dimensional  compressive  the  stress  mining  field.  structures  Typically  there  are are  usually one  in  a  vertical  three  and  two  horizontal compressive stresses, together with three shear stresses, with a total of six  independent  possible  to  original  stress  o i>o 2^0 . 3  components.  define  a  field.  They  However,  stress  field  The three are  the  only  from with  elasticity three  theory  components  orthognal components  normal  stresses  to  are  the  the  [26], to  it  is  represent  principal  three  always  principal  the  stresses, planes  respectively, on which there is no shear stress.  For  a structure of isotropic and homogeneous  material, its strength is  the  same in all directions. Its stability can then be determined by shear stress r on the bigger half circle defined by o y and a  on Mohr's diagram. Thus the stress  3  field has treated  as  only two in two  normal  components  dimensions,  figure  a y and a 3.4.  If the  shear strength, is above the circle, it is stable.  3  correspondingly, and can be  line  the  with a 1.  major principal  stress  a ^,  or the  the  Otherwise failure takes place. In  the latter case, the normal to the failure plane makes with  OP, which represents  an angle  failure plane makes  of a = 4 5 ° + #/2 an angle of  0  Failure of a Massive Rock /  35  Fig.3.4 Schematic showing shear failure plane. Because  shear  stresses  are  conjugate  and  /3 + a = 9 0 ° ,  the  failure  planes  make angles of 0 with  the  plane  of  =  ±(45°  -  0/2)  (3.1)  major principal  stress.  This  rock  usually  makes  specimens  explains an  the  angle  phenomenon of  about  that  4 5 ° with  the  failure  the  axial  load.  However, contains would can  more  possibly  in  nature,  perfect  or less joints take  place  intact  material  or weaknesses  along  these  of  is  lower  weaknesses.  rare.  Rock  strength.  usually  Therefore  Obviously, the  be either a pre-existing weakness or a fractured surface,  mass  failure  plane  depending upon the  orientation of the weakness and its strength with respect to the rock mass. often, rock failure takes place along some weakness.  failure  More  Failure of a Massive Rock / 36 3.6.  SUMMARY  The results from analysis in this chapter can be summarized as follows: 1.  A  rock  mass  is  a  kind  of  anistropic, nonhomogeneous  material, which  is  brittle, especially on a short term base. 2.  As stress  reaches  some  level,  the  process  of rock  failure is  a  matter  of  fracture development until the strength point is reached. The development of fractures can be divided into two  stages: stable  fracture propagation, which  can be stopped by stopping loading and unstable fracture propagation, which is self-maintained and cannot be stopped by stopping loading only. 3.  These  micro-fractures  when  load is  initiate  randomly  throughout  the  body  low and concentrate in a zone which has  the  of  the  highest  rock stress  as load increases. 4.  As unstable from  fracture propagation is  fracture  direction.  development  The forking  makes  process  approached, the  the  will  existing  eventually  extra  fractures lead  to  energy  fork the  available  in the  weak  formation of  a  macro-fracture surface on which the final failure takes place. 5.  After  the  similar  to  formation shear,  so  of  the  any  macro-fracture surface,  sudden  increase  of  shear  the  failure  force  or  process any  is  sudden  decrease of shear resistance can cause violent failure. 6.  Accompanying the  fracture development,  acoustic  emission  occurs,  which  is  characterized by higher frequency for smaller events and by lower frequency for larger events. 7.  Results from fracture  studies by . Mogi and Scholts have  development  and the  associated  in laboratory tests and in the field.  acoustic  shown  that the process of  emission  are  similar both  CHAPTER  4. F A I L U R E  4.1.  GENERAL  The  failure behavior on surfaces  studying  violent  failure  along  a  failure  geological  is  discussed  will be  because  previous intact rock in the  BY A PROCESS  such  a process  of shearing.  previously,  the  can  vicinity of the face  weakness,  obviously  fracture  SHEARING  an important aspect to be  rockbursting  as  OF  a  originate  as  (Spottiswood  fault.  For  For the  development  the  will  shear  1984) case  case of  analyzed in failure  of  and can occur of  a  fault,  a massive  eventually  lead  the  rock,  as  to  the  formation of the final failure surface.  Shear failure has been considered by seismologists to be the mechanism of shallow earthquakes be  the  result  along geological  of shear  failure on a  large amount of energy. in  terms  mechanism possible  of is  seismic  Because  to  fault  of the  emissions  assumed  faults. This kind of earthquake is thought to  and  apply  for  to  a sudden  slip can release  similarity of rockbursts and  the  manner  rockbursts  in  as  which  well.  they  Therefore,  a  earthquakes occur, it  this  may  be  to describe rockbursts occurring on a fault as well as in a massive rock  mass by shear failure and consequently rise  because  violence.  As  such  it  is  to derive the  worthwhile  to  study  conditions the  which may give  characteristics  of rock  during shearing.  Shearing respect  usually  implies  that  two  contacting  to each other under a pair of forces  universal phenomenon  in earth engineering,  37  surfaces  tend  to  move  parallel to these surfaces.  such as  It  with is  a  landsliding, slope sliding and  Failure by a Process of Shearing / wedge failure of a slope. on rock surfaces,  4.2.  study  shearing  resistance.  process,  the  friction  arise  on  between  friction  on  the  failure,  first  the  friction  all  scales:  opposing  The simplest  from  surfaces  model for study  with  an approximate plane  force  P  and pulled by  will never move motion:  a  surface shear  until F reaches  F = MX,  the  must be some resistance This resistance  microscopic  of  body  minute  of contact F,  scale  the  4.1.  some critical value.  should  in  which  cracks  move  once  between the contact  one  are pressed  figure  the  major  to  friction  is  macroscopic  bodies  together by a normal  Obviously, the  upper body  However, by Newton's  F>0.  surfaces  in which two  This  means  that  law there  in the direction opposite  to  is called frictional force and is denoted by f here.  of the  viscosity  of rock, which is important in the  research,  because  from  occurred  at  time  and widely  shear  upon many factors,  roughness  relation:  is  [26].  material,  simplest  surfaces  Griffith  of friction is  force  This frictional force depends  the  contacting  studj' of friction is of greatest importance. The effects  scale of friction on joint or fault surfaces  F.  shear  shear strength and slip behavior should be examined.  Therefore, the  postulated  of  the  T H E L A W OF FRICTION  During  of  In order to  38  field of  surface,  observations,  rapid  stress  used form for the  such as properties of the  normal stress,  etc.  The time  effect of  long term period, is ignored in this it  was  change,  seen such  that as  rockbursts  during  maximum frictional force is  usually  blasting.  The  the Coulomb  Failure by a Process of Shearing /  39  P  F  M  x  \ \ \ \ \ \ f N  Fig.4.1 Simple model for shearing  f where  =  C  no  +  (4.1)  C is the cohesion, material property a is the normal stress ix is the frictional coefficient,  Obviously,  when  of  and  action  F is  constant.  less than this  reaction,  the  maximum frictional force, bj' Newton's  frictional  force  will  be  equal  to  F  acting  law  in  the  opposite direction.  It movement  has  been  begins,  the  motion. The simplest equation  (4.1)  observed  with  M' = M'(X).  many  frictional force way  a  value of fi' is expected i.e.  in  laboratory  experiments  that  drops and crucially controls  once  the  nature  to consider this effect is to replace the constant  lower  value  p.'—the dynamic  coefficient  to be less than u and to vary with the  of  shear  friction.  of  n in The  slip velocity X ,  Failure by a Process of Shearing / Unfortunately,  this  dynamic  relationship with the slip velocit3  based  on  the  figure 4.2  a slip-velocity dependent  laboratory data  u'  H =  little  of Scholz  coefficient  understood  and  its  of friction is derived here  and Engelder (1976)  are the original data. Based on the  formula is postulated  is  is not well known to date. In order to consider  r  this dynamic effect,  coefficient  40  [27].  The dots in  appearance of these, an empirical  as  a +  b/[7  +  log(X + 1 0 - ) ]  (4.2)  6  where a and b are constants to be determined.  These Constants column  data  a  were  and  (u,#l),  b  are  with  read  off  obtained  static  b3' by  digitizer nonlinear  coefficient  of  and  are  listed  regression  friction  in  analysis  / j = 0.805. g  table for  For  4.1.  data  in  comparison,  another formula M was  analyzed  =  a +  with  the  b/[6  +  log(X+10- )] 5  same  data.  r = 0.9157 and standard deviation (4.2)  is  chosen,  for its  the. curve in figure  lower  Sd  R  ^ = 0.55  ^  came  up  (X±0.148,  standard deviation,  as  with  correlation  ju±0.0105). the  coefficient  Finally,  equation  best fit represented  by  4.2.  Through linear scaling in figure case of  It  were estimated  4.2,  and listed  another  group of data for a typical  in column (u,#2) of table  4.1.  The  constants a and b were also obtained.  It means  the  can  be  seen  that  formula represents  the the  correlation  coefficient  laboratory data  very  r  is well.  above  0.9,  However,  which because  Failure by a Process of Shearing / 41  Fig.4.2 Velocity dependent friction. A , B and C refer to different experimental runs (data from Scholz et al, this  formula  is  derived based  on  limited data  with  value of p, or the population correlation coefficient r,  the  sample correlation coefficient.  on significance  level  given null hypothesis:  a = 0.05.  [23]) sampling points  the  is not necessarily so high as  In order to verify this  According to the  n=16,  testing  formula,  r is tested  theory in statistics,  for a  Failure by a Process of Shearing / 42  Table 4.1 regression analysis of velocity-dependent coefficient  #  logX  1 .2 3 4 5 6 7 8 9 10 11 12 13 14 15 16  -4.9359 1.159 3.1624 -4.5 5.2324 -4.2813 -3.7813 16.5463 6.0423 -4.2188 -3.8397 14.4644 -3.9359 11.5904 -4.0064 9.8557 -3.5513 28.0996 -2.9038 124.7958 -2.8077 155.7041 -3.2308 58.7760 -1.7756 1676.486 -2.0641 862.7799 -1.8718 '' 1343.383 -2.0820 827.9422  XXIO"  5  7 + log(X+10" ) 6  l/[7 + log(X+10- )] 6  2.100 2.5135 2.7269 3.2213 2.7883 3.1633 3.6783 2.9980 3.4502 4.0966 4.1926 3.7699 5.2244 4.9359 5.1282 4.9181  .47618 .39785 .36671 .31043 .35864 .31613 .32596 .33356 .28984 .24411 .23852 .26526 .19141 .20259 .19500 .20333  nonlinear regression  U = 0.6859+ 0.1192/[7 + log(X +10" )] with M = M(0) = 0.805  for  correlation coefficient r = 0.9214 standard deviation Sd X + 0.08168, n-1  for  of friction  u,#.l  u,#2  .7445 .7312 .7347 .7300 .7279 .7226 .7224 .7212 .7165 .7209 .7159 .7087 .7118 .7088 .7089 .7100  .485 .4712 .4753 .4700 .4682 .4623 .4623 .4612 .4565 .4609 .4556 .4488 .4517 .4476 .4477 .4476  6  S  u,#l  M±0.01057  M = 0.4245+ 0.1235/[7 + log(X+10" )] with M = M(0) = 0.55  u,#2  6  G  correlation coefficient r = 0.92 standard deviation Sd • X + 0.07797, M + 0.01050 n-1  from |r|>r  if  |r|^r ,  the  table  a  H of  is accepted. Otherwise, H  0  critical  . Therefore, H  0  correlation  is rejected.  coefficient  This  zero. If we wish to set a confidence should be used, which requires n&50. reasonable representation of these data.  0  means  is rejected. In our case, n=16, [28], p*0  r  = a  0.4973.  Obviously,  and appears greater  interval on p, the Fisher's testing  than  method  Therefore the empirical formula (4.2) is a  Failure by a Process of Shearing / 43 Equation  (4.2)  variation of static other possible ju  4.3.  SHEAR  The  shear  coefficient  as  of friction,  the  law  constants  of friction.  a's  and b's  by linear interpolation and listed in table  g  strength  is  the  maximum shear  To consider  the  are estimated  for  4.2.  stress  required  It varies with rock type, surface roughness,  of temperature,  surfaces  later  STRENGTH  rock surface. conditions  will be used  vary  from  the  smoothest planar cleavage  pore  pressure,  roughest  rock  loading rate, joints  formed  For rock,  intrusive  surface found in slates. The simplest  used shear failure criterion is the  cause  slip  on  Coulomb criterion where the  the  rocks  shear to  strength  normal  of  pressure,  envelope  rock this  surface  strength  is  not  a  decreases  straight  to  zero.  line  but  the  and most widely strength  envelope  is a straight line. However, it has been commonly accepted that the envelope shear  a  confining pressure and  etc. in  to  curvilinear.  A t high normal  of  A t low  pressure,  curves downwards. It is not impossible but difficult and unnecessary  this to  describe this envelope with an exact formula. From laboratory results, such as in figure 4.3  [29], it is found that this envelope can be represented very well by a  multilinear line. The common practice is to use a bilinear envelope with the first part  for  low  normal  pressure  passing  through the  origin of the  r - a coordinate  system.  At  low  normal  pressure,  many  authors  [29]  suggested  the  following  equation for the peak shear strength for non-planar shear surface: T where  =  atg(<t>  +  i)  (4.3)  <j> is the basic angle of friction, i is the dilation angle, or the effective  roughness.  Failure by a Process of Shearing / 44  7  T  1  :  1  r  6  o o  =4  °2  I 0  Fig.4.3 Friction strength of sawcut and fault surfaces of variety of rock types under different conditions of temperature(to 400 degree Celsius), rate and amount of water (after Stesky, £ 2 9 J )  Table 4.2 # M  s  a b  Constants for empirical formula of slip-velocity dependent friction  1  2  3  4  5  6  7  .35  .505  .55  .65  .75  .805  .95  .2235 .1265  .381 .1241  .4245 .1235  .528 .1218  .63 .12  .6859 .1192  .8333 .1166  Many experimental indicate that for  i  data  reported in  most rocks have  value  Schneider(1976)  is  rather  literature  <j> between  scarce,  but  can  for  2 5 °3~ 5° be  [31] gave an empirical formula as  blasted [30].  and  sawcut  surfaces  Unfortunately, the data  determined  by  shearing  test.  Failure by a Process of Shearing / 45  with  k,  R  i  =  i  i  =  R log(a la)  being  0  exp(-ka), or  the  empirical  constants.  Barton(1973)  extracted from literature gave some values of i between  [30]  from  information  6.2°—30.1°.  At high normal pressure, since most of the irregularities would be sheared off  and the  amount  of dilation would decrease,  the  term of frictional  resistance  would dominate the shearing characteristics. In this case, the Coulomb relationship  = C + atg0  r  (4.4)  would be valid.  Usually, defined of  as  the  the  irregularities  considered But  crushing strength  of the  of rock surfaces,  there  Vesic  at and  high and low  asperities. is  no  a  [33]  Clough  [34]  form  found  considerably this  to  be  below  due for  to it.  the  is  variety  Barton  [32]  at the brittle-ductile transition.  found no dilation during sliding on  normal pressure  normal pressures  However,  general  this critical normal pressure to be that  Byerlee(1968)  granite  critical value between the  this  (5~10)X10  7  a sawcut  transition Pa  for  for  surface that  medium  to  in  rock. fine  grained sands.  In by  contrast,  Barton [30]  { where  another  empirical envelope  for  the  shear  strength  was  given  for rough-undulating joints: r  =  atg70°,  if a / a > 100  r  =  atg[JRC-log(JCS/a) +  (4.5)  c  <j>], if 100>a / a a l  7 0 ° is used to replace (0 + i) in equation (4.3),  Failure by a Process of Shearing / 46 0" is the unconfined compressive strength, c  JCS  is the  effective  joint wall compressive strength. J C S = a  c  if the joint  is unweathered. JRC  is the joint roughness coefficient,  rough-undulating joints,  with a value of 20,  smooth-undulating joints  and  10 and 5 for  smooth-nearly  planar  joints  are  purely  respective^. For  a basic angle of friction 0 0  +  i =  rWo  It  should  empirical  and  curvilinear criterion. and  =  is  thing  here  which  much safer  A bilinear envelope  (4.4)  0  we have  and consequently,  emphasized  only  envelope  28.5 ~31.5°,  50 to 200 are suggested.  be  the  64°~76°  =  is  that is  all  certain  the is  above  that  for  values rock  and more realistic than the  therefore  used  and is  surfaces,  simple  the  Coulomb  given by equations  (4.3)  will be used, because most data available from past shear tests given  is (C, 0) parameters, ie.  T =  atg(i  ^T = C + where  the  constant  strength envelope  B  +  <p), if  rWasB  atg0, if B > a / a > l  (4.6)  c  should be  determined such  that  is maintained at the point o = oJB.  the  continuation of  If i is known, then  the from  equation (4.6), B is given by  B  = [tg(i + <(>) -  tg0]a /C c  (4.7)  or if B is given, then  1 = tg- [C-B/a !  If  + tg0]  data from shear tests were given as  -  0  (4.8)  (C, u ) parameters, equations (4.6)-(4.8)  Failure by a Process of Shearing / 47 become  {  r  op. ', if a / a > B s ' c  T  C  (4.6a)  on , if B > a  +  /a>l  B  (4.7a) B-C/a  Then  the  When  shear  slip  strength  begins,  for  +  c  a  n  (4.8a)  s  of rock joints  will  given  stress  normal  appear a,  as  the  shown shear  in  figure  strength  4.4a).  will  va^  along a vertical line within the shadowed area of figure 4.4b).  4.4. As  EFFECTS OF ENVIRONMENT mentioned before, the behavior of the shearing process and the shear strength  also  depend on conditions of confining pressure, temperature, pore pressure,  etc.  A brief review and discussion related to mining situations is given below.  4.4.1. N o r m a l The  normal  bilinear  Pressure  pressure  relation  higher normal  is  with  obviously  shear  dominant during  strength  pressure, the coefficient  as  discussed  of friction  shearing in  the  process.  It  previous  has  section.  a At  decreases more or less due to  the crushing of asperities on the shear surface.  In high  laboratory studies,  normal  pressure,  although the  typical  example  process  is considered to be the  on  natural  of  it is commonly found that stick-slip is  faults  testing  [11].  results  This  sliding is is  shown  stable in  at low  figure  4.5  mechanism of generation  implies  that  the  normal  normal [29].  dominant at pressure. A  This  of shallow  pressure  is  stick-slip  earthquakes a  significant  Failure by a Process of Shearing / 48  fig. 4.4 a) Postulated bilinear shear strength; b) the effect of slip velocity  Failure by a Process of Shearing / 49 factor in rockbursting as well.  It envelope  should  be  developed  compressive  noted  are  strength  only  of the  that  all  good rock  for  the  normal  the  Under  friction  strength  these extreme  arguments  pressures  in question. A t pressure  for certain rocks) or at temperature above and  above  becomes  conditions, the  less  400°C,  dependent  up  and to  above  equation (4.6) on  the  friction strength is  normal  supposed  the the  strength unconfined  10 Kbars  (less  no longer holds pressure  [35].  to be equal to  Fig.4.5 Sliding characteristics of stick-slip (curve A) and stable sliding (curve C) on sawcut surfaces (after Christensen et al, [57])  Failure by a Process of Shearing / the compressive  50  strength of intact rock.  4.4.2. Temperature The  role  of  temperature  seems  friction strength increases water  [36]  shear  strength  In  general,  shown  in  or  due  to  to  be  complicated.  Under  some  conditions,  with temperature either due to the removal of absorbed  the  formation  of glass  [37].  Under  other  conditions,  either is unchanged or decreases with increasing temperature  the  strength  figure  4.6.  the  envelope The  is  friction  valid  at  temperatures  behavior  seems  also  up to  to  [35].  400 °C  change  temperature. The stick-slip phenomenon is enhanced by low temperature  this  as with  [35].  m 6  XL  1>' .60°  r =0.7+ 0.6  to  CO  CT <-  V  *o 700'  •  < X CO  2-  O • a A  25°C too 200 300 400 • 500 O 600 • 700  JO  2  4 6 NORMAL STRESS, c\ KBAR  a  8  10  Fig.4.6 The effect of temperature on the friction strength of dry gabbro Stesky, [29])  (after  Failure by a Process of Shearing / 51 4.4.3. Pore Pressure The  presence  of water  in a rock joint leads  to several mechanical and chemical  effects. The most important of them probably is the reduction of effective  normal  pressure. This certainly leads to the reduction of shear strength.  The  effect  mineralogy of the coefficient the  of  rock  of massive  presence  of  water  pressure  and the  on  surface  crystal structures  water.  In  the  shear  roughness. such  other  strength  cases,  as  In some  the  frictional  and decreases on the shear change  tensile surface  depend  cases, the  upon  frictional  increases  coefficient  when wet.  in  for larger  However, these  increases.  In addition to the effect of reduction of effective strength increases  to  quartz and calcite  lattice structures such as mica and chlorite decreases effects diminish as the surface roughness  seems  normal stress,  or remains unchanged for smooth, polished surfaces  the  shear  when  wet,  for non-planar rough surfaces due to the adverse effect of moisture and compressive tends  the effective  to  strength of rock  enhance  normal  the  stick-slip  [30]. stress  The presence drop  [38],  stress at which the transition between  of water on but  does  not  stick-slip and  stable sliding takes place.  4.4.4. T i m e Dependency The effect of time includes two aspects:  the time of loading to failure or loading  rate and the time duration in which stationary contact remains.  It  was  found that  there  is  some  strength  reduction in both tension  and  Failure by a Process of Shearing / 52 compression,  when comparing the  high "instantaneous  long term strength (2 — 4 weeks).  loading" strength with  the  This is thought to be from the creep effect. By  extending this result, it is probable that normal laboratory shear tests might give an over-estimate of strength [30].  Another those  aspect  experiments  constant  for some  of  by  time  dependency  Dieterich(1978)  time  and then  the  is  [39],  from  the  shear  the  stationary  stresses  stress  r  T was  and  contact.  a  were  In held  increased rapidly  to  the critical level required to cause slip. It was found that the static coefficient of n  friction  s  However,  increases  the  with  magnitude  of  the  logarithm  of  the  the  time-dependency  time  effect  of was  compared with both the uncontrolled variability of u between  stationary found  to  contact, be  small  stick-slip events and  the often observed overall increase in n with displacement. Therefore even though the time dependency of ju is  a general characteristic of rock friction,  may be easily masked by other  The  time effect  is  this  effect  effects.  mainty brought about by the  creep of asperities. The  asperity creep depends on absorbed water. Therefore it is expected that the time dependency free  effect  environment.  small, factors.  this  time  would be Besides, effect  reduced if experiments because  can be  the  duration  ignored as  being  were  conducted  rock  burst process  of of  less  in  a  water  is  very  important than  other  Failure by a Process of Shearing / 53 4.5. It  STICK-SLIP is  well  experiments  PHENOMENON  known of  that  metallic  regular  friction.  relaxation  Similarly,  oscillations  these  frequently  phenomena  were  occur  also  observed  during studies of rock friction [29,39,40]. The sliding behavior on a shear may  occur as either of two types of motion. If the  small  fluctuations  it  called  is  in velocity  "stable  rapid  slips  with  called  "stick-slip".  when  the  shear  sliding takes  period  little  motion  gives  a  a  Figure  of 4.5  in  surface  sliding is smooth with only  stress reaches  sliding". If the  in  place  some critical  by  between,  typical example  a  series  the of  of  value,  discrete,  sliding behavior  stick-slip  is  phenomenon  from laboratory recordings.  The very  complex.  presence  of  laboratory when  a  under  which  Experimentally, the water,  studies,  surface it  has  increase  critical  been  of normal  normal  either  found  and that  at  sliding  figure  which  the  sick-slip on  other  behavior  stable 4.5.  or  depends  possibly the  For example,  pressure,  pressure  stable  sliding behavior  properties  loading condition varies.  with the be  conditions  transition  [29].  become  suggests that would  take  are  pressure,  sliding will  sliding can This  normal  factors  of  occurs  From change  stick-slip  there place  may given  certain other factors. This transition normal pressure is considered by some to be the  minimum normal  But  stick-slip is also observed at normal pressures below that level  The  roughness  pressure  of  the  behavior. On rough surfaces, observed  with smooth  to  cause  shear  asperity  surface  seems  the sliding is stable.  or polished surfaces  indentation  [40].  also  to  and ploughing  [39].  [29].  affect  the  sliding  On the contrary, stick-slip was By reworking the  shear  surface  Failure by a Process of Shearing / 54 to a different roughness, the  point  of  view  of  the stick-slip behavior could be inhibited. However, from rock  mechanics,  this  stick-slip  due  to  roughness  is  not  considered to be important since a high degree of surface finish is rare except in some slickensided or natural cleavage  Stick-slip is system  [39].  machine low  machine  transition  also reported to be  The  stiffness  tendency as  roughness,  stiffness.  enhanced low  of  observed  normal stress  stick-slip is  surfaces.  dependent  stick-slip  in metal.  Figure  4.7  versus  machine  decreases  shows  some  stiffness  of  testing  machine  the and  stiffness  with  Similarly, this  by high normal stress,  stiffness  on the  the  increase  tendency  typical  of the  is  factors.  absence of gouge, the  presence  of  of  enhanced  laboratory  and other  testing  results  the by of  In  general,  low  surface  strong,  brittle  minerals such as quartz and feldspar.  Among the two types of slip behavior, stable sliding can not cause violent failure  because no extra energy  case of stick-slip, energy at  slip.  A  understand  sudden the  slip  can be  stored  in the  system.  However,  for  the  can be accumulated during the stick period and released will give  conditions  rise  bringing  to  violence.  stick-slip.  Therefore, it  From  the  above  condition may be a combination of many factors not a single studied in the following chapter.  is  important to  discussion,  this  factor and will be  Failure by a Process of Shearing / 55  500  ...  1  J.  MINIMUM NORMAL STICK - SLIP  FOR «00 — in  <  a  Li VI J 0 0 e  • / /  STRESS IN  WESTERLY  GRANITE  0  *600  SURFACE  •  *2«0  SURFACE  /  _  /  —  —  in _, < o z  200  100  / /  — /*  —  --Si  0  -  •  —  *  •  /H  /' » •^  •  o  ^  \  10 STIFFNESS  \  \  \  20 ( KBARS/CM )  1 30  Fig.4.7 Transition from stable sliding to stick-slip as a function of normal stress, stiffness and surface finish, (after Dieterich, [39]) 4.6.  SUMMARY  Shear behavior on rock are 1.  surfaces  and the  The law of friction is introduced, and the coefficient  on  following  results  A bilinear envelope  and  of friction  for which an empirical formula  is  is found to  derived based  the previous testing data.  strength  3.  been investigated  found:  be slip-velocitjr dependent,  2.  has  of  rock  is used  surfaces,  as with  the most reasonable representation of shear the  first  part passing  through the  origin  the second part having a nominal value of cohesion.  Environmental  factors,  such  as  normal  have significant effects on shear strength.  pressure,  pore  pressure  and  time,  Failure by a Process of Shearing / 56 4.  Stick-slip is  an  sudden  due  slip  important phenomenon to  the  release  of  because energy  it can cause accumulated  violence  during  by a  the  stick  period. 5.  Stick-slip  is  roughness  and  usually low  enhanced  stifffness  by  of the  high testing  normal machine.  pressure,  low  The conditions  cause stick-slip appear to be complex and need further study.  surface which  CHAPTER  5. T H E O R E T I C A L  SHEAR  MODEL:  CONSTANT  FRICTION  Because it is impossible both economically and technical^ to carry out a complete study  of  shear  failure  by  experiments  under  a  variety  of  conditions  and  observation from one situation may be different from another situation [29,39], a model  is  during rock  developed  shearing.  and  along  in this  In a  research  order to fault,  the  two  behavior  on  the  may  the  causes of violent  chapter  8.  and this  failure  Stick-slip  to  violent  process  surface.  from  as  give  rock  failure process  previous  be  chapters  study  so  a  full  failure  for both  analysis  occurring in  cases  is  seems to be closely  Sudden  A  shearing  and  stable  Figure  5.1  model  associated  [26].  With  It  with  the  later in  able  this  consists  to  be  affected  by  many  etc.  to  simulate  intention, of  a  the  phenomena  spring-mass  a block of  mass  system  of both is  load P and is connected  which  moves  with  mass,  the normal force P and the shear force F are self-explained.  speed  of  by a spring of stiffness  V . The  spring represents  the  stick-slip  suggested  M which rests on  under normal  a  the  failure. Sudden loading will be discussed seems  stick-slip  in  MODEL  should be  sliding.  massive  discussed  factors, such as rock type, normal pressure, surface roughness,  5.1. M A T H E M A T I C A L  a  shearing  discussions  and  stick-slip  during  previous  loading  of  a  in  surface  X to a support, elasticity  of  In the  rock given  coordinates, the system  is stable  in Y direction due to the balance of the normal  force  reaction  force  (P + Mg)  and  its  N.  In  X  direction,  by  Newton's  law  of  motion, we have MX  =  F  +  f  (5.1)  where F and f are the shear force and the resistance,  57  respectively.  Theoretical Shear Model: Constant Friction /  58  P  V  F  M  x f  Fig.5.1 Simple shear model  about t=0.  If  we  begin  to  move,  Let the  Then  the  to  the  count  driving  the  time  at  the  moment  support would have  moved  when a  the  mass  distance  £  0  is at  just time  contact area between the mass and the surface concerned be unit.  shear  force  and  normal  force  will  be  equal  to  the  corresponding  stresses.  At  any  moment  t,  the  shear  force,  which is  a  function  of  time  t  and  displacement X of the mass M , is given by F(t,X) where  =  X(£  0  +  Vt -  (5.2)  X)  X is the stiffness of the connecting spring, X  is the displacement of mass M , a function of time,  V is the moving speed of the support.  The seismic  resistance  radiation. For  includes simplicity,  frictional only  the  force,  resistance  frictional force  from is  viscosity  considered  at  and the  Theoretical Shear Model: Constant Friction / 59 moment. The viscous effect is ignored because and  the rockbursting is a quick  it occurs only in long term failure  action. The seismic  radiation will  be introduced  in a more sophisticated model in next chapter. Then, the maximum resistance  is  just the shear strength, f(0)  =  ( P + Mg)M  C +  (5.3)  g  where C is the inherent cohesion, u  s  P  is the static coefficient of friction, and ' is the normal force acting on the mass M ,  f is the frictional resistance, a function of slip velocity: f(X).  When  the  shear  reaction,  the  friction  opposite  direction.  If  force  f  is  F  F  is  obviously  is  bigger  less  than  equal than  to  f(0), F  f(0),  by  the  in its  the  mass  law of  value,  action and  pointing  begins  to  to the  move.  As  discussed before, the frictional resistance varies with the slip velocity of the mass M.  To further  moving process than  the  static  simplify  this  model,  we  assume  a  by introducing a dynamic coefficient coefficient,  /z'<M g  The complete  constant  friction  of friction  function  u\  during  the  which is less  of the friction  would  thus look like  p- CC f =VC  '(P M'(  +  •X(£ C  AM'(P  -  R  +  0  +  + Mg), if X > 0 ,  + Mg), if X < 0 , Vt -  M (P + G  (5.4)  X), if X = 0 and |F(t,X)|<f(0), Mg), opposite  to F in direction, if X = 0 and  |F(t,X)|>f(0). where M > M ' are the static and dynamic coefficients G  of friction, respectively,  £ o is the initial value of compression in the spring, and  Theoretical Shear Model: Constant Friction / 60 % o = f(0)/X = [C + M ( P + Mg)]/X. s  To  study  With equation  the  slip behavior, only  (5.2) MX  and the upper two parts of (5.4), (5.1) =  X(£ +Vt-X)  +  + fC + M'(P.+ Mg)]  =  [C + M ( P + Mg)]  0  becomes  *  + [C + M'(P + Mg)]  G  X + XX/M = X  condition of X * 0 needs considering.  the  +  X(Vt-X),  [C + y ( P + Mg)]/M + [C + M'(P + Mg)]/M  +  g  a X  =  2  b +  where a and b are constants,  X > o r < 0 , or  XVt/M,  X>or<0  a Vt  (5.5)  2  given by  a = X / M , and 2  b = [C + ju (P + Mg)]/M + [C+ju'(P+Mg)]/M, X > or g  The  ordinary  second  order  differential  equation  (5.5)  is  <0. a  non-homogeneous  vibration equation with an inciting force of (b + a V t ) . 2  The  initial  conditions for equation  (5.5)  is  the slip velocity of the mass are zero at t=0, X(0)  5.2.  SOLUTIONS  The  differential  speed  V  of  the  =  X(0) =  moving  displacement  in  (5.6)  (5.5)  support  and  ie.  0  TO T H E DIFFERENTIAL  equation  that both the  can should  be  EQUATION  solved  be  a  exactly,  function  of  if  V  time,  is  known.  because  in  The the  situation of mining, the rate of stress change  varies during redistribution. A t and  right  after  V  after  excavation  excavation, when  *note: the sign + is  the  resembling  a new  speed  should  increase.  state of stress equilibrium is  — when X > 0  and is  +  when X < 0 .  A  while  later  about to be reached,  Theoretical Shear Model: Constant Friction / 61 V  should decrease.  However, it is difficult to simulate this rate of stress change  exactly. A binomial function is introduced here: V where V  0  =  V  +  0  wt >  0  (5.7)  is a constant,  u> is  the  rate of speed  stress increasing; w<0 Substitute (5.7)  change,  a constant.  CJ>0 means  acceleration, for  means deceleration, for stress relaxing.  into (5.5), we have  X  +  a X  =  2  b +  a V t 2  0  +  a tot 2  The general solution to the homogeneous  (5.8)  2  equation corresponding to (5.8)  is  a trigonometric function, given by X''' where  A  and  =  Acos(at +  \p are  constants  specific solution to (5.8) X"'  =  \p)  (5.9)  to  be  determined from  the  M  B +  =  conditions. A  has the same form as the right hand side of (5.8), i.e. Gt +  Dt  (5.10)  2  where B, G and D are constants, determined as followings. X  initial  Because  2D  (5.11)  by substituting (5.10) and (5.11) into (5.8), we have 2D  +  a (B + Gt+Dt ) 2  (2D + a B ) 2  Comparing  the  =  2  +  a Gt  +  a Dt  each  term  2  coefficients  of  b + 2  2  a V t  +  2  0  a t J t , or 2  =  b +  a V t  on  both  sides  2  +  2  0  of  a wt 2  above  2  equation,  we  obtain a D = a w, 2  2  a G=a V , 2  2  0  D=co, G = V , 0  2D + a B = b , or 2  B = ( b - 2 D ) / a =(b-2cj)/a 2  2  (5.12)  Theoretical Shear Model: Constant Friction / 62 The  real solution to equation (5.8)  is the  sum of the  specific  solution and  the general solution corresponding to its homogeneous equation, i.e. X  * X . +  =  Considering equations X  =  X  **  (5.9), (5.10) and (5.12), we  Acos(at  + \fj) + cut  have  +. V t  2  +  0  B  (5.13)  The first order differentiation of (5.13) gives X Taking the  =  -aAsin(at  initial conditions  +  xp) +  (5.6)  2ut  +  V  (5.14)  0  into consideration, we  can obtain the  constants  A and X(0)  =  Acosi//  X(0)  =  -aAsim//  cos^  =  — B/A  sinv// =  {  +  V /aA,  B = +  With  MODEL above  =  0  0,  or  0  ^  =  tg-'C-Vo/aB)  A  =  V /asin^  =  tg- [-aV /(b-2cj)] ,  0  (5.15)  0  (5.13) and (5.15) are the solutions  5.3.  V  0  to the differential equation (5.8)  of our model.  RESULTS solution,  the  slip  behavior  of  this  following, a few commonly used parameters are  model  can  be  described.  In  the  reaches  the  discussed.  5.3.1. Slip T i m e Slip  time  maximum  is  the  duration of  resistance  according to  equation  f(0),  a  the  (5.14).  slip, mass  Due  to  Once begins the  the to  shear move.  movement  stress Its  of the  F(t,X) slip mass,  velocity the  varies  stress in  Theoretical Shear Model: Constant Friction / 63 the spring is relaxed in turn. After time T , , the mass stops moving, i.e. X(T,)  The  explicit  =  - a A s i n ( a T , + \//) + 2wT, + V = 0  ;  o  solution  of  obtained numerically. For the  T,  is  not  obtainable  -aAsin(aT,  + =  although  it  can  be  simple case of uniform rate of stress redistribution, i.e. CJ = 0. Then (5.16) becomes  the moving speed of the support is constant,  sinCaT, +\p)  here,  (5.16)  V  =  0  0  V /aA 0  Vo Vo JL2/(J12/ i ^) = i ^ , (5.17) a a Note sin(aT ,+*//) = sin(aT j )cosi// + cos(aT! )sin\//, therefore, above equation becomes =  sin(aT,)ctgtf  S  =  1 -  The solutions  =  n  (5.18)  2sin(aT ^ /2)cos(aT, /2)ctg\I/  2sin (aT,/2) 2  Substituting them back into (5.18), we tg(aT,/2)  S  cos(aT,)  In the above equation, left side = right side =  n  ctg^  =  have  -aB/V  (5.19)  0  to (5.19) are infinitive and are given iaT, T,  where k= 1, 2, 3,  =  kir +  = -kTr a  as  tg~ ( - a B / V ) , or 1  0  -^-tg- (aB/V ) a 1  0  all positive  integers.  From the physical meaning of our model, it is known that only the solution is valid, i.e. k = l T,  = i-ir a  The value of T^ for k = l and to slip back.  and X > 0 . -tg-'liu-u') a s and X < 0  Considering (5.12) and  (5.5)  (P+Mg)/(V /XM)], X > 0 0  first  (5.20)  is the time the mass takes to slip forward  Theoretical Shear Model: Constant Friction / 64 5.3.2. Slip Distance By  the  time  T,  when  maximum distance X X,  1  .=  Note cos(aT,+i//)  moving  is  ceased,  the  X(T,) =  Acos(aT  =  ±/l-sin (aT  =  +j/l-sin \//,  +  1  V  have  moved  a  0  T , + B  2  z  1  (see eqn. ( 5 . 1 7 ) )  ±cos\jj.  First, consider cos(aT, +\p)= — cos\p and equation sini//)cos\//  (5.15),  VQT, +  =  -(V /a  =  -  =  V  =  V Q T ,  +  2B  =  VQT,  +  2 [ ( M - M ) ( P + Mg)]/X,  0  o  +  — ctg,J/ + Si  VoT, +  -  f B)  T  l  o  at time T  1  ;  B  B  +  B  ,  s  X>0  (5.21)  gives rise to X ^ V Q T , ,  Similarly, cos(aT,+\p)=cos\// ignored because  would  which can be determined from equation ( 5 . 1 3 )  }  =  X,  mass  the mass  which is invalid and  must have moved a distance  X ^ V Q T , ,  the displacement of the support during time T , , so that the stress in the spring can  be released.  5.3.3. Stick T i m e After  the  mass  mass-spring resistance. potential  has  system Because  energy  in  moved  is  a  lowered.  distance This  the  support  the  connecting  X , , the  total  drop of energy  still  moves  spring  with  begins  reach the maximum values the mass-spring system  a to  potential  was  in  the  consumed against  the  speed build  V, up  energy  the  again  force  and  until  they  can hold. During this period,  Theoretical Shear Model: Constant Friction / 65 the  whole  system  is  stable  and the  duration of this  period T  2  is  called stick  time and can be determined as following:  At the moment the mass is about to move, or at t=0,  the total potential  energy is the energy stored in the spring, E  suppose at  the  the time  po  * «o  =  X  mass  were  t=T +T 1  to stay when  2  at the  the  maximum distance after each slip. Then  mass  is  about  to  move  again,  the  potential  energ3' reaches E  p2  =  i  * °  M  +  V  "  t  X  l  )  Obviously, at the two moments, t = T , same,  2  and t = T + T , 1  2  the energy should be the  i.e. E  =  po  iX£o So  Remember neglected.  p2  =  2  Q  iMU  =  do  +  Vt -  2  So  E  that  +  +  £  vt  Vt X,  is  0  -  x,)  2  X , ) , or 2  =  ±$o  the  initial  compression  and is  positive.  So - £  0  is  Then, Vt t =  -  X, (-V  = 0  V t + 0  + 1  2  =  X,/V  Alternatively, because  2  -  /Vy+4t3x7)/2cj  If cu=0, from (5.22a), t = X / V , T  wt  0  0  -  X,  = 0  (5.22a) (5.22b)  then  T,  (5.22)  in this simple model, the only external force is from  Theoretical Shear Model: Constant Friction / 66 the  support,  the  stick  time  T  can also be  2  obtained from force accumulation in  the spring. A t time T , , the shear force is, from F(T,,X,)  X(£o +  =  VT, -  After time T , the shear force reaches 2  F(T,+T ,X ) 2  X,)  the maximum resistance  X[? +V(T,+T )-X ]  =  1  (5.2)  0  2  =f(0)=  1  f(0) X$ >  (see  0  eqn.  (5.4)), or V(T,+T )  -  2  the same as  X,  =  0,  (5.22a). However, the energy method can be used in any conditions.  In the case that the mass may slip back due to the elasticity stay  at  a  distance  given  by  equation  rock  mass.  The  less  than  (5.22).  high  X,,  This  the  stick  situation  restriction  may  time  may  stop  not  the  will  be  exist in  motion  in  and finally  less  than  the  highly  less  than  the  value  restricted one  cycle,  although it may slip back a bit.  5.3.4. C o m p a r i s o n with Laboratory Results In order to verify the validity of this model in simulating the slip behavior, modelling results typical  are compared with laboratory tests. Figure 5.2  laboratory  recordings  characterized by the oscillation stable  sliding  of  figure  5.2a).  from as  shear shown  For  a  tests. in figure  close  up,  The  stick-slip  5.2b) one  shows [40]  the  some  phenomenon  in comparison with  cycle  of  the  is the  stick-slip  is  enlarged in figure 5.3a), which clearly indicates the force buildup during the  stick  time  and the  stick  time  are illustrated in figure  force drop at slip. Correspondingly, the  stick time and increases  5.3b), where  suddenly at slip.  the  slip distance  displacement  is  and the  unchanged during  Theoretical Shear Model: Constant Friction / 67  Fig.5.2 a) Load-displacement for a shearing test, surface roughness 180 micro in; b) the oscillation of load with displacement on a magnified scale, surface roughness 35 micro in (after Hoskins et al, [40])  Theoretical Shear Model: Constant Friction /  68  A  Fig.5.3 a) One cycle of the oscillation of Figure 5.2b) on an enlarged scale; b) the same showing displacement against time (from Hoskins et al, [40])  For 5.5.  comparison,  The detail  typical  slip  resistance change testing  of  of  are  time,  results, figure  shear  results force,  illustrated  varies the  the  with shear  slip force  of this  resistance,  in  figure  velocity. with  In  slip  model are slip  5.4,  distance  where  figure  5.5,  distances  are plotted. Obviously, they have  plotted  and  the the of  in figures  and  slip  shear  the  similar patterns  velocity  force  overall slip as  5.4  for  drop  picture  and  of  distance  a  and the with  the laboratory  5.3.  It can be seen that this shear model can reproduce the laboratory results and  simulate  the  stick-slip  well.  Therefore, it  can be used  to further study  the  Theoretical Shear Model: Constant Friction / 69  time Fig.5.4 Model results  (micro seconds)  showing changes of slip parameters with time  shearing process under various conditions of normal load P, surface roughness M > g  driving  speed  V  and  stiffness  X  and  to  search  for  the  transition  conditions  between stick-slip and stable sliding. This will be discussed in next chapter.  5.4.  DISCUSSIONS  In the only  previous  small  velocity.  chapter,  fluctuation  the  stable sliding is  in velocity.  described as  Therefore it is  For stable sliding, slip will not change  the  important to examine direction. For stick-slip  slip may do. From (5.14), we know X m i.n = 2 c j t + Vo - |iA a l | 0  <  X <  2 c j t + Vu+ | a i A |i= X m a x ', or 0  smooth  slip the  with slip  however,  Theoretical Shear Model: Constant Friction / 70  slip  time T, =1.5  X 10  5 S  .  s I  o n S QJ O  i—i  XI  2L  a, cn  •H  T3 10 b)  15  20 time  25  30  35  40  (seconds)  Fig.5.5 Model results: a) force-displacement curve; b) displacement-time curve  Theoretical Shear Model: Constant Friction / X If  X  one  . =2cJt+V -V /|sin\//| < min ' ' 0  . min of  and  X  them  is  have  max  X <  0  zero,  occurs. However, this  the  same  stick-slip  2o)t+V  signs °  occurs.  is not free  +V /|sin^| = X ' ' max  0  If  (5.23).  0  simultaneously, • the}'  have  vibration. As time  the  opposite  continues,  71  sliding occurs. signs, the  If  vibration  vibration will  damp off very quickly for low driving speed before the next slip begins.  For  the  case  V /|sini^| — V >0. 0  of  vibration occurs if  m i n  m  equations  is always  possible.  For  a  gives  X  m a x  >0;  note  |sin\//|<l,  m m  >0, and  exists.  V (l/|sim//|-1)  >  0  (5.24a)  V (l/|sin^|-1)  >  0  (5.24b)  0  =  0  or 2ut (5.24b)  >  V (l/|sin0|-1) o  only  exist  In other words,  as  >  0  temporarily. long  as  (5.24c) As  the  u>>0, stable  time sliding  (5.23) becomes  . = V ( l - l / | s i m / / | ) < X < V ( l + l/|sin //|) = X mm ° ° max 0  —0,  x m  or 2a>t <  = 0, or 2a)t  the case of o> = 0, X x  . =0. min  For  n  (5.24a)  (5.24c) always  when X  i  sliding occurs if X  continues,  Obviously,  <0>  x  stick-slip occurs if X  Obviously,  (5.23)  In this case,  0  stable  a)>0,  1  r  0  |  r )|  1  and 1 —l/|sin\^|<0, or X - < 0 . m  n  Therefore, stick-slip  happens  Otherwise damping vibration occurs,  the case of CJ<0,  cot is: wt> — V . 0  X  (5.23) gives X - < 0 . m  Therefore,  max  =  V ( l + l/|sinv//|) + ' • '  2cot  >  V ( l + l/|sin(//j)  -  2V  =  V (l/|sin\//|  1)  >  0  0  0  -  0  0  n  By (5.7), the lower limit for  Theoretical Shear Model: Constant Friction / 72 the same happens as when o> = 0.  Therefore,  when o;>0,  as time continues,  When to<0, stick-slip occurs if V before  the  next  slip.  Because  0  of  stable  possible.  is small enough for the vibration to damp off the  high  restrictions  vibration can last very little time and the mass to stay  sliding is always  in  the  rock  mass,  of this model can be suggested  at the maximum displacement. Therefore only stick-slip exists when  In  conclusion,  if  the  change in the rock mass  driving  speed  which  is zero, the system  the  resembles  is stable  the  rate  if there was  o;<0.  of  stress  no potential  problem before. On the other hand, for the case of nonzero driving speed, if the rate of stress change is decreasing, the slip behavior will eventually be stick-slip and the  system  is constant, If the  will also become  stable  after  the rate reaches  zero. If the  rate  the process will probably be stick-slip, depending on other conditions.  rate is increasing, the  system  will  be  unstable  and stable  sliding occurs  eventually.  It can be seen that the of shearing. This  means  the  driving speed  is  importance of the  very important to the behavior stress change  rate  to rockburst.  It should be pointed out however that in the above discussion, only driving speed is  analyzed,  only  the  and there  static  are  loading is  some  other factors  considered here  influencing the  and the  behavior.  dynamic effect  is  into consideration. A l l these will be discussed in the following chapters.  Besides  not taken  Theoretical Shear Model: Constant Friction / 5.5.  SUMMARY  1.  A  mathematical  stick-slip  Using stick  of  shearing,  which  and stable sliding, is developed  coefficients 2.  model  this time  can  show  using constant  phenomena static  of  73  both  and dynamic  of friction in order to analyze the slip behavior. model, are  the  obtained  slip  parameters,  theoretically,  and  such their  slip  distance,  results  are  slip  time  and  compared  with  laboratory recordings and similar patterns are found between them. 3.  By comparison, this model is reasonable to simulate the shearing process.  CHAPTER The in  6. S L I P  BEHAVIOR UNDER  VARIOUS  CONDITIONS  slip behaviour of stick-slip in terms of slip distance, force and energy drops each  slip, stick  time  in between,  etc.,  is  very  important in studying violent  failure and determining the conditions which may give rise to violence. The model developed in the previous chapter where a constant friction was used to analyze the friction  with  the  assumed will be  slip behavior under various conditions. Here the variation of  slip  velocity  and  the  seismic  radiation,  which  is  the  signal  detected directly by a seismic monitoring system, will be taken into consideration.  6.1. In  SUMMARY OF ROCK order to take  important  into  parameters  PROPERTIES  account  as  representing  many the  practical  rock  publications. The data listed in table 6.1  situations  properties  are  as  compiled  The  few from  [26].  Coefficient surface is the maximum resistance  when the block is  at rest and varies with the rock type and surface roughness.  In general, harder  rock  static  here  a  are the results of laboratory tests and  field measurements, most of them are from Jaeger and Cook  6.1.1. Frictional  possible,  and  friction of rock  rougher  surface  surface. For instance, 0.62 — 0.75,  have  sandstone  dolerite as high as  higher  friction  than  softer  rock  has a value of as low as 0.51, 0.95.  The coefficient  M  g  to the maximum friction resistance or the shear strength.  74  in table  and  smoother  marble between 6.1  corresponds  Slip Behavior under Various Conditions / 75  Table 6.1  summary of rock properties  index  general range  most of rocks  representative rock types  static frictional coefficient a s cohesion C elastic modulus E uniaxial compressive strength  0.45-0.95  0.5-0.8  sandstone, quartz, — marble, dolerite  0.3-1.1 M P a ' 7 - 1 0 0 GPa  0.3-0.45 M P a 4 0 - 1 0 0 GPa  35-570 MPa  70-570 MPa  granite, trahyte — marble sandstone, granite — diabase sandstone, marble — granite  note:  1 K P a =10 P a ,  1 M P a = 1 0 Pa,  3  6  1 GPa=10 Pa 9  6.1.2. Cohesion Cohesion  is  defined  zero.  In the  load.  However,  as  the  maximum frictional  case of rock, as  this resistance  discussed  before,  the  resistance  when  normal  is usually nearly zero at null strength  envelope  for  represented by a bilinear curve passing through the origin of the system.  When  normal  load  becomes  load  higher,  this  curve  is  rock  is  normal can  be  T-O coordinate  characterized  by  a  lower slope and a nominal value of cohesion. The corresponding data is given in table  6.1  The and  cohesion  therefore  varies  also with  comes the  from  rock  the  type.  viscosity Again,  such as granite of 0.3 M P a , marble of 1.1 M P a .  between  harder rock  the has  grain  particles  higher value,  Slip Behavior under Various Conditions / 76 6.1.3. Elastic Modulus Elastic modulus, a measurement type  and  is  defined  compression before stand  as  the  of the  the  strength  stress per unit change  slope  elasticity  of a material, varies with rock  of  stress-strain  the  point. It actually indicates  curve  the  of  uniaxial  ability of rock to  of strain. Usually, the higher its value, the harder  the rock. A typical value of sandstone  is 9.5  GPa, granite is 55 to 83 GPa and  diabase up to 99 GPa. More is listed in table  6.1.4. U n i a x i a l Compressive  6.1.  Strength  This is one of the most important indices of rock property. It is defined as  the  maximum  ability  one  dimension  load.  of  rock  Due  to  to  sustain  different  external  minerals  stress  contained  without in  a  failure  rock,  under  this  value  a c  varies  widely, ranging from 34 ~ 586  For example,  M P a . Generally, soft rock has  a typical value for sandstone  is  37 M P a , marble is  lower  value.  76 — 150 M P a  and granite up to 586 M P a .  Under  the  condition of multiaxial loading, the  not only with the  rock type,  is  difference  defined  as  the  but also with the between the  compressive  strength  confining pressure.  major and minor principal  This  varies relation  stresses by  Hoek's empirical formula [42], a, where  a,,  a  3  =  a  3  +  \/mo <y c  m, s are empirical constants, relation is  +  sa  (6.1)  2  are major and minor principal stresses  is th uniaxial compressive  This  3  valid only  if the  strength given in reference maximum effective  [42] normal stress satisfies  the  Slip Behavior under Various Conditions / 77 condition: o < o . c  6.2.  SEISMIC  In the  EFFECT  study of rockbursting, seismic  radiation is  a very important factor to be  considered. It is a mechanism in nature whereby energy can  be  removed  tremendous center  in  energy a  form  from  the  neighborhood  is  released,  of seismic  seismic monitoring system.  of  the  bursting.  part of which is energy,  which is  and  propagates  motivates  out  energy  is  particles  consumed  spherically. When  the  adjacent to  against  is over or until the seismic other  medium, such  source  to  the  them.  as  It starts  the  burst  be  occurs,  the bursting  detected  by a  at the energy  seismic  waves  reach  In this  resistance  of  process,  In the  a  vibration  continues  part of the and  until  part  the  point  latter  case,  a  seismic  of  seismic  waves reach some boundary between air.  release  begin to vibrate. This vibration in  transmitted to adjacent particles. This process  and  a  The radiation of seismic energy is a process of chain  around this centre, the particles of rock mass turn  As  radiated out from  reaction among the grain particles of rock mass. centre  released during a burst  of  is  energy  the rock  reflection  it  the  mass shock  wave occurs as a tensile wave with disastrous effects on the free surface of the mine  excavation.  Intense  slabbing  and  spalling occurs  within  milliseconds  filling  the opening virtually instantaneously with broken rock.  6.2.1. F o r m u l a t i o n of Seismic Radiation The  process  of seismic  radiation itself is  very  complicated. No attempt  is made  in this research to study this process in detail. Here we are trying to use simple  way  by  which  this  process  can be  introduced into  the  some  shearing model.  Slip Behavior under Various Conditions / One  way  model  [43]  unduly  in  which  radiation  complicated  is  to  the rock mass in such a way which propagates  along the  effects  attach  a  can  be  simulated  semi-infinite  string  particle  M at X = 0,  to  making  each  particle  the of  that motion of the particle excites an elastic wave  string.  This idea is diagrammatically shown in figure 6.1. the  without  78  and is  fixed  A string is attached  at X = =>. Any motion of the  particle  to can  induce a longitudinal wave in the string.  Suppose length. figure string, this  this  Consider 6.2.  string  has  an infinitesimal  an  area  A  and  elastic  modulus  E  element of dX between sections X  in  certain  and X + dX,  Obviously, the stress at any point is a function of its position on  i.e.  element  a(X).  If the  will be  stress  moved  to  is the  a  1  at  position  section  X  bounded  and by  a  2  the  at  dashed  P  V  semi-infinite string F  section  X= X  Fig.6.1 Simulating the effect of seismic radiation  lines  the  X + dX, under  Slip Behavior under Various Conditions / 79 the  differential  instantaneous  force  (a,—a )A. 2  movement (a,  -  By  Newton's  motion  law,  the  force  and  the  u would be related in the following way,  a )A  =  2  where A is the section  AjpH-ydX dt  (6.2)  area of the element  p is the density of the string pAdX is the mass of the element dX.  From the definition of first derivative, we da dX  _  p(X + AX)  =  ~  g  2  dX  ~  From elasticity  a(X)  (~Q~l)  dX  have  _  P i  —  P  (6.3)  2  dX  theory, a  =  Ee  =  E % dX  Differentiating equation (6.4) da — dX  (6.4)  with respect to X leads to  r-,d u = E—— dX 2  Fig.6.2 A n element of the semi-infinite  (6.5)  string  Slip Behavior under Various Conditions / 80 Substitute equations (6.3) —  -  2  (P/E)-^,  9 u 2  _  P 1 X »  ~  V  into (6.2), we have or  2 3 u 2  u  -  It*  and (6.5)  _  ,„ . a  0  (  -  6  6  )  2 where V^ = Elp, the p-wave velocity.  force.  Equation (6.6)  is the classic one-dimensional wave equation without exciting  If  force  an exciting  is  applied at  such as at X = 0 in figure 6.1, side of  the  centre  where  the  wave  originates,  another term should be added to the right hand  (6.6) *pL 3t  -  v -^HP 9X  =  2  2  2  *(t)  (6.7)  2  where 3>(t) is the exciting force, a function of time. Any  function u(X,t) satisfying  the  such solution to the homogeneous u(X,t)  To included  =  u(t  consider  the  (6.8),  which  in  -  above  equation will  equation of (6.7)  be  a  solution to  has the general form [44]  X/Vp)  (6.8)  exciting  force  #>(t)  would  have  the  in  (6.7),  same  another  form  as  function  $(t).  should  Let u(t)  be  be  a  would be the sum  and u(t)": u(X,t)  This  as (6.8)  particular solution to (6.7). Then the complete solution to (6.7) of  it. One  solution  =  can be  u(t verified  X/Vp) + by  u(t)'"  (6.9)  f  differentiating  (6.9)  with  respect  to  time  and  substitute it into (6.7), which leads to [u(t)*]£ Equation  =  *(t)  (6.10) is the requirement for u(-t)  (6.10) to be the  particular solution, which  Slip Behavior under Various Conditions / 81 can be obtained by solving the force  p in  the  string  area A . From (6.4)  section  will be  the  AE*J|  =  AE{[u(t -  X/Vp)]^ +  =  AE{u'(t -  X/Vp>^  =  -  ^ | u'(t Vp  -  +  p  at  other  sections,  important.  From  particle M is  the  at  force  the  end  at  any of  could  depending  it  that  of the  slip velocity of the particle,  The  force  6.1,  as  be  on  and the  a  the  position on the string.  tension  at  some  deformation.  sections  However,  can  be  seen  that  the  and  for  this  at X = 0,  displacement  of  the  is  the  string is  the  string end, or X = u(0,t), and so  X=u(0,t).  moment  the  the  (6.11)  only the force at the end of the string, i.e.  figure  same  times  X/Vp)  this  time,  stress  0}  at  same  corresponding  [u(t)'']^  is a function of time  the  (6.10) if <t>(t) is known. The  have  =  model of shearing process,  force  equation  Obviously, the force  compression  is  X  and (6.9), we  p(X,t)  Even  at  differential  string,  exerting which  on can  the  particle by  be  obtained  the  by  setting  X= 0  in  equation (6.11),  where  Eo  string is  =  p(0,t) =  "  ^  AE/Vp.  This  7  Vp  u(0,t) =  means  proportional to but in the  that  - E o u(0,t) the  opposite  force  (6.12) exerted  by  direction of the  the  slip  semi-infinite  velocity  X of  the particle M .  Thus,  the  seismic  radiation effects  can  be  easily  taken  into  account  by  Slip Behavior under Various Conditions / adding one term as (-EoX) to the resistance f(X)*  =  +f(X)  -  EoX, X >  equation discussed or <0  82  in chapter 4, ie. (6.13)  where f(X) is the frictional resistance Eo is the coefficient  of seismic radiation  X is the slip velocity of particle M . The general picture of f(X) for X > 0  slip velocity  Fig. 6.3  Shearing resistance  is shown in figure 6.3.  (logX, cm/s)  as a function of slip velocity  and seismic radiation  Slip Behavior under Various Conditions / 83 6.2.2. Characteristics of Seismic Radiation Coefficient The  coefficient of seismic radiation Eo is defined in equation (6.12) as Eo  By  AE/Vp.  (6.6), Vp = E / p , Eo  where  In  we have = A/Ep~  p is the material density A  is the cross section area of the semi-infinite spring.  general, the variation of density  elastic of  =  p of rock is negligible compared with that of  modulus E . Therefore the coefficient  Eo is proportional to the  square root  elastic modulus E , Eo  =  ky/E"  (6.13a)  where k is a constant.  6.3.  MATHEMATICAL  The  model  postulated  in  slip-velocity dependent shown  in  figure  MODEL chapter  friction  5.1,  the  5  and the  motion  will  be  completed  effect of seismic  equation  and  other  here  by  radiation. relevant  introducing the For the model expressions  are  rewritten here again for convenience. MX  =  F(X,t) where  F + =  f'  X(£  (6.14) 0  +  Vt -  X)  X is the stiffness of the connecting spring, X  is the displacement of the mass M ,  V  is the moving speed of the support,  £ =f(0)/X, o  the initial compression in the spring,  f(0) is the shear strength.  (6.15)  Slip Behavior under Various Conditions / 84 The  resistance force will be as described by (6.13) /•-f ^f  -  E o X , if X > 0 E o X , if X < 0  (6.16)  - F ( X , t ) , if X = 0 and |F|<f(0) •f(0)sign(F), if X = 0 and |F|>f(0) where sign(F) = - l if F < 0 , sign(F)= + l on', f  =  if o / f f ^ B  {  c  C O  if F > 0  +  (6.17)  on, if B > a  c  lo>l  is the uniaxial compressive strength  C  C is the cohesion o is the normal pressure n is the coefficient of friction and is given by equation (4.2): n = a, For  (6.18)  a + b/[7 + log(+X+10- )], X < or >0 6  b are constants, given in table 4.2  a given B , n' = B C / a + c  (6.19)  n  where B is an empirical constant and is given, or calculated by B  =  o (n'  ~  M)/C  equations  (6.15)  c  (6.20)  if n' is known. Considering  and (6.16),  from  (6.14)  we  have  the differential  equation r(F X  f -  EoX)/M, X > 0  =^(F + f -  EoX)/M, X < 0  0, V  [f  -  X = 0 and |F|<f(0) + f(0)sign(F)]/M, X = 0 and |F|>f(0)  (6.21)  Slip Behavior under Various Conditions / 85 The initial conditions are X(0)  =  Considering  X(0)  =  equation  0  (6.21a)  (6.18)  where  denominator, it is obviously impossible way  to  do  discussed  it  is  to  later  in  this  convenience  6.4.  find  an  the  to solve equation  approximate  chapter.  Therefore  solution  (6.21)  of  X  occurs  (6.21) exactly.  in  the  The only  numerically.  This  will  be  be  it  for  the  will  left  as  is  in programming.  ENERGY  In the introductory chapter, rockbursting was energy it  logarithm  is  release. Part of this energy very  energy  important to  look  at  defined as  is radiated out as  the  behavior  of the  a phenomenon of violent seismic shear  energy. model  Therefore  in terms  of  change.  It is known that, in a force system, the system is equal to the increase can be expressed dE  of the total energy  within this system.  This  as: =  (6.22)  FdS  where dE is the total energy F  the work done by external forces on  increase  is the total external force  dS is the distance increase over which work is done by F and along F .  For the model shown in figure 5.1, external forces which actually do work on the  system are the  resistance  of equation  (6.16) and the  driving force F of  Slip Behavior under Various Conditions / 86 (6.15) from the  mass  the  moving support. The total energy  M and the  potential  energy  in  the  includes the  connecting  kinetic energy of  spring.  Therefore, by  equation (6.22), the energy equation for this system is dtiMX ^[|MX  +  2  2  The  =  2  0  +  iX(£ +Vt-X) ]  +  Ep) =  or, 4- CE, dt  iX(£ +Vt-X) ]  0  =  2  0  We -  X(£ +Vt-X)Vdt-|f(X)X|dt-EoXXdt VX(£ +Vt-X)-|f(X)X|-EoX  (6.23)  2  0  W, -  Wr  (6.23a)  i  K  physical significance of each term in above equation is as following: E^  =  fMX ,  the kinetic energy of the  Ep  =  \ X( £ o  +  VX(£  +  2  system,  V t — X ) , the potential energy  in the connecting  2  spring, We  =  0  V t — X), the rate of doing work in moving the  support against the spring and being of order V , Wj.  =  |f(X)X|,  the  rate  at  Wr  =  EoX ,  the  power  which  work  is  done  against  friction,  positive, 2  radiated  along  the  semiinfinite  string,  positive.  For  a  given  period A t = t  2  ti,  the  total  work  done  by  external  forces  should be the integration of the right hand side in equation (6.23) over At. Thus, by integration equation (6.23a) becomes AE In  k  +  AEp =  We -  W  f  -  Wr  (6.24)  the numerical solution to be described later, the total energy radiated Wr will  be computed as  Slip Behavior under Various Conditions / 87 Wr  =  f ^ W r dt =  f^EoX dt 2  <* Eo X,± At J= 1 J J where n is the number of sampling points for the period At.  (6.25)  2  From sufficiently  (6.24),  it  small, so  the moment when  can  that  be  seen  that,  we  let  We=*0 and note X = 0 at  the  the  loading  speed  onset of a slip  V  be  and at  slipping ceases, so E^—0. Then the loss of potential energy in  the  system is approximately equal to the  and  the energy radiated during the slip, ie  Furthermore,  if  AEp  =* - W  we  can  -  f  see  sum of the work done  against friction  Wr  the  (6.24a)  loss  of  potential  energy  is  proportional  to  the  energy radiated, ie. AEp = - W r , this can be seen in the modelling results of chapter 12, figure  6.5.  NUMERICAL  For  an  ordinary  explicitty, a  few  linear and  its  12.4b).  SOLUTION differential  equation  such  as  (6.21),  which  is  not  soluble  approximate solution can be found by numerical method. There are  numerical methods multi-step  method  disadvantages.  Due  available,  such  as  Euler  and Adams' method. to  the  accuracy  and  Each high  method,  Runge-Kuta method,  of them speed  has of  Runge-Kuta method [45] is chosen here for our particular case.  its  advantages  convergence,  the  Slip Behavior under Various Conditions / 88 6.5.1. Introduction to Runge-Kuta Method  6.5.1.1. First Order Differential Equation Assume that the solution to a first order differential equation Y'(X)  =  f(X,Y)  (6.26)  with Y(Xo) = Yo exists and is unique. Based on the value of Y on step n, the approximate value of Y on step n +1 Y  is estimated by Runge-Kuta method as  n+1  =  n  Y  +  [  k  l  +  2 ( k  2  + k  a)  +  k„]/6  (6.27)  where k , = h « f ( X , Y ) n' n 1  k = h - f ( X +h/2, * n  Y +k,/2) n  k =h-f(X  Y +k,/2) n  2  3 J  k =h«f(X fHt  +h/2,  n n  1  +h,  i  Y  n  +k ) J3  h is the increment of X between step n and step  We  can  consider  this  approximate  value  Y  n  +  n+1.  ^ as  a  substitute  of  the  exact value Y(X , J , ie. n+ 1 Y(X By  n + 1  )  « Y  n  +  1  , (n = 0,  1, 2,  ...)  doing this, the error introduced is of the order of h error  =  5  and is expressed  as  0(h ) 5  6.5.1.2. Simultaneous Differential Equations Again, if solutions to a set of first order differential equations  {  Y'(X)  =  f(X,Y,Z)  Z'(X)  =  g(X,Y,Z)  (6.28)  Slip Behavior under Various Conditions / 89 with Y(Xo) =  Yo, and Z(Xo) = Zo  exit and are unique, the approximate values  of Y ( X  n +  ^) and Z ( X  n +  ^)  are given  by Y  n  Z  f  {  +  i  +  1  =  Y  1  =  Z  +  [k,  +  2(k +k )  +  [m,  +  2(m +m )  n  n  2  + k„]/6  3  2  3  +  m„]/6  (6.29)  where k , = h • f(X , Y , Z ) n n' n 1  m , =h-g(X , Y , Z ) n n n 1  b  k =h-f(X z2  n  +h/2,  Y + k , / 2 , Z +m,/2) n n 1  m = h . g ( X +h/2, z ° n  Y +k</2, Z +m,/2) n n 1  2  k,=h-f(X  n  m =h«g(X 3  8  k«,=h.f(X  n  +h/2, n  2  2  Y +k /2, Z +m /2) n ^ n 2  Y +k , n 3  m „ = h - g ( X +h, * ° n  with n = 0,  Y +k /2, Z +m /2) n * ' n ^  +h/2,  +h,  1  Y +k , n 3 J  2i  Z +m ) n 3  Z +m ) n J 3  1, 2, .... The error resulted from the approximation is also 0 ( h ) . 5  6.5.2. A p p l i c a t i o n to the N u m e r i c a l Model The  differential  obvious  from  equation its  given  physical  in (6.21) is  meaning  that  of second the  solution  order and nonlinear. It is to  (6.21)  exists  and  is  unique. To appfy the Runge-Kuta method, we First introduce a new function Z in such a way  that the second order differential equation can be reduced to a first  order equation. Let  X(t) = Z(t),  then X(t) = Z(t).  (6.21) becomes  Slip Behavior under Various Conditions / 90 r  X(t)  =  Z(t) (F  -  f -  EoX)/M,  X>0  =/(F  +  f -  EoX)/M,  X<0  f  ^Z(t)  0,  X = 0 and |F|<f(0)  IF and  (6.30)  +  f(0)sign(F)]/M, X = 0 and |F|>f(0)  from (6.21a), the initial conditions are: X(0)  Equations  (6.30)  and  (6.30a)  have  the  Therefore, the approximate solutions  same  =  Z(0) =  form  as  0  (6.30a)  those  given  in  (6.28).  in (6.29) can be directly applied to (6.30), if  one keeps in mind that f(t,X,Z) = Z and g(t,X,Z) is a multi-function Z(t).  6.6.  PROGRAMMING  The  execution  computer due written  for  of  numerical  to the  this  huge  purpose.  solution amount  to  were at  of each  Figure 6.4  written  the  parameter,  in F O R T R A N  U B C computing  and  language and  6.5  be  accomplished  are  the  flow  charts  by  for  running on the  are  listed  T. is the instant time I  X. is the slip distance at T. l * I is the slip velocity at T.  F . is the driving force at T. F . is the total force at T. ti l x  F ~ is the frictional force at T. fi I  in  in the following:  1  been  sensitivity  these  M T S computer  appendices  a  of program  for  Programs corresponding to  variables used in these programs are specified  X.  only  and of program M O D E L 2  respectively.  center  can  of calculation. Computer programs have  M O D E L 1 for typical numerical solution analysis  (6.30)  and  2.  charts system Some  Slip Behavior under Various Conditions / 91 T,  is the time length of slip duration  T  is the stick time between two adjacent slips  2  X,  is the maximum distance of a slip  Ep  is the potential energy  Wr  is the energy radiated  Wj. is the energy consumed against friction Wl  is the total potential energy drop after a slip.  Program M O D E L 1 calculates time  T i according to  calculates these  these  solutions  Runge-Kuta  solutions at  at  Ti + A T  A T is further decreased.  is  satisfied.  MODEL 1  and  prints  numerical solutions  method.  Ti+AT  The above  gives  X  1  the  is  and  it  of X . , X . , F . , F , . at r r I ti  increases  Ti + AT/2. more  to  T i by  At and  difference  between  pre-specified  accuracy  If the  than the  computing is repeated  printout  T,  ;  Then  and at  and Ti + AT/2  e,  computation  the  T  2  of  at  Xi,  the  X., 1  Fi,  end  of  until  the accuracy  F .  at  Ti  during  running.  A  typical  Xil-  printout is attached to appendix 1. Program M O D E L 2 as M O D E L 1 . and  X= 0  energy  for sensitivity  analysis does the work in the same  However, it prints out X i , X . , F i ,  during  running.  parameters.  By  A t the  changing  end each  of of  running, the  g  driving  running,  we  speed are  V , and  at  the  same  time  it  prints  X = maximum  X,, T,,  controlling factors  such as the static coefficient of friction M , elastic and  only at T = 0 ,  way  in  T  the  2  and  model,  modulus E , normal pressure P  keeping  able to obtain approximate values  others  of X , T r  unchanged 1  ;  T  2  during  and energy  parameters under various conditions. A typical printout is attached to appendix 2.  Slip Behavior under Various Conditions /  (  choose  )  5 t a r t  input  92  data  function for shear change C and u  strength  c a l l SUB2 t o c o m p u t e i n i t i a l forces p r i n t d a t a and i n i t i a l solutions  call  SUB1 t o  call  set  control  loop  begins,  compute  SUB2 t o  variables  Ti=To,  X i and  Xi  compute  1=1  controlled  forces  Fi  and  by  accuracy  Fti  Ti=Ti+AT  print  I,  T i , Xi,  yes compute  Xi,  F i ,  Fti  yes and  (  print  stop  T l , T2,  XI  ^  Fig.6.4 Flow chart for program M O D E L 1 : numerical solution to the model  shearing  £  Slip Behavior under Various Conditions / 93  ( start  input  choose  call  data  function for shear c h a n g e C a n d <u  SUB2 t o  store  compute  strength  initial  forces  set c o n t r o l variables T i , X i , X i , F i , F t i in  loop  begins, •i  call  ^  SUB1 t o c o m p u t e X i a n d c a l l SUB2 t o c o m p u t e  I  Ti=To,  arrays  1=1  >  Xi controlled f o r c e s F i and  Ti=Ti+AT,  summarize  energies  replace  array.s with  T i , Xi,  compute  print T i , X i , and p r i n t T I ,  y stop  Xi, T2,  by a c c u r a c y g Fti  ;  X i , F i , Ft.i  Fi,Fti XI and  energies  j  Fig.6.5 Flow chart for program M O D E L 2 :  sensitivity  analysis  Slip Behavior under Various Conditions / 94 6.7.  NUMERICAL  RESULTS  By  program M O D E L 2 ,  the sensitivity  of this shear model to each factor, such as  C,  M , X , P and V is extensively studied under a wide range of possible g  listed in table 6.1.  values  The slip behavior is represented by the following parameters:  max  ^  AF —  E  M  A  X  '  M  U  M  S  N  P velocity during a slip  total force drop after a slip  A E — total potential energy drop after a slip Wr — energy radiated during a slip and  T  1  ;  X  1  ?  T  as described in previous  2  section.  6.7.1. Effects of Major Factors The  effect of each factor on the slip behavior of this shear model can be clearly  seen when other factors is an efficient of  the  when  way to examine  system. it  is  are kept unchanged.  It  is  very  impossible  how a factor in a system influences  useful  both  This method of sensitivity  when  combined with  economically  and  model. The effect of each factor is discussed  a  analysis  the behavior  numerical method  technically  to  study  a  and  physical  below.  6.7.1.1. Effect of Cohesion Cohesion is an inherent property of a rock mass. is  plotted  MPa,  in figure  which covers  6.6. most  As can be kinds  slip behavior at all because the 6.2,  the  indicates  data  give  some  a value of 1.00  seen,  of rocks,  Its effect on the  within the the  cohesion  slip parameters  numerical concept  of  range has  slip behavior  of C = 0.1 no  influence  Pa to on  1 the  do not change  with it. In table  these  The  changes.  last column  for the ratio of maximum/minimum of each parameter.  Slip Behavior under Various Conditions / 95 This is  probably because that  Therefore  its  presence  cohesion  only  is constant  increases  the  before,  maximum  during and after  shear  stress  slip.  required  to  initiate the slip.  6.7.1.2. Effect of Frictional Coefficient The  coefficient  of  friction  is  proportional to  This internal characteristic is significant  before  effect on the slip behavior after slippage j/  g  increases  increases  from  slightly,  0.35  to  figures  0.95,  6.7.  maximum  slip  velocity  ^  m  maximum  displacement  X  1  a  At  stick  }  n  s  Generally,  is  indicated by  a  J  rougher  surface  behavior is hardly affected  logC (Pa) T , (0.1ms) X , (mm) T (ms) Xmax(100m/s) A F (100 MN) A E (10 K J ) Wr. (J)  - 1 . 0  a  material.  initiation of slip. However  a  total  bit.  T , 2  potential  time,  the  Other  total  slip  energy time  its  drop  drop  AE  and  the  as  the  T^  parameters,  force  higher  coefficient  of  by the surface roughness  Table 6.2  of  such  AF  and  energy  with  value has  the  same  time  the  strength  M . This little change of each parameter s of near 1.0 in the last column of table 6.3.  radiated Wr, are hardly changed with  the  shear  is initiated seems less important. When  only  fluctuate  x  the  friction.  Therefore  the  slip  within the analyzed range.  effect of cohesion C on slip behavior 1  2  3  4  5  6  max/min  .13405 .19473  " "  " "  " "  " "  " "  " "  .13405 .19473  1 1  .22822 .10710 .52859 .40159  " " " "  " " " ."  " " ' "  " "  " " 2 "  " "  .22822 1 .10710 1 9 1.037 .40159 1  2  note: the symbol (") means  .  5 "  8  5 "  having the same value as the data to the left of it  Slip Behavior under Various Conditions / 96  .6 .55 AE (10KJ)  .5 .45  Wr (J)  .4  V =10 m/s Aa =0.65 P=50 MPa E=55 GPa - 7  o  .35  s  .3 .25  X (lOOm/s) max  .2 X, (mm) , T (ms) 2  .15  , T. (0.1ms)  .1  AF  (100MN)  .05 0 -1  1  2  3  cohesion  4  ( l o g C , Pa)  Fig. 6.6 Variation in slip behavior parameters with cohesion  6.7.1.3. Effect of Elastic Modulus The  elasticity  E  of  a  material  is  represented  by  the  stiffness,  proportional to the elastic modulus, of the connecting spring in this  which  is  shear model.  It controls the rate of force transmission and energy buildup. The elastic  modulus  is constant for a given material but varies with different materials. The value of E  ranges  from  10  GPa to  100  G P a for various kinds of rocks. For some  soft  Slip Behavior under Various Conditions / 97  2 r  V =10 ' m/s E=55 GPa P=300 KPa o  1.8 ~ 1.6 h  ns)  T, (10  1.4 1.2 X max  (0.1.m/s)  1  AF (dyn)  AE (J)  W_ (K erg)  a!  — • X,.(10 jum) ,  T T (100s) z  -.2 .4  .5  friction  .6  .7  .9  coefficient  Fig.6.7 Variation in slip behavior parameters with friction coefficient  materials  such as coal, this value is even smaller, less than  of E on the shown  slip behavior is significant.  in figure 6.8.  5 GPa. The effect  The general trend of each parameter  is  Except for the total force drop A F which is unchanged, all  other parameters tend to decrease quickly as E goes up while E s 2 0 G P a .  The whole  picture of slip behavior can be divided into two  parts  in  this  Slip Behavior under Various Conditions / 9.8  W  (100  r  JU =0.65 V =10"7 /s P=300 KPa  erg)  s  o  \ax  T, (10  X, (10 Jam) ,  m / s  m  >  us)  Tj(100  s)  -1 2  3  4  elastic  5  modulus  6  7  10  (10 GPa)  Fig.6.8 Variation in slip behavior parameters with  elasticity  graph. In the first part, these parameters decrease as the elasticity the  second  curves  part,  the  slip  behavior  can be divided at E = 20  changes  little.  For  AE, X ,  GPa, whereas for T , , X  increases. and  T , 2  In the  and Wr, at E = 40 IT13.X  GPa from  under the given conditions of modelling. The amplitude of the 10  to  more  than  100  as  shown  in  effects of the elastic modulus on the energy  the  last  column  of  change table  drop A E , the maximum slip  varies  6.4.  The  distance  Slip Behavior under Various Conditions / 99  Table 6.3 effect of friction coefficient M  s  T ,(10*18) X,(10Mm) T ( 1 0 0 s) Xmax(0.1m/s) A F (dyn) A E (J) Wr (K erg)  n  on slip behavior  0.35  0.505  0.55  0.65  0.75  0.805  0.95  max/min  1.4978 .11297  1.498 .1107  1.348 .1143  1.3423 .1108  1.4992 .1071  1.4994 .1063  1.3517 .1059  1.12 1.08  1.3383 .629 .1527 .13822  1.298 .6095 .18225 .12968  1.235 .5891 .20939 .15092  1.225 .5846 .22557 .14861  1.2366 .5824 .27089 .11791  1.09 1.08 3.24 1.43  2  1.3024 1.276 .6214 .60887 .08353 .13402 .16861 .16173  Table 6.4 effect of elastic modulus E on slip behavior E  (10 GPa)  T ,(10/18) X , (lO/an) T ( 1 0 0 s) Xmax(0.1m/s) A F (100 K N ) A E (J) Wr (100 erg)  1  2  4  6  8  4.451 1.226  3.146 .6112  2.227 .3038  1.574 .1514  1.286 .1006  4.33 .6132 2.016 15.326  3.055 .6112 1.005 6.885  2.145 .6077 .5002 3.762  1.512 .6054 .2493 1.689  1.231 .6034 .1658 1.183  0.1  0.5  9.945 6.193 9.785 .6193 10.159 75.63  10  max/min  1.115 .0752  .9965 .0601  10 103  1.062 .6016 .124 .9208  .9507 .6013 .0992 .6379  10 1 102 118  2  X,  and the stick time T  In general,  the value  2  are most significant,  of E is above  next to the energy radiated Wr.  40 G P a for most  kinds of rocks. In this  case, the elastic modulus seems not to affect the slip behavior very much.  6.7.1.4. Effect of Normal The  Load  normal load is one of the parameters indicating the state of stress. In the  field,  it can be determined  from the in situ  orientation of the failure surface. of  the  above  conditions  would  stresses,  mining conditions  and the  Therefore it varies with conditions. A n y change result  in  a  change  in  the  normal  load.  This  Slip Behavior under Various Conditions / change  could in turn  change  shown  in figure  As can be  T,  6.9.  the  slip  behavior during shear  seen,  all parameters,  process,  100  in a  way  slip  time  a  clear  except for the  which is not changed, increase with the increase of the normal load.  Note relation,  those  graphs  are  empirical formulae  plotted  of  these  on  logarithmic  changes  and Wr, are obtained by linear regression given  in  maximum  table  6.5.  The  slip distance  force  X,  drop  change  for  scale.  some  To  typical  show  parameters, A F  based on the numerical data and are  AF,  stick  time  in a similar way  T , 2  peak  and have  and increase  with P .  This means  2  and  a linear relation  with the normal load P. The total energy release A E and seismic proportional with each other  velocity  energy  Wr are  that the normal  load P is one of the most significant factors in controlling the slip behavior.  6.7.1.5. Effect of Loading Speed The loading speed  V , or the  driving speed  in this  model represents  the  rate of  stress redistribution in the field. This rate can be ignored for virgin stress. When mining activity takes place, the virgin stress field is disturbed and stress changes significantly  around the  right  mining  after  excavation.  activity.  As  The maximum rate  time  continues,  this  of  rate  stress  change  decreases  and  occurs finally  ceases. However, the exact process of stress change is not well known.  In this model, constant  driving speed  slip behavior within the range numerical results affected  in figure  by  6.10, the  of V o = l 0  - 1 0  Vo was to  10"  used 1  m/s  for simplicity and the is studied. From  it can be seen that only the change  of  stick Vo,  time T  the 2  is  with  Slip Behavior under Various Conditions / 101  -lOT  normal load  (logP, Pa)  Fig.6.9 Variation in slip behavior parameters with normal load  T, /T .=10 , 'max ^ min 8  2  represented  table  6.6.  They have  by T = c / V o , where 2  c is  a reverse  a constant.  A l l other  change  with Vo if Vo is less than some value.  This  loading  conditions  be  and rock  chapter of transition analysis.  properties  and will  relationship,  which  can be  parameters  do not  critical value discussed  varies  in detail  with  in the  Slip Behavior under Various Conditions /  loading  speed  102  (logV , m/s) Q  Fig.6.10 Change of slip behavior parameters with loading speed  6.7.2. T h e V a r i a t i o n of Slip Behavior The characteristics of the slip behavior, described by parameters: T , , X , , T ,  A F , A E and Wr, have  the  slip  2  time  T,  microsecond, and the  which  is  been explained previously. Among these extremelj'  small,  maximum slip velocity  the slip behavior and will not be discussed  X  m  in a  x  the  order  parameters,  millisecond  to  seem to be not significant  to  in the following.  of  in 3.x  Slip Behavior under Various Conditions / 103  Table 6.5 effect of normal load P on slip behavior  4  logP (Pa) logT, (s) logX, (m) logXmax(m/s) logT. (s) logAF (N) logAE (J) logWr (J) 2  -4.829 -7.469 -2.416 - .4692 3.271 -3.7232 -7.8374  5  n  6  -4.871 -6.44 -1.372 .5597 4.2999 -1.6999 -5.8664  -4.873 -5.426 - .3566 1.5745 5.315 .31216 -3.8273  8  -4.873 -4.415 .654 2.5849 6.325 2.3206 -1.8056  9  -4.8729 -3.4074 1.6615 3.5926 7.333 4.3269 .21015  lg" max/ lg" min 1  1  -4.871 -2.4014 2.6676 4.5986 8.337 6.3317 2.2223  1.107 1.1684 1.2123 1.169 1.164 1.135 1.147  X10 xio X10 xio X 10 XIO 5  5  5  5  1 0 10  Empirical formulae for A F and Wr AF:  logAF = -0.65698 + 0.99673 logP, or A F = C , P r = 0.99978, Sd • P+2.45, A F ± 2 . 4 4 n-1 '  Wr:  logWr = - 1 5 . 2 7 + 1.93 logP, or Wr = r = 0.99899, Sd P+2.45, W r ± 4 . 7 2 n-1  C P  2  2  Table 6.6 effect of loading speed on slip behavior logVo (m/s)  -10  -8  -6  -5  -4  -3  -2  T,  (10 s)  X,  (IOMHI)  1.3423 1.108 1.2984 4.0446 .60941 .18225 .12968  1.3423 1.108 1.2985 2.0446 .60941 .18225 .12968  1.3423 1.108 1.2985 .0446 .60941 .18225 .12968  1.3424 1.1082 1.2986 -.9554 .60941 .18227 .12970  1.343 1.1094 1.2997 -1.955 .60941 .18247 .12994  1.3489 1.218 1.309 -2.955 .6096 .18449 .13230  1.4077 1.2511 1.4052 -3.955 .6108 .20563 .15742  M  Xmax(0.1m/s) logT (s) A F (100 K N ) A E (J) Wr (K erg) 2  *  -1  *  1.8917 3.0254 2.6648 -4.955 .6236 .50859 .84933  max/min 1.049 1.129 1.082  io»** 1.002 1.128 1.214  note: the last column logVo = — 1 is not included in computing the ratio max/min for the reason that Vo exceeds the critical value, see chapter 7 for detail. **: this value is from log' max/log" min. 1  6.7.2.1. Maximum  Slip Distance  1  Slip Behavior under Various Conditions / The  maximum  slip  distance  X,  is  a  measurement  of  the  extent  of  104  damage  caused by shear failure. The farther the slippage is, the bigger the damage could be.  For a  given  energy  release  shown  in  the  amount  rate,  figures  then  6.6  the  former and is  the elasticity  more  through  normal load and the  with  of energy  released,  violence 6.10,  elasticity  shorter  at  the  failure.  slip  of the  slip distance From  distance  larger  numerical  results  seems to  material. It  approximately in reverse  the  means  has  a  vary only  linear  proportion to the  is above 20 GPa, or for hard rock, X ,  tends to be  with  relationship  latter. When  constant.  6.7.2.2. Stick Time The parameter T , which is the 2  measurement  peace time between two consecutive  slips,  is  a  of the slip frequency. For a given condition and given time, shorter  peace time means more slips, and then higher slip frequency.  From  the  numerical  results,  T  2  seems  to  be  loading conditions. It is very sensitive to the change load. It has loading  a linear relation to the  speed.  °~ decrease  of  This the  means  loading  that,  speed,  strictly  controlled  of loading speed  normal load and a reverse  if  other  the  stick  conditions time  are  and  the  and normal  relation to  unchanged,  increases  by  slip  with  the the  frequency  decreases.  A s we radiated small  out  scale.  generate  know, each slip releases some amount of energy, as  seismic  Therefore,  more  acoustic  energy, high  slip  activity.  which  is  frequency This  is  called at in  acoustic  high perfect  part of which is  energy  driving  speed  agreement  because will with  of  its  certainly the  field  Slip Behavior under Various Conditions / observations, after  the  change.  where  rate  mining activity,  As time  equilibrium. frequency acoustic  the  of  such  continues,  rock as  the  noise  is  blasting,  found  because  stress changes  of  slowly  The rock noise decreases and eventually  does not necessarily emission  is  not  mean less acoustic from  a  single  to  change  of the  complicated  during  the  stress  concentration  relationship lower  between  normal  normal load on  period  zone T  2  pressure  of  and and  decrease  P,  means  stress  Figure 6.9, higher  slip  of stress  new  because the  but  from  in the It  state  of slip  source of  many  local  is  increase  zone.  loading  frequency  rock mass  may  relaxing  if the  a  right  10.  surface  the  high rate  reach  activity,  fracture  a  sharply  dies out. However, low  redistribution. in  the  to  micro-fractures as observed in laboratory tests, chapter  The  increase  105  in  the  By  the  linear  is  the  same,  lower  shear  speed  because  verj'  of  strength which requires less time for the shearing force to build up.  The When the  effect  of  elasticity  on  the  elastic modulus E is below  same,  has the  higher slip  elasticity. frequency  This is  activity too than for soft rock.  may  higher  time  T  some value, T  of E . When E is above this value, T rock  stick  2  2  impressive,  decreases as  Figure the  6.8.  increase  remains at a low level. In general, hard  imply that for  is  2  hard  if all other  rock  and  conditions  probably more  are  the  acoustic  Slip Behavior under Various Conditions / 106 6.7.2.3. Force Drop The force drop A F after a slip is a measurement of the change of slip potential and  seems  to be  mainly  affected  by the  normal  load.  They  have  a  linear  relationship, figure 6.9. Obviously, higher normal load requires higher shear force to initiate the slip. Therefore, this same,  the time  required  could mean  for the shear  force  that  if the driving speed  is the  to reach the strength is longer at  high normal load than at low normal load, just as indicated by T . 2  6.7.2.4. Energy Release The energy release  is a very important parameter, The more energy is released,  the bigger the failure and the damage could be. By equation (6.24a), during each slip, the total energy released is approximately the sum of the energy consumed against the  friction  external  energy  force  release  not possible monitored  and the energy is  ignored.  and the energy  to estimate  as  seismic  them.  energy.  radiated, if the small amount of work done by In the  field  monitoring of rockburst,  consumed against Only  friction  represent the total energy release  are unknown and it is  a small portion of total energy  Whether the seismic  the total  energy  released is  Wr can be used to  A E depends on the way they change which is  not clear.  According  to  the numerical  normal  load or driving  figures  6.6  speed  results,  changes,  as  remains nearly the same,  the  A E and Wr change  and 6.8 to 6.10. A slight difference  as the friction coefficient  any of  cohesion,  elasticity,  in the same way,  in the way they  change occurs  varies, figure 6.7. In this case, the seismic energy Wr whereas  the total energy  release  increases  slightly as  the  increase  of  y .  However  g  Slip Behavior under Various Conditions /  107  difference  the  this  is  relatively  seismic energy Wr may represent the total energj' release.  small.  Therefore  This will be shown in  the energy results generated from an acoustic simulating model in chapter 12.  In  figures  effects on the  6.6  energy  to  6.10,  release,  varies proportionally with P  the  normal  whereas  load  and  elasticity  other factors have  have  significant  little effect on it. Wr  and nearly reversely with the elasticity. Apparently,  2  a high normal load represents a high stress field, which causes large amount of energy  to be  released  stored in the  at failure. When  rock  the  structures. Consequently more energy  elastic  modulus is  low,  would be  Wr decreases dramatically  with the increase of E . When E is above some value, the change of Wr is very small. same  This  may  indicate  that  the  energy  released  in each  slip  is  nearly  the  from hard rocks and is less than from soft rocks. It should be noted that  the total amount of energy released in a given period is not necessarily less in hard rock than in soft rock because the slip frequency is higher in hard rock.  6.7.2.5. Average Energy Release Rate and Energy Release Ratio The  total  energy  release  can  indicate  the  possible  extent  of  failure  and  the  damage caused by the failure, whereas  the rate of energy release may show the  violence  of  given  released,  the  more  estimate  the  energy  failure.  Obviously, violent  the  released.  for  a  time  failure could be. However,  from  period,  the  In practice, it  the  energy seems to represent the total energy release of seismic energy radiation can be used to estimate  above  more is  discussion,  energy  is  impossible  to  the  seismic  quite well. Therefore the rate the violence of failure.  Slip Behavior under Various Conditions / 108 The  instantaneous Wr  =  seismic energy rate is defined in equation (6.25) as  EoX . 2  There will be some difficulty in determining Wr in practice. Usually, the average rate over a period can be used instead. For a given period At, if there slips,  each of which has released  energy  W ., the total energy  released  are N in that  period will be  W, tr  =  N I ,W I 1 ri  (6.31)  =  Then the average rate of energy release can be estimated as  • W avg As  = W. /At = tr  N (l/At).E , W I 1 n  (6.32)  =  can be seen from above numerical results, the slip time T , usually is much  shorter than the stick time T . If T 2  average better  energy way  rate  to  cannot  do this  is  2  indicate to  look  is extremely high compared with T , , the the at  real  the  rate  energy  of  energy  during  the  release slip  well.  time  A  only.  Therefore, the average energy released per event, also called energy release ratio, can  be used as an alternative, which can be estimated as 1 = W /N = - ^ . Z . W . tr N i = 1 • ri N  W  avg  Therefore both W avg  (6.33)  :  and W should be used together avg °  in practice in order to r  estimate the rate of energy release with a higher confidence.  6.8.  SUMMARY  1.  In order to take into account all possible conditions in field during analysis of  shear behavior, several important parameters of rock properties, such as  cohesion,  coefficient  of  friction,  elastic  modulus  and  uniaxial  compressive  Slip Behavior under Various Conditions /  109  strength, are compiled from the previous publications. 2.  3.  The previous model has been completed by friction and seismic  effect.  The seismic  is  model  effect  and  the  considered  derived force  by  introducing slip-velocity  attaching  from  seismic  a  semi-infinite  radiation is  dependent  string to  the  proportional to  the  slip velocity but pointing to the opposite direction. 4.  The energj' changes  5.  To  analyze  Approach  is  purpose.  By  during a slip is calculated.  the  sophisticated  used  and  these  computer  programs,  environments is extensively 6.  model,  a  numerical  programs  the  are  sensitivity  method,  written  of  this  Runge-Kuta  specifically shear  model  for  this  to  the  analyzed.  According to the numerical results, the cohesion C has no effect on the slip behavior, normal  the  effect  load P  is  of  frictional coefficient  most  significant,  the  ju  is  g  elastic  negligible,  modulus E  the  effect  and the  of  driving  speed Vo rank in between. 7.  During each slip, the maximum slip distance X , the  normal  load,  does not change frequency  approximate  normal  load.  energy  release,  relation with  the  elasticity,  2  changes  and elasticity,  The  reverse  a linear relation with  with other factors. The stick time T , which indicates  of slippage  driving speed coefficient.  an  has  total  and is independent from cohesion  force  The seismic increases  linearly with normal load, reversely  drop energy, with  reversely with the elasticity.  only which  the  increases has  square  as  the  of  the  normal  the the  and frictional  increase  similar pattern  with  and  as  of the  pressure  the total and  Slip Behavior under Various Conditions / In stick-slip  summary,  this  under various  model  is  useful  practical conditions  in  studying  the  and consequently  slip  behaviour  provides  tool to find the conditions which may give rise to violent failure.  110  us  with  of a  CHAPTER  7.1. In  7. T R A N S I T I O N C O N D I T I O N S A N D V I O L E N T  FAILURE  GENERAL study  of  violent this  rock  research,  failure,  objective  of  violence.  These conditions are associated  stick-slip and stable the  conditions  defined  as  slippage.  time  Obviously, the  find  the  major  interests,  conditions  which  which  is  may  give  the  first  rise  to  with stick-slip and the transition between  the  slip  sense  to  stick-slip by examining  between  adjacent  smaller T  When  T =0,  make  much  peace  period between adjacent  2  to  of  sliding. From previous discussion, it is now possible to derive  which cause  the  is  one  is,  2  number may measure  the  slips the  slips  and  infinite.  behavior  actually  stick  time  indicates  more slips  become slip  the  by  does not  T , which  is  frequency  of  2  the  for a given  In  this  slip  time period.  case,  it  number,  exist.  does  not  because  the  The nature  of slip  has been changed and stable sliding occurs.  7.2. T R A N S I T I O N From by  CONDITIONS  previous chapter, it is known that the  stick time T  is strictly controlled  2  the loading conditions. It is very sensitive to the change of loading speed and  normal load. The elasticity  of rock has a close relation to T  2  as well. The other  indices of rock property seem not to have much effect on it, figures  6.6 through  6.10.  Any T . 2  The  change of the factors mentioned condition  under  which  T  2  above will introduce some change  becomes  zero  is  from stick-slip to stable sliding and vice versa. Because  111  critical  for  the  to  transition  the stick time is  affected  Transition Conditions and Violent Failure / by  more  than  any of the  one  factor,  this  critical condition  influencing factors,  such as  the  is  not  unique  loading speed  112  and varies  with  V , normal load P, or  the elastic modulus E .  To  study  MODEL3, Figure  the  possible  transition  conditions,  appendix 3, has been written in F O R T R A N  7.1  shows  the  program flow  chart.  In  sliding is considered to occur when T < 1 X 1 0 ~  this  the  the  approximation of the  analysis,  followed  in  modulus  E,  only  this  using  increased  and T  a 2  does not converge  and  set  to  major  modelling  frictional  calculated  actually  the  is  influencing that  coefficient  given  /u  initial  for  and  g  group  speed  is  slip velocity  of  data  load  P,  Vo.  X never  If  T  5  The  stable  the 2  value  is  too  in the  program instead  repeated  again.  Finally  a  of  elastic  of  T  big,  seconds. If the  of zero),  critical  During principle  consisting  decreases to zero  m/s  1 3  the  cost.  included.  2  or the  called  for this purpose.  computing  were  normal  loading  program  numerical model,  and the  factors  any  language  is calculated again until T < 1 X 1 0 ~  1X10"  computation  numerical solution  computer  seconds instead of zero, because  5  2  of  a  Vo  is is  solution  (this value  Vo is  loading  2  is  decreased  speed  Voc  is  obtained corresponding to T =*0 for the given condition. 2  Then one  of the  factors  conditions  are analyzed. During this analysis,  between  range 10  between 0.1  between Pa and and  0.95.  obtained.  or P is  another  the  is  g  sequence  in  Voc  E, M  1 GPa and 10  9  Pa, the  This  100 static  changed  process  and following  continues  until  all  the elastic modulus E is  GPa, the coeffecient  normal load of friction  P M  g  the  same  possible  considered  in the  range  in the  range  Transition Conditions  (start  ,  :  choose  call  SUB1  to  to  data  I compute  control  compute  113  )  function for shear change C and u  SUB2  set  call  input  and Violent Failure /  strength  initial  variables;  X i and  Xi  1=1  controlled  print  call  forces  initial  SUB2  print  by  solutions  *  to  accuracy  calculate  final solution print T 2  (  stop  Fig.7.1 Flow chart for program M O D E L 3 : transition  r  forces  ^  analysis  £  Transition Conditions and Violent Failure / The numerical results from program M O D E L 3 is  surprising to  effect  on the  roughness other  note that  transition  on the  factors.  the  frictional coefficient  condition.  stick  time  is  This is negligible  The effect of surface  are given  of  a  shear  in figure 7.2. surface  probably because the  effect  when  the  roughness  compared  with  has of  It little  surface  effects  on stick-slip observed  may be due to the asperity or unevenness of the  114  from  in laboratory  surface.  As expected, the loading speed V , the normal load P and the elasticity have  significant  given linear  value  of  effects on the transition. As can be seen from figure 7.2, E , the  relationship.  loading  In  order  speed to  V  give  a  and  the  clear  normal load  idea,  an  P  for a  have  empirical  E  a close  formula  is  obtained for this relation by linear regression based on the numerical data. Voc  =  (7.1)  kP  with correlation coefficient  r>  and a constant k:  k = 4.267X10~ ,  when E = l GPa,  k = 0.843X10" ,  when E = 5 GPa,  5  5  k = 0.100X10" , 5  The  0.998  upper part in  when E =  figure  7.2  40~100  represents  GPa.  the  stable sliding and the  part the stick-slip. If the conditions of loading speed and normal load fall  lower within  the lower part, the slip behavior will show stick-slip, otherwise stable sliding. The maximum value of Vo or the minimum value of P for stick-slip to occur can be read off on this transition chart. For instance, of  E =  10  GPa, under  a  normal load  found from point A in figure 7.2  P=10  6  if the elasticity Pa,  the  to be logVoc = 0.78.  of the material is  critical loading This means that  speed  is  stick-slip  Transition Conditions and Violent Failure /  TRANSITION  stable  115  CHART  sliding  CO  o > 60 O  i—I  T3 QJ CD  -2  CO 00 C •H CO O  -3 -4  -  <u =0.65 same curves f o r other nis s  -5 -  2  3 normal  Fig. 7.2  can  only  happen  4 load  5  10  6  (logP,  Pa)  Transition conditions for stick-slip and stable sliding  if logVo<0.78,  or Vo<6  m/s.  Otherwise,  stable  sliding  would  appear.  It should be pointed out that the slip behavior determined from figure applies  only  of the  shear  after  slippage  sj'Stem  is  is initiated.  Before  still controlled by  the  the  initiation of slip, the  shear  stress  and  shear  7.2  stability strength.  Transition Conditions and Violent Failure / Besides,  this  chart is  obtained  based  on  general  analysis.  Therefore  it  can  116 be  used only as a guide line in practice and would apply for a particular case only if it has been calibrated correspondingly.  It is E,  say  figure  E<40 7.2.  transition little  interesting to notice the effect of the elasticity.  When  line  effect  clearly,  GPa, this is  exists.  on  the  E  T  data  effect  in  above  This  when  2  is  apparent and a 40  GPa,  is  actually  effect E>40  figure  7.2  this  transition zone  effect  clear  in  GPa. In order to is  replotted  in  For a low value of is  disappears figure  show  another  6.8,  formed in  and  only  where  E  a has  this  effect  of E more  way,  figure  7.3,  stable  sliding and  where  there is no change in the V - E curves when E > 4 0 GPa.  7.3.  SLIP B E H A V I O R  IN S H E A R  TEST  Slip behavior in shear tests generally falls into two categories: stick-slip. factors  The  [29],  characteristic  the  most  of  sliding  important of which are  the testing  machine and loading speed.  occur  complex  are  depends  and  are  derived  in  a  the  complex  normal  way  pressure,  on  many  stiffness  of  The conditions under which stick-slip will in  previous  section.  These  conditions  are  combinations of above factors.  The  modelling results  given  in figure 7.2  show  very well the  phenomena  observed in laboratory tests. In experiment the stick-slip is generally enhanced by higher  normal  testing  machine  For  instance,  pressure [39].  [29],  lower  The effect  surface  of normal  at points B and C, the  roughness pressure  and lower is  stiffness  of  confirmed in figure  the 7.2.  slip behavior is different at two levels of  Transition Conditions and Violent Failure /  TRANSITION  117  CHART  u =0.65 same curves for other u< s  logP=8 logP=7 logP=6 logP=5 logP=4 logP=3 logP=2 logP=l  3 4 5 6 7 elasticity E (10 GPa)  Fig. 7.3  normal load,  10  Transition conditions showed as loading speed against  pressure  stick-slip  when  other  conditions  occurs. On the  unchanged.  contrary, stable  A t point  11  elasticity  B  of  sliding takes place  high  normal  at point C of  lower normal load.  The verified.  fact  that  lower  machine  stiffness  will  enhance  the  stick-slip  can  be  For a given loading condition and rock specimen, which correspond to a  Transition Conditions and Violent Failure / position in the transition chart, say point A in figure 7.2,  if the testing  118  machine  is "soft" with stiffness of 1 GPa, apparently point A falls into the lower part of the  transition  very  high,  sliding  chart.  say  would  50  Then  stick-slip  GPa, point  happen.  occurs.  A jumps  However,  the  On the into  effect  the  of  contrary, if the upper  stiffness  part.  stiffness  Therefore  disappears  The change  effect  of the  surface  is found in this  roughness M  model when  is probably because of the following 1.  The effect when  of  surface  compared with  varies,  be  as  verified  here  E  is  for rock.  because  indicated in figure 7.2.  no This  reasons:  roughness the  G  cannot  stable  when  above 40 GPa until 100 GPa, the possible maximum value of elasticity  is  is  very  of  the  effects  small  within  the  normal pressure  modelled  and the  range  machine  stiffness. Therefore, it may be shadowed by the latter. 2.  The approximation in the numerical solution may bury this small effect.  3.  This effect observed unevenness of the  In  addition,  in experiments  become  asperity and  surface.  significant  effects  research. As shown in figure 7.2, eventually  may be actually from the  stable  sliding  from  loading  speed  are  observed  in  for a given normal load, the slip behavior will if  the  loading  speed  continues  to  increase.  other words, stick-slip can always occur if the loading speed is sufficiently  It conflict  is  further  noticed  with  the  conclusion  time-dependent  stick-slip  that  occurs  this  the  by only  numerical  Engelder if  the  and  results  in  figure  Scholz(1976)  normal load  is  7.2  [46]  sufficiently  In  low.  are  that large  in the to  Transition Conditions and Violent Failure / cause  cracking during static  contact  and  that  the  normal  stress  at  the  119  stable  sliding to stick-slip transition corresponds to the minimum normal stress to cause asperity  indentation  Dieterich's  (1978)  and  [39]  ploughing.  conclusion  pressure if both the loading speed words,  as  long  as  the  However,  that  the  stick-slip  OCCURRENCE  combination  of  the  to  because  the  OF VIOLENT  even  will eventually failure  energy in  a  release  massive  lead to the  takes place.  can  agree  occur  loading conditions  at  with  normal  low.  In other  the  specimen  possible.  FAILURE  at  failure  rock,  as  and  discussed  rock or along a fault is can  be  associated  before,  the  slip behavior  with  fracture  formation of a macro-fracture surface  Therefore the  well any  and  stick-slip is always  Violent rock failure occurring either in a massive related  results  and the stiffness are sufficiently  properties falls in the lower part of figure 7.2,  7.4.  these  on a surface  closely  stick-slip  development  on which final  may  be  a key  to  violent failure.  As  slip takes place,  determined stress there  from  remains is  a  more  chart  whether like  or less  no extra energy  stable sliding or stick-slip will occur can be  figure the  7.2.  same  For stable as  slip  continues,  accumulated during the  is controlled by the loading speed.  sliding, because figures  sliding process.  the  4.5  shear  and  The slip  Therefore, violent failure is not possible  5.2, speed  unless  the loading condition is changed.  For during the  stick-slip, the  situation is completely  quiet period is released  different.  The energy  accumulated  at slip. A sudden slip or any change  of the  Transition Conditions and Violent Failure /  120  loading conditions can cause violent failure. According to the physical conditions of the shear process and the transition chart, violent failure is expected  to occur in  the following 3 cases:  Mode I. The violence The  more energy  that  is  is from  released  a sudden slip under high normal  at  each  slip, the  damage could be. This energy released increases P  as  P  increases,  figure  6.9.  It  is  also  bigger  with the  noticed  that  the  failure and the  square of normal load the  higher  the  energy release, the more violent the failure is. When the loading speed the  stick time decreases,  energy rate the  is  released  of energy normal  during a  release  load  or the slip frequency increases,  and  given  increases the  time  period.  figure 6.10.  By equation  (6.32),  correspondingly. Therefore the  loading  speed  could increase  pressure.  the  rate  of  increases,  Then more the  average  increase  of both  energy  release  and  release rate and consequently  increase the incidence of violent failure. When both  the  loading speed  normal  load P  and the  V  are  low,  the  failure may  by  seismologists  be  not  interpret  the  violent at stick-slip.  The  I  violence  has  shallow  earthquakes  along a  natural  be  to  due  earth  mode  sudden  intends  to  slips initiate  in the  been  fault  crust.  relative  used  [11].  Because  movement  in  These the the  to  quakes  are  interior stress crust,  the  gradually builds up, which may be a result of many decades of  movement  along  a  fault.  When  this  crust, it is released by a sudden slip.  energy  can  no  longer  considered to field strain  or even be  held  in  the  energy centuries in  the  Transition Conditions and Violent Failure / Mode II.  The violence  comes from  the  transition from  stick-slip to  121  stable  sliding. For a given situation of stick-slip, if a change of any factor results in a transition suddenly from stick-slip to stable  sliding, extra energy will be released.  This energy has to be released at the transition point in order to keep up with the  sudden  change  of  could  happen,  as  case  suddenly  drops  which  loading  speed  goes  figures  10.6  to  acoustic  emission  conditions, shown  consequently  in  means  figure  the  7.2,  sudden  up abruptly. Typical  10.17,  where  this  when  examples  transition  in  either  reduction of  tests of shear experiment  uniaxial compressive test was  resulting  will  effect  violent the  shear be was  failure.  normal  pressure  resistance,  given  This  or  the  in chapter  10,  observed  during  the  and a bang similar to that from  a  experienced when the normal pressure was reduced  suddenly to zero at the initiation of slippage. The corresponding acoustic emission peaks up sharply at this transition.  In the field of mining, excavation may cause stress increase in some part and  stress  exists  in  decrease the  in  other  vicinity,  the  part of  mode  II  the  rock  violence  mass. may  If  occur  a  major discontinuity  as  a  result  of  this  transition. This will be discussed more in chapter 9.  Mode  III.  The violence  the slip behavior is stable  occurs under sudden loading. No matter  whether  sliding or stick-slip, violent failure is bound to happen  if a shear force much higher than the shear strength is suddenly applied to the system.  Because  Obviously, example  the  extra  higher  the  potential extra  energy shear  is  force,  always the  available  more violent  mentioned above of quick reduction of normal pressure  in the  this  case.  failure. The  at the initiation  Transition Conditions and Violent Failure / of  slippage  example  can  be  would  be  considered the  as  violent  a  kind  failure  of  indirect  sudden  of rock specimen  loading.  in uniaxial  122  Another  compression.  More about this will be given in next chapter.  It usually shear  should be  the  noticed  that  case in experiments  stress  has  reached  if the  shear  and in practice,  or exceeded  the  are only two possible modes of violence,  only  occurs  excavation stress  special  conditions,  such as blasting can create  state  on  a  fault  starts  from zero,  failure happens  shear  there  under  stress  strength.  which  only  when  Under this  the  condition,  namely Modes I and II. Mode III  which  do  exist  in  mining.  A  this kind of situation, especially  or a joint plane  is  in the  vicinity  sudden  when  of excavation  the  changes  abruptly.  7.5.  SUMMARY  1.  Based and  on  the  stable  numerical model,  sliding  are  studied  the by  transition examining  conditions the  case  between of  zero  stick-slip  stick  time  under a variety of conditions and a transition chart is obtained. 2.  Significant normal  pressure  coefficient 3.  From can  effects  the be  on and  the  transition  elastic  are  modulus  of  found the  from rock,  the but  loading little  of friction. The condition for stick-slip varies with above transition defined:  chart and physical conditions,  Mode  I  is  from  the  sudden  three slip  from  the  factors.  modes of  under  speed,  high  violence normal  pressure, Mode II comes from the transition from stick-slip to stable sliding and Mode III occurs under sudden loading. 4.  According  to  these  modes,  the  violent  failure  of  rock  both  in  laboratory  Transition Conditions and Violent Failure / tests  and  in  the  field  location and rock type.  can  be  adequately  interpreted  regardless  of  123 the  CHAPTER  8.1. In  8. E F F E C T O F S U D D E N  GENERAL studying  the  three  modes  effect  of sudden  conditions  of violence  which  defined  "sudden loading" here  from  zero  applied  to  its  instantly  strength  is  may  give  in the  loading. This effect  term  the  LOADING  is  rise  previous  to  violent  chapter  examined  to the  case where  maximum value  in an  extremely  available.  case where In the  a shear  following,  force  the  in this  time  which is  effect  of  the  from  the  chapter. The  a shear force  short  one  exclusively  in detail  refers  and the  is  failure,  or  is  increased  this  force  much higher  of sudden  is  than  loading on  the  slip behavior is discussed in detail.  In maximum  laboratory tests, value  the  shear  and can never  force  exceed the  is  usually  shear  initiated.  However, in mining, it may happen that  such as  at the  the  shear  force  stress adjustment is  applied from  strength a force  zero  much when is  strength  at the  the  sliding is  applied very  near. an opening right after blasting,  much higher than the  to  fast,  and that  sliding initiation, such  as at the stress change on a geological fault due to mining, or at the failure of a  rock  specimen  more than from  in compression. the  static  In these cases,  the  loading and dynamic effect  cause a change of the slip behavior.  124  effect  of a  appears.  shear  force  This effect  is  may  Effect of Sudden Loading / 8.2.  THE EFFECT  Suppose  the  OF EXCESSIVE  shear  strength  of  a  125  LOAD system  is  f(0).  The  minimum  shear  force  required to initiate the slip would be Fo = f(0).  If the shear force F < F o = f(0),  the  slip  previously.  slip  behavior  will  be  the  same  as  discussed  If  Fo/f(0)al,  the  behavior can be analyzed using the numerical model.  During the numerical analysis, the effects of various ratios of Fo/f(0) tested using the computer program M O D E L 2 . Fo  were  By changing the initial shear force  into various values for a given f(0) during different runs, we can look at the  change  of  all  respectively. seismic  slip  The  energy  parameters.  final  results  Wr, etc.  are  The  of  ratio  all  plotted  slip in  of  Fo/f(0)  was  parameters,  figure  8.1.  such  set as  The effect  to stick  of  1  to  time  this  11 T , 2  ratio on  each parameter can be clearly seen.  It is interesting to notice that the slip time T , ratio of Fo/f(0)  at all. All other parameters  most  have  of  release  them varies  linear  relations  approximately  with  with  does not change with the  are very sensitive to this ratio and Fo/f(0),  [Fo/f(0)] .  By  2  whereas nonlinear  the  seismic  regression  energy of  the  numerical data, an empirical formula for Wr is derived: Wr  «  -0.150  with correlation coefficient standard deviation Sd  It would  is  expected  increase  when  x  +  r =  0.046[Fo/f(0)] 0.999, and  : Fo/f(0)±6.73,  that the  (8.1)  2  Wr±2.08.  the  energy  release  ratio  Fo/f(0)  goes  up.  and  the  But the  maximum slip speed  of  distance  increase  for  Effect of Sudden Loading /  V =10 m/s u =0.65 =300  126  7  Q  s  3  AE(J)  * T , (0.1ms)  3  4  5  r a t i o of  6  10  7  11  12  13  14  F /f(0) o  Fig.8.1 Variation of slip parameters with the ratio of initial shear force over the shear each  parameter  corresponding The  seismic  is  curve. energy  different.  This  The steeper increases  the right hand column of table 8.1  Therefore  when  a shear  speed  the  with  strength  slope  is  indicated  of  the  by  curve, the  an increasing speed,  figure  the  slope  higher 8.1.  the  of  the  speed.  The data in  shows the actual slope of each parameter.  force  greater  than the  shear  strength  is applied  Effect of Sudden Loading / to  a  shear  The  system,  changes  it can result  of these  shear force over the the  same  much  way  as  portion of the seismic  change  in the  shear behavior.  slip parameters are much larger than the strength, or Fo/f(0). The seismic  the  higher speed.  in tremendous  total energy  This  energy  means  released  release  during the  shear process  dynamic effect.  It is  this dynamic effect which produces vibration in the  At  time,  particles.  Therefore,  the  seismic higher  and consequently  case,  more  it  becomes  total energy  energy  the  vibration will be  Fo<f(0).  sudden  the  conservative  is  to  This  has been converted into  in the  the  is  probably due  propagated loading  more seismic use  at a  case of excessive loading, a larger  than  same  where  of the  it changes  energy  the  case  change  energy does not change in  any more. Instead,  that in the  127  the  is,  through the  energy  seismic  energy  the  system.  vibration  of  intense  the  radiated. In  this  more is  to  to  estimate  the  release.  Table 8.1 effect of sudden loading on slip behavior Fo/f(0) T , (0.1ms) X , (lO/jm) T (100s) Xmax(m/s) A F (MN) A E (J) Wr (0.01J)  1  2  3  7  5  9  11  slope  *  .13506 .11076  .13402 .82216  .13402 1.5315  .13401 2.9493  .13041 4.3669  .13401 5.7843  .13401 7.2016  0° 35.34°  .12969 .06062 .18226 .00129  .96282 .45219 1.3828 .07152  1.7947 .84231 2.5095 .24838  3.4566 1.6221 4.8359 .92130  5.1181 2.4018 7.1686 2.0201  6.7795 3.1814 9.5083 3.5444  8.4408 3.9609 11.856 5.4943  39.73° 21.30° 49.37°  2  note: the slope is obtained from linear regression of each curve. * : Wr varies with the square of [Fo/f(0)] 2  **  Effect of Sudden Loading / 8.3.  OCCURRENCE  As  OF SUDDEN  occur.  In  applying way,  Therefore, it is  laboratory a  large  such  initiate.  as  For  shear  force  releasing the  the  former  shear  force  manually  as  gradually quickly  sudden  suddenly  loading  to  the  normal pressure  case,  a  large  can  be  obtained  by  system,  or  through  some  quickh' when  force  applied  same  as  be  occurring  at  that  violent  the  in  other  is  other  about  to  obviously  same effect can be  achieved.  tests were carried out during the acoustic  emission  reached  latter case, the  as  the  possible.  from conventional  presented  slip  simply  The shear tests were done as usual. However when the  same time, a sharp increase will  the  suddenly  strength, This  the  normal  pressure  was  carried out  experiment  levels of normal pressure. In each test, violent failure was the  sudden loading can  will  For the  this research, some shear  study of rock specimens.  sudden loading has significant effect on the  important to understand when  experiments,  produce a dynamic effect. In  LOADING  can be seen from above analysis,  slip behavior.  128  of acoustic  figures  stages  unconfined  are  10.16  activity was  and  completely  compressive  10.17, masked  failure is not unique to the  was  released  at  different  observed  test was  and a bang  heard.  At  the  recorded. These test results  where by  this  conventional  all  acoustic  increase. unconfined  emissions  This  means  compressive  test and it can also happen in the shear test. Further more, it implies that the violent failure of a rock mass is not only determined by the internal property of the rock mass, but also by the loading conditions.  In such  as  excavated  mines, blasting.  sudden  loading  Because  in  may  this  happen  case  the  right load  after  an  previously  abrupt  excavation,  supported  by  the  rock mass has to be undertaken by the rock mass around the opening  Effect of Sudden Loading / before built  a new  equilibrium of  up instantaneously  stress  in some  can  be  area.  reached.  If  this  Therefore  stress  is  a  high  129  stress  much higher  is  than  the  strength of the rock mass and if the loading is finished in a very short time, a dynamic effect appears and violent failure may be caused.  This is  specially  true  if a geological weakness , such as a fault exists in the vicinity of excavation, or a  newly  fractured  mining  activity  during  the  very  low  surface  could  stress level,  is  result  formed  in  a  the  sudden  redistribution, the  consequently  in  causing  highly  excessive  stressed  loading  normal  pressure  is  sudden  reduction of  rock  mass.  that  surface  on  reduced the  abruptly  resistance  The if,  to  on  a  that  surface.  8.4. To  OCCURRENCE apply above  Violence  is  specimens. tests  a  results,  the  commonly  FAILURE  IN C O M P R E S S I V E  violent failure of rock specimens  observed  in  the  conventional  is  a  close  such  as  during compressive  relationship  noise  considered here.  compressive  and  and  that  fracturing  at  they  both  failure.  exhibit  The  of rock  and shear  similar  occurrence  physical  of  violence violent  before.  If a weakness exists in a rock specimen, it is very possible  the  test  testing can be interpreted in the same way as for the  shear failure discussed  take  TEST  From this research it has been found that the compressive  have  reactions  OF VIOLENT  for failure to  place along that weakness if the direction of loading is not perpendicular to weakness  plane.  weakness  is  to  the  possibility  of violence  The failure  failure  may  surface  exists. This  as  be  gradual  determined  is because if the  or in  violent. figure  The  3.4,  closer  this  the  less  the  weakness coincides  with  the  Effect of Sudden Loading / predicted surface, at  static  the  loading on  whole that  process  may be  weakness.  In  controlled by the  this  case,  as  the  130  shear mechanism  load  increases  from  zero, the failure may take place as a smooth sliding or small stick-slip.  If existing be the  the  rock  weakness  different.  specimen plane is  A t the  specimen.  does  not  contain  any  perpendicular to the  the  stress reaches  some  will  level,  build  Once  a  process  occurs  macrofracture surface  and  the  failure  is  mechanism  or  formed  in  discussed  if  the  result  will  up uniformly within  fracturing  accompanied by acoustic emission. As loading continues, formed.  weakness,  loading direction, the  beginning of loading, stress  When  major  initiates,  which is  a fracturing zone will be the  fractured  before  will  zone,  apply.  On this  newly fractured surface, the shear stress is close to the maximum shear figure 3.4  and makes an acute angle of /3 = ± ( 4 5 ° — 0/2),  shear  stress,  equation (3.1), with the  major principal stress a ^. Because the rock specimen is not perfectly intact, this angle actually would vary somewhat from the theorectically predicted value of p\  Upon  the  formation  of  the  fracturing  surface,  the  instantaneous  and shear stresses acting on it can be estimated by these two equations  normal [26]:  a = a, cos a + a sin a 2  2  3  ^  r  =  -i(o^  -  0 )sin2a  (8.2)  3  A t the very moment when the failure surface is formed, a ^ equals to the peak compressive strength.  This the  shear stress  fractured  surface  at  r  can be much higher than the  this  moment.  It  is  the  shear  excessive  strength  shear  stress  T  G  on  which  Effect of Sudden Loading / introduces  the  effect  excessive  stress  of  dynamic  loading.  is not extracted quickly  Violent  failure  takes  enough at that moment.  place  131  if  this  The mode III  violence of failure defined in chapter 7 refers specifically to this type of failure.  As chapter  an example of above statement,  10  breakage  are used  angle  cylindrical  for  in the each  specimens,  possibly  caused  breakage  angle  the  #2  following. The mechanical properties and the actual  specimen  are  and  specimens  #3  disparity of  of the  the testing results to be presented in  the  given  in  table  had  some  compressive  failure surface. It is  10.1.  Among  tiny  strength  the  three  microcracks which  o~  c  and changed  very difficult to determine the  the real  frictional coefficient ju on the failure surfaces of those specimens. However, these g  surfaces  are similar to the chisel-cut surface. Then the results from direct shear  test of breakage  surface  given  in table  10.2  can be  used,  where  the  empirical  formula for the shear strength of the natural breakage surface is T  G  =  0.0144 +  0.58323a  (8.3)  During the uniaxial compressive test, a = 0  and by figure 3.4,  3  Using  the  stress  T on the  strength  of  data  the  in  table  10.1,  failure surface failure  surface  the  instantaneous  this  ratio,  the  intact, it has  more  a  and  shear  calculated by equation (8.2). The shear  can  estimated  be  all ratios of T/T  &  violent  stress  can be  results for the three specimens are listed in table  As can be seen,  normal  a+/3 = 9 0 ° .  the  failure.  by  equation  (8.3).  The  final  8.2.  are above one. Apparently, the higher Because  the  specimen  #1  is  almost  the highest compressive strength. Its  real  shear stress is as high  Effect of Sudden Loading /  Table 8.2 specimen No. 1 2 3  as  stress estimation on failure surface of rock specimen in compression  /3  a  deg.  deg. 64.244 43.943 50.492  25.756 46.057 39.508  3.55  times  the  "1  ° 3  a  ksi  ksi  ksi  ksi  S ksi  3.4855 5.8688 6.006  7.2242 5.6561 4.6335  2.0329 3.4228 3.5029  18.459 11.320 9.44  estimated  0 0 0  shear  strength  definitely made the failure violent. In fact, of this specimen. Even for the the ratio T / r However,  g  failure  was  much  less  are clearly indicated by the  and  where  8.5.  on the  T  T/T  failure surface,  a big bang was  specimen #2  phenomena  specimen #1  |r|  and #3,  s  3.5540 1.6525 1.323  which  have  heard at the failure  which are not very intact,  is also above one, which can still increase the violence at failure.  their  10.3,  132  both  event  rate  than from specimen  strong acoustic  and  energy  than  the  emission rate  specimen  #1.  These  shown in figures are  much  higher  10.2 from  #2.  SUMMARY  This chapter specifically deals  with the problem of sudden/excessive  loading. It is  found from above analysis that: 1.  Violence always  occurs at failure of rock if a large shear  force is applied  to the shear surface suddenly. 2.  The effects extensively  of sudden loading on the  shear  behavior during each  studied using a computer program.  In all cases,  the  slip are  change of  slip behavior is much greater than the change of ratio of shear force over the strength.  Effect of Sudden Loading /' 133 3.  The conditions which are likely to give rise to sudden loading in laboratory tests and in situ are discussed and a typical example of a compressive test is given to show the violent failure due to sudden loading.  CHAPTER  9.1. In  9. T H E N A T U R E  OF  ROCKBURSTING  GENERAL the  previous  conditions above field.  which may  results From  happen  will  be  previous  under  conditions  chapters,  three  consist  of  a  mechanism  give  rise  used  to  analysis, modes, rock  to  of  violent  rock  failure  failure are  interpret the  violence  it  found  has  been  of  which  each  properties,  occurs  loading  with them. Violent rock failure may therefore  speed  is  postulated  obtained.  of rock  this  the  chapter,  failure occurring in  that  violent  in  certain  and  In  and  rock  failure  conditions.  stress  state  can  These  and vary  happen in any mine rock as long  as the critical conditions are present.  The  most  important factors  normal pressure on the  contributing to  rock surface,  rock. The previous modelling results increases  when  the  normal  stress  the  critical  the loading speed  conditions  are  the  and the elasticity of the  indicate that the possibility of violent failure goes  up  speed becomes very high (Mode II violence)  (Mode I violence),  when  the  loading  or when sudden loading occurs (Mode  III violence). Therefore violent rock failure would be more likely to happen in the stress  concentration  excavation this  face  problem  is  or  zone, any  expected  such  as  a  pillar,  irregularit}' where in  places  close  the  stress to  corner  of  concentration  geological  an  opening,  exists.  structures,  the  Similarly,  such  as  a  natural fault, dyke, intrusion, etc. A t the same time, rapid stress change brought about, for example,  by blasting and sudden change of stress state induced, such  as,  also  on a fault will  increase  the  incidence of violence.  In the  following,  issue of how violent rock failure occurs in given conditions is discussed.  134  the  The 9.2.  VIOLENT  When  ROCK  a major  FAILURE  geological  ALONG  discontinuity  the rock mass will be completely  Nature of Rockbursting /  A NATURAL  exists  in  the  135  FAULT  rock mass,  controlled by it. The geological  the  stability  of  discontinuity can  be a big natural fault, or a small joint, even a bedding or foliation.  A loading  big natural fault can be considered condition  is  stress  components  stress  T on the  very  can  similar  always  fault plane  indicated in figure overburden, the  9.1.  to  be  that  These  of  resolved  no matter  as a shear model. In this case, the shear  into  what  stresses may  test.  the  The  three-dimensional  normal stress  a  and  shear  the orientation of this plane be very  high due to the  residual stress from tectonic movement  is,  as  gravity of  or mining induced stress  concentration.  As elasticity  an opening is excavated and rock mechanics  [47],  state of stress will be created. zones, figure  9.2,  i.e.  the  in the  rock mass, according to the  the virgin stress field is disturbed and a new  This new  stress field  the undisturbed zone III. In zone I, the  the  supporting  higher  than  the  on the  virgin  can be  divided into  three  stress relaxing zone I, the stress concentration zone II  and  force  theory of  opening  stress  and  face. is  stress decreases due to removal of  In zone II,  possibly  above  the the  stress becomes much strength  of  the  rock  mass.  and  Any  mining activity  shear  stresses due to  case of stress change  close to the  the  fault  will probably increase  stress concentration.  due to mining activity  near  Figure 9.3  the  shows  a natural fault.  normal  a typical  Because  the  The Nature of Rockbursting /  Fig.9.1  Stress components on a natural fault in the rock mass  136  The Nature of Rockbursting /  137  Fig.9.2 Stress redistribution after excavation of an opening in the rock mass  rock mass is usually interrupted and cut into blocks by joints  and fractures,  the  high shear stress may cause shear failure along this fault by pushing the highly stressed  block(s)  towards  a  free  surface,  figure 9.4.  As discussed in chapter . 7 ,  I  may  violence  occur,  because  stress. In this case, a single energy. openings.  Successive  slips  may  a  or  towards  the  low  when the failure occurs as  violent  slip  can  happen  stress  direction,  stick-slip, Mode  under  high normal  slip may give rise to violence due to the release of result  in the  complete  destruction  of underground  The Nature of Rockbursting /  138  .HI/' / 1  ... , / / / /  / / / / / /  Fig.9.3  Streamline  of stress change due to mining activity  adjacent  to a fault  The Nature of Rockbursting /  139  sliding direction  Fig.9.4 Possible sliding of highly stressed blocks  Even  if the  concentration, excavation within decrease  violent  passes  the  and  the the  not high enough  failure is  through  stress  Consequent^,  stress is  or  relaxing shear shear  still is  zone,  stress stress  possible  very the  to cause failure during the at  close  to  normal  may  increase  would be  the  stress  the  fault  stress at  relatively  on  process  of stress redistribution, the  total  shear  so  that  the  the  same  high  with  being relatively low, in which case, the frictional resistance the  relaxation.  When  the  fault time, the  stress the  fault  is  plane  will  figure  9.5.  normal stress  would drop. If during  resistance  dropped far  below  the shear stress, failure would occur. If this drop were big enough to induce the transition  from  stick-slip  excessive  shear  stress  to  being  stable  sliding  available  to  as cause  discussed a  sudden  in  chapter slip,  the  7,  with  effect  of  The Nature of Rockbursting /  Fig.9.5  Stress change due to the existence of an opening around a fault  140  The  Nature of Rockbursting /  141  dynamic loading appears and the Mode II violence could happen.  9.3.  R O C K B U R S T I N G IN A M A S S I V E  ROCK  MASS  If the rock mass does not contain any major weakness, mass the  still depends  strength  on  of the  the  rock  mass  However,  due  to  the  minor joints,  rock mass will be lower or much lower than that of intact  rock. For heavily jointed rock mass, homogeneous  itself.  the stability of the rock  isotropic system  containing some joints,  its  the  rock  mass can be  with much lower strength  strength  treated  [42].  similarly as  For the  will be between that of the  rock mass  intact rock and  that of the heavily jointed rock mass.  When the mass,  stress in the  rock mass has  reached the  strength  of the  rock  failure will occur along a failure surface, which as discussed in chapter 3,  can  be  a joint or a newly  the  failure  massive  process  rock,  corresponding violence  is  is  controlled  however, shear  most  the  surface  is  shear  strength  where  on  from  known.  the  shear  force  the A  A t the  is  failure  initiation of this  mechanism. usually  surface.  typical example  In  much  the  is  the  case  higher  Therefore  surface,  the  of  a  than  the  Mode  III  loading of a  rock  the loading condition is very similar to that of uniaxial  compressive test. The shear strength  by  likely to happen.  pillar, figure 9.6,  shear  fractured surface.  stress can be estimated  equation  For a  (4.6a)  hard  rock  if the with  from equation  frictional coefficient high  elasticity  and  (3.5)  and the  the  fracture  for high  compressive  strength, the ratio of the shear stress over the shear strength will be far above one.  As discussed  uniaxial  in chapter  compression  8,  the  rock  and failed violently,  specimen #1, has  a value  which was of above  tested under  3 of this ratio,  The table the  8.2.  In  excessive  this  case,  shear  Mode III  stress  at the  violence  Nature of Rockbursting /  occurs due  to  the  142  dynamic effect of  initiation of failure. Obviously, the  higher the  compressive strength and this ratio are, the more violent the failure would be.  The 9.7  can  case of a be  stope face  treated  development,  a  pair  as  a  or sidewall of an opening as  semi-infinite  of conjugate  shear  pillar.  surfaces  As may  a  shown  result  be  in figure  of  developed  fracturing first,  which  would make a V-shape in the section view. Under the high internal stress field, this  wedge of rock  outwards.  may  be  suddenly  As this block of rock moves  pushed  out by  out, the  very  fast  and  produces  geometry, the finite element  violence  resultant  resistance  to the loosening of contact on the failure surfaces. process  a  force pointing  decreases  quickly due  This would make the failure  according]}'.  For  the  more  complex  and boundary element methods will be a big help in  estimating the stresses.  9.4. I N F L U E N C E The  OF OTHER  GEOLOGICAL  STRUCTURES  presence of regional geological structures results in uneven distribution of the  stress field. results  In the vicinity of a fold,  of tectonic  movement,  the  an anticline or a syncline, which are the  stress may be higher than in areas far away  from them due to the possible residual tectonic stress and some be heavily crushed. A zone gets which  close will  to  these  increase  areas,  of crushed rock can act as the  both the  along those crushed zones.  mining openings  difficulty  may  rock mass  a weakness.  undertake  of supporting and the  a  may  As mining heavy  violence  load,  of failure  The Nature of Rockbursting /  Fig.9.6 The loading and the failure path of rock pillar  143  The Nature of Rockbursting /  Fig. 9.7  On the as  the  The loading and the possible failure path of a working face  other hand, if the country rock contains  dyke, sills or any other  them  during the intrusive  144  stress  just  from excavation.  acts  intrusive, the  redistribution as  a  "stiff  some hard inclusive,  stress will certainly  due pillar"  Therefore even though the  to  mining activity.  and  undertakes  concentrate As  the  in  around  figure  9.8,  load  first  extra  average stress estimated is  low, violent failure can still happen. In this case, the  such  relatively  failure is characterized by  \  The Nature of Rockbursting / a  large  number  chapter  13.  of  The failure  The first instance stress.  events  under  continuity  which  is  condition can  discussed happen  will be  the  failure occurs.  same  as  more  in two  detail  possible  stress field, the  intrusive  will  be  probably  fail  near  loaded to  by or  inclusion. As excavation  a  at  case will be  inclusion.  country rock which is  relatively the  high  interface  in  ways. high  a rock pillar and Mode III  The second  rock in a verj' thin zone around the  of the  in  will be the failure of the inclusion itself under extremely  is expected when  country  [48],  this  The process of failure  violence the  seismic  145  stress  In fact,  the  failure of due to  much weaker  within  between  the  this  country  thin  the  than  the  zone  and  and  the  rock  reaches the inclusion, failure may take place first in this  thin zone.  In  general,  any  regional  geological  structure  concentration will increase the possibility of violence  9.5.  INFLUENCE  O F MINING  As discussed before, important discussed  factors  violent  In  longwall  pillar between two very  short,  stress  rock failure.  In  and the loading speed are  addition to  the  geological  conditions  significantly.  and Size of a P i l l a r  In mining, pillars are opening.  to  CONDITIONS  above, mining conditions can also affect these factors  9.5.1. The Shape  rise  at failure.  the normal pressure on a surface  in  giving  usually  used  to  mining, pillars longwall  in square  faces.  are  support the usually  overburden or to  very  long,  such  as  In room and pillar mining, they  or rectangular  shape.  Because  of the  variety  protect  an  the barrier are of  usually  geological  The Nature of Rockbursting /  146  Fig.9.8 Stress redistribution due to mining around a hard intrusive  conditions reducing  and rock t3'pes, it is difficult to the  unnecessary  If  the  the  entire  area  the  pillar the  stress  rockbursting.  is  so  matter  large will  concentration  pillar  undertakes  supporting  of  The  the  pillar design  of  thumb  rule  is  in terms  to  reduce  of the  stress concentration.  concentration, that  incidence  specify  ability  will be higher and  that  not  be  its  center  is  not  important.  If  the  on both sides overla}'  important.  Apparently, the  stress and is easier energy  storing  to fail  capacity.  affected pillar  each pillar  by  size  other, with  the is  the  so  a  large  smaller  pillar  small  behavior of  and larger pillar has  When  stress  loading higher  fails,  the  The Nature of Rockbursting / failure  may  be  more  which is completely  violent  than  a  applied on the  small  pillar  because  failure surface  of  at the  higher  total  147 load,  initiation of failure. It  might be better in this sense to replace a large pillar by two  small pillars with  the same loading area in designing temporary pillars.  Pillars pillar  has  width  of  with  higher a  same  loading  stability  than  pillar  is  above  areas a  some  can  also  taller  pillar.  value,  say  behave  When 2,  differently.  the  the  ratio  pillar  of  can  A  shorter  height  over  behave  in  a  completely different way. In this case, the pillar may act as a bar and is likely to fail  in buckling. If a large  load is  available  at buckling, violent  failure can  also be expected.  9.5.2. M i n i n g  Rate  The  loading speed  this  loading  continuous  is  speed  another  is  excavation,  directly the  a result of self-adjustment speed zone  of excavation will just  stress  may  factor  related  in violent  to  the  rock  speed  stress concentration zone  failure. During mining,  of  stress  ahead will  forward higher  smoothly  and  higher  stress movement, as as  the  the  excavation  excavation  of  failure.  If  a  move  large  stress  is  available  under  the  During  forward as  If the  advance  stress concentration  continues.  advances  failure takes place. In this case, the higher the excavation risk  change.  and the creep effect of the rock mass.  is lower than the  move  become  key  speed, high  and  Otherwise, eventually  the higher the loading  speed,  violent failure can arise.  In  the  case  of drilling  and blasting,  the  excavation  is  discontinuous  and  The failure  usually  happens  at blasting.  loading on the adjacent rock mass. mass,  Here, each  Nature of Rockbursting /  blasting  results  If a blasting involves  in  148  instantaneous  a large amount of rock  a very high load will be transferred to adjacent rock mass and the  speed  of stress change will be very high as well. Then failure can be violent. In order to  reduce  the  risk  of rockbursting, it may  area on several separate  occasions  therefore  be better  to blast  a  large  than in a single large blasting.  Similarly, if two mines have the same daily production but the number of mining faces is different, the mine with less mining faces would have  the higher  possibility of violent rock failure due to higher advancing speed of mining.  9.6. In  ESTIMATION order  should three  to  be  prevent  estimated  major causes.  each cause  The pressure.  first.  control As  a  rockburst,  discussed  Therefore the  Mode I violence If the  in  the  possibility  chapter  possibility  7,  the  of occurrence  of  its  violence  occurrence comes from  can be estimated  for  is from stick-slip on a major discontinuity under high  failure appears  as  stable sliding, violence  cannot be built up. Even if the  not necessarily is  and  FAILURE  respectively.  extra energy  slip  OF POSSIBLE VIOLENT  will not occur because  stick-slip happens, the violence will  occur under low pressure because the energy released  relatively  small.  Only  when  the  pressure  is  very  high,  during each the  energy  released at a slip can cause violence.  As  we  can  see  here,  the  term  "violence"  is  ambiguous.  How  high  the  The Nature of Rockbursting / pressure  is  enough  violent in the related  to  to  the  damage a  and table 6.5, Wr  pressure  applied  to  the  for Mode  figure 7.2  failure  released.  caused.  can  the  be  seismic  If  on  This  energy  the  energy  specified energy  depends  from  release  what  release  we is  release  the  and the  usually is  damage  think  used  of  closely as  observed.  normal  pressure,  as  a The  figure  is given as =  Therefore, normal  it  level  relationship between  seen  violent  amount of energy  measurement,  6.9  cause  149  C P . 2  2  for an amount of energy can  be  determined.  specified  When  transition chart of  figure  I  particular  violence.  For a  this  7.2,  the case,  as  violent,  value  of  violence the  the corresponding  normal zone  pressure  can  is  be clearly  transition chart given in  should be calibrated by the test results of the rock mass concerned.  When the conditions determined from the rock properties, mining conditions and  stress  possible. zone.  state  falls  in  the  violence  zone  of  this  chart,  violent  failure  is  Then above factors should be changed to avoid getting into this violence  The  stress  state  can  be  obtained  from  in-situ  stress  measurement  or  numerical modelling.  Mode II violence occurs when the normal pressure drops quickly to a level low enough or when the loading speed from  stick-slip  to  stable  sliding.  increases  Therefore,  when  fast  enough to cause transition  mining  is  close  to  a  major  discontinuity, if the stress redistribution causes this big increase of stress rate or large drop of normal pressure on it, violence  will be possible.  However, in this  The case,  the  because  stress  by  the  change time  cannot  the  be  stress is  determined measured  Nature of Rockbursting /  by  the  in-situ  violence  stress  of  working  in  similar  conditions  or  more  measurement  would have happened.  This stress change after the mining activity can be estimated experience  150  in advance by the  accurately  by  numerical  modelling such as finite element or boundary element method. Thus better mining design  should be  adopted to  avoid the  sudden increase  of stress rate and large  drop of normal pressure.  Mode  III  violence  refers  to  the  failure in a massive  rock,  caused  by a  large excessive shear stress on a joint plane or the newly fractured surface. This fracture surface can be determined from the  stress state and the shear  strength  envelope.  It is easy to determine the shear stress by Mohr's circle if the  state  known.  is  Similarly,  determined if the  frictional  the  shear  coefficient  fracture surface of the rock mass  strength is  of  the  obtained by  concerned. Again,  fracture  shear  surface  testing  stress can  on the  be  fresh  the stress state for a given  mining condition can be estimated by in situ stress measurement or by numerical modelling.  A more conservative  way to estimate  the possible  violence under this  condition is to use the uniaxial compressive strength to calculate the approximate shear  stress on the fracture surface and compare it with its  shear strength. A n  example has been given in chapter 8.  9.7. P R E V E N T I O N As  previously discussed,  different of  OF VIOLENT the  FAILURE  violence  of failure can  occur in  three  modes,  and  methods of prevention should be utilized for each case. From the point  view of mining technology, the preventive methods can be adopted at different  The Nature of Rockbursting / stages  of  the  mining  process,  such  as  at  mining  design,  during  and  151 after  excavation.  9.7.1. M i n i n g  Design  The fundamental method of rockburst prevention is to optimize the mine planning so  as  to  reduce  the  possibility  of  unnecessary  stress  concentration  to  the  maximum extent. Good mining design would cause stress change uniformly during the  stress  redistribution  after  excavation.  This  is  extremely  important  in  minimizing all the three modes of violence.  Mine  planning  is  technical  and economical  methods  suggested  instance,  the  round  as  roadway the  a  factors  in this  corner pillar  represented should be  by  rather  complex  have  to  problem,  be  concerned.  section  can be  at  the  intersection  the  dot  line  avoided. In retreat  mining sequence can be adjusted  in  used  as  a  because  man}'  However, guideline  of  two  openings  figure  9.9.  The  the in  geological, preventive design.  should be  sharp  For made  turning of  a  panel mining or in recovering of pillars,  to reduce the  stress concentration.  Leaving  larger areas behind at the mined out zone is better than leaving a smaller area, figure 9.10. fault from  When mining across a big natural fault, it is better to approach the the  upper panel than  from  avoid the stress concentration shown  the  lower panel, figure  in figure  9.3.  9.11  The best way  in order to  if possible,  is  to start mining at and move away from the fault.  However,  when excavation  is near a fault,  whether  the  advance direction  The Nature of Rockbursting /  152  Fig.9.9 The intersection at two roadways should be made round as shown by the dot line in order to reduce stress concentration of  the  excavation  depends  on  excavation. violence  should  be  the  normal  whether If  would  the  normal  possibly  parallel, stress  stress  happen.  is In  inclined on  the  released this  case,  or fault  perpendicular will  quickly to the  be a  advance  to  released  low  the after  level,  direction  design that  by  numerical method,  can be chosen the  axis  of  an  such  as  finite  accordingly. A t the same opening  should  be  element  modelling  and  the  Mode  II  should  be  adjusted. This stress change due to the mining activity can be estimated advance  fault  well in a  better  time, it should be kept in mind  parallel to  the  direction  of  principal stress in order to minimize the induced stress on the opening.  the  major  The Nature of Rockbursting /  main roadway a).  not  good  b).  better  main roadway  Fig.9.10 Adjusting mining sequence to achieve better stress condition  The Nature of Rockbursting /  154  fault  <£---advance direction  a). plane view  mining face  not good  b). section view  Fig. 9.11  When mining across a fault, it is better to approach it from the panel in order to reduce unnecessary high stress  upper  The Nature of Rockbursting / 9.7.2. No  155  Destressing  matter  how  good  concentration will other  factors.  the  mining  occur after  The second  planning  excavation,  method,  is,  it  is  inevitable  that  stress  sometimes due to technical problems or  which can  be  used  during  the  excavation  to  prevent violent failure, is to destress an area concerned if stress has been built up, or to pre-destress  The  an area before the possible stress concentration occurs.  destressing  pre-conditioning [18],  techniques  destressing  have  been  widely  used  blasting or infusion [19],  in  practice,  etc.  such  In pre-conditioning,  it is intended to induce some fractures in the  rock mass by blasting before  stress  strength  concentrates  strain energy. zone,  the  there  When  so  the  as  to  reduce  destressing  fractured zone  is  widened  its  blasting is and  the  used  and  at  extra  its  the  load  as  ability  to  the store  stress concentration  is  distributed  over  a  larger area. A t the same time, the stress concentration is moved further into the rock  mass,  then  high  pressure  the  water  possible is  problem can be  supposed  to  have  reduced.  a purpose  In  the  have  available in the detail. caution  last  However, should  problems.  been  it  be  few  many years.  should taken.  Especially in the  be The use  publications  about  No attempt pointed over  out  is  fractured  of infusion, the  the  made  that  mass  presence  will  also  technique  to describe  using  It  energy.  destressing  here  when  rock  of infusion,  similar to blasting.  reduces the elasticity and the ability of the rock mass to store  There  use  them in  these  techniques,  cause  supporting  of water  will bring in  its special effects. Obviously, the water pressure existing on a fault or any other failure  surface  will  reduce  the  effective  normal  pressure  and then  increase  the  The Nature of Rockbursting / incidence  of  This  should  burst control. Meanwhile, as  discussed  in  surface  wet  either  of  polished,  Mode  rock  II  violence.  mass  smooth  when  surface,  would be advisable  or  will  decrease  to test the  be  chapter  for  water  not  allowed  4,  the  increase  rough  happen  shear  or  in rock  strength  remain  natural  effect on the  to  156  of  the  unchanged  for  surface.  Therefore,  it  rock mass in the laboratory  before infusion can be applied for a particular mining condition.  9.7.3. Support Effective most  support  efficient  can  way  reduce after  the  load  excavation  on is  the  protected  finished.  But  structures. it  seems  This  is  the  impossible  to  eliminate the danger of rock burst by support only, as far as the limited ability of  a  support  and  the  tremendous  energj'  released  from  a  concerned. However, according to the  suggested failure hypothesis  obtained  in  in  this  research  support  when  it  expected  in  other  is  as  applied occasions.  given at  the  As  chapters  right  discussed  burst  and the  of  normal pressure  or  due  effective support which increases on a failure  surface  failure. In this  to  8,  the  role  and  right  time  is  more  chapter  7,  Mode  II  violence  sudden  increase  the normal pressure  can certainly  sense, the  the  at least reduce  support applied either  of  loading  and/or the  the  violence  of  figure  is  sudden An  shear  resistance  of any  mode of  parallel or perpendicular to  for other mining purposes,  a  than  speed.  failure surface if it is known is much more effective than applied in other provided this is acceptable  results  through  caused by the transition from stick-slip to stable sliding either due to the drop  are  6  place in  rock  the  ways,  9.12.  As to the heavily loaded structure, such as a pillar or places close to the  The Nature of Rockbursting /  157  Fig.9.12 proper support in advance can reduce the incidence of violent failure  The Nature of Rockbursting / excavation sudden  area,  an  excavation,  surrounding  such  reduction  yielding  as  rock mass,  applied in advance, The  effective  effect  yielding effect  support  may  blasting,  which  the  will be  it can reduce  the  of  the  loading  speed  of  the  support  and  prolongs  to the structure  the  should  be  yielding  9.8.  designed to be  especially  is  failure.  load will be at  a high  expected creep  suddenly  speed. well  here  effect  Because  If  as  a  the  because  of  the  during which the  in  applied  to  support  is  actual load.  of  the  rock  actual  mass.  whole load is  This  applied  with  its  protected. place  of the  In general,  will  help  in  rock mass. In this case, the  speed  less  effective  the  than  the  supports  reduction  of  creep  speed  applied before  incidence  of  support of  the  violent  on  obtained  stress failure,  when a critical condition occurs.  the in  numerical previous  analysis  chapters,  of  conditions  rockbursts  which  may  occurring  in  cause field  violent are  failure  adequately  interpreted and their possible occurrence is extensively discussed.  Specifically,  1.  slip or from  rockbursting on transition  a natural fault  induced by  a sharp  is  Rockbursting  in  a  massive  either  increase  reduction of normal stress on the  3.  the  SUMMARY  Based  2.  the  and at the same time the load is distributed over a larger part creep  takes  violent  loading speed as  loading time  rock mass due to the  change  extra  loaded  the  of the  structure  avoid  158  of  rock  is  rate  or a  the  sudden  of  rock  and is caused by the excessive shear  force  available  upon the formation of the failure geological  stress change  similar  in uniaxial compression  from other  a sudden  fault.  specimen  The effects  from  structures  to  violent  failure  surface. and mining conditions,  such  as  The Nature of Rockbursting / a hard inclusive, mining speed, etc. 4.  Various measures  are examined.  are suggested to reduce and to eliminate the possibility of  violent failure, such as optimizing mining design to avoid unnecessary concentration, structure.  159  destressing  and  providing  efficient  support  to  the  stress  unstable  CHAPTER  10. L A B O R A T O R Y S T U D Y  O F ACOUSTIC EMISSION A T R O C K  FAILURE  10.1. The  INTRODUCTION second objective of this research is to find precursory signals for violent rock  failure  in terms  dollars  has  of acoustic  been  spent  on  emission. field  As stated  research  into  progress has been reported. This is because precursor  to  research  has  violent  rock  attempted  laboratory testing  failure. to  In  identify  of rock specimens.  even  though millions of  rockburst prediction,  little  real  of the difficulty in finding a reliable  order to a  before,  avoid  reliable  expensive  precursor to  The obtained results  field  violent  trials, faillure  this from  will be compared with  field measurements.  As the  discussed  failure  of  propagation  in chapter 3, extensive microfractures are developed prior to  rock.  These  micro-fractures radiate  through vibration of  "acoustic  emission".  radiated,  for  as  The more the  unstable  rock  intense  particles. the  This  acoustic  development  acoustic  of  energy  process  activity,  fractures  is the  during  their  referred to  as  more energy  is  occurs,  the  acoustic  emission will be most active. Detecting these signals with suitable instrumentation provides us with a unique method to study the fracture development process and the rock behavior prior to the failure.  Acoustic defect  of  a  emissions  structure,  have  such  as  been in  engineering and related fields, they  widely aircraft  used  in  frames  are usually used  160  many  or  an  fields oil  to  detect  a  tank.  In mining  to monitor the  stability of  Laboratory Study of Acoustic Emission at Rock Failure / structures of geological  materials, such as  It  the  has  been  found  in  abnormal  increase  laboratory  experiments  However, failure  of  past  acoustic on  that  rock  violent  activity.  acoustic  slopes and underground openings. rock  failure  Similar  emission  161  of  was  phenomena rock  preceded by  were  specimens  an  observed  in  in  compression.  the information obtained seems inadequate to interpret the violent rock  properly and  experiments,  the  to  give  effect  of  a  reliable warning. Besides,  testing  method  on  acoustic  in previous laboratory  emission  was  not  full}'  studied.  In examine  order to correlate the the  effect of shear  rock specimens to  take  full  were  prepared study and  from  the  a  effect  potash  and compressive  of  with violent rock failure and to  loading, further laboratory tests on  the  available  resources.  and an acousti equipment were hard  rock  of rock  were  signals  conducted during this research. A n attempt has been made  advantage  loading equipments  acoustic  sample  type  tested. It  for  both  on acoustic  is expected  used. A few  compression  emission,  that  Shearing  the  and  and  compressive  specimens shear  were  tests.  To  one  sample  of each of coal  testing  results  can show  some  precursory signals for violent rock failure.  10.2. T E S T The from  PROGRAM  laboratory tests were mainly designed to study the acoustic emission pattern specimens  compression  same  and shear.  servo-controlled acoustic  of  testing  activity  It  type was  machine  being  of  rock  desired to and a  monitored  under  different  test the  rock  standard shear throughout  the  loading specimens  conditions on  test equipment, tests.  The  of  an M T S with  the  acoustic  Laboratory Study of Acoustic Emission at Rock Failure / instrumentation  should be  able  to  record  the  acoustic  information as  fast  162  as  it  occurs during the test and to re-process these data in several ways afterwards.  10.2.1. Specimen Preparation All  test specimens  were prepared from the same  sample of a Metamorphic rock,  which consists of Schist, Biotite, Quartz, Chlorite, Actinolite and some submetallic sulphide  and  specimens  oxide.  were  cut  For compressive with  a  tests,  three  1.58  length/diameter  ratio  of  prepared by cutting the core in the  same  in.  2:1.  diameter The  cylindrical  specimens  were  direction. The sample ends were  then  ground parallel to each other and perpendicular to the core axis. For shear tests, four  3 in. diameter specimens  each  half  shear  was  surfaces  specimen was specimens the  cast  in  created  cement by  were with  breaking  prepared. These specimens mould formers. the  cut with a diamond saw.  specimen Care was  Three with  a  of  were  halved and  them  chisel.  had their The  taken during casting that the  were in alignment and the shear planes horizontal. Access was  cement  fourth  left in  cast for mounting accelerometer, or the acoustic transducer, which is  recoverable after each test. The cement  was  completely dry when the  specimens  were tested.  10.2.2. E q u i p m e n t A  servo-controlled M T S hydraulic testing machine was used for compressive tests.  This machine can control loading either by loading speed or displacement rate. It is  also  able to record data of load and displacement on disk and to plot them  against testing time throughout the test. Its technical specifications are as follows: name: M T S 810 hydraulic testing machine load frame model: 312.41  Laboratory Study of Acoustic Emission at Rock Failure /  163  load frame capacity: 250 K N cell model: 661.23A-01 cell capacity: 250 K N control mode: load/stroke  A  manual-loading device  was  used  for shearing tests.  It  consisted  load gauges, two hand pumps and two mould formers. The technical  of  two  specifications  include: name: Potable Shear Box load frame model: EL77-103 load frame size: 460X250X600 mm hand pump capacity: 50 K N mould former model: EL77-103/4  In selecting  the  acoustic  equipment,  small amount of energy associated  the  broad band of frequency  and  the  with acoustic events, and the background noise  were considered. The frequency of acoustic events from rock falls in the range of 100  Hz to  KHz  [49].  up  500  The energy  detectable with heard.  to  K H z [15] released  with  the  in such  an  sensitive geophones to  Generally,  larger  background noises are  due to electricity  from something  16 — 32 barely  that can be physically felt and  lower  frequencies  and the  [15,23].  The  major  mechanical vibration of various  frequencies  compared with the major frequency range of acoustic events. Therefore  response  acoustic  laboratory.  within  in  desired  around the  event varies  something  have  concentration  equipment  the  and  events  largest  equipment  should  However  have  high  these  noises  sensitivity,  have  very  low  wide  frequency  (Physical  Acoustics  and ability to cut off the background noises.  The  selected  Corporation) system,  acoustic  equipment  is  the  PAC  which has a four-channel data recorder and a processor. The  Laboratory Study of Acoustic Emission at Rock Failure / PAC  system  changed noises  by by  transducer,  can  record  setting  the  changing  the  acoustic dead  time  with  each  other.  at  very  between  amplification  preamplifier and the  are compatible  signals  and  processor  high  signals the  speeds  and  threshold  of this  eliminate levels.  system came  The P A C equipment  which  had the  as  164  can  be  background Because  the  a unit,  they  following  technical  specifications: name: P A C 3000 A E system channels: 4 A E amplifier: noise: 4.5 juV R M S RTI gain: 0 - 6 0 db bandwidth: 10 K H z 1.2 M H z threshold: . 1 ~ 8 V input impedance: 50fl @ 120 pf A E in: 10/uV 10VAC A E out: 0 - 1 0 V A C into 50S2 and 470pf A E preamplifier: P A C 1220A noise: < 2 M V R M S RTI gain: 40 or 60 db selectable bandpass: 10 K H z - 2 M H z selectable input impedance: 10Kfi//120pf output voltage: 20 Vpp into 500 input: single or differential selectable power: +28 V D C A E sensor: P A C R15-1123 Piezoelectric Crystal sensitivity: voltage 0.06 v/g capacitance: 15pf resonant frequency: 150 K H z frequency response: 10 — 300 K H z  During the sensor be  and cabled  both  displayed  tests, signals to  the  on the  of acoustic  processor screen  after  emission 40db  system,  which has  up to 20  pre-amplification. These  and recorded on  They can also be played back and processed  were picked up by the A E  the  on the  disk  at  the  same  time.  micro-processor of the P A C  ways of plotting graphs. Hard copies are  at the printer of the P A C system on request.  data can  available  Laboratory Study of Acoustic Emission at Rock Failure /  165  10.2.3. Test Procedure Following specimen between setup  preparation, by the  is  acoustic  tape.  an  acoustic  emission  transducer  An  acoustic  couplent  was  transducer  and the  specimen  surface.  given  in  equipment  figure  10.1.  was  calibrated.  threshold  were  carefully chosen  could  be  picked up and any  with  a  1 — 2V  Before  threshold  by  the  In  test  particular,  trials  so  that  background noise  and  49~55db  used  was  to  attached  make  a  good  The diagram of of  each the  as  cut  in gain. The dead  off. time  testing  the P A C  amplification signals  the  contact  the  specimen,  many  could be  to  and  as  This  the  possible came  between  up  events  was set to 1ms for the shear tests and 3~6ms for the compressive tests.  During the compressive test, the M T S testing machine was programmed to give a constant rate of displacement. To shorten the testing time, this rate set  was  to 0.0001 mm/s at the beginning of tests and 0.00001 mm/s when the load  reached  approximately  40%  of  the  uniaxial  compressive  strength.  Load  and  displacement data were recorded by the M T S system. The P A C system monitored the  acoustic  loading was  information  as  well  as  the  load.  During  applied manually. Normal pressure was  set  the to  direct 1,  2,  shear  tests,  3 and 4.5 ksi  respectively for each specimen. Load and displacement data were taken by hand.  The the  acoustic  signals  recorded by the P A C system  were both displayed on  screen and stored on disk. These data can be replayed back and plotted in  different formats.  Laboratory Study of Acoustic Emission at Rock Failure / 166  P - load mount platen AE-transducer . specimen  a).  u n i a x i a l compressive  test  P  - normal force  AE-transducer upper specimen  cement casting  shear force cement casting lower specimen  loading frame  b).  d i r e c t shear Fig.  test  10.1 Loading diagram for acoustic emission test  Laboratory Study of Acoustic Emission at Rock Failure / 10.3. T E S T Results  RESULTS  from  presented  167  the  below.  uniaxial Despite  compressive  the  limited  tests  and  number  of  the  direct  specimens  shear  tested,  tests  these  are  results  provide useful information for analyzing rock behavior before its failure.  10.3.1. Acoustic E m i s s i o n from Compressive The  identification  and  mechanical  properties  Tests  of  the  under uniaxial compression on the M T S testing The  disparity  of  compressive  specimen  #2  and  #3  Therefore  the  actual breakage  strength  contained  is  some  surfaces  specimens  loaded  to  failure  machine are listed in table  probably  micro-cracks are away  due which  from as  to  the  fact  initiated  10.1.  that  the  determined by  the  failure. Mohr's  circle for intact rock, i.e. away from ± ( 4 5 ° — (j>/2), equation (3.1).  During the tests, an attempt as possible. events  specimen No.  1 2 3 average  set  to  3ms,  which  resulted  10.1  Identification  length  diameter D  area A  (in)  (in)  in  3.2835 3.0041 3.1272  1.5842 1.5845 1.5858  1.9711 1.9718 1.9752  L  made to acquire as much acoustic  A t first, in the test of specimen #1,  was  Table  was  0  in  use  the dead time between adjacent of  six  disks  for  and mechanical properties of compressive  2  strength a c 10 psi  modulus E  failure strain  10 psi  e  18.459 11.32 9.44 13.073  1.955 2.426 2.045 2.142  .00947 .00636 .00548 .00710  3  data  6  * not: the breakage angle /3 is defmedd in figure 3.4a)  the  single  specimens breakage angle  0  25.756 46.057 39.508  Laboratory Study of Acoustic Emission at Rock Failure / specimen. In order to cut off some the  dead  time  was  changed  to  168  data without losing the basic characteristics,  6ms  instead  of  changing  the  gain  and  the  threshold. Then, two disks were enough for each specimen.  The release are  acoustic emission information is presented here as  rate  plotted  and energy in  release  figures  10.2  curves for specimen #1 data for specimen #3  Figures acoustic  activity.  However, was  10.2  this  of this  and #2  together  with  the  load-displacement  only. Due to technical problems during the test,  show that at the start of the test, there was little  loading  during  high emission  10.5  They  are not complete and therefore have not been analyzed.  activity was  most intense  rate, energy  ratio against loading time and axial load.  through  and 10.3 As  event  continued,  minimal  acoustic  until the  emission  specimen was  slowly.  close to failure and  a period immediately preceding the  period is likely to vary  increased  failure.  The length  with the mechanical properties of  the specimen and with the loading speed.  During dies  down just  keeps release  this  active  before  period,  the  going up and shows  the  failure. a  peak  event  rate  A t the  same  at the  increases time,  failure.  the  rapidly energy  The tendency  at first,  then  release  rate  of the  energy  ratio, or the average energy released per event is similar to that of the  energy release rate. It shows a sharp increase before the failure.  The quiet period of emission corresponds to the perfect elastic phase up to  Laboratory Study of Acoustic Emission at Rock Failure /  B) .  TIME  (Sec)  FIRST ARRIVAL  Fig.10.2 Acoustic emission from uniaxial compressive test for specimen  #1  169  Laboratory Study of Acoustic Emission at Rock Failure /  Fig. 10.3  Acoustic emission from uniaxial compressive test for specimen  #2  170  Laboratory Study of Acoustic Emission at Rock Failure /  AXIAL  Fig. 10.4  fracture  initiation.  LOAD  Acoustic emission  As  fracturing  ( 2 5 KM)  vs axial load for specimen  propagates  more intense. The event number of emission increase  of  acoustic  energy  micro scale and the the  transition,  is  not  further,  significant  forking occurs  within  because remains the  #1  acoustic  increases  vibration of rock particles  or crack  activity  accordingly. the  fracturing  on the  unstable  to  specimen. due  to  low  a  failure  surface,  culminating  in  complete  the  coalescence because  above results,  of  micro-fractures.  macro-fractures it may  buildup of acoustic emission  be  But the  release  more  and that immediately  in  When  which join of  the  will decrease  energy  energy  than  will  increase \  micro-fractures.  rock failure,  preceding the  still  the  propagation,  failure  acoustic  suggested that prior to  is level.  fracture  During this period, the event number of acoustic emission  dramatically From  form  becomes  However,  coalescence of micro-fractures leads to the formation of macro-fractures, together  171  there is  a  failure, the event  Laboratory Stud.y of Acoustic Emission at Rock Failure /  AXIAL  LOAD  172  ( 2 5 KIM)  Fig. 10.5 Acoustic emission vs axial load for specimen  #2  number drops after a sharp increase and the acoustic energy increases suddenly.  Graphs of acoustic emission versus give  further indication of the  failure process.  the  fracturing  mechanism discussed  are  negligible  when  acoustic strength  emission for  the  load  is  #1  low  and #2  axial load, figures  at  and  10.4  and  10.5,  This data correspond very well to  in chapter  increased suddenly  samples  the  3.  In  increases  about  respectively.  these as  71% and A t this  tests,  loading  the  emissions  continues.  78% of the  The  compressive  point, event rate,  energy  rate and energy ratio all showed a sharp increase. This point may correspond to the  beginning of unstable  there  is  number  a  delay  shows  a  fracturing  between peak  the  before  development.  peaks the  of  energy  the  It is acoustic  release.  This  interesting to notice parameters. may  The  that event  correspond to  the  Laboratory Study of Acoustic Emission at Rock Failure / fact that at the fracture initiation, micro-fractures of low energy as  the  fracture propagation reaches  the  unstable  stage,  greatly intensified first by number and then by energy formed.  This  phenomenon  may  provide significant  the  173  are formed and  acoustic  activity  is  when macrofractures are  information in  analyzing rock  noise data measured in field.  During the  failure.  the  tests,  False  some  warning  peaks evidenced  of  acoustic by  a  emission  buildup  were  of  observed  rock  noise  monitoring may be due to this kind of phenomenon. However if the a  rock mass  in  before field  strength of  is known, and if only those buildups of rock noises at high stress  level are considered as warning signals  of an impending failure, the reliability of  a monitoring system could be adequately improved.  10.3.2. Acoustic E m i s s i o n from Direct Shear Tests Three specimens under  direct  with breakage surfaces and one with sawcut surface were tested  shear  and  at  properties are listed in table respect  to  the  normal  different  normal  stress  levels.  stress  were  obtained  statistically  with linear correlation coefficients  roughness  is accounted for by the  than  breakage  the  surface  mechanical  10.2. Empirical formulae of the shear strength with for  these  shear surfaces. For both, there is a good relationship between and the normal stress,  Their  and  consequently  has  a  lower  types  of  the shear strength  above 0.96.  friction angle. The sawcut  two  The surface  surface is friction  smoother  angle,  figure  10.6.  The basic information of acoustic emission, i.e. event rate, energy rate and  Laboratory Study of Acoustic Emission at Rock Failure /  174  3 p  2.5  0  1  2  normal Fig. 10.6  3  stress  (1  4  5  6  ksi)  Shear strength of sawcut and breakage surfaces  energy ratio, is presented as a function of testing time, figures 10.7 to 10.9 and shear  displacement,  figures  10.10  test, information for specimen #6 it  was  observed  not  possible  during  the  to  record  through  Due  to  problems  during  is not complete and not shown here.  the  test of the  10.12.  oscillation,  sawcut  the  phenomena  specimen #4,  of  especially  Although  stick-slip at  the  were  high normal  stress level.  The breakage surfaces were not ideally flat and had some undulations. The shear stress therefore still went up slightly after slip began and the appearance of slip is not very clear, as indicated by arrows in figures 10.7 and  10.8  Laboratory Study of Acoustic Emission at Rock Failure /  Fig. 10.7  Acoustic emission from breakage specimen #5 under direct shear test. Arrow indicates the beginning of slip  175  Laboratory Study of Acoustic Emission at Rock Failure /  Fig. 10.8 Acoustic emission from breakage specimen #7 under direct shear test. Arrow indicates the beginning of slip  176  Laboratory Study of Acoustic Emission at Rock Failure /  Fig. 10.9  Acoustic emission from sawcut specimen #4 under direct shear test. Arrow indicates the beginning of slip  177  Laboratory Study of Acoustic Emission at Rock Failure /  ROCK  5  SHEAR  IMS IV  55DB  shear displacement (0.005 normal p r e s s u r e 1.5 k s i  Fig. 10.10  ROCK  7  Fig. 10.11  1.2/23/85  15:47:30  in)  Acoustic emission vs shear displacement for specimen  SHEAR  IV  49DB  IMS  #5  1 2 / 2 3 / 8 5 ' 1 6 : 2 6 s 4 6  Acoustic emission vs shear displacement for specimen #7  178  Laboratory Study of Acoustic Emission at Rock Failure / At  the beginning of testing, there was  Table  normal stress 1 a(ksi) failure shear .32 stress r(ksi) shear T  strength  little acoustic emission. As the test  10.2 Mechanical properties of shear  sawcut surface #4  specimen No.  specimens  breakage surface #5 #6  2  3  4.5  1.5  3  1  .8  1.4  1.9  .94  1.8  .65  3  #7 2  3  1  1.6  1.97  T s = - . 0 9 9 + .45869a  T s =.0144 + .58323a  24.5°  30°  S  friction  angle  standard diviation Sd , n-1 correlation coefficient r  a±1.493,  r  ±0.69 s  0.99253  a + 0.8803,  r  s  ±0.5342  0.96117  ROCK 1A 3MS 2 V 55DB  12/23/85 16:59:30  shear displacement (O.Olin) normal pressure 3ksi Fig. 10.12  179  Acoustic emission vs shear displacement for specimen  #4  Laboratory Study of Acoustic Emission at Rock Failure / continued,  event  rate  began  to  increase.  When  slip  showed  up,  180  event  rate  reached a maximum value. Then it remained almost constant as sliding went on. However, energy  the  was  energy released  release during  interesting to look at the rate  of  displacement,  goes  up,  emission  especially is  compressive  slip  was  as  very  shown  small  in  until  figures  slip  10.7b)  began and  shear displacement and energy release  indicated  at  the  where  by  end  displacement-rate failure  rate  its  of  slope,  each  dependent  acoustic  increases  test.  for  This  shear  emission  the  may  failure  appears  suggest  more  10.9b).  most It  rate. When  energy  as  and  the  release that  rate  acoustic  compared with likely  to  is  be  the  stress  dependent.  Figures  10.10  to  10.12  against shear displacement. release  reached some  almost  constant,  the  acoustic  emission  and  shear  pressure  Slipping took place when both event rate and energy  critical  event  show  rate  values.  During  remained  the  unchanged  slip where as  for  shear breakage  pressure  was  surface,  or  remained constant at a lower level after a drop as for sawcut surface, and the energy release abrupt This  increase  may  remained almost constant for a period then went up sharply. This of  suggest  energy that  if  release slip  is  rate  due is  to  the  constant,  increase the  of  displacement  acoustic  activity  rate.  will  be  unchanged.  The acoustic emission from the sawcut surface is similar to that from the breakage  surface.  The only difference  is  that  the  magnitudes  of event  rate and  energy release for the breakage surface are bigger than the sawcut surface.  Laboratory Study of Acoustic Emission at Rock Failure /  181  During the test, the normal pressure seems to have a significant effect on acoustic  emission.  specimen #4  no  typical example,  emission  activity  existed  in this  period became  pressure does not change only  the  the  acoustic  emission  1000  for  the  sawcut  to 4500 psi is presented  and 10.14, Which indicate that at low normal pressure, little or  acoustic  acoustic  a  under normal pressure ranging from  in figures 10.13 even  As  magnitude  the  alters.  before  slip. As the  normal  more  It  active.  profile of event This  can  be  pressure increased,  seems  that  the  rate very much after  clearly  seen  in  normal  slip begins,  figures  10.13  and  10.15b).  The at  low  critical  normal  normal value.  stress  at  10.4.  pressure  Figures  normal  energy release  pressure is can  10.14d)  pressure  also related to the become and  stick-slip  10.15a)  way  of slip. A stable  when  show  of 4500 psi. These  two  normal  sliding  pressure  reaches  significant drops of  drops are  shear  accompanied by sharp  which are clearly seen in figures 10.14d) and 10.15c).  DISCUSSIONS  From  above  emissions failure  results,  there  seems  to  be  little  relationship  between  acoustic  from compressive and shear tests. However, from previous analysis, the  of  intact  similar  to  a  surface  is  first  fracturing of shear extremely  rock  under compression  conventional formed.  development.  compressive In  After  this this  test. Unfortunately, this fast  and cannot be  has  test  path,  two up  the  point, the  to  the  failure  The  in the  observed. This  first  point  process  failure path is  shear process  easily  stages.  where is  one  a  a a  the  path  failure  matter  similar to  compressive  is because  is  of that  test happens shear  stress  Laboratory Study of Acoustic Emission at Rock Failure /  ROCK ••  10  4 *  SHEAR •  IMS •  IV  49DB  •12/23/B5 * Normal  16: 45:  p r e s s u r e  1000  psi  2000  psi  182  13  OB  CC 06 u :>  04  L — J 4 ,  A).  'X>  1  40O  1  50 Normal  UJ  h<r  40  •  30  •  1  W 10  100  D)  .*  00  B)  .  pressure;  1  10'.".'  j »  1900  TIME  (Sec)  FIRST ARRIVAL  .10.13 Effect of normal pressure on event rate, specimen #4 arrow indicates the starting of slip  sawcut  surface,  Laboratory Study of Acoustic Emission at Rock Failure /  ROCK  4  SHEAR  IMS  IV  47DB  12/23/85 Normal  400  16:47:21 p r e s s u r e  1000  psi  >  <£. UJ Z hi A),  s - d i s p l _t?n. 00  I  r-~~*^  00 1600  *  .  .  *  •  100  I  *  200  .  U(.01 r a t e  i1  .  300  in)  400  I  * Normal  p r e s s u r e  2000  psi  a. a.  B).  c ID IE  C),  <t a:  >a cc u D).  Fig. 10.14  2020  214C time (second) f i r s t  arrival  Effect of normal pressure on energy release, specimen #4 surface, arrow indicates the starting of slip  sawcut  183  Laboratory Study of Acoustic Emission at Rock Failure /  toe u r-  or z u >  ise ice lie I2B laa  (T-4 5 a a  p  B  i  1/  b»  68 4B  ,  J - 3 a a a  p n  r  2D  ff-i  B).  7.3  aaa  18.5  t  ps I  13. a  f*<-ff-3BOepsi  tx a  >o a LJ z u  c).  a: >  D).  8  1  2  3  S  6  7  6  I  10  II  12  13  l«  15  l«  17  IB  SHERR DISPL (X .0.1 i n )  Fig. 10.15 Acoustic emission vs shear displacement at various normal pressure, specimen #4  184  Laboratory Study of Acoustic Emission at Rock Failure /  185  on the newly formed failure surface is much higher than the shear strength. A t the  same  normal  time,  the  pressure  shear  strength  acting on it  may  of  the  decrease.  failure  surface  Therefore, the  drops shear  because  the  failure occurs  immediately once the failure surface is formed.  This means that if a large shear load is suddenly applied to a specimen, the failure will occur extremely  rapidly. This has been successfully  proven during  tests by releasing the normal pressure quickly when the slip began and bursting phenomena emissions  were for  observed.  this  kind  of  Figures sudden  10.16  shear  and  10.17  failure. As  can  illustrate be  the  seen,  acoustic  the  acoustic  emission occurring prior to the slip had been completely shadowed b3' the peaking up of signals at the instantaneous failure. Because the load is reduced to minima instantly,  after  expected  if the  the  acoustic  shock  is  excessive  activity  is  hardly  observed.  load is  higher.  expected  to  Meanwhile, more From  increase  discussions with  acoustic  activity  is  in previous chapters,  loading  speed  because  high  loading speed will accelerate the process of fracture development.  As for the effect of rock type, one specimen of coal and one specimen of potash  were  tested  for  approximately the same can  be  seen  completely evidence  that  different  comparison.  these  speed. The acoustic results  acoustic from  During  the  emission  from  the  tests,  was  applied  are given in figure ductile  failure  brittle failure of coal and granite  of failure at all. Although coal is brittle, its  than granite because  load  acoustic  of  at  10.18. It potash  and it  has  is no  activity is higher  of more pre-existing cracks. This may suggest that acoustic  emission can show clear evidence prior to failure for brittle material but not for  Laboratory Study of Acoustic Emission at Rock Failure /  tt4  SAWCUT  SUDDEN  LOADING  1 2 / 0 3 / 8 5  1 5 : 2 9 : 4 9  time (second) first arrival B) .  Fig. 10.16 Acoustic emission from sawcut specimen at sudden shear loading, by releasing normal pressure at 1, 2.5 and 4.5 ksi level, respectively  186  Laboratory Study of Acoustic Emission at Rock Failure /  #7  BREAKAGE  SUDDEN  LOADING  1 2 / 0 3 / 8 5  18/  'IS": 4 1 : 5 9  time (second) first arrival B).  Fig. 10.17 Acoustic emission from breakage specimen at sudden shear loading, by releasing normal pressure at 1, 2.5 and 4.5 ksi level, respectively  Laboratory' Study of Acoustic Emission at Rock Failure /  188  ductile material.  10.5. A  SUMMARY  limited number of rock specimens were tested  under the  availabl resources. A  few important points may be noted from the testing results. For the uniaxial compressive test: 1.  At  the  beginning  of  loading,  acoustic  emission  was  very  low.  This  corresponds to the period of perfect elasticity during the brittle failure. 2.  As load reaches the rock tested, 8B  some  value, say  71%-78% of the compressive strength for  acoustic emission begins to build up quickly, in response  r  APPLIED LOHD/ULTIMRTE STRENGTH  Fig. 10.18  Effect of rock type on acoustic emission  to  Laboratory Study of Acoustic Emission at Rock Failure / 189 the  onset of unstable  fracture propagation and is most  active  in a period  immediately before the failure, during which fractures develop rapidly. 3.  During goes  active  up initially,  release the  this  keeps  fact  period, then  if displacement  drops  preceding  rate is constant, the failure.  the event  However  rate  the energy  increasing and shows a peak at failure. This may be due to  that  as failure is approached, events become  bigger in magnitude  because of the formation of macro-fractures. 4.  There is a short delay between with  event  rate  increasing  buildups of event  first.  This  is  rate and energy  probably because  release,  microfractures  develop first, which then join up to form larger fracture zones. 5.  During the ductile failure, acoustic emissions  do not show above signals.  For the direct shear test: 6.  A t low stress  level,  there  is little  acoustic  emission.  When  slip  begins,  acoustic activity reaches a critical level and remains more or less constant. 7.  Most energy is released during slip.  8.  During sharply  slip, the event rate remains constant, towards  failure, which  is accounted  displacement rate. It may suggest that  but energy  release  rate  for by the increase  acoustic  rises  of shear  emission in shear  mode is  more displacement-rate dependent than stress dependent. 9.  Roughness of shear surface does not change the pattern of acoustic emission very  much.  However  the magnitude  of emission  for breakage  surface  is  much higher than for the sawcut surface. 10.  In stick-slip, each  slip is accompanied by a drop of shear  increase of energy release,  stress  and  then followed by a drop of acoustic activity.  Effect of normal pressure on acoustic emission in shear test:  an  Laboratory Study of Acoustic Emission at Rock Failure / 11.  A t low  normal pressure,  little  acoustic  emission  exists before  slip. A t high normal pressure, emission  becomes more active  It  is  is  probable  micro-fractures,  that  a  macro-failure  each of which initiates  a  combination  the  190  onset of  in this period. of  many  local  at some local point. The higher  the  normal pressure, the more local micro-fractures initiate. 12.  After  slip  begins,  normal  pressure  seems  acoustic emission, except for increasing the 13.  not  to  change  pressure  may  change  the  way  of  slip.  normal  pressure  may  become  stick-slip  at  higher  chapter 7.  pattern  of  magnitude.  Normal  other conditions.  the  This agrees well with the  A  stable normal  sliding pressure  at  low given  transition conditions described in  CHAPTER  11. P R E C U R S O R Y  SIGNALS  IN C O M P A R I S O N  WITH  FIELD  MEASUREMENTS  11.1. G E N E R A L In  order to  verify  the  acceptability  of  acoustic  results  from laboratory tests,  comparison will be made with some field measurements. well  established  that  a  rockburst is  usually  preceded  a  In field monitoring, it is by  a  sharp  increase  of  microseismic activity. However, the reliability of prediction of an impending failure based on a sharp increase of either of event rate or energy release rate is poor because mines, the  few the  successful  predictions  introduction of  reliability [17].  have  spectrum  Unfortunately,  been  analysis  seismic  achieved  of  in  seismic  the  past.  waveform  data of potential  In  some  has  increased  successfully  predicted  rockbursts are very rare and in fact are only available from some South African mines.  11.2.  PRECURSORY  During the  SIGNALS  IN T H E L A B O R A T O R Y T E S T S  laboratory tests of this research, a limited number of rock  were tested under the  available  resources.  The testing  results  ignificant phenomena. The acoustic  emission  the final failure of the specimens.  During the acoustically  rate  increases  sharply at first,  Simultaneously,  the  energy  failure,  figures  10.2  to  event  rate.  released  10.3.  This  then  has  shown  some  is very low until some time prior to active period, the event  decreases immediately increases  steadily  The sharp increase been  have  specimens  described  and  of energy in  peaks released  the  abruptly  at  appears  to  lag  the  The  most important fact is that a sharp increase of the event number alone can  191  in detail  preceding the failure.  previous  chapter.  Precursory Signals in Comparison with Field Measurements / not indicate  an impending failure but the  simultaneous  peak  192  up of energy  rate  or energy ratio will be critical for violent rock failure.  11.3. In  PRECURSORY  microseismic  The  acoustic  event  monitoring  signals  may  SIGNALS  mean  of  IN F I E L D rockbursts,  MONITORING precursory  signals  are  are recorded as event number and energy little,  but  a  number  of  events  the energy  observed.  release.  occurring  indicate a "hot spot" where violent failure will take place [16]. failure, however,  also  A  single  successfully  can  To predict violent  released has to be considered and precursory signals  are needed.  11.3.1. P r e c u r s o r y Signals prior to R o c k b u r s t i n g It  is  found  that  a  sharp  predict a rockburst [21]. have  to be  found and the  of  fracture  of  the  event  rate  alone  is  not  enough  To improve the reliability, better ways of data  Seismic events can however beginning  increase  technique  of  data  acquisition  needs  an  event  has  small  analysis  to be improved.  be distinguished by their magnitudes  development,  to  [23,25]. A t the  magnitude.  As  failure  progresses,  the event magnitudes increase due to the formation of macro-fractures.  Therefore,  in  addition  to  the  sharp  increase  of  acoustic  events,  the  event  wave  forms,  magnitude should also be examined in judging an impending failure.  With the  the  frequency  found  that  therefore  spectrum of  the  the  introduction of the  pattern  waveform  of  the  technique  events  seismic  frequency  has  waves  of  been varies  recording seismic  analyzed in some mines. at  distribution of the  different  stress  It  levels  waveform should change  is  and as  Precursory Signals in Comparison with Field Measurements / failure  is  approached.  In  some  cases,  frequency, which will be discussed the  a  characteristic  parameter—the  et  al  responsible  [24]  for the  observed  frequency.  When  event  rate  energy  and  that  high-frequency  lower  rockbursting has been  the  microfracture  propagation  events and the  waveform  released  greatly  all  considered,  improved in some cases.  the  the  the  to and  mechanism have  increases  ability  to  of  predict  In particular, it is found  where the event energy is increasing and at  same time the corner frequency is shifting downwards,  can  beexpected.  still  is  distribution,  the  is  Scholz [23]  audible events at failure  frequency  are  that when the pattern is established  Experience  corner  later in this chapter, is also found to shift  lower band prior to a rockburst. In laboratory experiments,  Savage  193  being  obtained  a violent  in interpreting  rock failure  the  results  in  order to predict precisely when a failure will take place. The pattern of the rate at which acoustic events are emitted appears to be irrelevant.  11.3.2. T y p i c a l  Examples  In a South African rockbursts [17].  Example  mine,  some useful  Two case examples  15,  apparent.  1983, In  at  figure  1983  3.4)  03h37. 11.1,  were recorded prior to  are copied in the following.  1: rockburst on May 15,  A large rockburst (magnitude May  precursory signals  occurred on 101W1 panel, No.3 shaft,  A  the  concentration number  of  of  microseismic  microseismic  event  originating from the panel for the period 8th to 18th May, 1983 A  steady  before  increase  the  burst.  numbers  be  burst to almost  drop in the the  can  rate  300  larger  this and  from  approximately  events only  of microseismic  For in of  seen,  activity  particular case, smaller  events  24  was the  events  60  per  change  provided  in  events  the  6  hour  ratio  days  A sharp  immediately the  is  is plotted.  hours beforehand. measured  on  before  between  researchers  with  Precursory Signals in Comparison with Field Measurements /  194  additional information to make a reliable prediction.  Example 2: Rockbursts on October. 4 and 10,  1984  On October 4, a 2.6 magnitude rockburst occurred during shift time (10h31) on  110  frequency time  level.  Figure  and  average  window  11.2  shows  the  event  energy  as  22h00 to 04h00 every  event observed  alone the  made the  the  measurement  rock burst would  event rate  parameter  is  unreliable.  not  have  very  drop to  showed below the  Hz was  average  that  area  the  basis  anticipated to  the  on  indicated  preceding  a few  event  regular blasting 1.4)  at  4h39  and a relative  clear  before  conformed  and was  on October  and no  the  burst.  This  precursory indication of  energy  started  as  and a further  and the rockburst occurred at a relatively high energy  corner frequency  average  a  hours  days  expected  (magnitude  gave  11  rate  However the corner  rockburst.  rockburst  frequency  a blast  mining activit3'  previous afternoon.  the  October 4th,  pending  later,  for  of event  of  The  corner  from an external  behavior  5 days  corner  On the  been  a steady drop for the  600  from  27, inference  sensitive  blasting took place in that area the frequency  the  night. The symbol B indicates  during the previous afternoon. On September source  rate,  10th.  high event energy  to  what  was  level.  followed Again  a  by  a small  a relative  preceded the  low  burst. The  blasting the previous afternoon made the event rate unusually high.  11.4. There  COMPARISON is  a correlation between precursory acoustic  tests and the increases increase  field monitored data in example  1.  signals  recorded in laboratory  In both cases, the  event rate  sharply at first and drops immediately preceding the failure. The abrupt of  the  event  energy  corresponds  with  an  abrupt increase  of the  ratio  Precursory Signals in Comparison with Field Measurements /  Fig. 11.1  195  Microseismic event rate and relative energy plotted for one week before and three days after the May 15 event (after Brink et al, [17])  Precursory Signals in Comparison with Field Measurements /  Fig. 11.2  196  Event rate, corner frequency and event energy measured over a period of 25 days, covering two rockbursts (after Brink et al, [17]).  Precursory Signals in Comparison with Field Measurements / between the  numbers  of large  and  small  the  fracturing mechanism  discussed  due  to both  decrease  in  smaller  correspond  to  the  Larger  the  events  in  events.  chapter events  3,  and  development  This  of  the the  is  because,  increase increase  according to  in  this  ratio  in larger  macro-fractures,  197  is  events.  which  release  more energy.  In example  2, the behavior of event rate and energy  rockbursts are generallj' in agreement failure.  The  event  to  increase  continues burst  on  October  rate  drops  with laboratory tests prior to the  after  a  and shows a peak  10th,  when  the  release prior to the  sharp value  blasting  increase.  The  energy  at failure. Exception is  the  previous  specimen  afternoon  release the rock  caused  the  unusual high event rate.  In the precursory frequency  2nd example,  parameter, is  usually  the  corner frequency,  shifts  to  a  low  defined  as  the  level  frequency  which is introduced as  as  a  burst  corresponding  occurs. to  the  The  from  the  seismic  scheme  spectrum [5],  that  at  low  f  0  being the corner frequency.  frequency  level, and at high frequency band f > f , 0  frequency  corresponds  magnitudes,  figure  11.3  to  lower  band  f<f  0  the  magnitude.  Because  smaller  shows  It can be seen  amplitude  the spectrum decays.  corner  intersecting  point of the two asymptotes on the spectrum diagram [41,50]. Figure 11.3 a schematic  another  spectrum  This means events  have  is  higher lower  indicates that normally, small events are characterized by  high frequency and large events by low frequency [23,24].  When  f  0  shifts  to  the  lower  band, the  high frequency  amplitude  decays  Precursory Signals in Comparison with Field Measurements /  198  log A  high-frequency amplitude decay(~f" log Q  n  n=2  or 3)  0  - n log f  log f  log f  0  Fig. 11.3 Schematic seismic spectrum, clarifying: low-frequency amplitude level, corner frequency (fo), and high-frequency amplitude decay (after Bath, [51]) much  more.  events  occur  amplitude. released.  Thus at  fewer the  Therefore  events  low-frequency  with  the  This is in agreement  Meanwhile,  occur  many  at  the  band,  high-frequency  which  decrease in the  is  characterized  corner frequency,  with the top curve in figure  years  of  observation  band  of  seismic  more  =  a -  bM  is the magnitude of event,  a and b are constants, with  by  large  energy  is  events  has  found  an  [17,43]: (11.1)  where N is the number of events of magnitude M  more  11.2  inverse relation between the number of events and their magnitudes log N  and  b>0  > M,  Precursory Signals in Comparison with Field Measurements / This is illustrated in figure two axes. A small increase when the  the  event amplitude  corner  Thus  frequency,  the  event  indicated in the high  event  down-shift  event  which is  number  appearing on the  in  the  the  of the scale on the  second  accompanied  during  period  agrees  caused  with  by  both  11.2.  of  sharply.  accordingly.  blasting.  the  down-shift  decreases  bottom curve of figure  rockburst is  corner frequency  that  b.y the  recorded data drops  first rockburst at the  rate of  Notice the difference  of M can result in a great decrease of N . Therefore increases,  the  rate  11.4.  199  This  is  The unusual Therefore  decrease  of the  the  event  rate and the increase of event size or more energy release.  It  can  be  seen  generally  similar to the  preceded  by  may  decrease  intense  from  above  examples  measurements  acoustic  preceding the  activity. failure.  made  that  in the  The event  abruptly at some critical level,  In  by  fact,  research  Scholz  [23]  and  Mogi  laboratory  field.  rate  Simultaneously,  ratio will increase  the  are  Violent rock failure  will  the  results  increase  energy  is  sharply and  release  rate  and  indicating an impending failure.  [25]  have  of  acoustic  showed  that  laboratory  acoustic emission is similar to earthquakes.  11.5. In  SUMMARY  this  laboratory  chapter, tests  the are  precursive  phenomena  compared with  measurements  made  emission in the  observed  field.  From  in the  above discussion, these statements can be made: 1.  Acoustic emission  can be used  as  a precursive signal  rock mass in laboratory and in field.  for violent  failure of  200  ur  1A3  [NTS  Precursor}' Signals in Comparison with Field Measurements /  z 13 O  •  ;  10 ooo -  1A zz  1 000 "  3  >:  ; b  i i  I  2  100 -  1  10 "  '  \  1  1  1  1  0  1  2  3  MAGNITUDE M  Fig. 11.4 The relationship between size and number of seismic events. The wavy line shows measured typical data. The straight line is a best fit of the form logN = a-bM (after Brink et al, 2.  Before the by  a  violent failure, the event rate increases  drop  immediately  acoustic energy increases 3.  The  most  frequency  [17])  significant usually  preceding  the  failure,  sharply, usually and  at  the  followed  same  time,  steadily and shows a peak at the failure.  effect  decreases  measured prior  to  in the  the  field  failure.  is This  that is  the found  corner to  be  related  to  associated with the previous facts. 4.  Precursive  signals  monitored  in  the  laboratory  tests  can  be  violent rock failure in the field. When compared with field measurements,  a  Precursory Signals in Comparison with Field Measurements /  201  similar pattern of acoustic emission is observed, and these may be universal phenomena preceding specimen failure and rockbursts.  Most microseismic monitoring systems used in the field today cannot carry out the spectrum analysis. Data is usually displayed as event rate and energy the  release.  phenomena  associated  Reliability in predicting an impending failure can be  improved if  of decreasing  energy  taken into consideration.  corner frequency  and increasing emitted  are  CHAPTER  12. N U M E R I C A L  SIMULATION ROCK  Because  the  testing  number  of  specimens  studies,  the  behavior  may  affect  signals  it  is  obtained  results and  still  not  above  are  results,  emission  rockbursts emission  clear.  universal  were  predicted  taking into  Therefore it  emission  ACTIVITY  AT  FAILURE  acoustic  actual  of acoustic  understanding of acoustic above  of  OF ACOUSTIC  is  are  very  account  hard  phenomena.  obtained  In  to  on the  order  seismic  rare  other  a  limited  from field factors  that  saj' if the precursory to  behavior and to verify the  a numerical model based  on  obtain  a  better  acceptability of the  model by Burridge [ 43 ]  is developed to simulate acoustic emission under various conditions.  This Usually given  model  is  unique  modelling means  set  empirical  of  to  relationships  formula  and no simulate  of  available  that for  evidence  MATHEMATICAL  similar  a phenomenon  event.  the  There is  acoustic  entirely based on the proposed stick-slip  12.1.  of  the  level.  a  anistropic  matter  A macrofailure starts  fracture any  is  process,  movement  surrounding  such  as  of  however of  no  physical  law  or  rock.  This  model  is  particles.  rock  fracture  mass.  as  Before  development  a kind of violent the  strength  beginning  at  failure  point, some  the  stress  from some local microfractures. A t any stage of the  at the  of rock particles  rock  found.  MODEL  of  process  been  process.  described earlier, rockbursting can be considered  failure  has  or an event according to a  emission  As  nonhomogeneous  work  It  is  beginning, during fracturing, or during slipping, at a local area will induce vibration among the this  vibration  202  which  generates  acoustic  signals  Numerical Simulation of Acoustic Activity at Rock Failure /  203  and it is by this means that the seismic energy is radiated.  In  the  analysis,  the  discretized  same finite  way element  into individual  represented by  Because  as  with  or  other  numerical  boundary  elements.  element  The continuous  methods  method, system  in  the  stress-strain  rock  of the  rock  mass  is  mass  is  a discrete system of individual particles.  the  shear  process  takes  place  on  the  contacting  surfaces,  the  movement occurs only on the failure plane. Besides, two variables are enough to describe an exact location in a plane. This model is not however involved in the exact  description  of  location  of  an  element.  Only  the  behavior  of  an  element  during the movement is of interest, so only one degree of freedom is needed for the model.  This model is a multi-particle shear system simple  shear  connected  models  together  presented  in chapter 6.  It  by weightless springs, figure  and is a combination of many consists  12.1.  of a series of particles  The mass  of the material  is concentrated on the individual particles and the spring represents the elasticity of the  rock mass.  The driving force  is  applied at the  end of the  last particle  from a support which moves at speed V .  Let the mass of particle i be M . , the stiffness of spring i be distance  between  beginning,  all  coordinate  system  adjacent  particles shown  are  two at  particles rest,  in figure  be  with  12.1a)  a.  Further  particle  N  assume at  the  Xj and the  that  at  the  origin  of  the  and all springs are unstressed  except  Numerical Simulation of Acoustic Activity at Rock Failure / 204  A; i—* fA}.j Y^rty i r v -Onn  ilj «79—  V \ a)  - * f.  i-1  1  R  b)  Fig. 12.1 Diagram of acoustic activity model  the  last  spring  N . Then  the  initial  conditions  of position  and speed  of  each  particle are  {  X.(0)  =  (N -  i)a  (12.1)  X;(0)  =  0, i = l , 2,  N.  Because we are interested in the slip behavior of the whole system of the model, we further assume the driving support has moved a distance  £  0  at t=0,  or the driving force in spring N has reached the static friction of particle N : X So  =  Ho =  (C +  N  f* (0)  =  N  C +  M ff)/X g  UL <J, or S  N  where f(0) is the static frictional resistance, M  G  is the coefficient of static friction,  (12.2)  Numerical Simulation of Acoustic Activity at Rock Failure /  205  o is the normal stress.  Therefore at any time t, the motion equation of particle i can be obtained by  the  equilibrium of forces  acted  on that  particle, as  in figure  12.1b),  in  the  horizontal direction. M.X.  =  F. -  11 i  F. ,  I  =  I  1, 2,  -  ff  '  (12.3)  I  N  where M . is the mass of particle i, X. is the acceleration of particle i, F. is the driving force from spring X. behind, F . . , is the resistance force from spring X j . , ahead, * f. is the total resistance force from particle i.  The forces vectors  in the vertical direction are in balance. According to the  in figure  12.1b), springs  on both  sides of particle i are  force  in compression.  The distance between two adjacent particles at any time t will be AX.  =  I  X. — X. , + 1  I* '  I  i = l , 2,  N.  Obviously, the compression in spring i is |. i  =  s  a -  AX. = I  a +  X. i  ++ 11  -  X.. I  Therefore, the force F . in spring i is F. i  = =  X.$. = 1, 2,  X^a +  X.  -  + 1  X.)  (12.4)  N-1.  Similarly, the force in spring i-1 is F.., l  1  =  X..,(a I  1  +  X. I  X..,) I  1  (12.5)  Numerical Simulation of Acoustic Activity at Rock Failure / 206 Note  in above  two equations,  values to Xo and X ^  The  +  i= l,  2,  ^ , see equations  resistance  * f.  force  N  can be used  by assigning  (12.8) and (12.9).  includes  the  frictional  force  f.(X-)  I  component terms. /j , g  of seismic  The frictional  radiation EoX- and is  referring equations  Substituting  ]  force fj(Xj)  special  is in turn  and  the  i  a combination function  of the two  a combination function of C, a and  (6.16) through (6.20) for detail.  equations  (12.4)  and (12.5)  into  (12.3),  the  motion  equation  becomes X.  [X.(a + X .  =  I  i  I  =  + 1  i  1, 2,  +  -X.)  X.. (a + X . - X . . ) 1  I  -  1  I  1  I  f.(X-) I  1  EoX-]/M I  r  (12.6)  i  N  where Eo is the coefficient E  -  I  l  of seismic radiation, E o ^ k / E ,  is the elastic modulus of rock concerned,  k is the material constant.  For all  the simplicity of programming, equation (12.6) will be left as it is. If  springs have the same stiffness  equation (12.6) X. i  In required. F  1 V  =X (^ M  [X(X.. -2X.+X. 1  1, 2,  order It 0  M , then  becomes  = =  X and all particles have equal mass  to  can be  + Vt—X ). M  + 1  )  -  f.(Xj) -  EoXjj/M  (12.7)  N  solve seen  equation from  (12.6)  figure  or  12.1  (12.7),  that  if  boundary i=l,  Substitute Fo into equation (12.5),  Fo = 0  conditions  are  and if i = N ,  Numerical X (a +  0 =  X  0  X =0,  Then either  Simulation of Acoustic Activity at Rock Failure / 207 - Xo).  X  or Xo = a+X^.  o  In later programming, both X  0  and Xo have  to be assigned values. For convenience, we set X  =  0  =  a +  0  (12.8)  { Xo  Similarly, substituting X X  (£  X  R  into equation (12.4) results in  0  +  Vt  -  X )  N+1  =  So  +  Vt -  N  N  X (a  =  N  +  X  N  +  1  -  X ),  or  N  a  (12.9)  Obviously, by equations (12.8) and (12.9), we have X  *N  {  Therefore,  X,  =  0  +  1  =  V  (12.10) -  with equations  (12.8)  to (12.10),  i varies  from  1 to N in equations  (12.4) and (12.5).  12.2. The  ENERGY energy  simple  ESTIMATION  changes  during  slip can be estimated  in the same  shear model. The basic energy calculation for a single  way as for the  particle model has  been achieved in chapter 6. As previously described, this multi-particle model is a combination  of  system  be the sum of energies  will  many  single  particle  models, stored  and so  the  in all particles  energy  of  the  whole  and all springs. By  equation (6.23a), the energy change rate is given as -V-(E, dt  +  Ep) =  k  v  '  We -  W, -  Wr  (12.11)  f  where E ^ is the total kinetic energy, and N  .L T I M . X -  E,  =  Ep  is the total potential energy stored in all springs, and  2  (12.12)  Numerical Simulation of Acoustic Activity at Rock Failure I 208  N,  Ep  =;.2 iX (X.  We  is the rate of doing work in moving the support against  1  -  i  X.  + 1  )  •  2  (12.13) the spring  being of order V , and We  =  F  N  - V=  X ('$o  +'Vt - X )V,  N  N  Wj. is the rate at which work is done against friction, positive,  Wr  is the power radiated along the semi-infinite string, positive,  • Wr  =  N .1 , E o X . . 1 = 1 !  For  a  time  2  period  At— t —t\, the 2  total  energy  change  in the  system  would be the integration of both sides in equation (12.11) over At. That is ; ^(E t  AE where A E ^ = AEp  =  We  W  f  =  + Ep)dt =  k  + A E p = We -  k  E (t )  -  E (t,),  Ep(t )  -  Ep(t,),  k  2  2  ; ^[X V(^ t  N  - ^ i t l M  =  J^(We -  t,  W  f  -  W  f  -  Wr)dt, or  Wr  (12.14)  k  + Vt -  0  (  * i  )  ]  d  X )]dt  (12.15)  N  t  N  = -r* x ,f(x.)dx. Wr  =  "X! 1 = 1 1 1 & [ £ .EoX. ]dt ti l — i i  1  2  During the period At, the total energy dominant method  function later,  these  given  in  energies  equation can only  (12.6)  loss is Wj = W^.+ Wr. will  be  be estimated  solved  by  Because the  the  numerical  by approximation. In field  Numerical Simulation of Acoustic Activity at Rock Failure /  209  microseismic monitoring of rock burst, the recorded energy is the only information available and is just a small part of the total energy radiated out to some distance estimate  the  radiated  from  comparison. rectangle [45],  item  Wr in  the  In  which  can  this  source.  later  away  modelling,  Items  programming, be  found  from a source. which  represents will  to  in mathematics  energy  Therefore it is significant  W^> and according  loss, which is the  the  textbook  the  also  seismic  to  energy  be  computed  for  numerical  integration  by  on  numerical integration  the approximation of these parameters will be : Wr  =  W,  =  n N . E o . Z , X ^X?.-At J= 1 i = 1 ij J  f  (12.16)  .1 , f . ( X . . ) « A X j= 1 i = 1 i  y  (12.17) ij  where At. is the time increment at step j , AX.j is the movement of particle i at step j , Xj. is the slip velocity of particle i at step j , n> 1, is the number of sampling points within time window At = t — t , . 2  In  each  succeeding  running  of  the  program,  the  period  At  will  be  specified. The sampling number n varies and is determined by the program itself, depending on the  time  step A t , which is  in turn controlled by the  accuracy  e  specified to the computer solution of slip velocity X .  12.3.  COUNT  In field seismic event  number  OF EVENT monitoring, in addition to the seismic energy released, is  another  important precursory  signal.  The event  the  rate  seismic indicates  the frequency of microseismic activity. In this acoustic activity modelling, both the  Numerical Simulation of Acoustic Activity at Rock Failure / energy  change  will  therefore  be  calculated  and  chapters,  the  the  acoustic  event  210  will  be  simulated and counted.  As  discussed  phenomenon  in  in  shear  previous  failure.  The  many discrete particles connected the  This  overcomes  force  the  elasticity  can  move  the  springs, these  If the  load is held at  vibration.  If the  load to  high  that  some point, the  load continues  adjacent a  until the  shear  particles along this  springs failure  surface  be  considered  particle  When  the move  for  load  is  significant  to  consist  of  formed  compressed  will begin to move.  as  due  if to  it the  and probably vibrate  particle moving will the  distance  removed,  back to  and chain reaction takes is  some  energies stored in the  to increase,  surface  a  first and a force is induced in each of  particles will  around their original positions  can  is  by springs. When a load is applied to  relevant  corresponding resistance.  of the  mass  together  model, some springs are compressed  them.  the  rock  stick-slip  springs damp off.  still possibly  induce  springs will transmit  place.  If the  discussed  in  A t the same  load is  chapter  time,  4,  so all  the vibration  becomes intense.  If every  any  slip  or  any  change  of  particle is considered to generate  activity prior to the expected  from  increase  significantly  vibration.  In  direction  as  tests the  following  of  failure  program,  acoustic is a  during  the  vibration of  an acoustic event, the history of acoustic  failure can be recorded during the  laboratory  the  moving  emission,  approached specific  program running.  the due  register,  the moving and the change of direction of all particles.  acoustic to  the  L , is  event more  assigned  As is should  intensive to  count  Numerical Simulation of Acoustic Activity at Rock Failure / 12.4.  211  LIMITS T O T H E M O D E L  The  physical  limits  conditions  and  certain  which should be considered  requirements  of  this  model  in programming. They are the  introduce  some  logical position  of each particle, the effectiveness of the spring and the stick-slip conditions.  12.4.1. The Logical This  model is  Position  concerned with the  stand in a line when  no shear  due to a shear force, they necessarily  in the  part  may  be  keep  in  the  problem of one dimension. All the  force  is applied to them.  particles  Once movement  starts  move one after another along the same axis but not  same direction. This can be pictured from the  in compression original  N  and  consecutive  some  part may  sequence.  In  be  in tension.  other  words,  fact  that  But  some  they  there  is  all no  superposition among particles, and so the following conditions must be satisfied all the time, X..,  >  i  1, 2,  =  X. >  X.  (12.18)  + 1  N.  12.4.2. T h e P h y s i c a l Condition The springs connecting adjacent particles are elastic only under normal conditions, i.e.,  the  the  elastic  limit.  If  load is  the  load  deformation  the  compression  not too high. Once the load reaches  load  be  the  continues  happens.  would  or  A t this  compression  to  point, the  transmitted  deformation, or very little.  increase,  through  of  the  the  the capacity of the spring,  spring reaches  elasticity  disappears  spring would act as it  to  the  next  its  maximum  and  no  more  a "stiff stick" and  spring with  no  further  Numerical Simulation of Acoustic Activity at Rock Failure / This extreme case can occur when the normal load acting on a particle so high that the the  frictional resistance is more than the  spring. Because the  elastic  force  of  the  normal load increases the  spring is  linearly  order to avoid this problem during running the be  limited under this maximum value  is  maximum elastic force in  friction force  proportional to  212  its  linearly  elastic  program, the  and  deformation.  the In  normal load should  corresponding to a particular spring which  is characterized by its stiffness Xj.  As particles  shown  in figure  12.1,  Apparently, compressed  if  no  the  distance between two  adjacent  =  I  the  stress  X. * is  , i=l, ' '  2, '  N.  induced  in  spring  + 1  i,  A X . = a.  As  the  spring  is  under load, the deformation will be £.  induced  particle i is o,  a -  force  AX., ( £ . < a ) I *i in  spring  i  is  F . = XJ£J.  ^  the  normal  pressure  on  the  the static friction would be  f.(0) • = The  time  is AX. = X. i i i  and  at any  C +  M (M. s  a).  +  maximum elastic force occurs in the spring when F. = f.(0), or X.£. r r  0  — C +  n (M. + s I  a),  Usually, the particle mass M . is extremely Then above equation  order  for  the  model  to  should satisfy the condition: X^jo  negligible.  s  s  in  a and  reduces to  i i° Therefore,  small compared with  s  C +  ju a, or g  function  properly,  the  normal pressure  a  Numerical Simulation of Acoustic Activity at Rock Failure / a  where  Xj  is  <  (X.SjO  the  C)/M  -  stiffness  of  213  (12.19)  S  spring  i,  which  is  proportional  to  the  elastic  modulus, £. C  is the allowed maximum deformation of spring i,  0  is the cohesion, and  a  is the static coefficient  of friction.  s  In purpose.  running  the  Therefore,  corresponding reassigned  Xj is  program once  later,  £. ^0.2a, to  a large  is  o  used  for  demonstration  AX. = a — £ . <0.8a  or  o  increased  £. =0.2a  occurs,  0  value  to  simulate  the  stiffening.  the Xj is  to its normal value when AX. > 0.8a.  12.4.3. Conditions for Stick-slip This  acoustic  when on  the  occurs,  model  slip begins,  either  assumption means  the  failure of the whole  The been  is  based  on  stable  in  principle of  shear  we  know,  sliding or stick-slip will occur. This model  works  of stick-slip of individual particles. movement  of  all  particles  and  process.  As  The stable sliding, once it is  considered  as  the  final  system.  stick-slip phenomenon  discussed  the  chapter  7.  only occurs under certain conditions, which have These  conditions  are  satisfied  if  the  loading  conditions of the model system fall into the lower part of the transition chart in figure  7.2.  For a  given  material,  stick-slip to occur, for each  its  elasticity  normal pressure  is  there  given, is  and in order for  the  a maximum loading speed,  or for any loading speed there is a minimum normal pressure.  Numerical Simulation of Acoustic Activity at Rock Failure / 214 Therefore,  in  order  for  conditions have be to satisfied  12.5. The order  NUMERICAL expression  model  to  function  properly,  all  the  above  and must be considered during programming.  SOLUTION  BY RUNGE-KUTA  METHOD  given in equation (12.6) is a set of ordinary multi-variable second  differential  solutions  this  equations,  cannot  be  with  found  unknown  due  to  in their  their  denominators.  complexity  and  we  Again, must  explicit look  for  numerical solution.  An  introduction to Runge-Kuta method has been given in chapter 6 and it  is applied to the second order differential' equation of one variable. By the same principle will  of extension,  be much  more  it can be applied to equation convenient  for discussion  implicit function. Let X = Y , then (12.6) 1  Y {  X  1  = f(t, X - , X , X  f  1  1  1  = g(Y*) =  i=l, Note,  1  the function  convenience  in the  superscripts  here.  2,  Y  i + 1  (12.6)  to express  of multi-variables. It  equation  (12.6)  as an  becomes:  , Y ) 1  (12.20)  1  N  f represents following Then  ,  from  the right hand side of equation all subscripts (12.1)  and  in  (12.6)  have  (12.8)  to  (12.10),  (12.6)  been we  and for  replaced by have  initial  conditions Y\0) { and  = 0  : X*(0) =  boundary conditions  (N -  i)a  (12.21)  Numerical Simulation of Acoustic Activity at Rock Failure / X°  x  {  =  Y  By  N  a  Vt =  Y  +  (12.22) |  -  0  a (12.23)  1  1 = V.  +  simple  expressed  +  1  -  N  Y° {  X  215  extension  of  equation  (6.29),  the  solution  to  (12.20)  can  be  as  {  X  n  Y  !  i where X ^  + 1  =  =  n + 1  =  n  +  + 1  *  ( m  Y'  +  n  1, 2,  and Y ^  + 1  X  +  (k ,  2 m  +  1  2  +  2k  2  2 m  +  3  +  2k  m  «  +  3  )  /  (12.24)  6  ki)/6  N are new values to be found for particle i,  X ^ and Y ^ are known from previous calculation at step n, each m  h  m,  =  k,  =  1  l  m 1  l  1  n  X  n > *n*>  Y  „)  = h-Y^  h « f ( t +h/2,  X j j - ' + m , ^ , X + m\/2, 1  X **+m\l2,  l  Y ^ + k',/2)  l  n  +h/2,  n  (12.25)  Xj- +m /2, 1  '  2  n  *  X +m /2,  X  1  '  n  2  *  '  1 + 1  +m /2, 2z  n  Y +k /2) 1  '  n  2  h-0^+^/2)  3  k,  =  1  I  mj, h  for particle i are calculated as following:  K-*>  h.g(Y )  = h-f(t  k,  Here  - « V  1  h.(Yj^4-^/2)  2  m  and k  1  is  h . f ( t + h, X ^ - ' + m i , x U m l , n  h-(Y the  x  n  X ^  1  +m ,  Y ^ + k ,) 1  3  +k ). 33  increment  of  determined according to accuracy  time  t  between  step  e for the solution.  n  and  step  n+1  and  is  Numerical Simulation of Acoustic Activitj' at Rock Failure / 12.6. A  216  PROGRAMMING  computer  solution  given  in  running  on  the  Hewlett-Packard computer. The flow chart of this program is given in figure  12.2  equation  and are  the  program  (12.24)  program  named  was  is  MODEL4  written  listed  in  for  the  BASIC  in appendix  4.  numerical  language  Some  for  variables used  in the  program  listed in the following. To and T. are start time and instant time, J h is the time step, varies, X.j and X.j are slip distance and velocity of particle i at time T., F.j is the total driving force on particle i at T., Fj-j is the total resistance from particle i at T. L T-  is the event counter, and n t  is  the  sampling window  At in  which  numerical sampling  is  taken. The sampling number n depends on the window At and the time step h.  This  program  starts  counting the  event  number from  the  beginning.  At  the same time, the work against friction Wp the seismic energy Wr, the energy ratio  W r / L and the  total  energy  loss  W l are  calculated.  A l l these  results  are  accumulated for a given time window Tj ^ and stored on file. The kinetic energy can  also be estimated at any moment.  iMimerical Simulation of Acoustic Activity at Rock Failure / (start  choose  function  prepare  for  file  input shear  on d i s k  set  control  loop  begins,  computations,  see  217  )  data strength,  to  change  store  C and  n  results  variables  J=l ,  Tj =To  figure  12.2b) yes h"  J=J + 1 Tj =Tj +h  1  yes  store  events  & energies  & reset  them  to  0,  To=T  yes  yes  yes  store  data  on  file  ( s t o p ^) Fig. 12.2a) Flow chart for program M O D E L 4 : acoustic simulation  Numerical Simulation of Acoustic Activity at Rock Failure /  call  search SUB1 f o r t i m e  choose  call  call  SUB2  RK1 t o  to  compute  f o r max. & m i n . X step h, accuracy c o n t r o l l e d  time  compute  compute  energy:  step,  Xj j ,  forces  W, ,  b y g.  h=min.  X;/,  Fij ,  Wr, W , t  i =l  Ff-j  t  energy  to  N  i =l  to  N  ratio  1 = 1  count  event,  L=L+1  no  yes  Fig. 12.2b) Flow chart of the computation part in program M O D E L 4  218  Numerical Simulation of Acoustic Activity at Rock Failure / 12.7.  MODELLING  RESULTS  12.7.1. Resemblance  to the  The  produced  acoustic  similar  to  model  those  results  computer results and  12.4.  acoustic way  the  activity  Results  fascinating during  failure took place  in terms  As failure is  results,  which  acoustic  emission  runs of program M O D E L 4 as  of event rate  as from tests, figures  generated.  Testing  recorded  from two  Before  219  10.2  approached, the  tests.  are given  indicated by the  and  and 10.3.  surprisingly  seismic  are  Some  typical  in figures  arrow, the  energy,  very  behaves  12.3  modelled the  same  A t the beginning, not much signal is  generated  signals  are  very  active,  both  the event rate and energy release increasing sharply.  field  In  chapter  data  and  signals  the  good  acoustic emission agreement  are realistically simulated  increases level  a  10,  sharply  immediately  as  failure  is  was  found  again by this approached  preceding the  from experiments between  was  them.  compared with  These  precursory  numerical model. The event rate  and then  drops  failure. Meanwhile, the  to  seismic  the  previous  energy,  both  low the  energy rate and the energy ratio, remains low when the event rate goes up and increases to  dramatically prior to the failure. The increase  fracture  propagation.  The  drop  of  event  rate  and  in event rate the  increase  corresponds of  acoustic  the  acoustic  energy indicate the formation of macrofractures.  Even  though  the  model  itself  has  no  direct  relation  to  Numerical Simulation of Acoustic Activity at Rock Failure /  I SB Ui \  ui rr Q:  event  80  rate  SB  failure  z Ul >  Ld  f .A  B  k  .035 .Bl .015 .02 .025 .B3 .035 .64 .045 .05 .055 .06 .0E5 .67 .075  .09  fa I lure  I Ul LO  (\J \  seismic  -energy  rate  Ul  y—  fX  oc >(J Q: UI  2B0  z•  —I  Ul  1  1  L_  1  i  i  L__  i — ' vn — /  .005 .01 .015 .02 .085 .03 .035 .04 .045  I  i  i  i  i  .05 .055 .06 .065 .07 .075 . 0B  •failure  O. (J > Ul  z ui  seismic  energy  ratio  600  .003 .01 .015 .02 .025 .03 .035 .04 .045 TIME  Fig. 12.3  .05 .055 .06 .OES .07 .075 .OB  (S)  Computer results from the numerical acoustic model  220  Numerical Simulation of Acoustic Activity at Rock Failure /  Pn-IGOO Pa V-. 1 iVs Us-.65 E-1E6  EVENT RATE  failure  after  0  .81  .82  .63  .34  .63  .66  .8?  .09  .89  shocks  . 1  .11  . 13  SEISMIC ENERGY RATE  I Ld  in LJ  slt.r  shocks  .1  .11  .IS  SEISMIC ENERGY RATIO  LJ  3B  LJ U)  13  .83  .04  .05  .66  .07  .1  .11  .12  TIME (S)  Fig. 12.4a) Complete pattern of acoustic activity prior to failure, showing after shocks  221  Numerical Simulation of Acoustic Activity at Rock Failure /  Fig. 12.4b) Complete pattern of acoustic activity prior to failure, showing the similarity between total and seismic energy  222  Numerical Simulation of Acoustic Activity at Rock Failure / emission,  its  field  results.  used  to  results  223  are in good agreement with both the experimental and the  This justifies  interpret violent  that rock  the  postulated  failure.  shear  failure mechanism  Acoustic emission  is  indeed  can be  a precursory  phenomenon for rock failure.  12.7.2. T h e Total E n e r g y Released versus the Seismic Energy  released  during  a  rockburst  is  complicated  Energy  and  cannot  be  calculated  precisely. In microseismic monitoring, the  monitored energy is only a small part  of  of  the  total  energy  released.  This  part  energy  is  radiated out  as  seismic  energy and is detectable by special sensor. It is not known what the relationship is between the seismic energy and the total energy released.  It is believed that the major part of the  energy released during a burst  is consumed against the resistance force including frictional force. In addition to a small as the  part  seismic seismic  transformed into heat,  the  energy.  energy  waves  If the  seismic  reach the  rest  is has  almost completely not  boundary between  damped off  the  rock  transmitted out completely  mass  when  and air, it  is  transformed into sound energy. If this sound energy is big enough, an air shock can be experienced.  A question arises about how accurate it is to estimate the pattern of the total energy through the detected  seismic energy, as is usually done in the field.  In other words, it is a question of whether the parameters  remains  the  same  throughout the  proportion of these two  failure process.  In this  energy  numerical  modelling, the total energy release is also calculated for comparison. Some typical  Numerical Simulation of Acoustic Activity at Rock Failure / results  are  given  in  figure  12.4b).  These  two  parameters  are  alike,  224  for  they  change in largely the same way throughout the process. This gives us confidence in the use of the seismic energy to estimate  The  seismic  rate and energy  energy  release  is  analyzed in this  ratio, which is  given time window. In the results parameters  show  a  similar  the change of total energy  the  from  modelling as  average  all the  behavior,  energy  both energy per event  runs of the  although  the  release.  release  during a  program, these two  energy  ratio  shows  the  anomaly more clearly.  12.7.3. After  Shocks  The  is  program  usually stopped  once  the  final  failure occurs because  each run  takes hours to finish. In some cases, an attempt was  made to run the program  until  completely  the  typical  energy  example  accumulated before is  given  in  were  generated.  many  after  shocks  lower  speed  than  released the  during the  curve  represents  of the  it  after  energy total  built  figure  up  shocks.  rate  and  the  failure  12.4a).  As  But the  before This above  amount of energy  the is  can  energy failure.  also the  has  be  seen,  release  damped off.  after rate  Obviously,  released.  This  axis, is  failure,  decayed  more  clearly shown by the  horizontal  the  A  in a  energy  is  area under  because  this  in agreement  with  area the  what was observed during the direct shear tests described in chapter 10.  After the failure, the energy ratio drops immediately, and so the anomaly of failure indicated by this parameter is well defined. During the period of after shocks, the event rate seems to build up again when the energy is about to be  Numerical Simulation of Acoustic Activity at Rock Failure / finished.  These  after  shocks  may  be  explained  in  such  a  way  that  initiation of failure, manj' microcracks are formed. As slip continues,  225  at  the  these cracks  are crushed and at the same time new cracks are formed.  According lead  to  the  takes place.  to  the  formation  fracturing of  During the  a  principle,  final  shear  failure  movement,  the  joining  surface,  on  some new  of  macro-fractures  which  the  shear  micro-fractures  will  process  are generated  and some micro-fractures  are crushed. Therefore, the  event rate will remain high  on  energy  small.  some levels but  the  consumed,  the  shear  rate  be  confused  may  movement with  release  involved  ceases.  the  A t this  the  major  failure induced from the build up of the the energy release rate  12.8. 1.  is  moment,  failure.  When the  This  the  energy  is  build up of event  mis-impression  of  a  event rate can be cleared by looking at  simultaneously.  SUMMARY A  numerical  simulate counted  acoustic  the by  acoustic  examining  model  has  activity the  been developed prior  to  slip and the  violent  based  on the  stick-slip  rock  failure.  Events  to are  change of slip direction and energy  release is estimated for each event. 2.  The limiting conditions position  of each  for this model  particle in the  are considered,  string, the  which  are  the  logical  physical condition for the  spring  to effect properly and the condition for stick-slip to occur. 3.  To do the  simulation, a numerical method is used and a computer program  has been written,  which has  reproduced results very similar to the  acoustic  signals recorded during laboratory tests of rock specimens and measured  in  Numerical Simulation of Acoustic Activity at Rock Failure /  226  field monitoring. The  simulated  results  show that  the  total  energy  release  and the  seismic  energy vary in similar way. After  shocks  may  be  generated  after  microcracks formed during slip but they These results  therefore  a. The process  the  failure  due  have very little  to  the  new  energy.  show that:  analogous  to  shearing can be  a fundamental  mechanism  at  the post failure stage of rock, b. The  acoustic  emission  is  indeed  model is  a useful  a  useful  precursory  signal  for  violent  rock failure, c. This  acoustic  tool to study  the  acoustic  activity  prior  to the violent rock failure. d. More are  importantly, probabty  the  universally  precursory acceptable  signals and  field interpretation of violent rock failure.  obtained the  method  during this can  be  research  applied  to  CHAPTER  13. A C O U S T I C  ACTIVITY UNDER  DIFFERENT  CONDITIONS  Because the behavior of acoustic emission is not clear for many conditions due to the  limited  results  from  emissions  under  numerical  model developed  various  used to simulate the allowed  violent  realistically. activity  laboratory  rock  conditions in the  acoustic  are  during violent  it  in  chapter.  associated us  measurements, this  This  research  acoustic  with  acoustic  activity  a  method  to  failure on computer.  Further  study  using a computer program M O D E L 4 under different situations.  field  are  using  model  to  be  study  For each condition to be simulated, this  modelled  as  realisticalty  as  possible,  of  the  program  should  be  tolerable.  In  parameters or the event rate and the seismic are  changed.  instead  of  The main interest its  absolute  value.  is  acoustic  in the  The  the  energy  following,  simulated  emission  they  emission.  have  the  results  are  to  be  convergence most  are examined as  pattern of change  out  program runs  acoustic  but  it has  carried  within the limits of the model given in the previous chapter and the speed  can be  the  was  the  simulated  to uncover the mystery of acoustic  under a given set of parameters and generates the associated  Conditions  acoustic  activity prior to violent rock failure because  provides  rock  and  studied  previous  failure and the  Therefore,  tests  of each  useful  conditions parameter  presented  in  the  following.  13.1. A C O U S T I C  EMISSION A S N O R M A L P R E S S U R E  VARIES  First, the effect of normal pressure on acoustic emission is examined. The normal pressure is set  to 500  program,  other  with  Pa, 1 K P a and 10 K P a respectively  conditions  unchanged.  227  The computed  for each run of the  results  are  plotted  in  Acoustic Activitj' under Different Conditions / figure  13.1a)  to  c).  The results  from  the  three  Before the failure, a sharp increase of the event  runs  have  a  228  similar pattern.  rate occurs and is followed by  a drop. The increase of energy occurs at a moment prior to failure. The pattern of  acoustic  event  rates  release  the  the  are  because normal  normal  is  in  increases  expected of  emission  the  the  with the  same  same  the  pressure  order  normal  energy  pressure,  under  also  of  10  normal per  4  pressure.  released  figure  all  6.9,  second,  The increase  considered.  although of  For  a  a  linear increase single  the  seismic  during each slip increases where  exists.  pressures  particle,  energy  energy  with the  of  stick  the  The  square  time  event  is  with  rate  is  approximately the reciprocal of the stick time. If however more than one particle exists,  as  in  this  acoustic  model,  the  event  factors, such as the mutual reaction between  rate  is  also  influenced  by  other  particles. The vibration effect should  also be considered.  This pressures of the  suggests  high  stress  will  increase  effect  the  pattern  release.  field  does  the  It not  can be believed change  fracturing  loading.  the  time  it  takes  the  energy  In addition, these results on  of  acoustic  if other conditions are the same.  energy  violent.  that  for  show the  is  The only difference that  during  process  and  emission  the  failure  to  normal occur  make  all  process,  a  propagation, but it the  failure  pressure has from  for  is the magnitude  fracturing  of fracturing  consequently  that  the  similar  the  more  not much  beginning  of  Acoustic Activity under Different Conditions / P n - 5 0 0 Ptx V-. I Us-.G5 E-IE6  n/t  EVENT RATE  LJ r—  rx  .21  .07  .02  .08  SEISMIC ENERGY RATE  u ^-  •02  .83  .B4  SEISMIC ENERGY  .05  .86  .87  RATIO  IT) Ul  TIME  Fig. 13. la) Numerical acoustic emission  (S)  at normal pressure  500  Pa  229  Acoustic Activity under Different  Fig. 13. lb) Numerical acoustic emission  at normal pressure  Conditions /  1 KPa  230  Acoustic Activity under Different Conditions /  •f ai lure  EVENT RATE  Pn-lt:4 V - l <i/s Us-.65 E-1E6  Ul >  B  H  78B  (S3  680  >->  —' UJ 1IX  . 0 8 5 .61 . 8 1 5  02 . 0 2 5  .93 . 6 3 5 . 6 4 . 8 4 5 .85 . 0 5 5  B5 .67 . B75 . BB  SEISMIC ENERGY RATE  588  <x  48B  UJ  3 OB  z  o s: U) Ul Ul  28B 180  -J J . 0 8 5 .61 . 8 1 5 . 0 2 . 0 2 5 . 8 3 .033 . 6 4 . B 4 5 .83 . 0 5 5 . 0 6 . B B 3 . 0 7 . B  I  38B 28B 2E8 248 22B 288  o — tt 1n cr 2 Ul  c_l  3:  01  > U— li Ui  188  SEISMIC ENERGY  168  RATIO  148 128 188 68 6B 48 28 8  . 6 8 5 .61 . 8 1 5 . 0 2 . 0 2 5 . B 3 . 0 3 3 . 0 4 . 6 4 5 .83 . 6 5 5 . 0 6 .863 . 0 7 . B 7 5 . 8 6  TIME («-)  Fig. 13.1c) Numerical acoustic emission  at normal pressure  10 K P a  231  Acoustic Activity under Different Conditions / 13.2. In  ACOUSTIC  chapter  slip  6,  the  behavior.  EMISSION  AS LOADING  loading speed  When this  speed  SPEED  VARIES  is  found to be another  is  above  a  232  important factor  critical level,  which  is  in  the  described in  chapter 7, the stable sliding will occur. When this speed is less than the critical level, the stick-slip behavior remains the same,  but the stick time has an inverse  relation with the loading speed T =c/V, figure 6.10.  The value of the constant c  2  is  very  small. In fact,  if V is  much higher than c, the  stick time T  will be  2  very short. The stable sliding corresponds to a near zero stick time.  During V = 0.01,  0.1,  13.2a) to the  c).  this  research,  1.0  m/s  the  acoustic  respectively.  When the  emission  The  loading speed  is  computed  is  results  relatively  particular condition modelled, both event rate  modelled  low,  for  are  loading  plotted  or when  and seismic  in  V<1  energy  clear precursory signal as observed before. The pattern of acoustic  the  relatively  increase  of  unchanged.  model,  figure  6.10.  loading  speed  will  loading These  This  speed,  are in agreement  may  increase  although  indicate  the  that  fracture  the with  energy the  during the  propagation,  indicate  it  rate  remains  of single  has  a  increases  fracturing process,  but  for  activity is not  release  results  figure  m/s  changed by varying loading speed, but the number of events per second with  speed  little  particle higher  effect  on  the energy release from fracturing.  When the loading speed is relatively high, say V = l ratio energy  indicates rate,  a  are  clear  anomaly.  ambiguous.  This  The is  other  two  m/s,  parameters,  probably caused  by  the  only the event  fact  energy  rate  that  for  and the  Acoustic Activity under Different Conditions /  EVENT RHTE f s i Iure  Ul  rx  a.  z ui >  .61  * B7  .82  .B9  .69  after  shocks  .68  .03  .1  failure  SEISMIC ENERGY RATE  ui — i (X  o: z Ul  u t-1  2:  U) *—i  u  .07  SEISMIC ENERGY  TIME  .1  RRTIO  (s)  Fig. 13.2a) Numerical acoustic emission  at loading speed 0.01  m/s  233  Acoustic Activity under Different 2B •  Conditions /  Pn-50B  Pa  25 Ua-.6S E-IES  24 22  EVENT RBTE  29 IB  a.  .81  .82  .63  .84  .83  .88  .87  .88  . B7  SEISMIC ENERGY RATE  LJ rx  •8'  -82  83  .B4  .83  1  1  1  .88  .89  .1  SEISMIC ENERGY RATIO  a. z  TIME  (S)  Fig. 13.2b) Numerical acoustic emission  at loading speed 0.1  m/s  234  Acoustic Activity under Different Conditions /  61  SEISMIC ENERGY I Ul  rvj u t—  .BJ2  .614  ,G16  .018  .02  .BIG  .BIS  . QJ  RHTE  Pn-308 Pa V - l m's Us-.63 E-IE6  (X  .eaa  . e x  .eee  .eae  .ei  .BIS  . e n  TIME (S)  Fig. 13.2c) Numerical acoustic emission  at loading speed 1.0  m/s  235  Acoustic Activity under Different Conditions / particular  condition  given in section  modelled,  12.4.  this  If the  loading  speed  loading speed  is  close  to  the  becomes higher, the  236  limit boundary stable sliding is  going to happen instead of stick-slip.  Figure the  beginning  13.2  also  shows the  of  loading  and  effect  the  of loading speed  failure.  At  higher  loading  should be shorter. This effect can be clearly seen from the  13.3.  ACOUSTIC  The  elasticity  and  hence  EMISSION A S E L A S T I C I T Y  directly  influences  were  studied  modulus  with  E  =  on 10 ,  10  8  are plotted in figure  observed,  figure  the  the  When the elasticity clearly  time  speed,  between  this  time  results.  VARIES  of rock mass has a close relation to its capacity of energy storage  emission  results  on the  behavior  model 6  ,  of  failure.  program under  10 , s  3X10*  Its  effects  different  Pa  on  values  respectively.  acoustic of  Some  elastic typical  13.3a) to d).  is high, the previously described precursive signals 13.3a)  and  b).  Both  event  rate  and  seismic  are  energy  release indicate a well defined anomaly. It can also be seen that the event rate and This  energy release rate increase increase  of energy  release  in magnitude with the increase of the may  indicate  that  higher  elasticity  elasticity.  of the  rock  mass can make the failure more violent.  However, precursive  when  phenomena  the tend  elasticity to  is  disappear.  low  as  Both  in  figure  event  rate  13.3c) and  and energy  d),  the  release  Acoustic Activity under Different Conditions / Pn-338 Pa V-.I m/s Us-.63 E-IE8  EVENT RRTE 3.5 3 2.5  -  tr  Bl  . BB2  .863  •8B5  .BOG  SEISMIC ENERGY RRTE  rvj  ft  LO  u  SI  .0B2  .883  .684  SEISMIC ENERGY  a  .003  .D06  .887  ,3BB  .66?  .BBS  .D89  . B1  RRTIO  3  a.  z  .694  TIME  .BBS  .B86  ts)  .13.3a) Numerical acoustic emission  at elastic modulus  100 M P a  237  Acoustic Activity under Different Conditions /  EVENT RHTE  Ul  a.  z UJ >  .81  .82  .83  .88  .84  .87  SEISMIC ENERGY RHTE  UJ I-  SEISMIC ENERGY RATIO o t<x  Q: z  TIME  (S)  Fig. 13.3b) Numerical acoustic emission  at elastic modulus  1 MPa  238  Acoustic Activity under Different Conditions / Pn-5B0 Pa E-IE5 Us=.65  event  rate  in \  energy  ratio  d a. a. u  2  1A  L I  / l , T I M E CS)  Fig. 13.3c) Numerical acoustic emission  at elastic modulus  100 K P a  239  Acoustic Activitj' under Different  Fig. 13.3d) Numerical acoustic emission  at elastic modulus  Conditions /  30 K P a  240  Acoustic Activity under Different Conditions / showed  broad signals.  However,  all  these  probably  because  plasticity.  Hence  event  number  Even  at the  parameters  the  rock  during the  consequently  moment of failure, the  are  mass  characterized with  very  failure process, decreases.  by  low  anomaly is not clear.  low  to  the  small. In this case, when failure occurs, the damage  occur  from  increases.  the  13.3,  This  possesses  fracturing becomes  capacity of storing strain energy is lowered. Energy released  From figure  magnitude.  elasticit3'  Meanwhile, due  241  less low  is  higher  intense,  the  elasticity,  the  during the failure is  and danger will be little.  it can be seen that the time it takes for the failure to  beginning  of  loading  decreases  dramatically  as  the  elasticity  This may also indicate that fracturing and energy release will be more  intense at failure when the elasticity becomes higher.  13.4.  ACOUSTIC  Rock  masses  EMISSION U N D E R M U L T I P L E  usually  consist  of  different  kinds  ELASTICITY of  rocks.  Each  different mechanical properties. In various conditions, the acoustic different from that in a massive to  study  the  effect  of the  modelled for cases in which middle of a massive  of  them  has  activity may be  rock mass consisting of a single layer. In order  anistropy of the  rock mass,  a thin harder or softer  acoustic  rock is  emissions were  intercalated in the  rock mass. This is done by assigning different elastic  to the springs of the model shown in figure  12.1.  values  Acoustic Activitj' under Different Conditions / 13.4.1. A H a r d First,  the  computed different  Intercalation  case of a hard  intercalated  242  layer  is  results  one  are  intercalation order of  given  is  modelled.  magnitude  in figure  The elastic  higher  13.4.  than  the  Apparently, the  from that as in the matrix rock alone  modulus matrix  acoustic  shown in figure  of  this  rock. The activity  is  13.3b). A large  number of events are generated before the failure, which surprisingly agrees with observations  [40]  made in the field because more seismic  events were recorded in  this condition.  As  can  be  expected,  when  the  shear  force  is  transmitted  to  the  hard  layer which seems to behave like a barrier, the acoustic emission begins to build up sharply. From then, the seismic  energy  equivalent  in  failure.  to  the  maximum  Meanwhile,  the  anomaly.  The precursive  rate,  not  is  situation, failure.  the  signals  event rate,  if  rate  signal  described  though  alone  the  are  figure  event  even  event rate  However,  precursive the  unique,  value  still  varies  greatly  until and  or the  occurs  prior  a misleading  energy  observable.  13.3b),  before,  one  can give  seismic  rate remains on a level of magnitude  is  A  and a peaking up of the  dramatic jump  shows up to  more  than  and down the  at  failure.  sharp increase release  the  same  followed  by  can be  The layer.  As  increase  of acoustic  harder rock usually  activity has  is  higher  one  In  this  the  final  time, a  the  drop of  seen prior  the failure. However, the magnitude of the event rate and of the energy is much higher than in the country rock, figure  at  of event  interpretation to  examined  energy  a  to  release  13.3b).  caused strength  by  the  presence  and fails  at  of the  higher  hard  level  of  Acoustic Activity under Different Conditions /  Id trx  3  1  •  E  '  - ° «  5  .B35  .83  .84  .B45  .05  . 845  . 83  SEISMIC ENERGY RATE  u cn  fail  .883  .81  .813  .82  .825  . 63  . 835  . 04  24  22 °  SEISMIC ENERGY  RATIO  IB  .AM 315  .82  .825  ii .83  .835  .04  .B45  .83  TIME ( s )  Fig. 13.4  Numerical acoustic emission  with a "hard intercalation  243  Acoustic Activity under Different Conditions / stress.  When  developed time  the  in the  stress  reaches  a  certain  matrix rock. But the  microfracturing develops  in the  level,  microfractures  layer  may  hard  hard  layer, the  may  remain intact.  stress will have  244  well  be  By  the  reached a  higher level. During this period, the microfracturing in the matrix rock will have become rate  more  intense  increases  increase  due  sharply.  of energy  to  But  the  increase  because  the  may be not significant.  of  at  Correspondingly, the  fracturing is  still  in  micro  event  scale,  the  As the stress continue to increase,  hard layer may dominate the failure process released  stress.  until it fails.  failure will be much higher because the  Obviously, the  existence of the  the  energy  hard layer  has enabled the stress to reach a higher level.  In  addition,  due  to  the  existence  of  takes for the failure to occur is decreased  the  hard  intercalation,  the  time  it  as compared with that given in figure  13.3b).  13.4.2. A Soft  Intercalation  Similarly, the case of a soft intercalation is modelled. The elastic modulus of this intercalated  layer  is  one  The computed results seen,  these  figure become  results  13.3b).  of acoustic  are  However  much higher.  order of magnitude  quite the  emission  similar to magnitudes  The warning time  though the event rate drops after  lower  than  the  matrix rock  are plotted in figure those of by  from  event  the  rate  event rate  a sharp increase,  13.5.  matrix  and  As can be  rock  energy  seems very  mass.  shown  in  release  rate  short.  Even  it drops not long before  the  failure. However a precursive signal is well developed by the energy release. The event rate and energy release together can still work to indicate the failure.  Acoustic Activity under Different  ^  see  Conditions /  4 a i lure  SEISMIC ENERGY RRTE  43B 4BB 35B 38B 23B  Ul u ui  .82  .83  .64  .D3  .87  I .86  if .39  I .t  i .11  i .12  .13  .14  .IS  700  Pn-SBO Pi  SSB  1 Uc«.65 E=IE6 tE5~E£=lE5>  68B 35B  see  EVENT  RRTE  45B 4BB . 33B  3BB 23B 2BB t5B  .Bl  .82  .83  .04  ,B3  .B6  .87  SEISMIC ENERGY RRTIO  TIME C s )  Fig. 13.5  Numerical acoustic emission with a soft  intercalation  245  Acoustic Activity under Different Conditions / 246 presence of a soft layer will obviously affect the failure behavior of  The the  rock  mass  because  this  layer  closely related to its orientation  has  a  failure behavior  maximum  different.  angle  to  In this case,  in the  soft layer first.  figure  13.3c).  As  delay  the  failure  A t this time continues,  as  can  be  in the the  the failure process  failure  The soft layer failure  may  will  as  be  place  shown  and propagate act  has  take  is small as  may  time  If it  behavior  and deformation  acoustic emission  the  however  soft layer alone.  fracturing will initiate  seen by  is  loading direction. If this soft  direction,  the  effect  in  the  a bumper  in figure  13.5  in  and  which  is  13.3b).  SUMMARY  In this chapter, numerical  model.  mass and the previously  sharply  The  drops  changes  anistropy  exist  and  approached.  acoustic emissions under  described  measurements  rate  shear  failure occurs.  shorter than in figure  13.5.  same as  microfracturing initiation  loading  matrix rock until the  major  Its  direction, it will dominate  will be the the  strength.  with respect to the  layer is parallel to the major shear and the  lower  drops the  of  in to  all a  energy  While the  pressure,  were introduced.  precursive  signals  cases.  low  level  release  profile  various  loading  prior  speed,  were studied elasticity  of  on the  the rock  Except in some extreme conditions,  obtained  Before  conditions  the to  increases  from  laboratory  failure,  the  the  failure.  dramatically  of acoustic emission  tests  event  At the when  is not changed,  rate  and  field  increases  time  the  the  failure  the  the  event is  magnitude  does vary with the change of conditions.  As number  the  pressure  of event  rate  increases, the  magnitude  and  for  the  time  failure  of energy release increases. The to  take  place  remain  more  or  Acoustic Activity under Different Conditions / less unchanged.  When the  loading speed  the  of  rate  magnitude  shorter.  However  event  gets higher but below  becomes  the  energy  release  the  magnitude  is  not  the  affected.  In  much higher, and the failure time becomes significantly  time  case of a hard intercalation, the  which  surprisingly  with  field  results  than  in  the  country  failure,  disappears release  which and the  can  conditions  is  time  give  mass;  if  a  Here  delay seen.  some  the  softer  precursive  between the  The  of  time  mass  number  of  with events  shorter.  show a large number of and  give  layer  is  more  than  intercalated  in  release are observed signal  from  the  of event rate  acoustic  activities  the  becomes  rock  increases  simulated  explanation  critical level,  release and a shorter failure  in the event rate and energy delayed.  hardly be  may  rock  the  measurements,  anomaly in event rate, higher value in energy  massive' rock, increases the  agrees  and  the  become  one  release  failure  elasticity,  events  energy  and  higher  In the  of  higher  the  247  problems  under  encountered  before  event and  a  rate  energy  these  two  microseismic  monitoring in the field, in that sometimes anomaly is not followed by failure and sometimes failure occurs without anomaly [21,52].  critical  Under  the  level,  or  extreme a  conditions,  rock mass  may not be well developed.  with  such very  as low  a  high  elasticity,  loading  speed  above  those precursive  the  signals  CHAPTER  14.1.  14.  CONCLUSIONS  CONCLUSIONS  During has  this  been  project,  a basic  postulated.  A  mechanism  process  of violent  analogous  to  rock  failure  shearing  is  and rockbursting  considered  to  basic mechanism of rock failure under all conditions. Even with massive shearing  process  development a  ultimately  the  of extensive microfracturing will  fracture surface  been  determines  on  which the  final  post-failure eventually  behavior  This  force  as  a  failure is  result  of  a  gradual process  small  stress  sliding. However if large stress differences applied  suddenly  resistance  or  makes  at  high  speed,  or  at  low  speed  and therefore if  a  sudden  a sudden slip, the stored energy  has  condition and  which occurs when  differences  the  formation of  location of an underground opening. According to this hypothesis,  non-violent rock  rock, the  assumption  used to interpret violent rock failure occurring under any  any  the  because  lead to the  failure takes place.  be  at  the normal  a  low  results  pushing  in  smooth  a high pushing force is  reduction  of  the  will be released  shearing  suddenly and  the resulting failure will be violent.  Based was  developed,  involved the  on  were  stick-slip by  that  which  examined.  frictional coefficient  the  takes  place  effects  Cohesion has is  negligible.  on  during the  no effect  The effect  shearing,  slip  a  behavior  numerical model from  on slip behavior. of normal  all  factors  The effect of  pressure  is  the  most  significant factor and all slip parameters increase with the normal pressure. The effect of elasticity  is great when it is relatively low but becomes less important  when it is high. Loading speed it hardly changes  other  has  an inverse  slip parameters  when  level. 248  relation with the it is  below  the  stick  time but  critical transition  Conclusions / Transition were obtained elasticity.  conditions  and they  From  the  of  slip  behavior  are combinations  transition  conditions,  between  stick-slip  and stable sliding  of normal pressure, violent  failure  is  249  loading speed  expected  to  and  occur in  the following three cases: Mode I. Violence is the  result of stick-slip under very high normal  pressure because of the large amount, of energy released at each slip. Mode TI. Violence comes from the transition from stick-slip to stable sliding due to the extra energy Mode III. behavior  is  available at transition.  Violence occurs under sudden loading. Whether the  in stable sliding or stick-slip,  violent  failure is  bound to  shear  occur if a  shear force much higher than the strength is instantly/suddenly applied.  A  rockburst  explained by example  along  a  natural  Mode I and II violence.  of Mode II violence.  fault  or  a  major  discontinuity  Violent failure during shear  Mode III violence  can be used  can  testing  is  be an  to interpret clearly  the violent failure of a rock specimen in conventional compressive  testing and the  results can also be applied to describe rockbursts occurring in a massive rock.  Acoustic conditions  emissions  from  rock  and some important results  shearing  process  acoustic  emission  is  considered  to  in compression  be after  specimens  were  also  studied  in laboratory  were obtained. Acoustic emission during the a  continuation  the  and  formation of the  an  expansion  failure  of  surface. For  warning purposes, the most significant information is the precursive signals the  formation  of  the  failure  surface  under  compression.  In  this  the  case,  before  after  an  initial quiet period, which corresponds to the perfect elastic deformation, the event  Conclusions / rate increases  rapidly initially when  stress has  reached a certain level  and then  may die down immediately preceding the specimen failure. A t the same energy  released  increases  steadily  and reaches  a peak  as  250  time,  the  failure is approached.  It is proposed that the increase of acoustic activity corresponds with a process of unstable peak  fracture propagation. If this is so then the drop of event rate and the  up of the  energy  release  indicate  phenomena are in good agreement  the  coalescence of microfractures. These  with the  fracture process discussed  in chapter  3.  A  numerical acoustic  model based  upon the  stick-slip during the  shearing  process is developed to simulate the acoustic activity prior to violent rock failure. It has realistically simulated the acoustic numerical  acoustic  signals  are  an  laboratory tests and measurements  The compared agreement. violent  results with  laboratory  both  measurements  This  rock  from  suggests  failure  indicate  and a  accurate  laboratory  the  that  method  reproduction of  acoustic  signals  from  made in the field.  made  that  activity during violent rock failure. The  in  a  proposed  acoustic by  tests  mine  numerical  and  mechanism  signals  which  and  they is  obtained  rockbursts  are  valid in  can  modeling largety  in  for interpreting  this be  are  way  in  predicted  the with  satisfactory reliability.  study as  Further  research  acoustic  emissions  normal  pressure,  was  carried  under  loading  out  using  the  various conditions.  speed,  elasticity  and  numerical acoustic The influence anistropy  of  model  to  of  factors  such  rock  mass  were  Conclusions / extensively transition change,  analyzed. value  and  In  general,  if  the  the  loading  is  emission  Significant signals  intercalation can increase  the  elasticity  the pattern of acoustic  are observable.  if  not  speed  too  low,  changes little  are obtained when  magnitude  is  of energy  less  when  than  the  critical  above  factors  precursive  signals  the  and the  251  an inclusion exists. A hard  release,  decrease  the  time  it  takes for failure to occur and generate a large number of events and more than one anomaly in event rate. A soft intercalation can increase both event rate and energy release and delay the failure. This information may interpret the problems faced in microseismic  monitoring in the  field  that  sometimes violent  rock  failure  occurs without warning and sometimes an anomaly is not followed by failure.  In  results  of this  research  can  occur in any mine rock as  long as  the  and  that  rock  conclusion,  acoustic  the  emission  failure, in terms  show  conditions  can provide precursive  of event rate,  energy  that  violent  for violence  signals  rock are  satisfied,  for warning of  release rate  and the  failure  violent  down-shift  in  corner frequency, in particular the latter two factors.  14.2. R E C O M M E N D A T I O N S Although  this  research  has  amount of laboratory testing  FOR FURTHER achieved that  RESEARCH  satisfactory  was  possible.  results,  it  In order to  was  limited  in  the  apply the principals  used and the results obtained in this research to the practice of rockburst control and  microseismic  extended locate  to  monitoring in  a burst-prone mine  "hot spots" in the  mine  the with  field,  is  felt  a microseismic  and then  from these potential rockburst sites.  it  that  the  work  should  monitoring system that  monitor the  seismic  energy  be can  emanating  Conclusions / As described in this thesis, precursive signals specific  rock  specimens.  mass  should  be  obtained  from  the  of acoustic  emission  laboratorj'  testing  252  from a of  small  After being calibrated with data obtained from monitoring in the same  mine, these results  should provide a sound method of predicting which rocks in a  mine would be likely to burst if the geological conditions, stress state and mining activity  are  clearly  known.  pattern of acoustic  energy  It  will  emitted  need  experience  prior to  in  assessing  the  changing  a major event in order to  establish  limits that will allow reliable prediction.  The length of the period during which the acoustic emission can  be  obtained  statistically  from  tests  or  monitoring in  a  is most  active  particular mine  so  that an accurate time of warning for a violent failure can be provided.  In  order  microseismic technique  to  give  monitoring  of  its  data  a  reliable  system  prediction  needs  acquisition.  of  improving in both  For instance,  to  axial direction and cannot detect signals  that  direction. Energy should be  common  reference  varies  with  energy  is  signals  first,  accurate waveform  point  distance usually  for  and use  because  estimated  the  and properties  of the  estimated  at  the  measured  in  data  comparison.  the  energy  It  is  in use  the  existing  analysis  geophones  today  and  the  should  be  is only sensitive  coming in the plane perpendicular at  the  signal  attenuation  can  source be  or at  some  significant  rock. But in monitoring in the  location for  data  multi-axial  used as transducers because the uniaxial geophone to its  rockbursting,  of  some  different  essential  should be conducted in order to study  that  geophone signals  which  are  spectrum  field,  receives  therefore  analysis  the frequency change  and  of  not the  as failure  is approached and to provide another important precursor to violent failure.  BIBLIOGRAPHY  1.  Blake, W. "Rockburst mechanism". Mines, vol. 67, 1972, 61 p.  Quarterly  2.  Curtis, J . F . "Rockburst phenomena in the Gold Mines of the Witwatersrand: a review". Trans. Inst. Min. Metall. vol. 90, Oct. 1981, pp.A163-A176.  3.  Mclaughlin, W . C . , Waddell G . G . & McCaslin J . G . "Seismic equipment used in the rockburst control in the Coeur d'Alene Mining District, Idaho". USBM tf/8138, 1976, 29 p.  4.  Heunis, R. "Rock bursts and the search Mining Magzine, Feb., 1977, pp.83-89  5.  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"A note on rockbursts considered as a problem of stability". S. Afr. Inst. Min. Metall. J., v 65, pp437-446, 1965.  14.  Taylor, J . T . M . "Research on ground control and rockbursts on Kolar Gold Field, India." Trans. Inst. Min. Metall. vol. 72, 1962-63, pp.317-338.  15.  Blake, W . "Microseismic application for mining — a practice Report, U S Bureau of Mines, 1982.  contributions to reference  mechanism 1963 b.  253  of  the  Colorado  for an early  warning  School of  system".  2, pp.A212-213.  "Stick-slip  in Ontario mines". The  "Soft,  stiff  and  as  a  mechanism  of rockbursts".  S.  Afr. Inst. Min.  guide".  for  Final  /  254  16.  Zou, Daihua & Miller, H . D. S. "Microseismic source literature investigation". Internal Report, Dept. of University of B C , 1984.  17.  Brink, A . V . Z. & Mountfort, Western Deep Levels, ltd." 1984, report No. RPP.21, 1985.  18.  Blake, W. "Preconditioning an entire stope Final Report, U S Bureau of Mines, 1980.  19.  Neyman, B., Zuberek, W. & Szecowka, Z. "Effective methods for fighting rock burst in Polish collieries". 5th int. Strata. Control Cong., 1972, pp. 1-9.  20.  Krawiec, A . & Stanislaw, T. "Rock bursting in Polish deep coal mines in light of research and practical observation". Society of Min. Eng., A I M E , vol. 262, 1977, pp.30-36.  21.  Leighton, F . "Growth and development of microseismics applied to ground control and mine safety". Mining Engineering, August, 1983, pp. 1157-1162.  22.  Bieniawaski, Z. T. "Mechanism of brittle fracture of rock, Part 1-111". int. J. Rock Meek. Min. Sci., vol. 4, pp.395-430, Pergamon Press ltd. 1967.  23.  Scholz, C. H . "Experimental study of the fracturing process in brittle rock". J. of Geophy. Res. vol. 73, No. 4, 1968, pp. 1447-53.  24.  Savage, J . C. "Radiation from a tensile fracture". J. 6345, 1963.  25.  Mogi, K . "Study of the elastic shocks caused by the fracture of hetero-geneous materials and its relation to earthquake phenomena". Bull. Earthquake Res. Inst. Tokyo Univ. 40, 125, 1962.  26.  Jaeger, J . C. & Cook, N . book, 1969  27.  Scholz, C. H . & Engelder, J . T . "The role of asperity indentation and ploughing in rock frictions: 1 asperity creep and stick-slip". Int. J. Rock Mech. Men. and Geomech. abstr. 13, 149-154, 1976.  28.  Pan, W . D . "Statistic  29.  Stesky, R. M . "Rock friction effect of confining pressure, pore pressure". Pageoph. vol. 116, pp.690-703, 1978  30.  Barton, N . "Review of a new Geology, 7, pp.287-332, 1973  31.  Schneider,  H.  J.  location technique — Mining Eng., The  P. I. "Rockburst prediction research at A Review Report for the Period 1981 -  block  G. W. "Fundamentals  for  rock  burst  Geophys. Res.  of rock  "The  friction  and  68(23),  mechanics", text  method", text book, Shanghai Publishing House,  shear-strength  control".  1980  temperature,  criterion for rock joints".  deformation  behavior  of  rock  and  Eng.  joints".  / 255 Rock Mech. 8, pp. 169-184, 1976 32.  Barton, N . "The shear strength of rock Mech. Min. Sci. 13, pp.255-279, 1976  33.  Byerlee, J . D . "Brittle-ductile transition in rocks". J. Geophys. Res., 73(14), pp.4741-4750, 1968  34.  Vesic. A . S. & Clough, G . W. "Behavior of granular materials stresses". J. Soil Mech. Found. Div. 94(SM3), 661-688, 1968  35.  Stesky R . M . , Brace W . F . , Riley D . K . & Robin P . Y . F . "Friction in faulted rock at high temperature and pressure". Tectonophysics 23, pp. 177-203, 1974  36.  Drennon, C . B. & Handy, R. L . "Stick-slip of lightly-loaded limestone". Pnt. J. Rcok Mech. Min. Sci. 9, pp.603-615, 1972  37.  Friedman M . , Logan J . M . & Rigert J . A . "Glass-indurated quartz gouge in sliding friction experiments on sandstone". Bull. Geol. Soc. Amer. 85, pp.937-942, 1974  38.  Engelder, J . T . "The sliding characteristics of sandstone fault-gouge". Pure and Appl. Geophys. 113, pp.69-86, 1975  39.  Dieterich, J . H . "Time-dependent friction and the mechanism Pageoph. vol. 116, pp.790-805, 1978  40.  Hoskins, E . R . , Jaeger J . C . & Rosengren K . T . "A medium-scale direct friction experiment". Int. J. Rock Mech. Min. Sci. vol. 5, pp. 143-154, 1968  41.  Spottiswoode, S. M . "Source mechanisms of mine tremors at Blyvooruitzicht Gold Mine". Proc. 1st int. cong. on Rockbursts and Seismicity in Mines, Johannesburg, 1982. pp.29-37. S A I M M No. 6, 1984.  42.  Hoek, E . pp.187-223,  43.  Burridge, R. & Knoppoff, L . "Model and theorectical seismicity", Bull, of the Seism. Soci. of America, vol. 57, No.3, pp.341-371, 1967.  44.  Dobrin, 1976  45.  Nie, T . J . "Engineering mathematics: Numerical method", National Defense Publishing House, 1982  46.  Engelder, J . T . and Scholz, C. H . "The role of asperity indentation and ploughing in rock friction: II. Influence of relative hardness and normal load". Int. J. Rock Mech. Min. Sci. and Geomech. Abstr. 13, pp. 155-163,  "Strength 1983  of  jointed  rock  and rock joints".  masses".  M . B. "Introduction to geophysical  Int. J. Rock  on  quartz  of stick-slip".  Geotechnique  prospecting",  under high  33, No.3,  text book, 3rd ed.  text book, China  /  256  1976. 47.  Coates, D. F . "Rock mechanics principles", text book, Methuen & Co Ltd, 1970  48.  Gay N . C . , Spencer D., V a n Wyyk J . J . & Van Der Heever P . K . "The control of geological and mining parameters in the Klerksdorp gold mining district". Proc. 1st Int. Cong, on Rockbursts and Seismicity in Mines. Johannesburg, 1982, S A I M M No.6, pp. 107-121, 1984  49.  Hardy, J r . H.R. "Emergence of acoustic emission/microseismic activity as a tool in geomechanics". Proc. 1st conf. on AEIMA in Geol. Structures & Materials, 1977, ppl3-31.  50.  Gibowicz, S. J . - "The mechanism of large mine tremors in Poland". 1st Int. Cong, on Rockbursts and Seismicity in Mines, Johannesburg, pp. 17-28, S A I M M No.6, 1984.  51.  Bath, M . "Rockburst seismology", Proc. 1st Int. Cong, on Rockbursts Seismicity in Mines, Johannesburg, 1982. pp.7-15, S A I M M No.6, 1984.  52.  Leighton, F . "Microseismic activity associated with outbursts in coal mines". (Report, USBM, Denvor Research Center, 1981, 11 p.)  53.  A'lheid, H . J . and Rummel, F . "Acoustic emission during frictional sliding along shear planes in rock". Proc. 1st Conf. on AEIMA in Geological Structures and Materials, 1977, pp. 149-155.  54.  Holcomb, D . J . and Teufel, L . W. "Acoustic emission during deformation of jointed rock". Proc. 2nd Conf. on AEIMA in Geological Structures and Materials. 1984, pp.37-45.  55.  Sondergeld, C. H . Granryd, L . and Estey, L . H . "Acoustic emission during compression testing of rock". Proc. 2nd Conf. on AEIMA in Geological Structures and Materials. 1984, pp. 131-147.  56.  Starfield, W . M . & Wawersikk W.R. "Pillars as structural components in room and pillar mine design". Proc. of 10th Symp. on Rock Mech. University of Texas, 1968, pp793-809.  57.  Christensen R . J . "Torsional shear measurements of the frictional properties of Westerly granite". Final Report, Dept. Nucl. Agency, Contract No DNA001-72-C-0026, 1973, 48p.  58.  personal contact with Dr. Miller,  H.D.S.  Proc. 1982.  and  APPENDIX  I. L I S T  OF FORTRAN PROGRAM MODEL 1 AND SAMPLE RESULTS  •j  p  2 3 4 5 6 y 8 9 10 11 12 13 14 15 1g 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58  c  *********************************************************  C C C C C Q  * * * * *  C C C  * * * * . . * *********************************************************  " MODEL 1 " typical shearing analysis by D a i h u a Z o u , 1985  . •  N u m e r i c a l s o l u t i o n : s i n g l e b l o c k model S l i p v e l o c i t y dependent f r i c t i o n : u=u(X') S l i p back p e r m i t t e d here  C C  T h i s program is w r i t t e n f o r of f i r s t order d e f f e r e n t i a l  Q  numerical equations  s o l u t i o n to the system by R u n g e - K u t t a m e t h o d  ********************************************************* IMPLICIT R E A L * 8 ( A - H . 0 - Z ) COMMON / B L K 1 / A , B , X X I C 0 M M 0 N / B L K 2 / T I , XI , H C0MM0N/BLK3/FM,FLAMD,VO.BTA COMMON/BLK4/A 1 , B 1 , C 1 , E 1 DATA U , P , C 0 , G / 0 . 6 5 D 0 , 1 0 . D O , 0 . D O , 9 . 8 0 6 D O / DATA T O . X X O . X O , E O , N / 0 . D O , 1 D - 1 1 . O . D O , O . 0 1 D O , 2 5 0 0 / V0=1.0D-7 BTA=O.DO FM=1.D0 FLAMD=100.D0 A 1 = .528DO*(P/FM+G) B1=.1218D0*(P/FM+G) C1=C0/FM E1=1.D0 E=E0 HO=.05DO TI=T0 XXI=XX0 XI=X0 11=0 A = F LAMD/FM B=U*(P/FM+G) A2=(A1+C1)*FM B2=B1*FM C2=B*FM CALL S U B 2 ( A 2 , B 2 , C 2 , E 1 , F I , F F I )  10  12  14  15  WRITE(6,10) F O R M A T ( 2 X , ' s o l u 1 t i o n s by R u n g e - K u t t a method f o r s i n g l e b l o c k 1 friction model',/,25X,'unit system: ***M-KG-SECOND***',/) WRITE(6,12)FM,P,FLAMD,G,U,VO.BTA F0RMAT(6X,'M=',F10.4, 6X,'P=',F10.4,2X,'LAMDA=',F10.4,6X, 1 ' G = ' , F 1 0 . 4 , / , 6 X , ' U = ' , F 1 0 . 4 . 5 X , ' V 0 = ' , F 1 0 . 8 . 3 X , 'BETA= ' , F 1 0 . 5 . / ) WRITE(6,14) F0RMAT(3X, ' N ' , 8 X , ' T ( I ) ' , 1 1X,'X->(I ) ' , 1 12X.'X(I)',12X,'F(I)',12X,'FF(I)',/) WRITE(6,15) 1 1 , T O , X X O , X O , F I , F F I FORMAT(1X.I4,1X.5F15.8) J=0  257  /  59 GO 61 62 G3 G4 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 1 10 11 1 1 12 1 13 1 14 1 15 116  J 1=0 K=0  45  DO 80 1=1,N H = HO CALL S U B 1 ( E , X X 2 , X 2 )  60  CHECK THE SIGN OF VELOCITY AT TWO ADJACENT POINTS I F ( D S I G N ( X X I , X X I ) . E Q . D S I G N ( X X I , X X 2 ) ) GOTO 65  40  C  C  63 65  70 80  100  105 107  INCREASE THE ACCURACY WHEN THE VELOCITY REACHES 0 I F ( J . N E . 0 ) GO TO 63 E=E/((IDINT(J1/10.D0)+1)*5.DO) J = J+1 J1=J1+1 GO TO 45 d =0 XXI=XX2 XI=X2 TI=TI+H CALL S U B 2 ( A 2 . B 2 , C 2 , E 1 , F I , F F I ) WRITE(6,15)I,TI,XXI,XI,FI,FFI I F ( D A B S ( X X I ) . L T . 1 D - 1 1 ) G 0 TO 100 CONTINUE GO TO 150 K=K+1 IF ( K . E Q . 2 ) G 0 TO 150 IF ( B T A . E Q . O ) G O TO 105 T O = ( D S Q R T ( V O * V O + 4 * B T A * X I ) - V O ) / ( 2*BTA ) GO TO 107 T0=XI/V0 T1=T0-TI  WRITE(6,110)TI,XI,T1 T1=',F15.5,' SECONDS' 1 10 F 0 R M A T ( / , 3 X , ' T H E S L I P TIME 1 / , 3 X , ' T H E S L I P DISTANCE X m a x = ' , F 1 5 . 5 , ' METRES', 2 / , 3 X , ' T H E S T I C K TIME T2=',F15.5,' SECONDS',/) TI=T0 XXI=XXO J=0 J 1=0 E = EO GO TO 40 150  STOP END SUBROUTINE SUB 1 ( E , X X 2 , X 2 ) IMPLICIT R E A L * 8 ( A - H . O - Z ) C0MM0N/BLK1/A,B,XXI C0MM0N/BLK2/TI,XI.H  5  CALL RK(TI,XXI,XI,H,XX1,X1) H=H/2.D0 CALL RK(TI,XXI,XI,H,XX2,X2) D1=XX2-XX1  258  117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 14 1 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174  15  20  D1=DABS(D1) IF ( D 1 . L T . E ) G O H=H/2.DO XX1=XX2 GO TO 5 RETURN END  TO 20  SUBROUTINE S U B 2 ( A 2 , B 2 , C 2 , E 1 , F I , F F I ) C  C A L C U L A T E FORCES IMPLICIT R E A L * 8 ( A - H . O - Z ) COMMON/8LK1/A,B,XXI COMMON/BLK2/TI,XI,H C0MM0N/BLK3/FM.FLAMD,VO.BTA  5 20 .30  FORS(T,X)=C2+FLAMD*(V0*T+BTA*T*T-X) FI=FORS(TI,XI ) FFI=O.DO IF ( X X I ) 5 , 3 0 , 2 0 FFI=FI+A2+B2/(7.DO+DLOG10(-XXI+1D-6))-E1*XXI GO TO 30 FFI=FI-A2-B2/(7.DO+DL0G1O(XXI+1D-6))-E1*XXI RETURN END SUBROUTINE  5  lO  RK(X,Y,Z,H1,YN,ZN)  IMPLICIT R E A L * 8 ( A - H , 0 - Z ) G(X,Y,Z)=Y H=H1 F1=H*F(X,Y,Z) G1=H*G(X,Y.Z) I F ( ( Y + F 1 / 2 . D O ) . L T . O . D O ) G O TO 10 F2=H*F(X+H/2.DO,Y+F1/2.DO,Z+G1/2.DO) G2=H*G(X+H/2.DO,Y+F1/2.DO,Z+G1/2.DO) I F ( ( Y + F 2 / 2 . D O ) . L T . 0 . D O ) G O TO 10 F3=H*F(X+H/2.DO,Y+F2/2.DO.Z+G2/2.DO) G3=H*G(X+H/2.DO,Y+F2/2.DO,Z+G2/2.DO) I F ( ( Y + F 3 ) . L T . O . D O ) G O TO 10 F4=H*F(X+H,Y+F3,Z+G3) G4=H*G(X+H,Y+F3.Z+G3) YN=Y+(F1+2.D0*(F2+F3)+F4)/6.D0 ZN=Z+(G1+2.D0*(G2+G3)+G4)/6.D0 I F ( Y N . L T . O . D O ) G O TO 10 RETURN H=H/2.D0 GO TO 5 END DOUBLE  P R E C I S I O N FUNCTION  F(X,Y.Z)  IMPLICIT R E A L * 8 ( A - H , 0 - Z ) C0MM0N/BLK1/A,B,XXI C0MM0N/BLK3/FM,FLAMD,VO,BTA C0MM0N/BLK4/A 1 , B 1 , C 1 , E 1 FR(Y)=A1+C1+B1/(7.DO+DLOG10CY+1D-6))  7 260  175 17S 177 178 179 180 181 182 183 184 185 186 187 188  FRO=FR(0) F=O.DO FD=B+A*(VO*X+BTA*X*X-Z) IF(DABS(Y).LT.1D-13) GO TO 30 I F ( Y . G T . O . D O ) GO TO 20 F=FD+FR(-Y)-EO*Y/FM GO TO 50 20 F = F D - F R ( Y ) - E O * Y / F M GOTO 50 30 I F ( D A B S ( F D ) . L T . D A B S ( F R O ) ) GOTO 5 0 F=FD-DSIGN(FRO,FD) 50 RETURN END  ******* solultions  M= U= N 0 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34  Results  of  typical  shearing  analysis  by R u n g e - K u t t a method f o r s i n g l e b l o c k u n i t system: * * * M --KG-SECOND***  1.0000 0.6500 T(I) 0 .0 0 .00625000 0 .00937500 o .01250000 0 .01562500 0 .01875000 0 .02187500 0 .02500000 0 .02812500 0 .03125000 0 .03437500 0 .03750000 0 .04062500 o .04375000 0 .04687500 0 .05000000 0 .05312500 0 .05625000 0 .05937500 0 .06562500 0 .07187500 0 .07812500 0. . 0 8 4 3 7 5 0 0 0 .09062500 0 .09687500 0 .10312500 0 .12812500 o .15312500 0. . 1 6 5 6 2 5 0 0 0 .17812500 0 .19062500 0. . 1 9 6 8 7 5 0 0 0 .20312500 0 .20937500 0. . 2 1 5 6 2 5 0 0  P= 10.0000 V0=0.. 0 0 0 0 0 0 1 0 X-(I) 0 .. 0 0 0 0 0 0 0 0 0. . 0 0 8 6 3 7 0 6 0. . 0 1 4 6 8 6 7 9 0 ..02077483 0 .. 0 2 6 8 7 7 2 0 o. 0 3 2 9 7 8 8 2 0. 03906826 o.. 0 4 5 1 3 5 9 3 0 .. 0 5 1 1 7 3 3 0 0 . .05717256 0 . 06312634 0. 06902764 0 . 07486972 0 . 08064606 0 ..08635033 0 .. 0 9 1 9 7 6 4 0 0 . 09751824 0 . 10297000 0 . 10832598 0 . 11872837 0 . 12868237 o. 13814715 0 . 14708405 0 . 15545674 0 . 16323126 o. 17037613 0 . 19212046 0. 20197600 0 . 20221003 0 . 19928646 0 . 19324564 0 . 18908214 0 . 18417628 0 . 17854644 0 . 17221376  LAMDA= BETA=  100.0000 0.0 X(I)  0. .0 0 .00002069 0 .00005712 0, . 0 0 0 1 1 2 5 3 0. .00018698 0. .00028051 0. .00039309 0 .00052466 0. . 0 0 0 6 7 5 1 5 0. . 0 0 0 8 4 4 4 5 0, . 0 0 1 0 3 2 4 3 0. .00123894 0. . 0 0 1 4 6 3 8 0 o:.00170681 0. .00196776 0. .0022464 1 0. .00254252 0. .00285581 0. ,00318599 0. .00389575 0. .00466916 0. ,00550326 0 . ,00639489 0. ,00734064 0. 00833686 0 . 00937972 0. ,01393437 0. .01888618 0. 02141563 o: 02392826 0. 02638479 0. 02757995 0. .02874677 0. 02988065 0. 03097713  ******* friction  G=  F(I) 12. . 8 7 3 9 0 0 0 0 12 .87 183064 12 . 8 6 8 1 8 7 8 0 12 . 8 6 2 6 4 7 5 8 12, . 8 5 5 2 0 2 1 3 12. . 8 4 5 8 4 9 4 8 12 .83459171 12 . 8 2 1 4 3 4 1 6 12 . 8 0 6 3 8 4 9 8 12 . 7 8 9 4 5 4 8 8 12 . 7 7 0 6 5 6 9 3 12, . 7 5 0 0 0 6 4 4 12, .72752088 12 . 7 0 3 2 1 9 7 8 12 . 6 7 7 1 2 4 6 7 12 . 6 4 9 2 5 9 0 3 12. . 6 1 9 6 4 8 2 6 12 .58831958 , 12, , 55530205 12 . 4 8 4 3 2 5 3 8 12, . 4 0 6 9 8 5 1 6 12 ., 3 2 3 5 7 4 5 0 12 ,23441136 , 12. . 139837 14 12. ,04021522 1 1 .93592941 . 1 1, 4, 8 0 4 6 3 8 2 10, .98528365 10, . 73233861 10, , 4 8 1 0 7 6 2 0 10. .23542327 10. .11590682 9,. 9 9 9 2 2 5 4 3 9 ., 8 8 5 8 3 7 4 7 9. ,77618898  model  9 . 8060  FF(I) 0, . 0 0 3 9 7 1 6 8 1 .91693689 1 .92904871 1 .93064338 1 .92644067 1 .91814024 1 .,9 0 6 5 4 5 9 5 1 .89210415 1 .87509176 1 .,8 5 5 6 9 5 9 0 1 ,,8 3 4 0 5 2 1 8 1 ,,8 1 0 2 6 4 9 4 1 .78441877 . 1 ., 7 5 6 5 8 5 4 0 1 .,7 2 6 8 2 8 0 6 1 .,6 9 5 2 0 4 2 8 1 ,.6 6 1 7 6 7 8 3 1 .62657001 . 1 ..5 8 9 6 6 0 5 8 1 ..5 1 0 9 0 2 0 6 1 ..4 2 5 8 8 0 5 5 1 .. 33498716 1 .. 23862125 1 .13719324 . 1 ,.0 3 1 1 2 5 9 0 0 . .92085455 0. . 4 4 6 8 5 8 4 9 - 0 . .05685437 - 0 . ,31000292 - 0 . 55872587 - 0 . .79915236 - 0 . 91508365 - 1 . ,02755954 - 1 . , 13614803 - 1 . ,24043428  H -H - i I I I m m m  momni<nuiuiui(iiuiuiuiuiuiuiti^^^^^^^^^(ouuuu 4*aro^otooo^mui4icoN3-^otooo^mui.c»coro-kOcooo--ioiui  </)c/)i/> H r- rt-l M h-l O "0 "0 7C O -I H >-< w >iaia)(ninuiuiuiuibj>A^uuuuuuioiouioMUMioioioio MUI j -^tocT>4i.^cooiji^cDOiA-ktooiA^comaicnuicfltnui-tk.c»cororo 2 Hm nuoouoouoiuDucouixiuoouiiiunuiNXDOiuoiiiO^OD^  oooooooooooooooooo oooo oooo ooo o  Z O m X 3 H U H ro X -«. II it II  U l U l U I O l U l U l U I U l i n U l U I O l U l U l U l U I U I O l U I M O ' - ' U ' M O - J W - J M - J OOOOOOOOOOOOOOOOOOOUIOUIOUIOUIUIUIUIUI  oOOOOOOOOOOOOOO OO OO O OOO OOOOO O O oooooooooooooooooooooooooooooo  OOOOOOOOOOOOOOOOOOOOOOOOOOOOOO O O O O O O O O O O O O O O O O O O O O - * - - - - ' - - - ' - - ' - - ' - - -  ^ o o o o o o o o o o o o o o o o M * « n o o - ' - ' H M i a 4 M i i i i i O OOOOOOOOOOOOOOO-IOOIJ^CDUIOOIOcnOOCD-JU! ui o O O O O O O O O O O O O - ' - o i A u i O O C i - k U i - 4 C Q O C o c o i o u i i o O O O O O O O O O O O O M i i l - i u O i o - l O - ^ - ' l D m O M a i M J O ro 000000000^0)o^u^uii»io*Oi-'iynJiiioiaiio-4io •-•• O • O • O oOo O o oOoOo O o OU M^ (t j D) c^ oO OOJ I^MO IMI Cl o - '- J- j' (- oJ oMj f0 f3 l( ^D i~- j '( ^o -a J) U M 'a O) cI nJ uU i ' ~- JdOl CD -I  O O O  0) u  Ul CD oo o i - j M Ul W m O aZ o t/i  S U I m m H o o 7} m z CO o (/I  o o O o O O O o o o o o O o O O o O O O O O O o o o O O O O o o O o O £h o o O o O Ul Ul Ul Ul Ul o o O O O IO ro M IO IO Si ro IO IO IO 01 01 01 01 01  O O o o o o o O o O o o O co O o o o o o o O o O o o 10 Ul Ul Ul Ul Ul Ul Ul Ul Ul Ul to O O o o o o O o o O 03 o 00 lo IO IO IO IO IO IO IO IO ro CO ro IO M M IO IO IO IO ro ro _k Ul O oo 01 01 01 01 01 01 01 01 Ul -4 M 01 CO  O CO 00 IO ro CO -J O  O O CO u ~j ~ i CO 01 IO o —k —k ro Ul  -k  O CO -4 ro 01 & 10 O  o CO 01 CO _b O 01 01  o CO 01 Ul  o CO Ul -1 Ul o CO 01 ~J ro CO - J - I O CO 0) _k Ul  O O O CO CO CO CO CO CD CO O O CO Ul CO O _k oo CO oo ro O  O CO ro O CO —k CO o  co 00 oo oo CD oo oo co CD 03 03 03 co oo oo CO 03 03 CO CO CO CO CO CO CO CO CO CO CO to co ro CO 01 oo  00 ro CO 01 co  I  oo ro CO 01 00  oo ro CO 01 oo  oo ro CO 01 oo  co O Ul O CO O O CO •b CO CO o O  _k _k  CO  co ro CO 01 oo  I  I  I  I  I  CD ro CO 01 -4 CO co  CO ro CO 01 -~i CO Ul CO  GO ro co 01 -~i CO CO  I  I  I  00 ro CO 0) -4 CO  03 ro CO 01 -j oo 00  CD ro CO 01 -a CD 01  oo ro CO 01 -~i oo  I  I  I  CO o oo I  oo ro CO 0) -4 oo  I  03 ro CO 01 oo 01 ro oo  I  00 03 03 ro CO - J CO Ul CO 01 Ul (0 _k -j. - 4 CO 03 Ul Ul ro 01  I  I  I  o Ul O 03 -I  Ul CO ro CO  I  _i _k _k ro _k 00 _i  CO ~J - 4 - 4 41 03 ~J -4 - 4 03 ro ro CO CD CO CO 03 «k  I  I  I  ro 00 CO 01  CO oo CO CO ro Ul - 4  I  I  ro ro CO *. Ul CD oo - J 03 03 CO CO Ul CO CO ro -1 CO _k Ul Ul 4V CO CO 03 IO CO CO -k - J  O  I  I  I  I  Ul 01 CO 00 ro ro Ul •b  I  01 -j  O _k _k  CO -*  AAAAj^jk4i>cococororoioroioroioio-k-k-k-k-k-^-k-k-k-k-k->  I  OOOOOOOcomcocoaiuicoro-kOOcooooocD'-i-i-JciiaiuiAco AA^ikji4i-k~jrocoroA-^-4-4cotouio~44i-'03uiiooooroco*> o i a i o i u i A C O O s c o a i - j C D a i co 03 o m o o . & . c o - 4 c o c o o o u i O - t > o i c o 4 i O roroooioorooorooui4iOo-»4iO*kUi-kCouiui-4corouirotocr)Uio uiAJiUioicoouicouiArotooio-'OCDOai-kOoaicDcocDOococoto uicoO)roAro003CO-kioO'-40o*>roro-^ooaitoouioiuicD-kCocDio c o o o ^ i u i - 4 ~ 4 03 0 3 r o - k ( o - k - - i - k O i c i i u i o t o u i u i * k C O u i u i a i - 4 u i r o - - j AUlAr001-403C003-J4i.rOOWUlfllCD03~4-40iaiCOO-4-4CO-~l-JCO  -•  to  APPENDIX  II. L I S T  OF FORTRAN PROGRAM MODEL2  AND SAMPLE  RESULTS  -|  Q  2 3 4 5 6  C C C C C  7  Q  8 9 10 11 12 13 14 15  ]Q 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4 1 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58  ********************************************************* * * * * *  * * * * *  " M0DEL2 " sensitivity analysis b y D a i h u a Z o u , 1985  *********************************************************  C C C  Numerical s o l u t i o n : s i n g l e b l o c k model S l i p v e l o c i t y dependent f r i c t i o n : u=u(X') S l i p back p e r m i t t e d here  C C  T h i s program is w r i t t e n f o r of f i r s t o r d e r d e f f e r e n t i a l  Q  numerical equations  s o l u t i o n to the system by R u n g e - K u t t a m e t h o d  ********************************************************* IMPLICIT REAL*8(A-H,0-Z) DIMENSION S(5) COMMON / B L K 1 / B . X X I C0MM0N/BLK2/TI,XI,H C0MM0N/BLK3/FM,F LAMD,VO,BTA C0MM0N/BLK4/A2,B2,C2,EO DATA U , P , G , C O / O . 6 5 D O , 3 . O D 0 5 , 9 . 8 O 6 D O , O . 0 D O / DATA T 0 , X X 0 , X 0 , E S P , N / O . D O , 1 D - 1 0 , O . D O , 0 . 0 1 D O , 2 5 0 0 /  2  C  C  4 6  PC=1.379D8 V0=1.OD-7 BTA=O.DO FM=1.D0 FLAMD=5.0D10 P1=PC/P EC=4.264D-04 I F ( P 1 . L T . 1 5 0 . D O ) G 0 TO 2 U=150.D0*C0/PC+U CO=O.DO EO=DSQRT(FLAMD)*EC E = ESP H0=.05D0 TI=TO XXI=XXO XI=XO 11=0 c a l c u l a t i n g s t a t i c shear force US=U*1.D0 B=US*(P/FM+G)+C0/FM estimate constants for f r i c t i o n U0=-.14D0+1.03DO*U B0=.133D0-.018D0*U P1=U0+B0-U IF(P1)6,6,4 U0=U0-P1*3.DO/5.DO BO=BO-P1*2.01DO/5.DO A2=C0+U0*(P+FM*G) B2=B0*(P+FM*G) C2=B*FM CALL S U B 2 ( F I , F F I ) FO=FI  262  coefficients  /  59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 1 12 113 114 115 116  XX=C2/FLAMD EP1=XX*XX*FLAMD/2.D0 WRITE(6,10) F 0 R M A T ( 2 X , ' s o l u l t i o n s by R u n g e - K u t t a m e t h o d f o r s i n g l e b l o c k Ifriction model',/,25X,'unit system: ***M-KG-SECCND***',/) W R I T E ( 6 , 12 ) C 0 , U 0 , B O , E O , F M , P , F L A M D , U , G , V O . B T A , E S P 12 F O R M A T ( 2 X , ' C O H S N = ' , E 1 0 . 3 , ' 3X, ' U0=',F9.6,4X,' B0=', 1 F9.6.4X,'RADIA=',F11.4,/,2X,' M=',F7.4,6X,' P=', 2 E 9 . 2 , 4X , ' L A M D A = ' , E 9 . 2 , 4 X , ' Us= ' , F 9 . 6 , / , 2 X , ' G=', 3 F7.4.6X,'DRIVO=',E12.5,1X.'DRI.AC',F9.6,4X,'PRECISN'.F9.5,/) WRITE(6,14) 14 FORMAT ( 3 X , - ' N ' , 8 X , ' T ( I ) ' , 1 1 X , ' X " ' ( I ) ' , 1 12X,'X(I)',11X,'F(I)',10X, 'FF(I ) ' , / , 12X,'sec',12X, 'm/sec', 2 13X,'m',14X,'N',13X,'N') WRITE(6,15) I I , T O , X X O , X O , F I , F F I 15 F O R M A T ( 1 X , 1 4 , 1X, E 1 4 . 8 , 1 X . 2 E 1 5 . 7 , 2 E 1 5 . 5 ) X1=0.DO T1=0.DO T2=0.D0 K=0  10  40  45  DO 80 1=1,N H=HO C A L L SUB 1 ( E , X X 2 , X 2 , H 3 )  60  CHECK THE SIGN OF VELOCITY AT TWO ADJACENT POINTS I F ( D S I G N ( X X I , X X I ) . E O . D S I G N ( X X I , X X 2 ) ) GOTO 65  C  C  63 65  C  d=0 J1=0 WF=O.DO WR=O.DO WE=O.DO BUF2=XXI BUF1=0.D0 S(1)=TI S(2)=XXI S(3)=XI S(4)=FI S(5)=FFI  INCREASE THE ACCURACY WHEN THE VELOCITY I F ( d . N E . O ) GO TO 63 E=E/((IDINT(J1/10.DO)+1)*5.DO) J=d+1 J1=J1+1 GO TO 45 d=0 XXI=XX2 XI=X2 TI=TI+H3 CALL S U B 2 ( F I , F F I ) WR=WR+XXI*XXI*H3 WE=WE+FI*H3 WF=WF+DABS(FFI)*(XI-S(3)) WRITE(6,15)1,TI,XXI,XI,FI,FFI  REACHES  0  263  /  117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174  70  75  80  81  100  C  I F ( B U F 2 . L T . B U F 1 . O R . B U F 2 . L T . X X I ) G O TO 75 11=1-1 WRITE(6.15)11,(S(L),L=1,5) BUF1=BUF2 BUF2=XXI S(3)=XI GO TO 80 BUF1=BUF2 BUF2=XXI 5(1)=TI S(2)=XXI S(3)=XI S(4)=FI S(5)=FFI I F ( D A B S ( X X I ) . L T . 1 D - 1 3 ) G 0 TO 100 CONTINUE WRITE(6,15)1,TI,XXI,XI,FI,FFI WRITE(6,81)I F O R M A T ( 2 X , 1 4 , ' t i m e s have been r u n , not GO TO 150  converge')  WRITE(6,15)1,TI,XXI,XI,FI,FFI K=K+1 X1=XI-X0 T1=TI-T0 TT=(DSQRT(V0*V0+4*BTA*X1)-V0)/(2*BTA) TT=X1/V0 T2=TT-T1 DF=F0-FI PCT1=DF/F0*100.D0  WRITE(6,110)T1,X1,T2,DF F 0 R M A T ( / , 3 X , ' T H E S L I P TIME T1=',E15.8,' SECONDS', .1 / . 3 X , ' T H E S L I P DISTANCE Xmax = ' , E 1 5 . 8 , ' METRES', 2 / , 3 X , ' T H E SLICK TIME T2=',E15.8,' SECONDS', 3 / , 3 X , ' T O T A L FORCE DROP DF=',E15.8,' NEWTDNS',/) XX1=XX+V0*TI-XI EP2=XX1*XX1*FLAMD/2.DO EK=FM*XXI*XXI/2.D0 WR=EO*WR WE=VO*WE DE=EP1-EP2 PCT2=WF/DE*100.D0 PCT3=WR/DE*100.D0 WRITE(6.121)WE,DE,WF,WR,PCT1,PCT2,PCT3 121 F O R M A T ( 3 X , ' W E = ' , E 1 5 . 8 , 1 2 X . ' D E = ' , E 1 5 . 8 , 2 X , 'Wf = ' , E 1 5 . 8 , 2 X , ' W r = ' , E 1 5 . 8 , / / , 2 3 X , ' F O R C E DROP FRACTION DF/FO=',F7.3,' %',/, 3 3 X , ' F R I C T I O N CONSUMPTION Wf/DE=',F7.3,' %',/, 4 3 X , ' R A D I A T I O N PORTION Wr/DE=' ,F7 . 3 , ' %',/) I F ( K . E Q . 1 ) G 0 TO 150 TI=TI+T2 T0=TI XXI=XXO XO=XI E=ESP WRITE(6,14) GO TO 40 110  264  /  175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232  150 151  W R I T E ( 6 , 151 ) FORMAT(2X,/) STOP END SUBROUTINE SUB 1 ( E , X X 2 , X 2 , H 3 ) IMPLICIT R E A L * 8 ( A - H , 0 - Z ) COMMON/BLK1/B,XXI C0MM0N/BLK2/TI,XI,H  5  15  20  CALL R K ( T I , X X I , X I , H , H 2 , X X 1 , X 1 ) H1=H2/2.D0 CALL R K ( T I , X X I , X I , H 1 , H 3 , X X 2 , X 2 ) D1=XX2-XX1 D1=DABS(D1 ) IF ( D 1 . L T . E ) G 0 TO 20 H1=H3/2.D0 XX1=XX2 GO TO 5 RETURN END SUBROUTINE  C  SUB2(FI,FFI)  CALCULATE FORCES IMPLICIT R E A L * 8 ( A - H , 0 - Z ) COMMON/BLK1/B,XXI C0MM0N/BLK2/TI,XI , H COMMON/BLK3/FM,FLAMD,VO,BTA C0MM0N/BLK4/A2,B2,C2,EO  5 20 30 50  FR(Y)=A2+B2/(7.D0+DL0G10(Y+1D-6) ) FRO=FR(0) FI=C2+FLAMD*(V0*TI+BTA*TI*TI-XI ) FFI=-FI IF ( D A B S ( X X I ) . L T . 1D - 13 ) GOTO 30 IF ( X X I . G T . O . D O ) GOTO 20 FFI = F R ( - X X I ) GO TO 50 FFI=-FR(XXI ) GOTO 50 IF ( D A B S ( F I ) . L T . D A B S ( F R O ) ) GOTO 50 FFI=-DSIGN(FRO,FI) RETURN END SUBROUTINE R K ( X , Y , Z , H 1 , H , Y N , Z N )  5  IMPLICIT R E A L * 8 ( A - H , 0 - Z ) G(X,Y,Z)=Y H=H1 F1=H*F(X,Y.Z) G1=H*G(X,Y,Z) IF((Y+F1/2.D0).LT.O.DO)GO TO 10 F 2 = H * F ( X + H / 2 . D O , Y + F 1 / 2 . D O , Z + G 1 / 2 . DO) G 2 = H * G ( X + H / 2 . D O , Y + F 1 / 2 . D O , Z + G 1 / 2 . DO) IF((Y+F2/2.DO).LT.O.DO)G0 TO 10  265  / 266  233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266  10  F3=H*F(X+H/2.DO,Y+F2/2.DO,Z+G2/2.DO) G3=H*G(X+H/2.DO,Y+F2/2.DO,Z+G2/2.DO) I F ( ( Y + F 3 ) . L T . 0 . D 0 ) G 0 TO 10 F4=H*F(X+H,Y+F3.Z+G3) G4=H*G(X+H,Y+F3.Z+G3) YN=Y+(F1+2.D0*(F2+F3)+F4)/6.D0 ZN=Z+(G1+2.DO*(G2+G3)+G4)/6.DO I F ( Y N . L T . O . DO) GO TO 10 RETURN H=H/2.D0 GO TO 5 END DOUBLE P R E C I S I O N  FR(Y)=A2+B2/(7.DO+DLOG10(Y+1D-6)) FRO=FR(0) F=O.DO FD=C2+FLAMD*(V0*X+BTA*X*X-Z) I F ( D A B S ( Y ) . L T . 1 D - 1 3 ) GO 'TO 30 I F ( Y . G T . O . D O ) GO TO 20 F=(FD+FR(-Y)-E0*Y)/FM GO TO 50 20 F = ( F D - F R ( Y ) - E O * Y ) / F M GOTO 5 0 GOTO 50 30 I F ( D A B S ( F D ) . L T . D A B S ( F R O ) ) F=(FD-DSIGN(FRO,FD))/FM 5 0 RETURN END  Solutions  N  results  from  sensitivity analysis  ********  by R u n g e - K u t t a method f o r s i n g l e b l o c k f r i c t i o n u n i t system: ***M-KG-SECOND***  0.0 1.0000 9.8060  U0= P= DRIVO=  T(I) sec  THE S L I P TIME THE S L I P DISTANCE THE S L I C K TIME TOTAL FORCE DROP  0.529020 B0= 0 . 1 2 0 9 7 8 0.30E+06 LAMDA= 0 . 5 0 E + 1 1 0.10000E-06 DRI.AC 0 . 0  X-(I) m/sec 0. 1000000E-09 0.1306933E+00 0.9748039E-13  O O.O 3 O.83923340E-05 84 0 . 1 5 6 1 2 0 9 7 E - 0 4  We=  F(X,Y,Z)  IMPLICIT R E A L * 8 ( A - H , 0 - Z ) C0MM0N/BLK1/B,XXI C0MM0N/BLK3/FM,FLAMD,VO.BTA C0MM0N/BLK4/A2,B2,C2,EO  *******  COHSN= M= G=  FUNCTION  T1= Xmax= T2= DF=  O.25432353E-06  FORCE DROP FRACTION F R I C T I O N CONSUMPTION R A D I A T I O N PORTION  X(I) m 0.0 0.5719526E-06 O.1189830E-05  O. 15612097E-04 0.11898298E-05 0.11898282E+02 0.59491410E+05  DE= O . 1 9 6 6 3 1 8 0 E + 0 0 DF/FO= Wf/DE= Wr/DE=  model  RADIA= 95.3459 Us= 0.650000 PRECISN 0.01000 F(I) N 0.19501E+06 0.16641E+06 0.13551E+06  FF(I) N - O . 19500E-I-OS -0.16465E+06 -0.13551E+06  SECONDS METRES SECONDS NEWTONS Wf= 0 . 1 9 6 0 1 1 5 7 E + 0 0  30.507 % 99.685 % 0.008 %  Wr= O . 1 6 5 3 4 9 6 5 E - 0 4  APPENDIX  III. L I S T  OF FORTRAN PROGRAM MODEL3  AND SAMPLE  RESULTS  1 2  c  3 4 5 6 7  C C C C C  3  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58  ******************************************* * * * * *  c  * * * * *  M0DEL3 transition analysis b y D a i h u a Z o u , 1985  ****************************************************  C C C  N u m e r i c a l s o l u t i o n : s i n g l e b l o c k model S l i p v e l o c i t y dependent f r i c t i o n : u=u(X') No s l i p b a c k p e r m i t t e d h e r e  C C  T h i s program is w r i t t e n f o r of f i r s t order d e f f e r e n t i a l  C  *****  computing  the  critical  numerical equations normal  s o l u t i o n to the system by R u n g e - K u t t a method  pressure  at  transition  IMPLICIT REAL*8(A-H,0-Z) DIMENSION S(5) COMMON / B L K 1 / B , X X I C0MM0N/BLK2/TI,XI,H C0MM0N/BLK3/FM.FLAMD,VO.BTA C0MM0N/BLK4/A2,B2,C2,EO DATA U , P , G , C 0 / 0 . 6 5 D 0 , 0 . 1 0 0 5 , 9 . 8 0 6 D 0 . 0 . 0 D 0 / DATA T 0 , X X 0 , X 0 , E S P , N / 0 . D O , 1 D - 1 0 , 0 . D O , 0 . 0 1 D 0 , 5 0 0 0 /  2  C  C  4 6  PC=1.379D8 V0=.10D-4 BTA=0.D0 FM=1.DO FLAMD=.10D10 P1=PC/P EC=4.264D-04 I F ( P 1 . L T . 1 5 0 . D O ) G O TO 2 U=150.D0*C0/PC+U C0=0.D0 EO=DS0RT(FLAMD)*EC E=ESP H0=.05D0 TI=TO XXI=XXO XI=XO 11=0 c a l c u l a t i n g s t a t i c shear force US=U*1.D0 B=US*(P/FM+G)+CO/FM estimate constants for f r i c t i o n U0=-.14D0+1.03D0*U B0=.133D0-.018D0*U P1=U0+B0-U IF(P1)6.6,4 UO=UO-P1*3.DO/5.DO BO=BO-P1*2.01DO/5.DO A2=C0+U0*(P+FM*G) B2^B0*(P+FM*G) C2=B*FM CALL S U B 2 ( F I , F F I )  267  coefficients  *****  /  59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 1 10 111 1 12 1 13 1 14 115 1 16  F0=FI FF0=FFI XX=C2/FLAMD EP1=XX*XX*FLAMD/2.D0  10 12  14 15 20 25  40  42 43 45  80 82  83 85 86  90  100  WRITE(6,10) F 0 R M A T ( 2 X , ' s o l u t i o n s by R u n g e - K u t t a method f o r s i n g l e b l o c k 1 f r i c t i o n m o d e l ' , / , 2 5 X , ' u n i t system: ***M-KG-SECOND***',/) F0RMAT(2X.'C0HSN=',E10.3, 3X,' U0=',F12.6,1X,' B0=', 1 F12.6,1X,'RADIA='.F12.6./.2X,' M=',F12.4,1X, ' P=', 2 E10.2, 3X,'LAMDA='.E10.2.3X,' Us=', F 12 . 6 , / , 2 X , ' G=', 3 F 12.4,1X, ' D R I V 0 = ' , E 1 2 . 5 , 1 X , ' D R I . A C ' , F 1 2 . 6 , 1 X , ' P R E C I S N ' , F 1 1 . 6 . / ) FORMAT ( 3 X , ' N ' , 8X . ' T ( I ) ' , 1 1X , ' X">( I ) ' , 1 12X, 'X(I)',11X.'F(I)',10X.'Ft(I)') FORMAT(1X,I4,1X,E14.8,1X.2E15.7,2E15.5) J=0 <J1=0 K=0 IW=0 1=0 1=1+1 IF(I-IW.NE.50) GOTO 43 IW=I WRITE(7.42)1,V0.XXI FORMAT(2X,15,2(2X,E15.8)) H=HO CALL S U B 1 ( E , X X 2 , X 2 , H 3 ) XXW=XXI XXI=XX2 XI=X2 TI=TI+H3 I F ( D A B S ( X X I ) . L T . 1 D - 1 3 ) G 0 TO 100 I F ( I . L T . N ) GOTO 40 I F ( X X W . G T . X X I ) GOTO 4 0 J = J+1 VO=VO/2.DO I F ( J . L T . 2 ) GOTO 108 I F ( d l . E Q . O ) GOTO 85 WRITE(6,83)XW,VW,TW F O R M A T ( 2 X , ' i n p r e v i o u s r u n : XXI= ' . E 1 5 . 8 . ' V0= ' , E 1 5 . 8 , ' 1.E15.8,/) WRITE(6,86)I FORMAT(2X,'run ' . 1 4 , ' t i m e s a l r e a d y , not converge!') WRITE(6,14) WRITE(6,15)11,TO.XXO,XO,FO,FFO WRITE(6,15)1,TI,XXI,XI WRITE(6,90)VO,P FORMAT(2X.'V0= ',E15.8,3X,'Pn= '.E15.8) GO TO 150 TT=XI/VO T2=TT-TI I F ( T 2 . G T . 1 D - 5 . 0 R . T 2 . E 0 . 1 D - 5 ) G 0 TO 105 WRITE(6,12)CO,UO,BO,EO.FM,P.FLAMD,U,G,VO,BTA.ESP WRITE(6,14) WRITE(6,15)II,TO,XXO,XO,FO,FFO CALL S U B 2 ( F I , F F I )  T2=  '  26S  /  117 118 1 19 120 121 122 123 124 125 12G 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174  WRITE(6,15)I,TI,XXI,XI,FI  105  108  110 150 151  , FFI  WRITE(6,110)T2 GO TO 150 d1=J1+1 XW=XXI VW=vO TW=T2 V0=V0*10 TI=TO XXI=XXO XI=XO I F ( d . L T . 2 ) GOTO 25 GO TO 20 F 0 R M A T ( / , 3 X , ' T H E SLEEP TIME  =',E15.8,'  WRITE(G,151) FORMAT(2X,/) STOP END SUBROUTINE SUB 1 ( E , X X 2 , X 2 , H 3 ) IMPLICIT R E A L * 8 ( A - H , 0 - Z ) COMMON/BLK1/B,XXI C0MM0N/BLK2/TI,XI,H  5  15  20  CALL R K ( T I , X X I , X I , H , H 2 , X X 1 , X 1 ) H1=H2/2.D0 CALL R K ( T I , X X I , X I , H 1 , H 3 , X X 2 , X 2 ) D1=XX2-XX1 D1=DABS(D1) IF (D1 . L T . E ) G 0 TO 20 H1=H3/2.D0 XX1=XX2 GO TO 5 RETURN END  SUBROUTINE C  SUB2(FI.FFI)  CALCULATE FORCES IMPLICIT R E A L * 8 ( A - H , 0 - Z ) COMMON/BLK1 / B , X X I C0MM0N/BLK2/TI,XI ,H C 0 M M 0 N / B L K 3 / F M , F L A M D , V O , BTA C0MM0N/BLK4/A2.B2.C2.E0  5 20  FR(Y)=A2+B2/(7.D0+DLOG10(Y+1D-6)) FR0=FR(0) FI=C2+FLAMD*(V0*TI+BTA*TI*TI-XI ) FFI=0.D0 IF ( D A B S ( X X I ) . L T . 1 D - 1 3 ) GOTO 30 IF ( X X I . G T . O . D O ) GOTO 20 F F I = FI + F R ( - X X I ) GO TO 50 FFI=FI-FR(XXI) GOTO 50  0  SECONDS')  269  /  175 17G 177 178 179 180 181 182 183 184 185 186 187 1S8 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224  30 50  50  IMPLICIT R E A L * 8 ( A - H , 0 - Z ) G(X.Y,Z)=Y H=H1 F1=H*F(X,Y,Z) G1=H*G(X.Y.Z) IF((Y+F1/2.D0).LT.O.DO)G0 TO 10 F2=H*F(X+H/2.DO,Y+F1/2.DO,Z+G1/2.DO) G2=H*G(X+H/2.DO,Y+F1/2.DO,Z+G1/2.DO) IF((Y+F2/2.D0).LT.O.DO)GO TO 10 F 3 = H * F ( X + H / 2 . D O , Y + F 2 / 2 . D O , Z + G 2 / 2 . DO) G3=H*G(X+H/2.DO, Y + F 2 / 2 . D O , Z + G 2 / 2 . D O ) IF((Y+F3).LT.O.DO)G0 TO 10 F4=H*F(X+H,Y+F3,Z+G3) G4=H*G(X+H.Y+F3.Z+G3) YN=Y+(F1+2.D0*(F2+F3)+F4)/6.D0 ZN=Z+(G1+2.D0*(G2+G3)+G4)/6.D0 I F ( Y N . L T . 0 . D 0 ) G 0 TO 10 RETURN H=H/2.D0 GO TO 5 END  5  10  DOUBLE PRECISION FUNCTION F ( X . Y . Z ) IMPLICIT R E A L * 8 ( A - H . 0 - Z ) COMMON/BLK1/B,XXI C0MM0N/BLK3/FM,FLAMD,VO.BTA C0MM0N/BLK4/A2,B2,C2,EO  20 30 50  solutions  by  FR(Y)=A2+B2/(7.D0+DLOG10(Y+1D-6)) FRO=FR(0) F=O.D0 FD=C2+FLAMD*(V0*X+BTA*X*X-Z) IF(DABS(Y).LT.1D-13) GO TO 30 I F ( Y . G T . O . D O ) GO TO 20 F=(FD+FR(-Y)-EO*Y)/FM GO TO 50 F=(FD-FR(Y)-EO*Y)/FM GOTO 50. I F ( D A B S ( F D ) . L T . D A B S ( F R O ) ) GOTO 50 F=(FD-DSIGN(FRO,FD))/FM RETURN END results  from  transition  T(I) 0 0.0 592 0 . 1 9 6 3 4 9 1 6 E - 0 3 TIME  analysis  ******  R u n g e - K u t t a method f o r s i n g l e b l o c k f r i c t i o n u n i t system: ***M-KG-SECOND*** .  U0= 0.529020 B0= 1.0000 P = 0.10E+05 LAMDA= 9 . 8 0 6 0 DRIVO= 0.10000E+01 D R I . A C  N  THE S L E E P  GOTO  SU3R0UTINE R K ( X . Y , Z , H 1 , H , Y N , Z N )  ******  COHSN= O . O M= G=  IF ( D A B S ( F I ) . L T . D A B S ( F R O ) ) FFI=FI-DSIGN(FRO.FI) RETURN END  270  X-(I) 0.1000C00E-09 0.9375850E-13 = 0. 24029417E-05  0.120978 O.10E+1O 0.0  X(I) 0.0 0.1987518E-03 SECONDS  model  RADIA= Us= PRECISN  F(I) 0.65064E+04 0.41037E+04  13.483952 0.650000 O.O1000O Ft(I) 0.68603E-01 O.O  APPENDIX  IV. LIST O F BASIC P R O G R A M M O D E L 4  AND SAMPLE  RESULTS 10 ! RE-STORE " M0DEL4: H7, 0, (3" 20 ! 30 ! ******»*«*******#*»****»*#*********»*»*****•»**«#**»*****»******* # 40 ! * " M0DEL4 " 50 ! * * acoustic activity simulation 60 ! * * by D a i h u a Z o u , 1986 70 ! * 60 ! * * n u m e r i c a l s o l u t i o n : m u 1 t i - p a r t i c 1 e f r i c t i o n model 90 ! * * s l i p v e l o c i t y dependent f r i c t i o n : U=U<X') 100 ! * * s l i p back p e r m i t t e d h e r e 110 ! * # 120 ! * 130 ! * D r i v i n g f o r c e i s a p p l i e d at t h e end o f t h e l a s t p a r t i c l e * 140 ! * * * T h i s program i s w r i t t e n f o r numerical s o l u t i o n t o t h e system * 150 ! 1£0 ! * o f s e c o n d o r d e r d i f f e r e n t i a l e q u a t i o n s by RUNGE-KUTTA method * 170 ! * * 180 ! ***************************#************************************** 190 ! OPTION BASE 1 200 INTEGER L , N , I , J , C n o , C c t , N < 1 0 ) , S y m b o l n o < 1 0 ) , E r r c o d e C 1 0 ) , L c t , L i n e n o ( 1 0 ) 210 SHORT S y m b o l s i z e < 1 0 ) , L i n e s i z e < 1 0 ) 220 230 ! 240 DIM S ( 2 , 1 2 > , W o k ( 2 , 1 2 > , W o k l < 2 , 1 2 > , X < 1 0 , 4 0 0 > , Y C 1 0 , 4 0 0 > 250 DIM T i t l e $ C 3 0 ] , X l a b e l $ C 3 0 : , Y l a b e l * C 3 0 3 , F I * C n 260 COM B , X x i , M 270 COM T i , X i l , X i 2 , X i 3 , H 280 COM F m , F 1 a m d < l l ) , V 0 , B t a 290 COM R 2 , B 2 , E 0 , K l , F a 300 DATA . 6 5 0 , 5 E 2 , 9 . 8 0 , 0 . 0 310 DATA 0 . 0 , - 0 5 0 , 2 0 0 0 0 , 1 0 READ U , P , G , C 0 , T 0 , E , N , M 320 330 ! 340 Bta=0 Tint0=2E-2 350 360 Flamd0=3E4 370 Pc=1.379E8 380 V0=1E-1 390 Fa=lE-l ! s p a c i n g between p a r t i c l e s 400 Fm=lE0 410 P1=PCP 420 IF P K 1 . 5 0 E 2 THEN L2 430 U=1.50E2*C0/"Pc+U 440 C0 = 0 450 L 2 ! E0=4.264*Sqrt<F1amd0> 460 Cl=C0/Fm 470 H0=5E-3 480 ! 490 FOR I=1E0 TO 2 S00 FOR J=1E0 TO 12 510 S<I,J>=0 520 UokCI,J>=0 530 W o k K I , J)=0 540 NEXT J 550 NEXT I 560 ! 570 FOR K=1E0 TO M 580 K1=K+1EB 590 FIamdCKl)=F1amd8 ! *<K*1E-1> 600 NEXT K 610 Flamd<lE0>=0 ! s e t lamda0=0 620 ! c a l c u l a t i n g s t a t i c shear f o r c e 630 Us=U*lE0 640 B=Us*<P/'Fm+G) + C l ! .99997662  271  /  650 ! 660 ! estimate constants for f r i c t i o n c o e f f i c i e n t s 670 U0=-. 14 + 1. 03*11 680 B0=1.33E-1-.01S*U 690 P1=U0+B0-U 700 IF P1<=0 THEN L6 710 U0=U0-Pl*3/5 720 B0=B0-Pl*2/5 730 L 6 : A2=C0+U0*<P+Fm*G> B2=B0*<P+Fm*G) 740 C2=B*Fm 750 760 ! 770 ! a s s i g n i n i t i a l v a l u e s f o r XI<0> AND X I ' < 0 ) 780 Ti=T0 790 Cct=5 800 Lct=0 810 FOR K=1E0 TO M 820 K1=K+1E0 830 Wok 1 < 1E0, K l ) (M-K > * F a 840 Wok 1 <2, K l ) = 0 850 NEXT K 860 I 870 Wok 1<1E0,lE0)=Wok1C1E0,2> + F a Wok 1<1E0,12)=V0*Ti 880 Wok l < l E 0 , 1 2 ) = V 0 * T i+C2/F1amcK + C 2 / F l a m d <11)ll)-Fa Wok 1 < 2 , l E 0 ) = W o k 1 ( 2 , 2 ) 890 Wokl<2,12)=V0 900 910 c a l c u l a t e i ni t i a l v a l u e s f o r f i and f i 920 FOR K=1E0 TO M 930 Km=M-K+lEB 940 Kl=Km+lE0 950 K2=Km+2 960, 970 Xil=Wok1<1E0,Km) 980 Xi 2=Wok1<1E0,Kl ) 990 Xi 3=Wok1<1E0,K2) 1000 Xxi =Wok1<2,Kl) 1010 CALL S u b 2 ( F i , F f i > 1020 S<1E0,Kl)=Fi 1030 S<2,Kl)=Ffi 1040 NEXT K 1050 1C60 GOSUB F i 1 e _ d a t a ?.070 L 1 0 : PRINTER IS 16 PRINT L I N C 5 ) . " 1030 EXECUTION BEGINS, PLEASE WAIT ! " , L I N < 3 ) 1090 I 1100 PRINTER IS 0 1110 PRINT L I N O ) 1 120 PRINT " RESULTS FROM RUNGE-KUTTfl METHOD FOR MULT I-PART I CLE SHEAR M ODEL" 1130 PRINT UNIT SYSTEM : * * * M-KG- SECOND * * * " , L I N < 1 E 0 ) 1140 PRINT USING L I S ; C 0 , U 0 , B 0 , E 0 , F m , P , F I a m d < 1 1 ) , U 1150 L 1 8 : IMAGE 2 X , " C O H S N = " , X , . 3 D E , 4 X , " U 0 = " , X , . 7 D , 6 X , " B 0 = " , X , . 7 D , 4 X , " S E I S M = " , 4 D . 3 D / ' 2 X , " MASS = " , X , . 3DE, 4X, ". P n = " , X , . 3 D E , 2 X , " L A M D A 1 0 = " , X , . 3 D E , 6 X , " Us=",X,.6D 1160 PRINT USING L 1 9 ; G, V0, Bt a, E , F a , T i nt 0, F1 amd<6) , Pc 1170 LI 9:IMAGE 6 X , " G = " , X , . 3 D E , 2 X , " D R I V 0 = " , X , . 3 D E , 3 X , " D R I . A C = " , X , . 3 D E , 2 X , " P R E C I S N = " , X , . 6 D , / , 6 X , " A = " , X , . 3 D E . 3 X , " T I N T = ",X,.3DE,3X,"LAMDA5=",X, .3DE.7X,"Pc = " , X , . 3 D E , 1 180 ! 1190 ! print i n i t i a l values 1200 1=0 1210 PRINT USING L 6 0 ; I , T i 1220 FOR K=1E0 TO M 1230 K1=K+1E0 1240 PRINT USING L66;K,Wok 1<2,K1),Wok 1 < 1 E B , K 1 ) , S < 1 E 0 , K 1 ) , S < 2 , K 1 ) 1250 NEXT K 1260 PRINT LIN<1E0> 1270 PRINT USING L t i t 1280 L t i t : IMAGE " II TI # RATE TOT EN SEISM EN EN RA TIO KINET EN" :  272  /  1290 1300 1310 1320 1330 1340 1350 1360 1370 1380 1390 1400 1410 1420 1430 1440 1450 1460 1470 1480 1490 1500 1510 1520 1530 1540 1550 1560 1570 1580 1590 1600 1610 1620 1630 1640 1650 1660 1670 1680 .690 1700 1710 1720 1730 1740 1750 1760 1770 1780 1790 1800 1810 1820 1830 1840 1850 I860 1870 1880 1890 1900 1910 1920 1930 1940 1950  Wr = 0 Uf=0 L=0 ! counter 12=0 ! key 10 = 0 li=0 H = H0 Tint=Tint0 lx=0  o f f i r s t s i i ps t o s t o p program  LOOP FOR TIME INCREMENT BEGINS, FOR I=1E0 TO N 11=0 13=0 14=0 15 = 0 M0=M+2 FOR K=1E0 TO 2 FOR J=1E0 TO M0 ! Check t h e l o g i c position IF K=2 THEN L21 IF <J=1E0> OR <J=12) THEN L21 J1=J+1E0 IF W o k l C l E 0 , J l > < = W o k l < l E 0 , J ) THEN L21 Wok 1 C 1 E 0 , J > = U o k 1 < 1 E 8 , J l > + 8 E - l * F a IF Wok 1<2, J X U o k 1 <2, J l ) THEN L20 GOTO 1570 L20: Wokl<2, J)=Uo'kl<2, J l ) L=L+1E0 L2l: Wok<K,J)=Wok1<K,J) NEXT J NEXT K  ENDS RT L80  |  ! s e a r c h f o r min & max s l i p speed L22: X1=0 X2=0 Kkl=Kk2=lE0 FOR K=1E0 TO M K1=K+1E0 IF R B S ( X l ) < A B S < : U o k l < 2 , K l ) ) THEN Lgo Kk 1=K Xl=Wokl<2,Kl) GOTO 1750 Lgo: IF RBSc;X2)>=RBS<Wokl<2,Kl>) THEN 1750 Kk 2=K X2=Wok1<2,Kl) NEXT K s e t up t i m e s t e p by t h e p a r t i c l e Kl=Kkl+lE0 K2=Kkl+2 Xil=Wok<lE0,Kkl) Xi2=Wok<lE0,Kl> X i 3 = Wok< 1 E 0 , K 2 ) Xxi=WokC2,Kl> CALL S u b l ( E , X 0 0 , X x 0 , H 3 ) Kl=Kk2+lE0 K2=Kk2+2 Xil=Wok<lE0,Kk2) Xi2=Wok<lE0,Kl> Xi3=Wok<lE0,K2> Xxi=Wok<2,Kl) CALL S u b l < E , X 2 2 , X x 2 , H h ) IF H3<Hh THEN 1960 H3=Hh Kk=Kk2 GOTO L c o n t  with  o r min  speed  273  / 274  Kk=Kk1 I960 1970 X22=X00 . 1980 Xx2=Xx0 1990 L c o n t : K l = K k + l E 0 2000 Wokl<lE0,Kn=X22 2010 WokK2,Kl)=Xx2 2020 i 2630 ! s o l v i n g t h e d i f f e r e n t i a l e q u a t i o n f o r X and X ' f o r e a c h p a r t i c l e 2040 FOR K=1E0 TO M ! Loop f o r each p a r t i c l e b e g i n s , ends at L45 2050 Km=M-K+lE0 2060 Kl=Km+lE0 2070 K2=Km+2 2080 Kkl=Kk2=0 2090 Xil=WokC1E0,Km) 2100 Xi2=Uok<lE0,Kl> 2110 Xi3=Wok<lE0,K2> 2120 Xxi=Wok<2,Kl> 2130 GOSUB L o g i c 2140 ! 2150 ! t i m e s t e p s a r e s e t t h e same as t h a t d e t e r m i n e d by max o r min spee d 2160 2170 2180 2190 2200 2210 2220 2230 2240 2250 2260 2270 2280 2290 2300 2310 2320 2330 2340 2350 2360 2370 2380 2390 2400 2410 2420 2430 2440 2450 2460 2470 2480 2490 2500 2510 2520 2538 2540 0 2550 4 + 1E0 2560 2570 2580 2590  L30:  ! L45:  CHLL R k l C T i , X i l , X i 2 , X i 3 , X x i , H 3 , X 2 2 , X x 2 ) WoklUE0,Kl>=X22 Wok1<2,Kl>=Xx2 IF Kk2<>0 THEN F1 amd<Kk2)=F1amd0 IF Kkl<>0 THEN Flamd<Kkl>=Flamd0 NEXT K  I  Wokl<lE0,lE0)=Wokl<lE0,2)+Fa Wok 1 < 1E0, 12> = V 0 * T i +C2--F1 amd< 1 O - F a Wokl<2,lE8)=Wokl<2,2) Wok 1<2,12>=V0 Ti=Ti+H3 ! To i n c r e a s e  time  step  c a l c u l a t e f o r c e s and e n e r g i e s FOR K=1E0 TO M Km=M-K+lE0 Kl=Km+lE0 K2=Km+2 Kkl=Kk2=0 Xil=Wokl<lE0,Km) Xi2=Wokl<lE0,Kl> Xi3=Wokl<lE0,K2> Xxi=Wokl<2,Kl> GOSUB L o g i c CHLL S u b 2 < F i , F f i ) S<lE0,Kl>=Fi S<2,Kl)=Ffi Wf=Wf+ RBS<Ffi *<Xi 2-Wok <1E0,Kl>> > Wr=Wr+Xxi*Xxi*H3  L4S  L44: L46:  Count e v e n t number f o r each s l i p o f any p a r t i c l e IF <Wok<2,Kl>=0> FIND <Xxi<>0> THEN L44 IF SGN<Wok<2,Kl>X>SGN<Xxi > THEN L44 GOTO L46 L=L+1E0 ! To count e v e n t # f o r each s a m p l i n g i n t e r v a l IF RBSCWoklC2,Kl)>>0 THEN I1=I1+1E0 IF RBS<Wok(2,Kl)>>0 THEN I5=I5+1E0 IF CRBS(Wok<2,Kl>)<lE-3> AND ( BBS < Wok 1 < 2, K1) ) > 1 E - 3 > THEN I3=I3+1E IF  <RBS(Wok<2,Kl)X1.00E2)  RND <RBS<Wok 1<2,K1)>>1.0@E2>  IF Kk2<>0 THEN F1amd<Kk2>=F1amd0 IF Kkl<>0 THEN F1 amd<Kk1>=F1amd0 NEXT K  THEN 14 = 1  / 275  2600 ! IF I=1E0 THEN L56 2610 2620 IF I - I 0 < 5 THEN L58 ! screen monitoring 2630 10=1 2640 L 5 6 : PRINTER IS 16 PRINT LIN<13> 2650 2660 PRINT USING " 1 0 A , 5 D , 2 < 2 X , r 1 . 7 B E > ' ; " < I , H , T I > " , I , H 3 , T i 2670 PRINT USING ' V , 1 5 A , 5 < X , M . 6 D E ) " ; " < X 1 0 , X 9 , X 8 e t c ) " , W o k 1<1EO, 11>,Wok1<1E0 , 1 0 > , W o k K 1 E 0 , 9 ) , W o k 1<1E0,8),Wok 1<1E0,7) 2680 PRINT USING 1 8 R , 2 < I X , M . 6 D E ) , 3 C I X , M . 5 D E > " ; " < X 1 0 , X 9 , X 8 , X 7 , X 6 > ' " , W o k 1C 2, U J . W o k l ( 2 , 1 0 ) , W o k l < 2 , 9 ) , W o k l < 2 , 8 > , W o k l < 2 , 7 ) 2690 PRINT USING 1 7 A , 3 < l X , M . 6 D E ) , 2 a X , M . 5 D E > " ; " < X 5 , X 4 , X 3 , X 2 , X1) ' " , Wok 1 < 2 ,6>,Wokl<2 , 5 ) , W o k l < 2 , 4 ) , W o k l < 2 , 3 ) , U o k l < 2 , 2 > PRINTER IS 0 2700 2710 ! 2720 ! i f a l l p a r t i c l e s a r e mowing, s t a b l e s l i d i n g ! 2730 L 5 8 : IF I1=M THEN L55 2740 L 5 9 : IF I3=M THEN L57 2750 IF I4=M THEN L57 2760 GOTO L70 ! RIGHT ? ? ? 2770 L 5 5 : IF 15=11 THEN L59 2780 I2=I2+1E0 ! I n d i c a t i o n of a l l p a r t i c l e s mowing 2790 L 5 7 : PRINT USING L 6 0 ; I , T i 2800 L 6 0 : IMAGE s,3X,"I =" , 5D, 8X, "t i me TI = " , X, . 8DE/-2X, "P#" , 9X, "XXI " , 14X, "XI " , 15 Fri " X,"Fi",14X FOR K=1E0 TO M 2810 K1=K+1E0 2820 2830 PRINT USING L 6 6 ; K , W o k l < 2 , K l ) , W o k l ' : i E 0 , K l ) , S < : i E 0 , K l ) , S < 2 , K l > 2840 NEXT K 2850 PRINT LIN<1E0) 2860 PRINT USING L t i t 2870 L 6 6 : IMAGE X , 3 D , X , 2 < 2 X , M . 8 D E > , X , 3 X , M . S U E , 3 X , M . 8 D E 2880 ! check t h e p r e - s e t t i m e i n t e r v a l TINT 2890 ! 2900 L 7 0 : T s = T i - T 0 IF T s < T i n t THEN L80 2910 Ii=li+1E0 ! To count s a m p l i n g p o i n t s 2920 2930 ! 2940 ! c a l i b r a t e r e s u l t s f o r f i x e d i n t e r v a l TINT 2950 Ts=Tint^Ts 2960 Wr=Wr*E0*Ts 2970 Wl=Wf*Ts+Wr 2980 F1=L*Ts 2990 IF F1=0 THEN L73 3000 Ratio=Wr^Fl 3010 GOTO L75 3020 L 7 3 : Rat i o = 0 3030 ! 3040 ! compute k i n e t i c e n e r g y 3050 L 7 5 : M0=M+1E0 3060 18 = 0 3070 FOR K=2 TO M0 3080 Dl=ABS<Wok1<2,K>> 3090 Ek=Dl*Dl 3100 IF <ABSi:WokC2,K))>Dl) AND <DK1E-13> THEN W o k K 2 , K ) = 0 3110 IF <HBS<Wok<2,K) )<1E-13) AND ( D K 1 E - 1 3 ) THEN W o k l < 2 , K ) = 0 3120 IF <Wok<2,K)=0) AND <Wok 1<2,K) = 0) THEN I 8 = I 8 + l E 0 ! T o c o u n t p a r t i c l es coming t o r e s t NEXT K 3130 3140 IF K 1 0 THEN L76 3150 IF I8<M THEN GOTO L76 3160 H=<C2-S(1E0, l l ) ) / - F l a m d 0 / V 0 ! DX=<C2-Fn)^LAMDA DT=DX/"V0 3170 Tint=H 3180 GOTO L77 3190 L 7 6 : H=H0 ! I f al 1 p a r t i c l e s stopped moving, change time s t e p 3200 Tint=Tint0 3210 L 7 7 : E k = T s * E k * F m ' 2 ,  l  / 276  PRINT USING L 7 8 ; I i , T i , F I , U l , W r , R a t i o , E k 3228 3230 L 7 8 : IMAGE 3 D , I X , . 1 0 D E , 5 < M . 6 D E > 3240 ! store data for f i l e 3250 ! FOR K=1E0 TO C c t 3260 X<K,Ii>=Ti 3270 NEXT K 3280 Y C 1 E 0 , I i > = F1 3290 Y<2,Ii)=Wl 3300 Y<3,Ii>=Wr 3310 Y<4, I i >=Ratio 3320 Y<5,Ii)=Ek 3330 3340 ! IF I i = 4 0 0 THEN L85 ! e x i t 2: computer o v e r f l o w 3350 3360 IF I2<1E0 THEN L79 Ix=lx+1E0 ! To count I i a f t e r a l l p a r t i c l e s come t o moving 3370 IF Ix=10 THEN L85 ! e x i t l : normal e x i t 3380 3390 L 7 9 : L=0 3400 Wr = 0 3410 Wf = 0 3420 T0 = T i e x i t 3: a b o r t i o n due t o t i m e l i m i t 3430 L 8 0 : NEXT I I=I-1E0 3440 3450 ! 3460 L 8 5 : PRINT USING L 6 0 ; I , T i FOR K=1E0 TO M 3470 K1=K+1E0 3480 3490 PRINT USING L 6 6 ; K , W o k l < 2 , K l > , U o k l < l E 0 , K l > , S a E 0 . K l > , S < 2 , K l > 3500 NEXT K 3510 IF Ix=10 THEN L140 3520 IF I i = 4 0 0 THEN L130 3530 PRINT USING L 9 5 ; I 3540 L 9 5 : IMAGE 2 X , 5 D , " r u n s , s p e c i f i e d c y c l e s not f i n i s h e d y e t !",•/'• 3550 GOTO L150 3560 L130:PRINT USING 2X, 5D, 6 7 A , V / ' / " ' ; I , " r u n s , work i s not f i n i s h e d y e t ! c a p a c i t i e s of array X & Y exceeded.". GOTO L150 3570 PRINT USING 2 X , 5 D , 2 8 A , ; I , " r u n s , j o b i s done ! ' 3580 L140 3590 ! store data into f i l e 3600 ! PRINTER IS 16 3610 L150 IF B*="N" THEN L160 3620 GOSUB P r e p 3630 STORE DATA ON F I L E , PLEASE WAIT ! " , L I N U 0 > PRINT LIN<10>, " 3640 GOSUB C r e a t e 3650 GOSUB E n _ e v e n t 3660 PRINTER IS 0 3670 PRINT USING L 1 5 5 ; F i l e n a m e * 3680 IMAGE X,"******** the e v e n t r a t e and e n e r g y r e l e a s e a r e s t o r e d i n 3690 L155 file: ",5A," 3700 PRINTER IS 16 3710 L 1 6 0 : PRINT L I N C 1 E 0 ) 3720 PRINT LIN<5>," EXECUTION TERMINATED" 3730 PRINT L I N C 7 ) , " GOOD-BYE ! " 3740 ! STOP 3750 END 3760 3770 ! 3780 ! ************************************************************* 3790 L o g i c : Check t h e l o g i c p o s i t i o n : XCi-1>-Xi>0.1A .3800 IF X i 2 - X i 3 > 8 E - l * F a THEN L5 3810 Kk2=Kl 3820 Flamd<Kk2)=lE13 3830 L 5 : IF X i 1 - X i 2 > l E - l * F a THEN S o i r t 3840 Kkl=Km 3850 Flamd<Kkl>=lE13 3860 S o i r t : RETURN  / 277  3870 ! 3880 ! **********************************»«****#*** 3390 ! preparing f i l e data 3900 P r e p : ! 3910 T i 11 e* = "NUMERICAL. RESULTS OF RE MODEL" 3920 Xlabel*="TIME " 3930 Y l a b e l *="EVENT 8. ENERGIES" 3940 Xorigen=0 3950 Yorigen=0 3960 ! 3970 ! s e a r c h f o r Xmax & Ymax 3980 Xextreme=l.0E-7 3990 Yextreme=lE-2 4000 FOR K=1E0 TO I i 4010 FOR Cno=lE0 TO Cct 4020 IF Y < C n o , K X = Y e x t r e m e THEN Lc 4030 Yextreme=Y<Cno,K> 4040 L c : NEXT Cno 4050 IF X C 2 , K X = X e x t r e m e THEN Lk 4060 Xextreme=X<1E0,K> 4070 L k : NEXT K 4088 Xdelta=<Xextreme-Xorigen>/20 4090 IF X d e l t a > l E 0 THEN X d e l t a = I N T C X d e l t a > 4100 Y d e l t a = < Y e x t r e m e - Y o r i gen)-'20 4110 IF Y d e l t a > l E 0 THEN Yde11a=I NT<Yde11a) 4120 FOR Cno=lE0 TO C c t 4130 Symbolno<Cno)=0 4140 Symbol size<Cno> = 3 4150 Lineno<Cno)=Cno 4160 Linesize<Cno>=4 4170 Errcode<:Cno>=0 4180 N<Cno)=Ii 4190 NEXT Cno 4200 Lineno<2)=Cct+lE0 4210 RETURN 4220 ! 4230 ! it********************************************************** 4240 ! 4250 F i l e d a t a : ! c r e a t e d a t a f i l e on d i s k 4260 GCLEflR 4270 E$="Y" 4280 INPUT "DO UOU WANT TO STORE RESULTS ON F I L E Y / N ? " , B * 4290 IF B*="N" THEN E x i t 4300 Device$=":H?,0,0" 4310 INPUT "ENTER DATA STORAGE DIVECE : H 7 , 0 , 0 ? " , D e v i c e * 4320 Filename*"" " 4330 INPUT "ENTER F I L E NAME ? " , F i l e n a m e * 4340 IF F i l e n a m e * ' " " THEN F i l e _ d a t a 4350 Fi1e*=Fi1ename*&Device* 4360 ON ERROR GOTO E r r o r 1 !to f i l e purging r o u t i n e 4370 GOTO T r y 4380 E r r o r l : IF ERRH=54 THEN E r r o r 2 4390 CALL MessCERRM*) 4400 GOTO F i l e _ d a t a 4410 E r r o r 2 : BEEP 4420 DISP " F I L E "&CHR*<129)&Fi1e*&CHR*<12S>&" a l r e a d y e x i s t s : do you want be d e l e t e d y / N ?"; 4430 A*="N" 4440 INPUT A* 4450 IF A*="Y" THEN GOTO Purge 4460 GOTO F i l e _ d a t a 4470 P u r g e : PURGE F i l e * 4480 GOTO E x i t 4490 T r y : CREATE F i l e * , l E 0 , 1 0 4500 PURGE F i l e * 4510 E x i t : RETURN 4520 !  it  / 278  4538 ! **************************************************************** 4540 ! 4550 GCLEflR 4560 C r e a t e : ! STORE DATA ON F I L E 4570 Bytetot=200+28*Cct+80*Lct+16*Ii*Cct 4580 IF I i > 3 8 5 THEN 4610 4590 CREATE F i 1 e * , 1 E 0 , B y t e t o t * 1.05 ! I F Cct = 5 , I i > 3 8 5 U S E : F i 1 e * , 2 , B y t e t o t * 1 . 1 • 2 4600 GOTO 4620 4610 CREATE F i 1 e * , 2 , B y t e t o t * 1 . 1 ' 2 4620 ASSIGN F i l e * TO #1E0 ! 4630 OFF ERROR 4640 PRINT # 1 E 8 ; T i 11e*,X1abe1 *,Y1abe1 * , X o r i g e n , X e x t r e m e , X d e 1 1 a , Y o r i g e n , Y e x t r erne,Ydelta,Cct,Let 4650 FOR Cno=lE0 TO C c t 4660 PR INT # 1 E S ; S y m b o l n o C C n o ) , S y m b o l s i z e < C n o > , E r r c o d e C C n o ) , N C C n o ) , L i neno <Cno>,Linesize<Cno> 4670 FOR K=1E0 TO N<Cno> 4680 IF E r r c o d e < C n o ) = 0 THEN PRINT # 1 E 0 ; X < C n o , K ) , Y C C n o , K > 4690 IF E r r c o d e < C n o > = l E 0 THEN PRINT # 1 E 0 ; X < C n o , K > , Y < C n o , K > , Y e r r < C n o ,K> 4700 IF Errcode<Cno>=2 THEN PRINT # 1E0; X < Cno, K >, X e r r < Cno, K ) , Y C C n o , K ) 4710 IF Errcode<Cno>=3 THEN PRINT # 1 E 0 ; X < C n o , K > , X e r r < C n o , K > , Y < C n o , K ),Yerr<Cno,K> 4720 NEXT K 4730 NEXT Cno 4740 FOR L n o = l E 0 TO L e t 4750 PRINT # 1 E 0 ; L a b e l * < L n o > , L a n g l e C L n o ) , L s i ze<Lno) , L o r g < L n o > , L x C L n o ) , L y <Ln0> 4760 NEXT Lno 4770 ASSIGN #1E0 TO * ! c l o s e f i l e 4780 RETURN 4790 ! 4800 En e v e n t : ! ************************************************** 4810 Nl=50 ! Count t h e e v e n t # a c c o r d i n g t o e n e r g y m a g n i t u d e 4820 Cctl=Cct+lE0 4830 Xmax=0 4840 FOR K=1E0 TO I i 4850 IF Xmax>=Y<4,K> THEN L e x t l 4860 Xmax=Y<4,K> 4870 L e x t l : NEXT K 4880 Xd=Xmax/Nl 4890 FOR J=1E0 TO N1+1E0 4900 X<Cctl,J>=<J-lE0>*Xd 4910 Y<Cctl,J>=0 4920 NEXT J 4930 XCCct1,Nl+lE8)=Xmax 4940 FOR K=1E0 TO I i 4950 FOR J=1E0 TO N1+1E0 4960 IF Y<4,K>>X<Cct1,J> THEN L e x t 2 4970 YCCctl,J>=Y<Cctl,J)+Y<1E0,K) 4980 GOTO L e x t 3 4990 L e x t 2 : NEXT J 5000 L e x t 3 : NEXT K 5010 Ymax=0 5020 FOR J=1E0 TO N1+1E0 5038 IF Y m a x > = Y < C c t l , J ) THEN L e x t 4 5040 Ymax=Y<Cct1,J) 5058 L e x t 4:NEXT J 5060 ! 5070 Cct=lE0 5080 X1abe1*="AVERAGE ENERGY/EVENT" 5090 Y1abe1*="FREQUENCY" 5100 X d e l ta=INTCXmax/-20) 5110 Ydelta=INT<Ymax/20> 5120 N<1E0)=N1+1E0  / 279  5130 F i 1 ename*=Fi 1 ename*Sc"0" 5140 F i 1 e* = F i 1 ename*8cDeui c e * 5150 CREATE F i 1 e S , 1 E 0 , < 2 0 0 + 2 8 * C c t +16*51*Cct>* 1.1 5160 ASSIGN F i l e * TO #1E0 5170 OFF ERROR 5180 PRINT # 1 E 0 ; T i 11e*,X1abe1 $,Y1abe1 $,Xori g e n , X m a x , X d e 1 1 a , Y o r i g e n , Y m a x , Y d e I t a,Cct,Lct 5190 PRINT #1E0;Symbol no<1E0>,Symbol s i z e < 1 E 0 ) , E r r c o d e < 1E0) , N(1E@), L i neno<1E9) , L i n e s i ze(1E0> 5200 FOR K=1E0 TO N<1E0> 5210 PRINT # 1 E 0 J X C C c t 1 , K ) , Y C C c t 1 , IO 5220 NEXT K 5230 ASSIGN #1E0 TO * 5240 RETURN 5250 ! 5260 ! ============================================================ 5270 ! 5280 SUB R k l < T , X l , X 2 , X 3 , Y , H , X 2 n , Y 2 n > 5290 ! 5300 F1=H*FNF<T,X1,X2,X3,Y> 5310 G1=H*Y 5320 F2=H*FNF<T + H / 2 , X l + G l / - 2 , X 2 + G l / 2 , X 3 + G l / ' 2 , Y+F1^2) 5330 G2=H*<Y+Fl/'2> 5340 F3=H*FNF<T+hV2,Xl+G2/2,X2+G2'2,X3+G2/-2,Y+F2/2) 5350 G3=H*<Y+F2'2> 5360 F4=H*FHF<T+H,X1+G3,X2+G3,X3+G3,Y+F3> 5370 G4=H*<Y+F3> 5380 X2n = X2+<Gl + 2*<G2 + G3) + G4>/'6. 0 5390 Y2n = Y+<Fl+2*<F2 + F3>+F4>/'6.0 5400 SUBEND 5410 ! 5420 ! =========================================================== 5430 ! 5440 SUB S u b l < E , X 2 2 , Y 2 2 . H 3 ) 5450 ! 5460 COM B , X x i , M 5470 COM T i , X i l , X i 2 , X i 3 , H 5480 ! 5490 CALL R k C T i , X i 1 , X i 2 , X i 3 , X x i , H , H 1 , X 2 1 , Y 2 1 > 5500 H2=Hl/2.0 5510 L 5 : CALL Rk <Ti , Xi 1, Xi 2, Xi 3 , Xxi , H2, H3, X22, Y22> 5520 D1=ABS<Y22-Y21> 5530 IF D U E THEN L20 5540 H2=H3/-2.0 5550 Y21=Y22 5560 GOTO L5 5570 L 2 0 : SUBEXIT 5580 SUBEND 5590 ! 5600 ! ============================================================== 5610 ! 5620 SUB S u b 2 C F i , F f i > 5630 ! 5640 ! calculate forces 5658 COM B . X x i , M 5660 COM T i , X i l , X i 2 , X i 3 , H 5670 COM Fm, FI arndC*) , V 0 , B t a 5680 COM A 2 , B 2 , E 0 , K l , F a 5690 ! 5700 DEF FNFr<Y>=A2+B2/<7.8+LGT<Y+1E-6>> 5710 Fr0=FNFr<0) 5720 Fi=F1amdCKl>*<Fa+Xi3-Xi2>-Flamd<Kl-l>*<Fa+Xi2-Xi 1 > 5730 L 1 5 : F f i = - F i 5740 I F A B S ( X x i X 1 E - 1 3 THEN L30 5750 I F X x i >0 THEN L20 5760 Ffi=FNFr<-Xxi> 5770 GOTO L50  5780 5790 5800 5810 5820 5830 5840 5850 5860 5870 5880 5890 5900 5910 5920 5930 5940 5950 5960 5970 5980 5990 6000 6810 6020 6030 6040 6050 6060 6070 6080 6090 6100 6110 6120 6130 6140 6150 6160 6170 6180 6190 6200 6210 6220 6230 6240 6250 6260 6270 6280 6290 6300 6310 6320 6330 6340 6350 6360 6370 6380 6390 6400 6410 6420 6430 6440 6450  L20:  Ff i =-FNFr<Xxi) GOTO L50 L30: IF RBSCFi)<RBS<Fr0) THEN L50 Ffi=-Fr0*SGN<Fi) L50: SUBEXIT SUBEND I  SUB Rk < T , X I , X 2 , X 3 , Y , H I , H , X 2 n , Y 2 n ) H = H1 F1=H*FNFCT,X1,X2,X3,Y) G1=H*Y IF Y + F l / 2 < 0 THEN L10 F2=H*FNF(T+H/2,Xl+Gl/2,X2+Gl/'2,X3+Gl/2,Y+Fl/2> G2=H*(Y+Fl/2> IF Y+F2/2<0 THEN L10 F3 = H*FNFCT+H/2, X1+G2/2, X2 + G2/-2, X3 + G 2 / 2 , Y+F2/-2) G3=H*CY+F2/2) IF Y+F3<0 THEN L10 F4=H*FNF<T+H,X1+G3,X2+G3,X3+G3,Y+F3> G4=H*<Y+F3) X2n=X2+CGl+2*<G2+G3)+G4)/6.0 Y2n=Y+<Fl+2*<F2+F3) + F 4 ) / - 6 . 0 IF Y2n<0 THEN L10 SUBEXIT L10 : H=H/2 GOTO L5 SUBEND  L5:  DEF  FNFCT,X1,X2,X3,Y)  COM B , X x i , M COM T i , X i l , X i 2 , X i 3 , H COM F m , F l a m d < * ) , V 0 , B t a COM R 2 , B 2 , E 0 , K l , F a  L15  L20 L30 L50  DEF FNFr<Y)=R2+B2-'C;7.0+l.GT<Y+lE-6)) F=0 Fd=Fl amd<Kl)*<Fa+X3-X2)-Flamd<K1-1)*<Fa+X2-X1) :Fr8=FNFr(0) IF R B S C Y X 1 E - 1 3 THEN L30 IF Y>0 THEN L20 F=<Fd+FNFr<-Y)-E0*Y)'Fm GOTO L50 : F=<Fd-FNFr<Y)-E0*Y)/Fm GOTO L50 : I F R B S < F d X R B S c F r 0 ) THEN L50 F=<Fd-Fr0*SGN<Fd> >/Fm : RETURN F FNEND  I  SUB M e s s < M « ) d i s p l a y message M$,beep and p a u s e s FOR K=l TO 2 DISP CHR*< 129)8." " 8,M*S,CHR* < 123) 8," BEEP WRIT 200 NEXT K PRUSE DISP " " SUBEND  CONT"  RESULTS FROM RUNGE-KUTTf) METHOD FOR MULT I-PART I CLE SHERR MODEL UNIT SYSTEM : * * * M-KG-SECOND * * * COHSN= MRSS= G= fl= I = p# 1 2 3 4 5 6 7 S 9 10  II 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 23 29 30 31  .080E+01 .100E+01 .9S0E+01 .100E+00  U0= Pn= DRIV0= TINT=  .5290200 .500E+03 .100E+01 .200E-03  BQ= LRMDR10= DRI.RC= LRMDR5=  .1209S0O .100E+07 .Q00E+01 .100E+07  t i me TI = .00000000E+01 XXI XI Fi .00000060E+01 .90000000E+00 .00000000E+01 .00000000E+01 .S0000000E+00 .00000000E+01 .70000000E+00 .00000000E+01 .00000000E+01 .60000000E+00 .00000000E+01 .00000000E+01 .50000000E+00 .00000000E+01 .00006000E+01 .00000000E+01 .40000000E+00 .00000000E+01 .30000000E+00 .00000000E+01 .00000000E+01 .20000000E+00 .00000000E+01 .00000000E+01 .00000000E+01 .10000000E+00 .00000000E+01 .00000000E+01 .33137000E+03 .00000000E+01  . SEISM= 100.060 Us= .650000 PRECISN= . 0 5 0 0 6 0 Pc= .13SE+09  0  TI . 2500000000E-02 .5000000000E-02 .520507S125E-02 .5410156250E-02 .5615234375E-02 .5836078125E-02 .6064453125E-02 .7470703125E-02 .7695312500E-02 .7929687500E-02 .S144531250E-02 .8359375000E-02 .S564453125E-02 .8779296375E-02 .9287109375E-82 .94921S7580E-02 .9697265625E-02 .9912109375E-02 .1016601563E-01 .1040039063E-01 .106152343SE-01 .1082031250E-01 .11054S8750E-01 .1132812500E-01 .1257S12500E-01 .1382S12500E-01 .1404296875E-01 .143164O625E-01 .1453125000E-01 .1503906250E-01 .153125O000E-01  # RRTE .000000E+01 .000000E+01 .975238E+00 .975238E+00 .000000E+01 .000000E+01 .000000E+01 .000000E+01 .890435E+00 .000000E+01 .186182E+01 .000000E+01 .000000E+01 .080000E+01 .393846E+00 .975238E+80 .000000E+01 .930909E+00 .787692E+00 .060000E+01 .930909E+00 .97523SE+00 .000000E+01 .731429E+00 .160000E+00 .64B060E+00 .186182E+01 .146286E+01 .930909E+00 .393846E+00 .731429E+00  299 time I = P# XXI .75042432E-03 1 .10884304E+01 2 .39475155E+01 3 4 .99103804E-01 5 -.42010O99E+01 .26953725E+01 6 .27459989E+01 7 -.14235112E+00 8 .44283326E-01 9 .20910145E+01 10  TOT EN .000000E+01 .000000E+01 .329092E-01 .12903SE+00 .251831E+00 .334237E+00 .50742SE+00 .855710E+00 .863949E+00 .729467E+00 .683037E+00 .729930E+00 .818275E+00 .874542E+00 .839188E+00 .857216E+00 .804863E+88 .743366E+00 .731826E+00 .746937E+00 .779501E+00 .829800E+00 .869750E+00 .870186E+08 .128649E+01 .141981E+01 .167901E+01 .160216E+01 .166460E+01 .189502E+01 .208521E+01  SEISM EN .000000E+01 .000008E+01 ,664624E-02 .467106E-01 .117341E+00 .200953E+00 .279191E+00 .522719E+00 .465958E+00 .360139E+00 .346300E+00 .367596E+00 ,401239E+00 .428885E+00 .450546E+00 .432640E+0Q .38408SE+00 .343475E+00 .324521E+00 .342090E+00 .356323E+00 •367966E+00 .367910E+00 .355570E+00 ,717787E+00 .899829E+00 . 846033E + 00 .805825E+00 .810811E+00 ,999174E+00 . 1 1 0 1 U E + 01  = .15390625E--01 Fi XI .35561871E+03 .90000001E+00 .27793600E+04 .80035562E+00 .34526046E+03 .70349060E+00 -.54023152E+04 .60697084E+00 .50504877E+00 .16359555E+04 .40476265E+00 .44482273E+04 .30892475E+00 -.34769843E+04 -.14368452E+03 .2096098SE+00 .20676451E+04 . U015131E+00 .33298382E+03 .12760397E-01  Fri .00000060E+01 .00000088E+01 .00000000E+01 .00000008E+01 .00000000E+01 .00000008E+01 .60000600E+01 .00000000E+01 .00000000E+01 -.33137080E+03  EN RATIO .000000E+01 .000000E+01 .681499E-02 .478966E-S1 ;000000E+01 .000000E+01 .000000E+01 .000000E+01 .523293E+00 .000000E+01 .18S001E+00 .000000E+01 .000000E+01 .000000E+01 .114396E+01 .443625E+00 .000000E+01 .363967E+06 .411990E+0O .000000E+01 .383306E+00 .377309E+00 .000000E+01 .486131E+00 .448617E+01 .140598E+01 .454412E+00 .550857E+00 .870983E+00 .253696E+01 .150543E+01  KI NET EN .000000E+01 .000000E+01 .47188SE+00 .186887E+01 .380543E+01 .554775E+S1 .639053E+01 .109948E+Q1 .294280E+01 .551392E+00 .853240E-03 . 3 3 9 7 U E + 00 .90S365E+08 .121787E+01 .406694E+00 .441347E+00 .638557E-01 .256305E-01 .269993E+00 .709617E+S8 . U 9 7 5 4 E + 01 .157998E+01 .153760E+01 .126239E+01 .126912E+00 .882286E-01 .104433E+00 .363832E-01 .351452E+00 .727129E+Q0 .160057E+S1  TI  Fri -.28560705E+03 -.27845912E+03 -.27781353E+03 -.27998036E+O3 .27778475E+03 -.27799459E+03 -.2779855SE+03 .27971747E+03 -.28061768E+03 -.27811961E+03  / 282  II 32 33 34 35 36 37 3S 39 40 41  TI .15546S7500E-61 .1576171875E-81 . 1599609375E-O1 . 16201171S8E-01 .1640625000E-01 .1662189375E-01 .16S2617183E-01 .1703125000E-01 .1723632S13E-01 .1744140625E-01  # RHTE .256000E+01 .930909E+00 .000000E+01 ,195048E+01 .000000E+01 .930909E+00 ,000000E+01 .975238E+00 .000000E+01 .97523SE+00  I 428 time p# XXI 1 .35310227E+01 2 .12036699E+01 -.28254492E+01 3 4 .59766717E+80 5 .36373164E+81 6 -.93781994E+80 7 -.17866348E-01 8 .1430412SE+01 9 .18343676E+01 10 .64467934E+80 428 r u n s ,  ********  job  TOT EN .201898E+01 .191592E+01 .176381E+01 .161290E+01 .16029BE+01 .164S95E+01 .174377E+01 .179037E+01 .180131E+01 .178040E+01  = .17441406E--01 XI •90259083E+00 .80540261E+00 .70471118E+80 .60314030E+00 .50679165E+00 .40869957E+08 .30910059E+08 .21087872E+08 .11344842E+80 .15078477E-81  SEISM EN .105111E+81 .940139E+60 .822377E+00 .747775E+88 .720753E+60 .737829E+88 .779496E+60 .821210E+60 .8488S8E+80 .S54754E+88  EN RATIO .4165S9E+00 ,188992E+01 .008868E+01 .3S3381E+00 .088888E+81 .791730E+06 .000000E+S1 .842861E+88 .0080B0E+81 .876457E+08  . K I N E T EN .17685SE+01 .160131E+01 .183S2SE+81 .781920E+88 .471792E+88 .2523S8E+80 .162857E+08 .127905E+88 .141631E+88 .282668E+08  TI  is  done  the event  rate  -. -. -. -.  -.  Fi 28117230E+04 35831491E+84 87945211E+83 52222208E+04 17434151E+04 15069106E+04 13771036E+84 79157652E+03 93964281E+03 10447093E+04  and e n e r g y r e l e a s e  are  stored  Fri -.27786562E+03 -.27840581E+03 .27797179E+03 -.27879584E+03 -.27784531E+83 .27854043E+83 .23148217E+83 -.27831377E+03 -.27S18558E+03 -.27875189E+03  in  file:  SYE11 * * * * * * *  

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