FRACOD Manual V11
Contents
1. 68 FRACOD User s Manual APENDIX IIT DETERMINATION OF THE DIRECTION OF SHEAR FRACTURES When a fracture propagates in shear at an existing fracture tip there are always two conjugate candidate propagation directions in which the shear stress reaches the maximum The same mechanism causes the phenomenon that two conjugate failure planes often occur during triaxial compression tests of rock samples in laboratory In most cases however shear failure takes place only in one direction because the friction resistant is much higher in the conjugate direction due to higher normal stress Figure Al shows such a situation where the candidate direction 1 is the sole fracture propagation direction due to its lower friction resistant than the direction 2 O l On Candidate shear failure ake direction 1 Existing fracture irection 1 Candidate shear failure direction 2 Figure A1 Two candidate directions of shear failure Direction 1 is the favourable one due to its low shear resistant In other cases shear failure could in theory occur in both directions if the shear resistant happens to be same or similar in the two directions of the maximum shear stress This is particularly true when the shear failure initiates from an open fracture tip see Figure A2 In Figure A2 the shear fracture can in theory propagate in both directions 1 and 2 since both directions has the same shear stress and2. The fractures are found to propagate toward each other in mode II and finally form borehole breakouts 2 Input data TITLE Fracture propagation to form borehole breakouts SYMMETRY Model symmetry 4 0 00 0 00 MODULUS Poisson s Ratio and Young s modulus 0 25 0 40E 11 TOUGHNESS Gic and Giic 50 1000 PROPERIES mat kn ks phi coh 1 0 10E 13 0 10E 13 30 0 0 00E 00 SWINDOW x1l xur yll yur numx numy 2 00 2 00 2 00 2 00 30 30 STRESSES sxx syy sxy 10 0E 06 50 00E 06 0 00E 00 FRACTURE nume xbeg ybeg xend yend kode mat 3 0 700 0 700 0 90 0 60 2 1 ARCH nume xcen ycen diam ang1 ang2 kode ss sn 15 00 0 0 2 0 0 0 90 0 1 0 00E 00 10 E 06 CYCL 1000 ENDFILE 63 FRACOD User s Manual 3 FRACOD model A Two Dimensional Fracture Propagation Code Pio e Pk aay a PALE iG Pxx 0 100E 08 Lyy 0 500E 08 Pxy 0 000E 00 elastic open eS LD 24 Two Dimensional Fracture Propagation Code Pxx 0 100E 08 Pyy 0 500E 08 Pxy 0 000E 00 elastic open Seel Ba 64 FRACOD User s Manual Example 11 Complex fracture propagation to form borehole breakouts 1 Problem definition A circular borehole with hoop cracks in the borehole wall in an infinite rock mass is subjected to compressive stress of 60MPa and confining stress of 30MPa The borehole is also free from any internal hydraulic pressure The cracks are parallel to the borehole w
3. 3 FRACOD model sete Two Dimensional Fracture Propagation Code Pxx 0 000E 00 Pyy 0 150E 08 Pxy 0 000 00 elastic o open Sp Ea Two Dimensional Fracture Propagation Code Pxx 0 000E 00 Pyy 0 150E 08 Pxy 0 000E 00 elastic open un 60 FRACOD User s Manual Example 9 Fracture propagation from a tunnel under internal hydraulic pressure 1 Problem definition A circular tunnel with four fractures in its wall in an infinite rock mass is subjected to internal hydraulic pressure of 50MPa The in situ stresses far field stresses in the rock mass are Pxx Pyy 10MPa The inclination of the fractures is 45 The material properties of the rock mass and fractures are the same as in Examples 1 and 3 The fractures are found to propagate in mode I in radial direction 2 Input data TITLE A tunnel with four fractures subjected to internal hydraulic pressure SYMMETRY Model symmetry 4 0 00 0 00 MODULUS Poisson s Ratio and Young s modulus 0 25 0 40E 11 TOUGHNESS Gic and Giic 50 1000 PROPERIES mat kn ks phi coh 1 0 10E 13 0 10E 13 30 0 0 00E 00 SWINDOW x1l xur yll yur numx numy 4 00 4 00 400 4 00 30 30 STRESSES sxx syy sxy 10 0E 06 10 00E 06 0 00E 00 FRACTURE nume xbeg ybeg xend yend kode mat 3 0 700 0 700 1 000 1 000 2 1 ARCH nume xcen ycen diam ang1 ang2 kode ss sn 10 0 0 0 0 2 0 0 0 90 0 1 0 00E 00 50 E 06 CYCL 1000 ENDFILE 61
4. TWO DIMENSIONAL FRACTURE PROPAGATION CODE VERSION 1 1 USER S MANUAL Baotang Shen FRACOM Ltd info fracom com fi FRACOD User s Manual ABSTRACT A two dimensional boundary element code has been developed to simulate fracture initiation and propagation in an elastic and isotropic rock medium The current version of the code is fully Window based and user friendly The code can simulate up to 10 15 non symmetrical and randomly distributed fractures This manual provides a the basic theoretical background of the FRACOD code and b a detailed instruction on how to use the code A number of simple examples are provided at the end of the report to demonstrate the applicability of the code The current version of the FRACOD code was developed based on a Ph D research Shen 1993 Further work on the code was conducted during 1998 2001 supported by SKB Fracom Ltd and Tekes It was designed to simulate fracture propagation in hard rocks FRACOD User s Manual TABLE OF CONTENTS ABSTRACT 1 INTRODUCTION 2 THEORETICAL BACKGROUND Sch DISPLACEMENT DISCONTINUITY METHOD DDM 2 1 1 DISPLACEMENT DISCONTINUITY METHOD IN AN INFINITE SOLID 210 NUMERICAL PROCEDURE 22 SIMULATION OF ROCK DISCONTINUITIES 23 FRACTURE PROPAGATION CRITERION 2 4 DETERMINATION OF FRACTURE PROPAGATION USING DDM 2 9 FRACTURE INITIATION CRITERION CODE STRUCTURE 4 PREPARE INPUT FILE 5 CONDUCT AND MONITOR THE CALCULATION REFERENCES ACKNOWLEDGEMENTS
5. 0 15E 08 0 00E 00 FRACTURE nume xbeg ybeg xend yend kode mat 5 0 700 0 700 1 000 1 000 2 1 ARCH nume xcen ycen diam angl ang2 kode ss sn 10 0 0 0 0 Zi 0 0 0 90 0 1 0 00E 00 0 00E 00 CYCL 1000 ENDFILE STOP The input data are defined by a command line such as TITLE The command line will if needed be followed by a line which defines the values Only the first four characters of a command e g TITL are to be read by the code and hence meaningful However it is always desirable to write the whole word e g TITLE to help in understanding the function of this command All commands can be written in pure capital characters or pure small characters or their mixture such as STOP stop or Stop Unacceptable commands cause no action in the code no warning or error messages will be given The commands used by the FRACOD code are listed below Note that the units used for the input are given in brackets TITL SYMM MODU TOUG PROP ROCK SWIN FRACOD User s Manual give a title to the problem words within 80 letters give symmetry conditions ksym xsym ysym ksym 0 no symmetry ksym 1 problem symmetrical against vertical line x xsym m ksym 2 problem symmetrical against horizontal line y ysym m ksym 3 problem symmetrical against point x xsym and y ysym m ksym 4 problem symmetrical against
6. Kemeny J M and Cook N G W 1991 Micromechanics of deformation in rocks In Toughening Mechanisms in Quasi Brittle Materials S P Shaw ed Klewer Academic The Netherland 155 188 35 FRACOD User s Manual Kemeny J M 1991 A model for non linear rock deformation under compression due to subcritical crack growth Int J Rock Mech Min Sci 28 459 467 Lajtai E Z 1969 Shear strength of weakness planes in rock nt J Rock Mech Min Sci amp Geomech Abs 6 299 515 Lajtai E 1974 Brittle fracture in compression Int J Fracture 10 4 525 536 Li V C 1991 Mechanics of shear rupture applied to earthquake zones In Fracture mechanics of rock Atkinson K B ed Academic Press London 351 428 Lockner D Moore D and Reches Z 1992 Microcrack interaction leading to shear fracture Proc 33rd U S Symp Rock Mech 807 816 Melin S 1985 The infinitesimal kink Report LUTFD2 TFHF 3022 1 19 1985 Division of Solid Mechanics Lund Institute of Technology Lund Petit J P and Barquins M 1988 Can natural faults propagate under mode II conditions Tectonics 7 6 1243 1256 Reyes O and Einstein H H 1991 Failure mechanism of fractured rock A fracture coalescence model Proc 7th Int Con on Rock Mechanics 1 333 340 Rao Q 1999 Pure shear fracture of brittle rock A theoretical and laboratory study PhD Thesis 1999 08 Lulea University of Technology Savilahti T Nor
7. 07 0 50E 08 0 00E 00 FRACTURE nume xbeg ybeg xend yend kode mat 15 1 000 1 000 1 000 1 000 2 1 CYCL 1000 ENDFILE 53 FRACOD User s Manual 3 FRACOD model 5228 Two Dimensional Fracture Propagation Code A L L X ile Plot Add cadre Color Cycle Save P I Prx 0 1505407 Pyy 0 500E 08 Pxy 0 000E 00 elastic open slip 1 Ss Two Dimensional Fracture Propagation Code Bisi x ile Plot AddStress Plot Windo lor Cocke Save P D F C Pxx 0 750E 07 Pyy 0 500E 08 Pxy 0 000E 00 Ls elastic open gt Sp 54 FRACOD User s Manual Example 6 Two inclined fractures subjected to uniaxial compression 1 Problem definition Two inclined parallel fractures in an infinite rock mass are subjected to uniaxial compressive stress of 50MPa The inclinations of the fractures are 45 The material properties of the rock mass and fractures are the same as in Examples 1 and 3 The fractures are found to propagate and coalesce in mode I 2 Input data TITLE Two inclined fractures subjected to uniaxial compression SYMMETRY Model symmetry 3 0 00 0 00 MODULUS Poisson s Ratio and Young s modulus 0 25 0 40E 11 TOUGHNESS Gic and Giic 50 1000 PROPERIES mat kn ks phi coh 1 0 10E 13 0 10E 13 30 0 0 00E 00 SWINDOW x1l xur yll yur numx numy 5 00 5 00 5 00 5 00 30 30 STRESSES sxx syy sxy 0 0E 07 0 50E 08 0 00E 00 FRACTURE nume xbeg ybeg xe
8. FRACOD User s Manual 3 FRACOD model Two Dimensional Fracture Propag ation Code Bigg Pxx 0 100E 08 Pyy 0 100 Y08 Pxy 0 000E 00 maximum displacement 1 430 ES PETT EA A meee A A A A oo ae ecg eee de We AAA Ee EE A if Mts hore E acini aller he Paw e So Ee 4 DH 4 M i H EE GEET LN ARN NR G Pace eee OH ee eo ey y OS Cy NONE te gu fe ict Ob ee ye EE my Ae a ee E E ira A A ar ae Le Git Oe In Outta Gai sant fe OS Hiss VS ah e aD Stan eta io gt Me Br Pde EE EE REES PE e er e BN LAA A compressive tensile Pxx 0 100 E 08 Pyy 0 100 08 Par 0 000 00 maximum displacement 1 680 Let Zeg S DE NS 4 gt het sista drat Cur A O AI 4 Pe ae Alen en DE DE Ee d e die diy day uy amr Cees Se A Te Gy ROA AH at iy oe ate ge Fee tee ee oe ns Cit eee D CRL compressive tensile 62 FRACOD User s Manual Example 10 Simple fracture propagation to form borehole breakouts 1 Problem definition A circular borehole with four fractures in its wall in an infinite rock mass is subjected to compressive stress of 50MPa and confining stress of 10MPa The borehole is also subjected to an internal hydraulic pressure of 10MPa The inclination of the fractures is 45 The material properties of the rock mass and fractures are the same as in Examples 1 and 3
9. Step 5 Output During the cyclic calculation the geometry of propagating fractures and other boundaries are shown on the screen so that the user can monitor the calculation process When the calculation is completed or is interrupted by the user stresses and displacements in addition to the fracture geometry can be plotted on screen using the available options provided in the FRACOD program window The screen plots can also be printed to printer or captured as image files which can then be directly pasted to other applications such as MS Word for reporting FRACOD User s Manual PREPARE INPUT FILE The FRACOD code reads the input data from a data file previously prepared with specified formats Therefore the user needs to construct the input file e g input dat before running the code This section gives a detailed instruction on how to prepare the input file for the FRACOD code The following is an example data file which defines a borehole with cracks in the borehole wall under uniaxial compression example data file TITLE A borehole with four cracks loaded in uniaxial compression SYMMETRY Model symmetry 4 0 00 0 00 MODULUS Poisson Ratio and Youngs modulus 0 25 0 40E 11 TOUGHNESS Gic and Giic 50 1000 PROPERIES mat kn ks phi coh T 0 10E 13 0 10E 13 30 0 0 00E 00 SWINDOW xll xur yll yur numx numy 3 00 3 00 3 00 3 00 30 30 STRESSES SxXxX SYyy SxXy 0 00E 07
10. User s Manual 3 FRACOD model weed Two Dimensional Fracture Propagation Code Pxx 0 150E 08 Pyy R 750E 08 Pxy 0 000E 00 A e elastic open ie Two Dimensional Fracture Propagation Code Pxx 0 150E 08 eae Pxy 0 000E 00 elastic open EE 58 FRACOD User s Manual Example 8 Fracture propagation from a tunnel in uniaxial compression 1 Problem definition A circular tunnel with four fractures in its wall in an infinite rock mass is subjected to uniaxial compressive stress of 15MPa The inclination of the fractures is 45 The material properties of the rock mass and fractures are the same as in Examples 1 and 3 The fractures are found to propagate in mode l in the direction nearly parallel to far field compressive stress 2 Input data TITLE A tunnel with four fractures subjected to uniaxial compression SYMMETRY Model symmetry 4 0 00 0 00 MODULUS Poisson s Ratio and Young s modulus 0 25 0 40E 11 TOUGHNESS Gic and Giic 50 1000 PROPERIES mat kn ks phi coh 1 0 10E 13 0 10E 13 30 0 0 00E 00 SWINDOW xll xur yll yur numx numy 4 00 4 00 4 00 4 00 30 30 STRESSES sxx syy sxy 0 00E 07 0 15E 08 0 00E 00 FRACTURE nume xbeg ybeg xend yend kode mat 5 0 700 0 700 1 000 1 000 2 1 ARCH nume xcen ycen diam ang1 ang2 kode ss sn 10 00 0 0 2 0 0 0 90 0 1 0 00E 00 0 00E 00 CYCL 1000 ENDFILE 59 FRACOD User s Manual
11. e elastic contact sliding or open The default fracture colour is Blue elastic fracture Green sliding fracture Red open fracture The colour can be changed by users in view plots setup The magnitudes of far field stresses are shown in the Legend plot window Here Pxx normal stress in the horizontal direction of the model Pyy normal stress in the vertical direction of the model Pxy shear stress The magnitudes of stresses are in Pa unless specified otherwise 26 FRACOD User s Manual Edit Copy to Clipboard BMP Copy to ClipBoard EMF Copy the screen plot to the clipboard in the BitMap Format BMP or in Window Enhance Meta Files format EMF The plot can be pasted to other Window applications e g MS Word 27 FRACOD User s Manual View Model Geometry Principal Stress Max Shear Stress Displacement Plot the stresses or displacements from a paused simulation on screen An ongoing simulation can be paused see Pause and the geometry stresses and displacements can be plotted on screen Geometry Plot on screen the geometry of the fractures and other boundaries such as tunnels in the model The geometry plot is set as default and is automatically shown on the screen during calculation Principal Stress Plot on screen the principal stresses in the rock mass within a defined window Two orthogonal principal stresses the major principal stress and the minor principal stress w
12. line x xsym and line y ysym m give elastic properties modulus of the rock medium v Poisson s ratio E Young s modulus Pa give critical energy release rates Gr and Gre Gio Gite Gi mode I fracture critical strain energy release rate J m Gr 1 V Kr E Ki Fracture toughness mode I Gi mode II fracture critical strain energy release rate J m Gue 1 V Kno E Kite Fracture toughness mode II glve fracture surface contact properties jmat ks kn phi coh jmat joint property ID 1 2 3 ks fracture shear stiffness Pa m kn fracture normal stiffness Pa m phi fracture friction angle degree coh fracture cohesion Pa give intact rock strength parameters for fracture initiation rphi rcoh sigt rphi Intact rock internal friction angle degrees rcoh Intact rock cohesion Pa sigt Intact rock tensile strength Pa define a window for plotting the geometry stresses and displacements xll xur yll yur numx numy 20 IWIN FRACOD User s Manual xll left border of the window m xur right border of the window m yll bottom border of the window m yur top border of the window m numx number of grid points along x direction numy number of grid points along y direction define an area window for detecting fracture initiation used only when once particular problem area is of interests STRE FRAC ARCH xmin xmax ymin ymax xm
13. s Manual Option Far field stress Boundary stress Change the magnitude of the far field stresses or boundary stresses Far field stress Increase or decrease the magnitude of far field stresses The value of increment or reduction is requested Note that the compressive stress is negative so that an increment in compressive stress should be given as a negative values This command is particularly useful in studying the change of the fracture growth path when the far field stresses are changed Boundary stress Increase or decrease the magnitude of boundary stresses or displacement if the boundary condition is specified by displacement The value of increment or reduction of shear or normal stress is requested Note that the compressive stress is negative so that an increment in compressive stress should be given as a negative value This command is particularly useful in studying the change of the fracture growth path when the boundary stresses e g hydraulic pressure in a borehole are changed 31 FRACOD User s Manual Tools Model design A pre processor which helps the user to set up the numerical model Details of the pre processor are given in Appendix I 32 FRACOD User s Manual Windows Standard Window management functions which enable users to arrange the multiple calculation Windows 33 FRACOD User s Manual Help On line user s manual and helping functions 34 FRACOD User s Man
14. screen the filled contours of stresses or displacement Legend Show the legend on the plot window Included in the Legend are e Far field stress Sxx Syy Sxy e Maximum values of the stresses or displacement appeared on the screen plot e Colour conventions View Zoom in Zoom out Full plot Zoom in Enlarge the plot in a specified window defined by dragging the Mouse Zoom out Reduce the plot the plot in a specified window defined by dragging the Mouse Full screen Return the plot size to the originally specified window full screen View Magnifier Plot setup Magnifier Magnify an area of the screen To do so locate the mouse cursor to the desired position and press down the mouse right button Plot setup Specify or change the plot setup including e Line colour e Line thickness e Plot range e Problem title e Axis setting 29 FRACOD User s Manual Run Run Pause Stop Run Start or continue a calculation A cycle number is requested One cycle often produces a fracture propagation of one element length Pause Pause the current calculation A paused calculation can be reactivated and continued by using Run Stop Stop the calculation This command triggers the termination of the current calculation A stopped calculation cannot be restarted Some calculation results stresses and displacement at the previously specified grid points can however still be shown 30 FRACOD User
15. 0 1 0 21 CYCL 1000 ENDFILE 47 FRACOD User s Manual 3 FRACOD model ACO Two Dimension al Det acture Propagation Code LoadFile Plot dStress indow Color Cycle Save Pause Quit Pxx 0 000E 00 Pyy 0 000E 00 Pxy 0 500E 08 Gie 1280 Um Gire 1000 Um gt ET R elastic open slip axa Two Dimensional Fracture Propagation Code ane Plot AddStress Plotw indow Color Cycle Save Pause Quit Pxx 0 000E 00 Pyy 0 000E 00 Pxy 0 500 08 Gie 1330 J m Gire 1000 Um le a elastic open slip 48 FRACOD User s Manual Example 3 Single inclined fracture subjected to uniaxial compresion 1 Problem definition An inclined fracture in an infinite rock mass is under uniaxial compressive stress of 5 0MPa The inclination of the fracture is 45 The elastic properties of the rock mass and the fracture critical strain energy release rates are the same as in Example 1 The contact properties of the fracture surfaces are K 1000GPa m K 1000GPa m b 30 c 0 The fracture is found to propagate in mode I in the direction nearly parallel to the far field stress 2 Input data TITLE Single inclined fracture subjected to uniaxial compression SYMMETRY Model symmetry 0 0 00 0 00 MODULUS Poisson s Ratio and Young s modulus 0 25 0 40E 11 TOUGHNESS Gic and Giic 50 1000 PROPERIES mat kn ks phi coh 1 0 10
16. 0 189784703 1 134231796 0 1102429 1 144703675 5 1 CYCL 10000 ENDFILE 65 3 FRACOD User s Manual FRACOD model Complex fracture propagation to form borehole breakouts A E E A ER E3 S 3 3 2 1 0 1 2 3 4 PE ES S 3 2 1 0 1 2 3 4 66 FRACOD User s Manual Example 12 Fracture initiation and propagation to form borehole breakouts A circular borehole in an infinite rock mass is subjected to compressive stress and confining stress The borehole is also subjected to internal hydraulic pressure No cracks exist in the borehole walls The material properties of the rock mass and fractures are given below Kic 1 08 MPa m Kuc Kic Tensile strength 7MPa Compressive strength o 28MPa Failure criterion 01 200 03 3 95 E 55GPa v 0 25 Insitu stresses oy 36MPa o 45MPa fluid pressure op 18MPa Fracture contact properties Ks Kn 10e4GPa m friction angle 0 degrees cohesion 0 Borehole Diameter 0 216m Fracture initiation is predicted inside the borehole wall The new fractures propagate and coalesce while new fractures continue to initiates The process continues and finally forms borehole breakouts see the plots below 67 FRACOD User s Manual S x x Es J Ki Ni A e Ee a bN j a ef Ee es SZ Ki id L 2 S Dy x Mi ga x A Kd WA Wel Da EN Ly Zi d d ees a if Bes ee
17. 0 5 00 30 30 STRESSES sxx syy sxy 0 10E 08 0 50E 08 0 00E 00 FRACTURE nume xbeg ybeg xend yend kode mat 15 1 000 1 000 1 000 1 000 2 1 CYCL 1000 ENDFILE 51 FRACOD User s Manual 3 FRACOD model SACH Two Dimensional Fracture Propagation Code MA x Plol d Color C gt c Pxx 0 100E 08 Pyy 0 500E 08 Pxy 0 000E 00 le elastic open E ISS Two Dimensional Fracture Propagation Code L OL x Plo ldStress Plotwindow Cole F 1 Pxx 0 100E 08 Pyy 0 500E 08 Pxy 0 000E 00 elastic o open ai 52 FRACOD User s Manual Example 5 Single inclined fracture subjected to biaxial compression 1 Problem definition An inclined fracture in an infinite rock mass is subjected to compressive stress of SOMPa and confining stress of 7 5MPa The inclination of the fracture is 45 The material properties of the rock mass and fractures are the same as in Examples 1 and 3 The fracture is found to propagate in mixed mode I and II in the overall direction about 70 to the confining stress 2 Input data TITLE Single inclined fracture subjected to biaxial compression SYMMETRY Model symmetry 0 0 00 0 00 MODULUS Poisson s Ratio and Young s modulus 0 25 0 40E 11 TOUGHNESS Gic and Giic 50 1000 PROPERIES mat kn ks phi coh 1 0 10E 14 0 10E 14 30 0 0 00E 00 SWINDOW x1l xur yll yur numx numy 5 00 5 00 5 00 5 00 20 20 STRESSES sxx syy sxy 0 75E
18. 97 Simulation of borehole breakout using fracture mechanics models In Rock Stress Sugawara amp Obara eds Balkema Rotterdam 289 298 Shen B and Rinne M 2001 Generalised criteria for fracture initiation at boundaries or crack tips Report prepared for SKB Sih G C 1974 Strain energy density factor applied to mixed mode crack problems nt J Fracture 10 3 305 321 Wang R Zhao Y Chen Y Yan H Yin Y Yao C and Zhang H 1987 Experimental and finite element simulation of X type shear fractures from a crack in marble Tectonophysics 144 141 150 Wong T F 1982 Micromechanics of faulting in Westerly granite nt J Rock Mech Min Sci 19 49 64 37 FRACOD User s Manual ACKNOWLEDGEMENTS This work is financially supported by Fracom Ltd Tekes and SKB The current FRACOD code is based on the original version developed under the supervision of Professor Ove Stephansson of KTH The author wishes to thank Professor Stephansson for his invaluable input in the code particularly in the early stages of development The author would also like to thank Mr Mikael Rinne for his collaboration in the code development and his careful checking of the code Dr Kennert R shoff for his initiation and coordination of the study Valuable comments from Mr Christer Svemar Mr Rolf Christiansson and Prof John Cosgrove are gratefully acknowledged 38 FRACOD User s Manual APPENDIX I HOW TO USE THE PREPROCESSOR TO
19. APPENDIX I HOW TO USE THE PREPROCESSOR TO SET UP MODELS APPENDIX II VERIFICATION AND APPLICA TION OF FRACOD APENDIX III DETERMINATION OF THE DIRECTION OF SHEAR FRACTURES pd GO Ch GA AA N 11 16 19 24 35 38 43 69 FRACOD User s Manual INTRODUCTION Fracture propagation code FRACOD is a two dimensional computer code that was designed to simulate fracture initiation and propagation in elastic and isotropic rock mediums The code employs the Boundary Element Method BEM principles and a newly proposed fracture propagation criterion for detecting the possibility and the path of a fracture propagation Shen and Stephansson 1993 The current version of the FRACOD code provides the basic functions needed for studying rock fracture propagation in a rock mass subjected to far field stresses The code is created for running on PCs with a MS Windows 95 98 NT 2000 platform It provides an easy to use user s interface that enables users to monitor and interrupt the calculation It also provides an independent pre processor to help users in preparing the input file for a given problem The capacity of the current version of the FRACOD code is limited to about 10 15 fractures depending upon the complexity of the fracture system and the excavation As a general estimate a fracture system with 10 non symmetrical fractures will requires about 24 hours of calculation on a PC 400MHz to get a reasonably accurate pr
20. E 13 0 10E 13 30 0 0 00E 00 SWINDOW x1l xur yll yur numx numy 5 00 5 00 5 00 5 00 30 30 STRESSES sxx syy sxy 0 0E 07 0 50E 08 0 00E 00 FRACTURE nume xbeg ybeg xend yend kode mat 15 1 000 1 000 1 000 1 000 2 1 CYCL 1000 ENDFILE 49 FRACOD User s Manual 3 FRACOD model baang Two Dimensional Fracture Propagation Code Pxx 0 000E 00 Pyy 0 500E 08 Pxy 0 000E 00 le elastic open Bible ien Two Dimensional Fracture Propagation Code Biel ES Pxx 0 000E 00 Pyy 0 500E 08 Pxy 0 000 00 R elastic open Els 50 FRACOD User s Manual Example 4 Single inclined fracture subjected to biaxial compression 1 Problem definition An inclined fracture in an infinite rock mass is subjected to compressive loading stress of 50MPa and confining stress of 10MPa The inclination of the fracture is 45 The material properties of the rock mass and fractures are the same as in Examples 1 and 3 The fracture is found to propagate mainly in mode Il in the direction about 60 to the confining stress 2 Input data TITLE Single inclined fracture subjected to biaxial compression SYMMETRY Model symmetry 0 0 00 0 00 MODULUS Poisson s Ratio and Young s modulus 0 25 0 40E 11 TOUGHNESS Gic and Giic 50 1000 PROPERIES mat kn ks phi coh 1 0 10E 13 0 10E 13 30 0 0 00E 00 SWINDOW x1l xur yll yur numx numy 5 00 5 00 5 0
21. SET UP MODELS FRACOD provides a pre processor to help users in setting up the numerical model The pre processor is a Window based interface which enables users to see instantly the geometry of the fractures and boundaries they have defined It also provides pop up windows to guide the input whenever values are needed After all the fractures and parameters are defined for a problem a FRACOD input data file can then be created in a format that FRACOD can read and it is equivalent to the data file created manually by using a text editor The pre processor can be activated by clicking Model Design in the display Window 1 e the default Window when FRACOD is activated A second Window i e the Model Design Window will then pop up The following key functions are included in the Model Design Window File Load SaveAs Print Load Open an existing FRACOD input data file The model geometry and mechanical properties defined by the file will be loaded into the memory and can be shown on the screen They can also be modified by the user SaveAs Save the defined model into a FRACOD input data file Print Print the current model geometry Edit Copy to Clipborad BMP Copy to Clipborad EMF Copy to Clipborad BMP Copy the current model geometry to Clipborad in BitMap format It can later be pasted to other Window applications such as MS Word 39 FRACOD User s Manual Copy to Clipborad EMF Copy the current model geomet
22. T ENDF FRACOD User s Manual Stop the calculation End of the input data file To help preparing the input file an input pre processor Model Design has been developed for the user s convenience The interface is fully Window based and is coupled with graphics to help in defining a model easily An instruction of how to use Model Design is described in Appendix I 23 FRACOD User s Manual CONDUCT AND MONITOR THE CALCU LATION The FRACOD code is created as an executable file Fracod2D exe To start the code FRACOD simply double click the executable file in Windows a dialog menu will appear on your screen You then need to open an input data file or a FRACOD save file by using the open file options If you are starting a new problem you need to load a input data file which has already been prepared either by using a text editor or by using the model design function provided in the code see Appendix I If you are restarting a problem which has previously been run and saved you then need to load the saved file sav FRACOD will automatically detect whether the file you are loading is an input data file or a save file Once a calculation has started it will continue until it is interrupted manually or the defined cycles finished or no more fracture propagation is found During the calculation the instant geometry of the modelled fracture network is always shown on the screen so that any growth of the fractures can be mon
23. alls The material properties of the rock mass and fractures are the same as in Examples 1 and 3 The cracks are found to propagate and coalesce in a complex pattern Both mode I and mode II failures are involved The fracture propagation finally forms breakouts which have a failure pattern similar to that observed in the laboratory tests and field observations 2 Input data TITLE Complex fracture propagation to form borehole breakouts SYMMETRY Model symmetry 4 0 00 0 00 MODULUS Poisson s Ratio and Young s modulus 0 25 0 40E 11 TOUGHNESS Gic and Giic 50 1000 PROPERIES mat kn ks phi coh 1 0 10E 13 0 10E 13 30 0 0 00E 00 SWINDOW x1l xur yll yur numx numy 2 00 2 00 2 00 2 00 30 30 STRESSES sxx syy sxy 10 0E 06 50 00E 06 0 00E 00 STRESSES sxx syy sxy 30E 08 0 60E 08 0 00E 00 ARCH nume xcen ycen diam ang1 ang2 kode ss sn 12 0 0 0 0 2 0 0 0 90 0 1 0 00E 00 0 00E 06 FRACTURE nume xbeg ybeg xend yend kode mat 2 1 144703678 0 11024287 1 134231801 0 189784673 5 1 FRACTURE nume xbeg ybeg xend yend kode mat 2 1 077165893 0 402757543 1 046463904 0 476878704 5 1 FRACTURE nume xbeg ybeg xend yend kode mat 2 0 936221036 0 667824956 0 887381223 0 73147424 5 1 FRACTURE nume xbeg ybeg xend yend kode mat 2 0 731474264 0 887381204 0 667824981 0 936221018 5 1 FRACTURE nume xbeg ybeg xend yend kode mat 2 0 476878732 1 046463891 0 402757572 1 077165882 5 1 FRACTURE nume xbeg ybeg xend yend kode mat 2
24. applied in the model Intact rock properties for fracture initiation 40 Shapes FRACOD User s Manual This is an interactive function allowing users to define the model geometry such as boundaries and fractures It includes the following options Arc Disc in Shapes Arc Arc Disc Define a disc or part of a disc The geometrical properties as well as the boundary conditions can be defined and altered in View Object Properties The disc can also be repositioned and resized by selecting and dragging the object using mouse A disc is defined by giving the coordinate of the centre point the diameter and the start and end angles default 180 to 180 The start and end angles have to be defined in clockwise Arc Hole in Shapes Arc Arc Hole Define a hole tunnel in a rock mass The geometrical properties as well as the boundary conditions can be defined and altered in View Object Properties The hole can also be repositioned and resized by selecting and dragging the object using mouse A hole is defined by giving the coordinate of the centre point the diameter and the start and end angles default 180 to 180 The start and end angles have to be defined in anti clockwise Edge in Shapes Line Edge Define an edge i e a straight boundary in a rock mass The geometrical properties as well as the boundary conditions can be defined and altered in View Object Properties The edge can also be repositioned and resiz
25. ch the F value reaches its maximum F 9 a max 217 3 When the maximum F value reaches 1 0 the fracture tip will propagate Le 2 4 FRACOD User s Manual F O 9 0 1 0 2 18 The F criterion is actually a more general form of the G criterion and it allows us to consider mode I and mode II propagation simultaneously In most cases the F value reaches its peak either in the direction of maximum tension Gr maximum while Gy 0 or in the direction of maximum shearing Gy maximum while G 0 This means that a fracture propagation of a finite length the length of an element for instance is either pure mode I or pure mode II However the fracture growth may socialite between mode I and mode II during an ongoing process of propagation and hence form a path which exhibits the mixed mode failure in general DETERMINATION OF FRACTURE PROPAGATION USING DDM The key step in using the F criterion is to determine the strain energy release rate of mode I Gz and mode Il Gr at a given fracture tip As Gr and Gu are only the special cases of G the problem is then how to use DDM to calculate the strain energy release rate G The G value by definition is the change of the strain energy in a linear elastic body when the crack has grown one unit of length Therefore to obtain the G value the strain energy must first be estimated By definition the strain energy W in a linear elastic body is l 2 1 W Mes g f 19 whe
26. ct rock The localised failure can be predicted by an existing failure criterion e g Mohr Coulomb criterion Other criteria widely used in rock mechanics and rock engineering can also be used such as Hoek Brown criterion etc A rock failure can be caused by tension or shear Hence a fracture initiation can be formed due to tension or shear For tensile fracture initiation the tensile failure criterion is used in FRACOD i e when the tensile stress at a given point of the intact rock exceeds the tensile strength of the intact rock a new rock fracture will be generated in the direction perpendicular to the tensile stress Figure 2 5 Critical stress of fracture initiation in tension Otensile 2 Ot Direction of fracture initiation in tension Oit O Gtensite 1 2 where Otensile 18 the principal tensile stress at a given point o is the tensile strength of the intact rock Oi is the direction of the fracture initiation in tension and O Gtensile is the direction of the tensile stress The length of the newly generated fracture is determined by the spacing of the grid points used in the intact rock In the current FRACOD version it is equal to the grid point spacing in the initiation direction The less the grid point spacing the shorter the new fracture However the closer the grid points the less different the stresses at the adjacent grid points and hence the more likely a fracture initiation occurs in the adjacent grid points sim
27. d displacements using DDM is performed based on the assumption that all fractures are in surface elastic contact This may not be true as fracture surfaces could be open or sliding as well Therefore a correction process is needed This is done in a cyclic checking process After obtaining the shear and normal stresses on each fracture element from the DDM calculation using the elastic fracture assumption the code will check whether the open criterion or fracture sliding criterion is met for each fracture element If yes the new fracture states as determined will be assigned to the fracture elements and a new cycle of DDM calculation will be performed using the updated states of fracture elements The calculated shear and normal stresses on the fracture elements will again be used to check whether any change of state of fracture contact is detected If yes a new cycle of DDM calculation will be performed again This cycling process will continue until no more change in fracture contact state is found and the results then are regarded to be the true results Step 4 Determination of fracture initiation and propagation After obtaining the stresses and displacements on fractures and boundaries the elastic strain energy W a is calculated By adding a small fictitious element Aa normally the same length of the tip element at a fracture tip say tip A in the direction of a new strain energy W a Aa can then be obtained Using the formulas
28. dlund E and Stephansson O 1990 Shear box testing and modelling of joint bridges In Rock Joints Barton amp Stephansson eds Proc Int Symp Rock Joints Norway 295 300 Schultz R 1988 Stress intensity factors for curved cracks obtained with the displacement discontinuity method Int J Fracure 37 R31 34 Segall P and Pollar D 1980 Mechanics of discontinuous faults J Geophy Res 85 B8 4337 4350 Segall P and Pollard D 1983 Nucleation and growth of strike slip faults in granite J Geophy Res 88 B1 555 568 Shen B and Stephansson O 1992 Deformation and propagation of finite joints in rock masses Jn Myer et al eds Fractured and Jointed Rock Masses 303 309 36 FRACOD User s Manual Shen B and Stephansson O 1993 Numerical analysis of Mode I and Mode II propagation of rock fractures Int J Rock Mech Min Sci amp Geomech Abst 30 7 861 867 Shen B and Stephansson O 1993 Modification of the G criterion of crack propagation in compression nt J of Engineering Fracture Mechanics 47 2 177 189 Shen B Stephansson O Einstein H H and Ghahreman B 1995 Coalescence of fractures under shear stresses in experiments J Geophys Res 100 B4 5975 5990 Shen B 1995 The mechanism of fracture coalescence in compression experimental study and numerical simulation nt J of Engineering Fracture Mechanics 51 1 73 85 Shen B Tan X Li C and Stephansson O 19
29. dure as described in Sections 2 1 2 4 is then used FRACOD User s Manual CODE STRUCTURE The FRACOD code performs the fracture propagation calculation in a process shown by the program flow chart in Figure 3 1 Details of the key steps in the Figure 3 1 are described below Input Step l Fractures and boundaries far field stresses properties etc Calculate stresses and displacements on fracture surface and boundaries Step 2 Determine the state of fracture contact elastic contact open slipping Fracture state changed Step 3 Determine fracture initiation More propagation Step 4 and propagation Output Step 5 fracture propagation path stresses and displacements Figure 3 1 Flow chart of the FRACOD code Step 1 Input The code will read the input data from a pre defined data file The input data includes the geometry of pre existing fractures and boundaries far field stresses elastic properties of the rock mass fracture toughnesses etc A comprehensive description of the input data and their format are given in Chapter 4 FRACOD User s Manual Step 2 Calculation of stresses and displacements This is a standard routine using the DDM formulas described in Section 2 1 to calculate the stresses and displacements on fracture surfaces boundaries and or any pre defined internal points in the rock mass Step 3 Determination of the state of fracture contact The initial calculation of stresses an
30. ed by selecting and dragging the object using mouse An edge is defined by giving the coordinates of the start and end points The start and end points have to be arranged in such a way that the negative side of the edge is the rock mass as shown below End point Positive side opening Negative side rock Start point Fracture in Shapes Line Fracture 41 FRACOD User s Manual Define a fracture in a rock mass The geometrical properties as well as the mechanical properties of the fracture can be defined and altered in View Object Properties The fracture can also be repositioned and resized by selecting and dragging the object using mouse A fracture is defined by giving the coordinates of the start and end points The definition of a fracture is not sensitive to the sequence of the start and end points 42 FRACOD User s Manual APPENDIX II VERIFICATION AND APPLICA TION OF FRACOD Eleven demonstration problems are listed here as part of the verification and application tests of the FRACOD code The data files are provided in the program package Example 1 Single fracture subjected to normal tensile stress 1 Problem definition A 2m fracture in an infinite rock mass is under uniaxial tensile stress of 50MPa in the direction perpendicular to the fracture plane The elastic properties of the rock mass are E 40GPa v 0 25 The strain energy release rate in mode I for this problem is ca
31. ediction of fracture propagation This user s manual provides some basic theoretical background of the code in Chapters 2 and 3 and a detailed instruction on how to use the code in Chapters 4 and 5 Appendix I describes a pre processor of the FRACOD code while Appendix II gives ten simple application examples of using the FRACOD code For those who may be only interested in knowing how to use the code rather than the theory it is recommended to ignore Chapters 2 and 3 and start reading from Chapter 4 2 1 2 1 1 FRACOD User s Manual THEORETICAL BACKGROUND The FRACOD code is based on the Boundary Element Method principals It utilises the Displacement Discontinuity Method DDM one of the three commonly used boundary element methods In the FRACOD code a newly proposed fracture criterion the modified G criterion Shen and Stephansson 1993 is incorporated into the numerical method for simulating fracture propagation This section describes in detail the numerical method DDM as well as the modified G criterion DISPLACEMENT DISCONTINUITY METHOD DDM A crack or fracture has two surfaces or boundaries one effectively coinciding with the other Conventional boundary element methods such as the Direct Integration Method therefore become inefficient in simulating this problem The Displacement Discontinuity Method DDM was developed by Crouch 1976 to cope with problems of this type The DDM is based on the analytical solutio
32. ement which are unknowns in the system of equations A rock discontinuity has three states open in elastic contact or sliding The system of governing equations 2 10 developed for an open crack can be FRACOD User s Manual easily extended to the case for cracks in contact and sliding For different crack states their system of governing equations can be rewritten in the following ways depending on the shear and normal stresses o and o of the crack i e For an open crack o o 0 therefore the system of governing n equations 2 10 can be rewritten as N S j de D II o j IM gt 17 i ltoN a II Ges II Mez Ss D Y A D ei j l lt Il e When the two crack surfaces are in elastic contact the magnitude of o and o ill depend on the crack stiffness Ks K and the displacement discontinuities D D E ta A II K 2 12 K a I ER where K and K are the crack shear and normal stiffness respectively Substituting Equation 2 12 into Equation 2 10 and carrying out the simple mathematical manipulation the system of governing equations then becomes N ij N i D 3 A B D 0 K D E Ge i to N 2 13 N j NG j i i 0 SAs D Am tech K D j l j l e Fora crack with its surfaces sliding Ee a p E o tang tK D tang 2 14 where is the friction angle of the crack surfaces The sign of o depends on the sliding direction Conseq
33. epresentation of a crack by N elemental displacement discontinuities The elemental displacement discontinuities are defined with respect to the local co ordinates s and n indicated in Figure 2 2 Figure 2 2b depicts a single elemental displacement discontinuity at jth segment of the crack The components of discontinuity in the s and n directions at this segment are J J donated as Ds and D These quantities are defined as follows j J J _ 4 7 D u TU 2 5 j j J D u u In these definitions A and A refer to the shear s and normal n displacement of the jth segment of the crack The superscripts and denote the positive and negative surfaces of the crack with respect to local co ordinate n The local displacements ls and it form the two components of a vector They are positive in the positive direction of s and n irrespective of whether we are considering the positive or negative surface of the crack As a consequence it follows from Equation 2 5 that the normal component of FRACOD User s Manual j displacement discontinuity D is positive if the two surfaces of the crack J displace toward one another Similarly the shear component D is positive if the positive surface of the crack moves to the left with respect to the negative surface The effects of a single elemental displacement discontinuity on the displacements and stresses at an arbitrary point in the infinite solid can be computed from the result
34. er than Gr Li 1991 Applied to the mixed mode I and mode II fracture propagation the G criterion is difficult to use since the critical value G must be carefully chosen between Gr and Gre A modified G criterion namely the F criterion was proposed Shen and Stephansson 1993 Using the F criterion the resultant strain energy release rate G at a fracture tip is divided into two parts one due to mode I deformation G7 and one due to mode II deformation Gy Then the sum of their normalized values is used to determine the failure load and its direction G and Gy can be expressed as follows Figure 2 3 if a fracture grows an unit length in an arbitrary direction and the new fracture opens without any surface shear dislocation the strain energy loss in the surrounding body due to the fracture growth is Gz Similarly if the new fracture has only a surface shear dislocation the strain energy loss is Gy The principles of the F criterion can be stated as follows G G s G oga Mu Co a b c Figure 2 3 Definition of G and Gy for fracture growth a G the growth has both open and shear displacement b G the growth has only open displacement c Gu the growth has only shear displacement 1 In an arbitrary direction 6 at a fracture tip there exists a F value which is calculated by GD Gg G G F 0 2 16 Ic IIc 2 The possible direction of propagation of the fracture tip is the direction 0 0 for whi
35. ill be plotted on screen as two orthogonally lines The directions of the lines are the directions of the two principal stresses and the length of each line is proportional to the stress magnitude Colours are used to distinguish the compressive stress with the tensile stress Blue compressive stress default Red tensile stress default The colour can be changed by user in View Plot setup The maximum magnitude of the principal stresses in the plot is given in the Legend window in Pa Max Shear Stress Plot on screen the maximum shear stress in the rock mass within a defined window The maximum shear stress in two orthogonal directions will be plotted on the screen as two orthogonal lines The directions of the lines are the directions of the maximum shear stress and the length of each line is proportional to the stress magnitude The maximum magnitude of the maximum shear stresses in the plot is given in the Legend window in Pa Displacement Plot on screen the displacement vector at specified grid points in the model Rock displacement at a grid point will be plotted on the screen as a vector with an arrow indicating the direction of the displacement and the length of the vector is proportional to the values of displacement The value of maximum displacement in the plot is given on the top of the plot window in metres 28 FRACOD User s Manual View Image Legend Image currently not functional Plot on
36. in left border of the window m xmax right border of the window m ymin bottom border of the window m ymax top border of the window m give far field stresses in the rock mass Pxx Pyy Pxy Pxx Far field horizontal stress Pa Pyy Far field vertical stress Pa Pxy Far field shear stress Pa Warning if Pxy is not 0 only ksym 0 or ksym 3 may be used define a fracture joint num xbeg ybeg xend yend kode jmat num number of elements along the fracture xbeg vbeg co ordinates of the beginning point of the fracture m xend vend co ordinates of the end point of the fracture m kode no function mat joint property ID defined before jmat 1 2 3 define an arch or a tunnel borehole num xcen ycen diam angl ang2 kode bvs bvsn num number of elements on arch boundary xcen ycen coordinates of arch centre m diam arch diameter m angl beginning angle of the arch clockwise degree ang2 end angle of the arch clockwise degree kode type of boundary condition 1 shear and normal stress boundary 21 EDGE CYCL SAVE STOP FRACOD User s Manual 2 shear and normal displacement boundary 3 shear displacement and normal stress boundary 4 shear stress and normal displacement boundary bvs boundary value in shear direction stress or displacement Pa or m bvn boundary value in normal direction stress or displacemen
37. into FRACOD to continue the previously interrupted modelling Movie File mvi Load a movie file to replay the progress of fracture propagation from previous calculations A movie file is a file containing plot data of a problem run previously by using FRACOD It is saved automatically during FRACOD calculation using the same name as the input data file but with the extension of mvi This function provides the possibility of replaying printing the fracture propagation process without re runing the model Save Run Plot Run Save the current status of calculation into a saved file The saved file can later be reloaded see Load into FRACOD to continue the modelling Plot Save the current plot into a file emf or wmf The file has a emf Window Meta Files or wmf Window Enhanced Meta Files format It can be copied to other Window applications e g MS Word 25 FRACOD User s Manual Print Print the current screen plot on a default output device printer Exit Terminate the current calculation and close the FRACOD Window Default screen output During simulation the geometry of fractures and tunnels etc will be automatically shown on the screen The picture will be updated after each calculation cycle to trace any fracture propagation In this way the whole process of fracture propagation can be monitored In the screen plot fractures are plotted with different colours to show the state of the fractures 1
38. ion at boundaries FRACOD User s Manual Fracture initiation at a boundary is not as a straight forward task as that in intact rock Because the boundary may be a straight boundary a curved boundary or a boundary with sharp corners significantly stress concentration may occur at the boundary Recent study by Shen and Rinne 2001 has highlighted the complexity of the fracture initiation at boundaries The initiation criteria suggested by Shen and Rinne 2001 may be suitable for the cases studied but not universally for all cases There is no simple and yet theoretically sound methods for the prediction of fracture initiation from boundaries To enable the simulation of fracture propagation at boundary using FRACOD an alternative approach is taken Instead of directly predicting the fracture initiation from a boundary we examine the fracture initiation from the intact rock very close to the boundary using the intact rock failure criteria as discussed before Once an intact rock failure is detected a fracture initiation is predicted to occur in the intact rock close to the boundary FRACOD then detects whether the newly formed fracture will link to the boundary by using the fracture propagation functions This treatment fully utilises the advantage of the fracture propagation functions built in the code and overcomes the lack of effective methods in handling fracture initiation from the boundary New grid points are arranged in the intact
39. itored The fractures are plotted in different colours to distinguish whether the fracture surfaces are in elastic contact open or sliding the colours can be specified or changed by users When a calculation is completed or is interrupted a number of screen commands are available to plot the stress displacement or to change the parameters etc These commands are provided as icons on the program window and can be easily activated by clicking the mouse The key functions of the available screen commands are shown below 24 FRACOD User s Manual File Load Save Print Exit Load Load an input data file dat or a saved file sav Input File dat Load an input data file and start a new problem which is defined in the input data file An input data file is a text file that contains commands and values to define a problem This is a file being prepared in advance by the user using any text editor following the format that FRACOD can recognise or using the pre processing functions Appendix I provided with the FRACOD code Saved File sav Load a saved file and continue the simulation which was previously interrupted A saved file is a file containing data of a problem run previously by using FRACOD The data in the file is computer coded and can only be read by FRACOD itself An ongoing fracture propagation modelling can be interrupted see Pause and saved see Save into a saved file The saved file can then be reloaded
40. lculated by using the FRACOD code with 30 elements along the fracture Gy rracop 190x10 J m The theoretical solution of this problem gives the stress intensity factor K7 as K oxa 50x 43 1416 x1 88 6MPa m where a half length of the fracture The theoretical strain energy release rate is then calculated as G Ma E K y L 0297 SEET x RH De OTT 184x10 J m x The difference between the numerical result and the theoretical result is approximately 3 In this example the critical strain energy release rates of fracture propagation are 43 FRACOD User s Manual Gre 50 Jm Gite 1000 W m As the fracture propagation is pure mode I along the fracture s original plane only the critical strain energy release rate in mode I Gr is useful The F value obtained from the FRACOD modelling is GO Gy 0 Gr Gre _190x10 0 50 1000 F 0 3800 The F value is by far greater than the critical value 1 0 Hence fracture propagation is detected 2 Input data TITLE Single fracture subjected to normal tensile stress SYMMETRY Model symmetry 0 0 00 0 00 MODULUS Poisson s Ratio and Young s modulus 0 25 0 40E 11 TOUGHNESS Gic and Giic 50 1000 PROPERIES mat kn ks phi coh 1 0 10E 14 0 10E 14 30 0 0 00E 00 SWINDOW xll xur yll yur numx numy 3 00 3 00 3 00 3 00 30 30 STRESSES sxx syy sxy 0 0E 07 0 50E 08 0 00E 00 FRACTURE nume xbeg ybeg xend yend k
41. listed in Sections 2 3 and 2 4 the code then calculates the F value for fracture tip A in the direction of 6 i e F 0 The direction is then rotated from 180 to 180 with an defined interval to cover the whole direction range The F value at these directions is hence obtained As described by the F criterion the direction where the F value is the maximum will be the direction of a potential fracture growth If the F value in this direction is found to be equal to or greater than 1 0 a fracture propagation is detected for fracture tip A Similar process will be performed for all fracture tips The tips found to have fracture propagation will at the end of the detection process be added with a new element in the direction of fracture propagation The element will have the same length as the tip element After adding a new element to each fracture tip found to propagate a new cycle of calculation in steps 2 3 and 4 will be performed to find whether further fracture propagation will occur If yes the calculation cycle will FRACOD User s Manual continue until there is no more fracture propagation or is stopped by the user Also detected in this step is the fracture initiation from intact rock fracture surface and boundaries If failure is detected at a grid points new fractures will be added The code will then treat the new fractures the same as other existing fractures and detects whether any fracture propagation will occur
42. n to the problem of a constant discontinuity in displacement over a finite line segment in the x y plane of an infinite and elastic solid Physically one may imagine a displacement discontinuity as a line crack whose opposing surfaces have been displaced relative to one another see Figure 2 1 Displacement Discontinuity Method in an infinite solid The problem of a constant displacement discontinuity over a finite line segment in the x y plane of an infinite elastic solid is specified by the condition that the displacements be continuous everywhere except over the line segment in question The line segment may be chosen to occupy a certain portion of the x axis say the portion x lt a y 0 If we consider this segment to be a line crack we can distinguish its two surfaces by saying that one surfaces is on the positive side of y 0 denoted y 0 and the other is on the negative side denoted y 0 In crossing from one side of the line segment to the other the displacement undergoes a constant specified change in value D Dy D We will define the displacement discontinuity D as the difference in displacement between the two sides of the segment as follows D uy x 0_ u x 0 2 1 Dy Uy x 0_ Uuy x 0 Because u and u are positive in positive x and y co ordinate direction it follows that the D and D are positive as illustrated in Figure 2 1 2 1 2 FRACOD User s Manual Figure 2 1 Constant displaceme
43. nd yend kode mat 15 0 000 0 500 1 000 1 500 2 1 CYCL 1000 ENDFILE 55 FRACOD User s Manual 3 FRACOD model Two Dimensional Fracture Propagation Code Bisi x Pxx 0 000E 00 Pyy 0 500E 08 Pxy 0 000E 00 R e elastic open basste Fo Two Dimensional Fracture Propagation Code Miel Lx Pxx 0 000E 00 Pyy 0 500E 08 Pxy 0 000E 00 R elastic open cian 56 FRACOD User s Manual Example 7 Two inclined fractures subjected to biaxial compression 1 Problem definition Two inclined parallel fractures in an infinite rock mass are subjected to compressive stress of 75MPa and confining stress of 15MPa The inclination of the fractures is 45 The material properties of the rock mass and fractures are the same as in Examples 1 and 3 The fractures are found to propagate and coalesce mainly in mode II Occasional mode I propagation is also observed 2 Input data TITLE Two inclined fractures subjected to biaxial compression SYMMETRY Model symmetry 3 0 00 0 00 MODULUS Poisson s Ratio and Young s modulus 0 25 0 40E 11 TOUGHNESS Gic and Giic 50 1000 PROPERIES mat kn ks phi coh 1 0 10E 14 0 10E 14 30 0 0 00E 00 SWINDOW xll xur yll yur numx numy 5 00 5 00 5 00 5 00 30 30 STRESSES sxx syy sxy 0 15E 08 0 750E 08 0 00E 00 FRACTURE nume xbeg ybeg xend yend kode mat 12 0 000 0 500 1 000 1 500 2 1 CYCL 1000 ENDFILE 57 FRACOD
44. nge mode It was found that Gre 1289 J m is the critical value for the fracture propagation to change mode When Gr lt 1289 J m the fracture propagates in mode I in the direction of about 70 from the original fracture plane when Gr gt 1289 J m the fracture starts to propagate in mode II in its own plane see Figures Use the relation between the critical stress energy release rate and the fracture toughness a SE Kite numerical Gr 1000 The critical toughness ratio for mode Il fracture propagation obtained numerically by using the FRACOD code is 1 135 very close to the analytical solution of 1 15 reported by Rao 1999 Using different number of elements along the fracture the comparison between the FRACOD results and the theoretical results is shown in the figure below 1 3 e BEMF 1 25 41 _ _ _ Theory Rao 1999 1 2 4 11 ee Ge wate Gas eee ay 2 S S x 3 2 ke oO 30 35 40 Number of element 46 FRACOD User s Manual 2 Input data TITLE Single fracture subjected to pure shear stress SYMMETRY Model symmetry 0 0 00 0 00 MODULUS Poisson s Ratio and Young s modulus 0 25 0 40E 11 TOUGHNESS Gic and Giic 1289 0 1000 PROPERIES mat kn ks phi coh 1 0 0E 0 0 0E 0 0 0 0 00E 00 SWINDOW xll xur yll yur numx numy 2 00 2 00 2 00 2 00 30 30 STRESSES sxx syy sxy 0 0E 06 0 00E 0 50 00E 06 FRACTURE nume xbeg ybeg xend yend kode mat 30 1
45. normal stress In reality however a fracture propagation in direction 2 is rarely observed simply because it creates a conflicting shear movement against the existing fracture A fracture initiation in direction 2 may occur but its odd shear direction prevents it from growing longer to form a sustainable shear fracture propagation In this case the true shear fracture propagation has to be the direction 1 in which the shear movement of the new fracture is consistent with the existing fractures 69 FRACOD User s Manual Figure A2 Two candidate directions for shear failure Only in direction 1 can a sustainable fracture growth occur To accurately model the shear fracture direction for the cases as shown in Figure A2 we need to ensure that the FRACOD code can intelligently select the physically correct direction 1 This has been done by introducing a criterion to eliminate the shear failure direction which creates conflicting shear movement against the pre existing fracture Without such a criterion the FRACOD code can only randomly select one direction between the two candidate directions and in some cases gives an incorrect prediction Figure A3 demonstrates the improvement after introducing the direction selection criterion in the FRACOD code In Figure 3 a borehole and two inclined fractures are modelled to study whether the fractures propagate to the borehole in uniaxial compression The two figures in Figure 3 show tw
46. nt discontinuity components D and D The solution of the subject problem is given by Crouch 1976 and Crouch and Starfield 1983 The displacement and stresses can be written as uy D LAY y DAL if ay 2 2 u D zur yf y Dy bur wf yy and Orn 2GD 2 xy H om 26D Es p Wyl Oy 2GD Yf xyy D 2GD Ea H um Oxy 2GD Wf yyy d 2GD hoy where fx represent the derivative of function f x y against x similarly as for y tw foxy ete Function f x y in these equations is given by e Pa aw 3 f x y min arctan a x a a A e Eje Lean den ay py y arctan 2 4 Numerical procedure For a crack of any shape such as curved we assume it can be represented with sufficient accuracy by N straight segments joined end by end The positions of the segments are specified with reference to the x y co ordinate FRACOD User s Manual system shown in Figure 2 2 If the surface of the crack are subjected to stress for example a uniform fluid pressure p they will displace relative to one another The displacement discontinuity method is a means of finding a discrete approximation to the smooth distribution of relative displacement that exits in reality The discrete approximation is found with reference to the N subdivisions of the crack depicted in Figure 2 2a Each of the subdivisions is a boundary element and represents an elemental displacement discontinuity Figure 2 2 R
47. o different paths of fracture propagation predicted by FRACOD before and after the introduction of the criterion Clearly the prediction obtained with the direction selection criterion is more reasonable as one would expect such fractures normally propagate to the borehole rather than deviating away from the borehole 70 FRACOD User s Manual 1 sxx 0 syy Tel tar Pa 0 MP 253 4MPo 0 syy 100 sxy SXX maximum principa field eld rf n d Stress elastic open slip e x gt o EE E le o 5 mio DT e 9 ecg oCd Try 1000MPa MPa 1000 UUU cjg SS m o DD 220 Sa oon e b after DCL SIESS e sca RUE a before Figure A3 Two fracture propagation paths predicted by FRACOD before and after the shear direction selection criterion was introduced 71
48. ode mat 30 1 000 0 000 1 000 0 000 2 1 CYCL 1000 ENDFILE 44 FRACOD User s Manual FRACOD model 3 FRACS Pei ES Two Dimensional Fracture Propagation Code Pxy 0 000E 00 2 0 500E 08 2 12100 Pyy ent 0 000E 00 maximum displacen Pxx tensile compress1ve Jot x i SS El o b J 5 v A a U E ol E ES ay ab ES a SE GE Be S gt om y o ol Uu 3 2 E r iS ag d ES 5 ta El E D ZE Ga Ee om on a 2 E 8 Di rad ch o E o SE r S Ei Sk 5 I S 5 be Si Es SI Gu ay o gt El 45 FRACOD User s Manual Example 2 Single fracture subjected to pure shear stress 1 Problem definition A 2m fracture in an infinite rock mass is under pure shear stress of 50MPa The elastic properties of the rock mass are the same as in Example 1 According to Rao 1999 a fracture in pure shear may propagate in mode I or mode II depending on the ratio of the fracture toughness of mode I and mode II Kyc Kue Only when Ky Ky gt 1 15 a mode II fracture propagation can occur The FRACOD code is used in this example to compare with the theoretical results 30 elements are used along the fracture The critical mode II strain energy release rate Gy is taken as 1000 J m The critical mode I strain energy release rate Gz is varied to obtain the critical ratio Gj Gu at which the fracture propagation starts to cha
49. on 2 23 Crack a Figure 2 4 Fictitious crack increment Aa in direction with respect to the initial crack orientation In the above calculation if we restrict numerically the shear displacement of the fictitious element to zero the result obtained using Equation 2 23 will be G 0 Similarly if we restrict the normal displacement of the fictitious element to zero the result obtained will be G1 6 After obtaining both Gu and G7 the F value in Equation 2 16 can be calculated using the given fracture toughness values Gy and Gyr of a given rock type FRACTURE INITIATION CRITERION In addition to the propagation of existing fractures new fractures cracks may initiate at the boundaries or in the intact rock This section describes the criteria used to detect fracture initiation Fracture initiation in intact rock FRACOD User s Manual Fracture initiation is a complicated process It often starts from microcrack formation The microcracks coalesce and finally form macro fractures Because the FRACOD code is designed to simulate the fracturing process in macro scale only we ignore the process of microcrack formation Rather we will only focus on when and whether a macro fracture will form at a given location with a given stress state The FRACOD code considers the intact rock as a flawless and homogeneous medium Therefore any fracture initiation from such a medium represents a localised failure of the inta
50. re oj and r are the stress and strain tensors and V is the volume of the body The strain energy can also be calculated from the stresses and displacements along its boundary 1 W Lieu O yu y ds 2 20 where 0 O Us Un are the stresses and displacements in tangential and normal direction along the boundary of the elastic body Applying Equation 2 20 to the crack system in an infinite body with far field stresses in the shear and normal direction of the crack eo and 0 0 the strain energy W in the infinite elastic body is W to 0 D 0 0 o D Jda 2 21 0 2 5 FRACOD User s Manual where a is the crack length D is the shear displacement discontinuity and D is the normal displacement discontinuity of the crack When DDM is used to calculate the stresses and displacement discontinuities of the crack the strain energy can also be written in terms of the element length a and the stresses and displacement discontinuities of the ith element of the crack The G value can be estimated by W W a Aa W a a Aa 2 23 G 0 where W a is the strain energy governed by the original crack while W a A a is the strain energy governed by both the original crack a and its small extension Aa Figure 2 4 In Figure 2 4 a fictitious element is introduced to the tip of the original crack with the length Aa in the direction Both W a and W a Aa can be determined easily by directly using DDM and Equati
51. riterion and the Minimum Strain Energy Density Criterion S criterion The principal stress strain based criteria are only applicable to the mode I fracture propagation which relies on the principal tensile stress strain To be applied for the mode II propagation a fracture criterion has to consider not only the principal stress strain but also the shear stress strain From this point of view the energy based criteria seem to be applicable for both mode I and II propagation because the strain energy in the vicinity of a fracture tip is related to all the components of stress and strain Both the G criterion and the S criterion have been examined for application to the mode I and mode II propagation Shen and Stephansson 1993 and neither of them is directly suitable In a study by Shen and Stephansson 1993 the original G criterion has been improved and extended The FRACOD User s Manual original G criterion states that when the strain energy release rate in the direction of the maximum G value reaches the critical value G the fracture tip will propagate in that direction It does not distinguish between mode I and mode II fracture toughness of energy Gr and Gre In fact for themost of the engineering materials the mode II fracture toughness is much higher than the mode I toughness due to the differences in the failure mechanism In rocks for instance Gie is found in laboratory scale to be at least two orders of magnitude high
52. rock along the boundary Figure 2 6 They are set to be at a distance of one element away from the boundary since the constant DDM method does not give accurate results very close to the element The grid points are effectively treated to be the same as other grid points in the intact rock and the same procedure is used to detect any possible fracture initiation from these grid points If a fracture initiation is predicted from any of the grid points close to the boundaries a new fracture is created at the grid point in the direction of failure The length of the fracture is a half of the length of the nearest boundary element The code then detects whether the fracture will propagate to the boundary If yes the fracture will link to the boundary and effectively form a fracture initiation from the boundary uae Fracture KS point SE BE K Figure 2 6 Modelling process of a fracture initiation from boundary FRACOD User s Manual An existing fracture is treated to be the same as a boundary The same procedure is used to detect if any fracture initiation will occur close to the surface of a fracture In case of a fracture grid points will be added to both sides of the fracture surface since both sides are solid rock The fracture initiation process does not apply to the tips of an existing fracture At a fracture tip stress singularity occurs and any intact rock failure criterion is no longer valid The fracture propagation modelling proce
53. ry to Clipborad in Enhanced Window Meta Format It can later be pasted to other Window applications such as MS Word View Model Properties Model Properties View the properties of a selected object fracture edge arc etc The geometrical properties will be shown immediately after this function is selected If the selected object is a fracture the mechanical properties shear and normal stiffness friction angle and cohesion can be viewed and modified by clicking icon Define Fracture Properties in the current Window If the selected object is a boundary edge hole etc the boundary conditions of the object can be viewed and modified by clicking icon Define Boundary Conditions SetUp Set Parameters Set Parameters Set up the model geometrical and mechanical parameters Options include Caption Give a title to the current model Symmetry Define the symmetry condition of the model XY Range Define the range of display for both Model Design and the fracture propagation modelling Properties Define the mechanical properties of intact rock Young s modulus Poisson s ratio critical strain energy release rates Gj and Gy and fractures shear and normal stiffness friction angle cohesion Up to 10 different fracture properties can be given each with a material index number 1 10 Different fractures can be assigned with different fracture properties Also should be given here are the far field stresses
54. s for section 2 1 1 provided we suitably transform the equations to account for the position and orientation of the line segment in question In particular the shear and normal stresses at the midpoint of the ith element in Figure 2 2b can be expressed in terms of the displacement discontinuity components at the jth element as follows fe Ss CR D d i l to N 2 6 A D n ns H Q II a ta SS y D Il a gt Du y where Ass etc are the boundary influence coefficients for the stresses The ij coefficient Ans for example gives the normal stress at the midpoint of the j ith element i e on due to a constant unit shear displacement discontinuity J over the jth element i e D 1 Returning now to the crack problem depicted in Figure 2 2b we place an elemental displacement discontinuity at each of the N segments along the crack and write from Equation 2 6 i Ng j Aj 0 Y A D 9 4 D e Po ULi ltoN 2A i N j j N j j 0 4 D AnD J j j If we specify the values of the stress os and o for each element of the crack then Equation 2 7 is a system of 2N simultaneous linear equations in 2N unknowns namely the elemental displacement discontinuity components J J Ds and Da We can find the displacements and stresses at designated points in the body by using the principle of superposition In particular the displacements along the crack of Figure 2 2a are given b
55. t Pa or m Warning For an excavation opening the arch angle starts from low to high e g 0 180 For a solid disc the arch angle starts from high to low e g 180 0 define a straight boundary line num xbeg ybeg xend yend kode bus bvn num number of elements along the edge xbeg vbeg co ordinates of the beginning point of the edge m xend vend co ordinates of the end point of the edge m kode type of boundary condition 1 shear and normal stress boundary 2 shear and normal displacement boundary 3 shear displacement and normal stress boundary 4 shear stress and normal displacement boundary bvs boundary value in shear direction stress or displacement Pa or m bvn boundary value in normal direction stress or displacement Pa or m Warning The beginning point and the end point need to be defined in a sequence that the positive side of the edge is always the excavation The positive side and the negative side are defined as xend yend Positive side opening Negative side rock xbeg ybeg chum start calculation cnum number of cycle to be performed one cycle often produces one step of crack growth for each unstable crack tip If cnum is not given the default cycle number is 1000 filename save the current state of calculation into a file Note the saved state of modelling can be restarted later using Window commands Stop the calculation 22 QUI
56. ual REFERENCES Crouch S L 1976 Solution of plane elasticity problems by the displacement discontinuity method Int J Num Methods Engng 10 301 343 Crouch S L and Starfield A M 1983 Boundary element methods in solid mechanics George Allen amp Unwin publisher Eordgan F and Sih G C 1963 On the crack extension in plates under plane loading and transverse shear ASME J Bas Engng 85 519 527 Griffith A 1921 The phenomena and rupture flow in solids Phil Trans R Soc London A221 163 198 Griffith A 1925 The theory of rupture Proc Ist Int Cong Appl Mech Delft 55 63 Hellan K 1985 Introduction to fracture mechanics McGraw Hill Book Company publisher Hoori H and Nemat Nasser S 1985 Compression induced microcrack growth in brittle solid axial splitting and shear failure J Geophy Res 90 B4 3105 3125 Horri H and Nemat Nasser S 1986 Brittle failure in compression splitting faulting and brittle ductile transition Phil Trans Roy Soc 319 1549 337 374 Hussain M A Pu S L and Underwood J 1974 Strain energy release rate for a crack under combined mode I and mode II Fracture Analysis ASTM STP 560 2 28 Am Soc Testing Materials Philadelphia Ingrafea A 1987 Finite element models for rock fracture mechanics nt J Num Ana Meth Geomech 4 24 43 Kachanov M L 1982 A microcrack model of rock inelasticity Part I and II Mech Mater 1 19 41
57. uently the system of equations 2 10 can be presented as 2 3 FRACOD User s Manual N i j N 0 AD YA D o VEK D tang tee i ltoN 2 15 N j j N ij i 0 4 D 4 mD 0 Je D j l j d d The displacement discontinuities D D of the crack are obtained by solving the system of governing equations using conventional numerical techniques e g Gauss elimination method If the crack is open the stresses o o0 on the crack surfaces are zero otherwise if the crack is in contact or sliding they can be calculated by Equations 2 12 or 2 14 The state of each crack joint element can be determined using the Mohr Coulomb failure criterion 1 open joint o gt 0 2 elastic joint o lt c lo ltan J AS n 3 sliding joint where a compressive stress is taken to be negative and c is cohesion If the joint has experienced sliding c 0 FRACTURE PROPAGATION CRITERION In modelling fracture propagation in rock masses where both tensile and shear failure are common a fracture criterion for predicting both mode I and mode II fracture propagation is needed The exiting fracture criteria in the macro approach can be classified into two groups the principal stress strain based criteria and the energy based criteria The first group consists of the Maximum Principal Stress Criterion and the Maximum Principal Strain Criterion the second group includes the Maximum Strain Energy Release Rate Criterion G c
58. ultaneously The newly formed short fractures link with each other to form a longer fracture This mechanism reduces the sensitivity of the modelling results to the grid point spacing FRACOD User s Manual Shear stress New fracture Tensile stress CH Grid point New fracture Figure 2 5 Fracture initiation in tension or shear in intact rock For a shear fracture initiation the Mohr Coulomb failure criterion is used in FRACOD i e when the shear stress at a given point of the intact rock exceeds the shear strength of the intact rock a new rock fracture will be generated Figure 2 5 Critical stress of fracture initiation in shear Oshear gt Ontan c Direction of fracture initiation in shear Ois 0 2 10 4 where Otensile 18 the shear stress in the direction of Ois On is the normal stress to the shear failure plane d is the internal friction angle of intact rock c is the cohesion and D is the direction of potential shear failure which is measured from the direction of the minimum principal stress Because there are always two symmetric shear failure planes at any given point two fractures are added in the model whenever a shear failure is detected Often one of the two fractures will propagate predominately in later simulation of fracture propagation The length of the shear fracture initiation depends upon the spacing of the grid points as discussed above for the tensile fracture initiation Fracture initiat
59. y expressions of the form 2 2 FRACOD User s Manual i Ny j Ny j Us Bss D Bsn Dn 1 j FN TEE 2 8 i NG j NG j Uy Bns Ds a Bnn Dy j l j l y where Bss etc are the boundary influence coefficients for the displacements The displacements are discontinuous when passing from one side of the jth element to the other so we must distinguish between these two sides when computing the influence coefficients in Equation 2 8 The diagonal terms of the influence coefficients in these equations have the values ij ij sn Bos 0 ij ij 1 1 2 9 Ba Bnn a gt YES gt 0_ The remaining coefficients i e the ones for which Zil are continuous and they can be obtained by using Equations 2 1 2 2 and 2 3 in Section 2 1 1 Displacements us and u in Equation 2 8 will exhibit constant L L discontinuities Ds and D as required SIMULATION OF ROCK DISCONTINUITIES For a rock discontinuity crack joint etc in an infinite elastic rock mass the system of governing equations 2 7 can be written as i N j N ij i 0 Y A D 4 D 5 a ene i to N 2 10 i Nu NOG i 0 A D AnD 9 o j l j l where o and o represent the shear and normal stresses of the ith element respectively 0 0 are the far field stresses transformed in the crack ij ij shear and normal directions A are the influence coefficients and Fl D D represent displacement discontinuities of jth el
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