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A practical model for load-unload-reload cycles on sand Dabeet, Antone E. 2008-12-31

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A PRACTICAL MODEL FOR LOAD-UNLOAD-RELOAD CYCLES ON SAND  by ANTONE E. DABEET B.Sc., The American University in Cairo, 2005  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQIURMENTS FOR THE DEGREE OF MASTERS OF APPLIED SCIENCE  in  THE FACULATY OF GRADUATE STUDEIES (CIVIL ENGINEERING)  THE UNIVERSITY OF BRITISH COLUMBIA  (VANCOUVER)  October 2008  © Antone Dabeet, 2008  ABSTRACT  The behaviour of sands during loading has been studied in great detail. However, little work has been devoted to understanding the response of sands in unloading. Drained triaxial tests indicate that, contrary to the expected elastic behaviour, sand often exhibit contractive behaviour when unloaded. Undrained cyclic simple shear tests show that the increase in pore water pressure generated during the unloading cycle often exceeds that generated during loading. The tendency to contract upon unloading is important in engineering practice as an increase in pore water pressure during earthquake loading could result in liquefaction. This research contributes to filling the gap in our understanding of soil behaviour in unloading and subsequent reloading. The approach followed includes both theoretical investigation and numerical implementation of experimental observations of stress dilatancy in unload-reload loops. The theoretical investigation is done at the micromechanical level. The numerical approach is developed from observations from drained triaxial compression tests. The numerical implementation of yield in unloading uses NorSand  —  a hardening plasticity model based on the critical state theory, and extends  upon previous understanding. The proposed model is calibrated to Erksak sand and then used to predict the load-unload-reload behaviour of Fraser River sand. The trends predicted from the theoretical and numerical approaches match the experimental observations closely. Shear strength is not highly affected by unload-reload loops. Conversely, volumetric changes as a result of unloading-reloading are dramatic. Volumetric strains in unloading depend on the last value of stress ratio (q/p’) in the previous loading. It appears that major changes in particles arrangement occur once peak stress ratio is exceeded. The developed unload-reload model requires three additional input parameters, which were correlated to the monotonic parameters, to represent hardening in unloading and reloading and the effect of induced fabric changes on stress dilatancy. The calibrated model gave accurate predictions for the results of triaxial tests with load-unload-reload cycles on Fraser River sand.  11  TABLE OF CONTENTS  ABSTRACT  .  ii  TABLE OF CONTENTS  iii  LIST OF TABLES  vii  LIST OF FIGURES  ix  LIST OF SYMBOLS  xvi  ACKNOWLEDGEMENTS  xix  1. 1.1.  Research Objectives  1.2.  Thesis Organization  2.  I  INTRODUCTION  4  LITERATURE REVIEW  6  Experimental soil behaviour 2.1. 2.1.1. Typical stress-strain behaviour of sand 2.1.2. The Critical State 2.1.3. The state parameter 2.1.4. Yielding of sands 2.2.  7 11 17 20  Triaxial testing  Soil constitutive models 2.3. 2.3.1. Elasto-plastic soil modelling 2.3.2. Simple soil models 2.3.3. Cam-Clay soil model  25 25 29 32  2.4.  37  tress-iiiiaiancy  45 47 50 52 53  The NorSand soil model 2.5. 2.5.1. Yield surface and flow rule 2.5.2. Hardening of the yield surface 2.5.3. Typical evolution of the yield surface 2.5.4. Elastic properties of NorSand  111  .53  2.5.5. Summary of the NorSand model  55 55 56 62 65  2.6. Soil behaviour in unloading 2.6.1. A Simple physical model 2.6.2. Thermo-mechanical approach 2.6.3. Unloading in NorSand 2.6.4. Summary  3. DILATANCY IN UNLOAD-RELOAD LOOPS: A THEORETICAL INVESTIGATION  66  3.1.  Micro-Mechanical perspective for dilatancy in unloading  66  3.2.  Micro-Mechanical perspective for dilatancy in reloading  71  3.3.  Summary  74  4. DILATANCY IN UNLOAD-RELOAD LOOPS: AN EXPERIMENTAL INVESTIGATION  75  4.1. Sands Tested 4.1.1. Erksak Sand 4.1.2. Fraser River Sand  75 75 76  4.2. Testing program 4.2.1. Erksak Sand Testing Program 4.2.2. Fraser River Sand  77 77 79  4.3.  Experimental observations  4.4.  Implications of experimental observations  5. A MODEL TO ACCOMMODATE UNLOAD-RELOAD LOOPS USING NORSAND 5.1.  93  96  Yield surface and internal cap  5.2. Flow rule 5.2.1. Flow rule in unloading 5.2.2. Flow rule in reloading 5.2.3. Potential surface in unloading  100 100 102 106  5.3.  Hardening in loading, unloading and reloading  109  5.4.  Comparison with other models  114  5.5.  Summary  120  6.  MODEL CALIBRATION  121  Monotonic calibration for Erksak sand 6.1. 6.1.1. Critical state parameters  121 122  iv  128 6.1.2. Elasticity parameters. 130 6.1.3. Plasticity parameters 132 6.1.4. Summary of Erksak monotonic calibration Monotonic calibration for Fraser River sand 6.2. 6.2.1. Critical State parameters 6.2.2. Elasticity parameters 6.2.3. Plasticity parameters 6.2.4. Summary of Fraser River Sand monotonic calibration  135 135 139 139 142  6.3. Unload-reload calibration to Erksak sand 6.3.1. Overview of Erksak Unload-Reload Calibration  142 146  6.4.  .ummary  7. PREDICTIONS OF FRASER RIVER SAND UNLOAD-RELOAD BEHAVIOUR  151  7.1.  Model parameters  151  7.2.  Model predictions  152  7.3.  Discussion of model predictions  154  7.4.  Summary  156  8.  160  SUMMARY AND CONCLUSIONS  8.1.  Context of Research  160  8.2.  Research Objectives  161  8.3.  Methodology  161  8.4.  Conclusions  161  8.5.  Suggestions for Future Work  163  REFERENCES  165  APPENDIX A: PREDICTION OF STRESS DILATANCY IN UNLOADING  170  APPENDIX B: RESULTS OF THE UNLOAD-RELOAD CALIBRATION FOR ERKSAK SAND 176 APPENDIX C: FRASER RIVER SAND MONOTONIC CALIBRATION RESULTS 183  V  APPENDIX D: STEPS TO IMPLEMENT THE LOAD-UNLOAD-RELOAD MODELINACODE  189  APPENDIX E: TRIAXIAL TESTING PROCEDURE  192  vi  LIST OF TABLES  Table 2.1. Summary of NorSand equations (modified after Jefferies and Shuttle, 2005).54 Table 2.2. Summary of NorSand parameters (after Jefferies and Shuttle, 2005)  55  Table 4.1: Index properties of Fraser River and Erksak sands  76  Table 4.2: Drained triaxial compression tests on Erksak Sand with load-unload-reload cycles (data from www.golder.com/liq)  78  Table 4.3: Undrained monotonic triaxial compression tests on Erksak sand (data from 78  Been et. al., 1991) Table 4.4: Drained triaxial compression tests with load-unload-reload cycles on Fraser River sand (data provided by Golder Associates)  79  Table 4.5. Monotonic triaxial compression tests on Fraser River sand (data provided by Golder Associates)  80  Table 4.6. Direction of volumetric changes in unloading for the load-unload-reload tests onES  81  Table 5.1. Equations used in the triaxial compression version of NorSand and their step by step implementation in an Euler integration code Table 5.2. Summary of the unloading part of the model  96 115  Table 5.3. Comparison between hardening in the proposed model and Drucker and Seereeram (1987)  119  Table 6.1. Typical ranges for monotonic parameters (same as Table 2.2, modified after Jefferies and Shuttle, 2005) Table 6.2.  122  using stress-dilatancy method for the unload-reload tests on Erksak sand. 126  Table 6.3. Summary of M, values for Erksak sand  127  Table 6.4. Summary of monotonic calibration for Erksak sand  134  Table 6.5. Summary of NorSand monotonic calibration to Fraser River sand  141  Table 6.6. Summary of the unload-reload calibration for Erksak sand  146  vii  Table 7.1. Parameters used for Fraser River sand unload-reload predictions  viii  152  LIST OF FIGURES  2  Figure 1.1. The behaviour of an elastic material in loading and unloading Figure 1.2. Results of a triaxial test on Erksak sand in volumetric strain vs. axial strain  2  (reproduced after Golder, 1987) Figure 1.3. Drained simple shear tests on Fraser River sand (modified after  3  Sriskandakumar, 2004) Figure 1.4. Cyclic direct simple shear test on Fraser River Sand (modified after  4  Wijewickreme et al. , 2005)  Figure 2.1. Schematic of typical results of a drained triaxial test on loose and dense sand samples (a) deviator stress vs. axial or deviator strain (b) volumetric strain vs. axial or deviator strain  9  Figure 2.2. Schematic of typical results of an undrained triaxial test on loose and dense sand samples (a) deviator stress vs. axial or deviator strain (b) pore pressure vs. axial 10  or deviator strain  Figure 2.3. Schematic of stress strain curves for different mean effective stress values at 11  constant initial void ratio Figure 2.4. Effect of sample preparation method (a) deviator stress vs. axial strain (b) volumetric strain vs. axial strain. (modified after Mitchell and Soga, 2005)  12  Figure 2.5. Results of simple shear tests on 1-mm diameter steel balls at constant normal effective stress of 138 kPa (reproduced from Roscoe et. al., 1958)  13  Figure 2.6. Drained triaxial compression tests on Chattahoochee River sand (reproduced 14  after Vesic and Clough, 1968) Figure 2.7. Critical State Line for Erksak 330/0.7 sand (reproduced from Been et al.,  15  1991) Figure 2.8. The projection of the critical state line (a)p’- q (b) e-logp’  ix  18  Figure 2.9. Stress paths for three undrained triaxial tests on Kogyuk 350/2 Sand (reproduced from Been & Jefferies, 1985)  19  Figure 2.10. Peak friction angle as a function of state parameter for several sands (modified from Been & Jefferies, 1985)  19  Figure 2.11. Projection of the yield surface inp’-q plane for Aoi Sand (reproduced from Yasufukuetal., 1991)  21  Figure 2.12. Family of yield envelopes for Fuji River sand (reproduced from Ishihara and Okada, 1978)  21  Figure 2.13. Schematic of the triaxial apparatus  24  Figure 2.14. An example of a yield surface  27  Figure 2.15. Definition of dilatancy (modified after Jefferies & Been, 2006)  27  Figure 2.16. Definition of normality  28  Figure 2.17. Example of the yield surface hardening  29  Figure 2.18. Tresca yield criteria in 3-D stress space  30  Figure 2.19. Normality to Tresca and Mohr-Coulomb surface  30  Figure 2.20. Mohr-Coulomb yield criteria in 3-D stress space  32  Figure 2.21. Parallel CSL and NCL in e-logp’ plot  36  Figure 2.22. Original Cam-Clay yield surface  36  Figure 2.23. Typical assembly of rigid rods. (a) stress conditions (b) deformation characteristics (reproduced from Rowe, 1962)  39  Figure 2.24. Forces acting on a rigid block sliding on an inclined surface (reproduced from Rowe, 1962)  40  Figure 2.25. Comparison between Rowe’s stress-dilatancy, Cam-Clay flow rule, and Nova’s rule  42  Figure 2.26. Dilatancy component of strength as a function of mean effective stress at failure and relative density (reproduced from Bolton, 1986)  45  Figure 2.27. Infinite number of NCL’s (reproduced from Jefferies and Shuttle, 2002)... 47 Figure 2.28. NorSand yield surface (modified after Jefferies and Shuttle, 2005)  51  Figure 2.29. Minimum dilatancy as a function of state parameter at image for 13 sands (modified after Jefferies and Been, 2006) Figure 2.30. The Saw Tooth Model a) loading phase b) unloading phase  x  52 56  Figure 2.31. Energy balance as introduced by palmer (1967)  58  Figure 2.32. Stress-dilatancy for Cam-Clay loading, Nova loading, and Jefferies (1997) 60  unloading Figure 2.33. Schematic representation of work storage and dissipation according to  61  Collins (2005) Figure 2.34. Movement of yield surface in NorSand: Case of unloading from a point on the internal cap  64  Figure 2.35 Movement of yield surface in NorSand: Case of unloading from a point before reaching the internal cap  65  Figure 3.1 Micro-mechanical representation of dilatancy for a uniform packing of rigid rods during both loading and unloading a) Minimum void ratio for ,8 = 60° b) Maximum void ratio for fi = 45° c) Minimum void ratio for fi = 30°  68  Figure 3.2. Two different uniform assemblies of rigid rods; the dashed rectangle represents the basic unit volume (reproduced after Li and Dafalias, 2000)  71  Figure 3.3 Theoretical expression based on grain to grain friction (q ) for the 250 uniform packing in Figure 3.1 a) compared with a drained triaxial test on Erksak 330/0.7 (p’= 100 kPa and e 0 = 0.653) in stress ratio vs. dilatancy space, b) Angle between the vertical direction and the tangent at the interface between grains  73  Figure 3 .4 Rowe’s stress-dilatancy relation based on grain to grain friction for the two 74  packings in Figure 3.2  Figure 4.1. Data from ES_CID_867 (a) stress ratio vs. axial strain (b) volumetric vs. axial strain (c) stress ratio vs. dilatancy  85  Figure 4.2. Data from ES_CID_867 in shear stress vs. axial strain  86  Figure 4.3. Results of FR_CID_02 in shear stress vs. axial strain  86  Figure 4.4. Zoom on loops 1 and 2 for test ES_CID_867  87  Figure 4.5. Zoom on the elastic zone in Figure 4. lc  88  Figure 4.6. Data from ES_CID_868 (a) stress ratio vs. axial strain (b) volumetric vs. axial strain (c) stress ratio vs. dilatancy  89  Figure 4.7. Comparison of ES_CID_870 and ES_CID_872 with similar e 0 and initialp’ but different number of U-R loops (a) axial strain vs. stress ratio (b) axial strain vs. 90  volumetric strain  xi  0 and initialp’ Figure 4.8. Comparison ofES_CID_861 and ES_CID_862 with similar e  but different number of U-R loops (a) axial strain vs. stress ratio (b) axial strain vs. 91  volumetric strain Figure 4.9. Stress ratio vs. dilatancy for pre-peak and post-peak reloading loops  92  (ES_CID_862)  92  Figure 4.10. Stress ratio vs. dilatancy for different reload ioops (ES_CID_867) Figure 4.11. Dmin VS.  i  93  at Dmin for first and second loading of Erksak sand  94  Figure 4.12. The saw tooth model (a) loading (b) unloading (Same as Figure 2.35) Figure 5.1. Yield surface and internal cap in NorSand, same as Figure 2.28 (modified  99  after Jefferies and Shuttle 2005)  Figure 5.2. Demonstration of interpreted elastic and elasto-plastic zones on the results of ES_CID_682 in stress ratio vs. dilatancy plot  100  Figure 5.3. Drained triaxial tests on Erksak sand with unload-reload loops plotted in the dilatancy vs. Figure 5.4.  ‘?L  102  space  and M for L3 and U3, respectively, for ES_CID_862  Figure 5.5. Correlation between M and  ij  103  from previous loading (drained triaxial tests 103  on Erksak sand) Figure 5.6. Predicted and measured stress-dilatancy for ES_CID_866  104  Figure 5.7. Change of M for different reloading loops (ES_CID_862)  106  Figure 5.8. The shape of the potential surface in unloading  109  Figure 5.9. Expanded scale view of U2/L3 for ES_CID_868 in Figure 4.6a  111  Figure 5.10. The direction of plastic strain increment ratios in unloading with the corresponding yield surfaces and internal caps  113  Figure 5.11. The direction of plastic strain increments ratios in unloading normal to the 114  potential surfaces Figure 5.12. Predicted and measured stress-dilatancy for ES_CID_866  117  Figure 5.13. Drucker and Seereeram model (reproduced from Drucker and Seereeram, 118  1987) Figure 5.14. Hardening according to Jefferies (1997) (same as Figure 2.35)  119  Figure 6.1. M 1 using Bishops method for Erksak sand  124  Figure 6.2.  using stress-dilatancy method (ES_CID_871)  xii  125  Figure 6.3. Range of M using the stress-dilatancy method from the last reloading loops 126  for the 9 tests in Table 4.2 Figure 6.4. CSL determination for Erksak sand from loose undrained tests  127  Figure 6.5. Enlarged view of the elastic part in L3 for ES_CID_866  129  Figure 6.6. The elastic bulk modulus from Equations 6.1 and 6.3 againstp’ for the elastic 130  zone in L3 for ES_CID_866 Figure 6.7. Trend lines through Dmjn vs.  çti  at Dmin for first and second peaks for Erksak 131  sand Figure 6.8. Best fit to Hvs.  çt,  132  for Erksak sand  Figure 6.9. Example fit to test ES_CID_867  133  Figure 6.10. Recommended procedure for obtaining NorSand parameters  135  Figure 6.11.  137  using Bishop method for Fraser River sand  Figure 6.12. Enlarged view of the dilatant zone for FR_CID_03  137  using stress-dilatancy method for FR_CID_04  138  Figure 6.13.  Figure 6.14. CSL for Fraser River sand  138  Figure 6.15. Peak dilatancy vs. çt’at peak for Fraser River sand  140  Figure 6.16. Best fit for Hfor monotonic triaxial tests on Fraser River sand  140  Figure 6.17. Example fit to test FR_CID_03  141  Figure 6.18. Model fits using different H values compared to laboratory data (a) U2 for 144  ES_CID_867 (b) U3 for ES CID 867 Figure 6.19. Model fits for different Hr values compared to L4 for ES_CID_867  145  Figure 6.20. Model simulation for a changing and constant Hr values  145  Figure 6.21. Model fits for constant and changing values compared to ES_CID_867. 146 Figure 6.22. Comparison between calibrated NorSand model and ES_CID_867  148  Figure 6.23. Zoom on the second loop of comparison between calibrated NorSand model with elasto-plastic unloading and ES_CID_867  149  Figure 6.24. Zoom on the second loop of comparison between calibrated NorSand model and ES_CID_867 with plastic unloading  149 150  Figure 6.25. Zoom on the first loop for ES_CID_867 Figure 7.1. Predictions for Test FR_CID_01 (a) q—i (b)  xlii  i —‘j  (c)  s—8j  157  Figure 7.2. Predictions for Test FR_C1IJ_02 (a) q— (b) Figure 7.3. Model simulation for Test FR_CID_02 in  ,  6—ej  —&j  (c)  158  .,—&j  with constant  ,‘  of 4.34.  ...  159  Figure A. 1. Predicted and measured stress-dilatancy for ES_CID_860  171  Figure A.2. Predicted and measured stress-dilatancy for ES_CID_86 1  171  Figure A.3. Predicted and measured stress-dilatancy for ES_CID_862  172  Figure A.4. Predicted and measured stress-dilatancy for ES_CID_866  172  Figure A.5. Predicted and measured stress-dilatancy for ES_CID_867  173  Figure A.6. Predicted and measured stress-dilatancy for ES_CID_868  173  Figure A.7. Predicted and measured stress-dilatancy for ES_CID_870  174  Figure A.8. Predicted and measured stress-dilatancy for ES_CID_871  174  Figure A.9. Predicted and measured stress-dilatancy for ES_CID_872  175  Figure A. 10. Predicted and measured stress-dilatancy for ES_CID_873  175  Figure B. 1. Load-unload-reload calibration results compared to laboratory data for 176  ES_CID_860 Figure B.2. Load-unload-reload calibration results compared to laboratory data for  177  ES CID 861 Figure B.3. Load-unload-reload calibration results compared to laboratory data for  178  ES CID 862 Figure B.4. Load-unload-reload calibration results compared to laboratory data for  179  ES CID 866 Figure B.5. Load-unload-reload calibration results compared to laboratory data for  180  ES CID 867 Figure B.6. Load-unload-reload calibration results compared to laboratory data for  181  ES_CID_868 Figure B.7. Load-unload-reload calibration results compared to laboratory data for  182  ESCID_873 Figure C. 1. Monotonic calibration results compared to tests data for FR_CID_03  183  Figure C.2. Monotonic calibration results compared to tests data for FR_CID_04  184  Figure C.3. Monotonic calibration results compared to tests data for FR_CID_05  185  Figure C.4. Monotonic calibration results compared to tests data for FR_CID_06  186  Figure C.5. Monotonic calibration results compared to tests data for FR_CU_0 1  187  xiv  Figure C.6. Monotonic calibration results compared to tests data for FR_CU_02  188  Figure D. 1. A diagram illustrating loading in NorSand  189  Figure D.2. Description of unloading in the model  190  Figure D.3. Description of reloading in the model  191  xv  LIST OF SYMBOLS  c  Mohr-Coulomb stress parameters representing cohesion  CSL  critical state line  D  dilatancy (8v/Sq)  Dr  relative density  e  void ratio  E  elastic young’s modulus  G  elastic shear modulus  H  hardening/softening modulus in loading, a NorSand model input parameter  Hr  hardening/softening modulus in reloading, a NorSand model input parameter  f1  softening modulus in unloading, a NorSand model input parameter dimensionless shear rigidity parameter (G/p’), a NorSand model input parameter  K  elastic bulk modulus  M  critical state stress ratio (q/p’ at critical state), a NorSand model input parameter  M  stress ratio at image state (image is the boundary between contraction and dilation)  M  stress ratio at D”= 0 for the case of unloading  N  volumetric coupling coefficient, a NorSand model input parameter  NC  normally consolidated  OCR over-consolidation ratio p  mean stress, for triaxial conditions p = (j+2o)/3  Po  mean effective stress under initial conditions  Pcap  mean effective stress on the internal cap  p  mean effective stress at first yield in unloading  Pref  reference pressure equal to 100 kPa (often assumed equivalent to atmospheric pressure)  xvi  q  shear stress invariant, for triaxial conditions q  v  specific volume, 1+ e  W  total work done  F  Altitude of CSL in e-log p’ space at 1 kPa, a NorSand model input parameter  8j  major principal strain (axial strain in a triaxial test)  83  minor principal strain (radial strain in a triaxial test)  6  volumetric strain, for triaxial conditions  q 6  shear strain invariant, for triaxial compression  xi  slope of the line relating Dmjn to çu at Dmin defined for the first peaks; is equivalent  =  -o1 ( ) 3  (61+ q 6  263)  2(6i  —  63)13  to usual usage of %2  slope of the line relating Dmrn to çu at Dmin defined for the second peaks stress ratio, i(q/p’)  1L  the last value of stress ratio in a loading/reloading phase  K  slope of the elastic swelling lines  jo 2  slope of CSL in e-logiop’ space slope of CSL in e-logep’ space, a NorSand model input parameter  çt’  state parameter, ,u  6  angle of dilatation  ç4,,,  constant volume friction angle  qj  Rowe’s mobilised friction angle  max  peak friction angle  q  grain to grain friction angle  v  Poisson’s ratio  p  soil density  o-j  major principal stress (axial stress for triaxial conditions)  cr3  minor principal stress (radial stress for triaxial conditions)  o,  normal stress on the plane of failure  t  shear stress on the plane of failure  (e-e)  xvii  Subscripts •  dot over a symbol denotes increment  c  critical state denotes image conditions  q  shear invariant  o  initial,  tc  triaxial compression  u  unloading  v  volumetric  Superscripts effective stress e  elastic  p  plastic  xviii  ACKNOWLEDGEMENTS  I would like to express my deepest gratitude to my supervisor Dr. Dawn Shuttle for her guidance, support and encouragement. Without her advice this work would not have been accomplished. I would like to thank my reviewer Dr. John Howie for his useful comments and my official supervisor Dr. Jim Atwater. The author would also like to acknowledge the help of Mike Jefferies, Roberto Bonilla, and Golder Associates for providing access to the laboratory testing on which this research is based.  Thanks to my professors and  colleagues at the Geotechnical group at UBC for their encouragement and useful discussions.  The financial support provided by the University of British Columbia  Graduate Fellowship and the Vancouver Geotechnical Society is highly appreciated. Finally, I owe an enormous debt to my family for their constant support during the pursuit of my Masters degree at UBC. This work is dedicated to my mother.  xix  1. INTRODUCTION  The behaviour of sands during loading has been studied in great detail. However, little work has been devoted to understanding the response of sands in unloading. This is surprising as the behaviour of sands in unloading is of great practical importance, particularly for earthquake engineering. An elastic material is expected to expand upon unloading in a conventional triaxial test as illustrated in Figure 1.1. The figure on the left hand side is a schematic illustrating the expected elastic trend of decreasing volume associated with increasing confining stress in a conventional triaxial test. The solid square represents the original element size before loading and the dashed square is the deformed element. According to elasticity, the element is expected to recover its original size upon removing the confinement, as shown in the figure on the right hand side. Drained triaxial tests indicate that, contrary to the expected elastic behaviour of increase in volume in unloading, sand may exhibit contractive behaviour when unloaded. Figure 1.2 is a plot of the results of a triaxial test on Erksak sand with a single loadunload-reload cycle. Positive volumetric strains denote contraction, i.e. decrease in volume, while negative volumetric strains denote dilations, i.e. increase in volume. During loading, phase a-b, the sample initially contracts. This trend is reversed at 2.2%.  j =  Upon unloading, phase b-c, significant amount of contraction is observed.  Finally, the trend in reloading, phase c-d, is similar to that of first loading.  Drained  cyclic  simple  shear  tests  show  similar  behaviour  in  unloading  (Sriskandakumar, 2004). The results of two identical drained simple shear tests on Fraser River sand are plotted in Figure 1.3. A cyclic shear stress of 50 kPa is applied. It can be  1  noticed that unloading is associated with contraction, in some cycles more than that in loading.  In drained simple shear tests, because the vertical effective stress remains  constant, the expected elastic volumetric strains are zero. This is contrary to the observed behaviour.  Jr  I I  I 4-  4-  -  ‘I  I Elastic loading  Elastic unloading  Before loading or after unloading loading or before unloading Figure 1.1. The behaviour of an elastic material in loading and unloading.  I  0 C-” >  —1  -2  6:  %  Figure 1.2. Results of a triaxial test on Erksak sand in volumetric strain vs. axial strain (reproduced after Golder, 1987). 2  75 50  25  o -25 -50  -75  Figure 1.3. Drained simple shear tests on Fraser River sand (modified after Sriskandakumar, 2004).  The tendency to contract upon unloading during an earthquake is one contributory factor in soil liquefaction.  The importance of contraction during unloading may be  observed in undrained cyclic simple shear tests. Figure 1.4 shows a cyclic simple shear test on Fraser River Sand reported in Wijewickreme et al. (2005). Vertical effective stress is plotted on the x-axis and the applied shear stress is plotted on the y-axis. A decrease in the vertical effective stress is associated with an increase in pore water pressure. It can be observed that, apart from the first two cycles, the increase in pore water pressure generated during the unloading cycle often exceeds that generated during loading.  3  30  a-b:Loading 20  b-a:Unloading  a’ kPa 10  Figure 1.4. Cyclic direct simple shear test on Fraser River Sand (modified after Wijewickreme et al. , 2005).  Observed soil behaviour from both drained and undrained testing clearly indicates that soil behaviour in unloading is not wholly elastic. A constitutive model that yields in unloading is needed to predict this soil behaviour, and is the topic of this thesis. A basic requirement of such a model is stress-dilatancy, i.e. the inter-relationship between stress ratio  ‘‘  and dilatancy ‘D’, where  i  qIp and D =  ‘‘q’  ( and  are the increments  of volumetric strain and shear strain invariant respectively).  1.1. Research Objectives The main objectives of this work are:  1. Develop theoretical understanding of stress-dilatancy in unloading.  This  investigation includes the interaction between soil fabric and stress-dilatancy. 2. Utilize the theoretical understanding to guide development of unload-reload behaviour, including yielding during unloading, into a constitutive model.  This work will include developing an expression for stress-dilatancy in unloading based on a discrete element approach, including the effect of fabric changes on dilatancy in  4  reloading, fabric represents “the arrangement of particles, particle groups and pore spaces in a soil” (Mitchell and Soga, 2005). Soil fabric is expected to change due to cyclic loading, consequently changing stress-dilatancy in reloading as compared to that for first loading.  A continuum model that yields in unloading is developed. The model uses the ideas from the theoretical investigation of stress-dilatancy in unloading and reloading. The work will involve calibration of the model to experimental data and using the calibrated model to predict the results of drained load-unload-reload tests. The introduced model utilizes the NorSand soil model, a critical state hardening plasticity model, as its starting point.  1.2. Thesis Organization The thesis is organized into 8 chapters. Chapter 2 provides an overview of literature relating to constitutive modelling for soils, with particular emphasis on soil behaviour in unloading.  The theoretical investigation into stress-dilatancy in both unloading and  reloading phases is investigated from a micro-mechanical point of view in Chapter 3. Chapters 4 through 7 review experimental data to develop an improved constitutive model for yielding in unloading and reloading. Chapter 4 presents drained triaxial data on Erksak sand and Fraser River sand which includes load-unload-reload cycles. Chapter 5 uses the findings of Chapters 3 and 4 to develop an extension to the continuum constitutive model, NorSand. Chapter 6 presents calibrations of the model. Monotonic calibration of NorSand is done for both Erksak sand and Fraser River sand.  Load  unload-reload calibration of the model is then undertaken on Erksak sand. The calibrated model predictions for load-unload-reload tests on Fraser River sand are presented in Chapter 7. The conclusions from this work are summarized in Chapter 8.  5  2. LITERATURE REVIEW  The behaviour of sands depends on many factors, including density and mean effective stress. Constitutive models are necessary to capture the effect of these and other factors on soil behaviour, and to predict this behaviour for real engineering problems. This chapter focuses on soil constitutive modelling with particular emphasis on soil behaviour in unloading. First, a brief description of the typical behaviour of sands as observed from laboratory data and the basics of triaxial testing is introduced. This is followed by a description of the fundamentals of elasto-plastic constitutive models and some of the commonly-used soil models are introduced, with emphasis on the critical state model, Cam-Clay. The interrelationship between stresses and dilatancy is then discussed. Then the NorSand soil model, used as the basis for the unloading/reloading development later in this thesis, is introduced. Finally, a review of conceptual models for soils in unloading is introduced.  2.1. Experimental soil behaviour Much of our understanding of soil behaviour comes from laboratory testing. The main advantage of laboratory testing is that the initial conditions and stress path can usually be controlled. Typical soil behaviour is explained in this section by a review of laboratory testing in the literature. The discussion includes selected factors which are observed from laboratory testing to affect stress-strain behaviour.  The critical state theory is also  introduced, together with a description of yield characteristics of sands.  6  2.1.1. Typical stress-strain behaviour of sand Typical schematics of stress-strain curves for dense and loose sand in drained tests and with the same applied stress conditions, starting from uniform all-around pressure, are shown in Figure 2.la.  In Figure 2.la the deviator stress, q, is  conditions. The axial strain is compression.  Both  sj  and  q 6  6j  and the deviator strain,  q 6  is 2(81  oj-o —  63)13  for triaxial for triaxial  are commonly used to plot stress-strain curves in the  literature. They give similar trends. Typical behaviour for dense sand shows a peak value of deviator stress before dropping to constant stress at larger strains. Conversely, loose sand does not show a peak but instead directly reaches the same constant value of stress as the dense sand at large strains for identical mean effective stress conditions. Figure 2. lb plots data in volumetric strain vs. axial or deviator strain. Volumetric strain,  6,  is defined as  6J+  263  for triaxial conditions. In this thesis, positive strains are  compressive. Therefore positive volumetric strains denote contraction while negative volumetric strains denote dilation.  It can be seen that dense sand contracts initially  during shear and then dilates until a state is reached where volumetric strain remains constant. Loose sand contracts during shear until it reaches constant volume conditions at large strains. Reynolds (1885) was first to show that dense sand dilates when sheared towards failure while loose sand contracts. Typical undrained behaviour of sand is shown in Figure 2.2.  As the undrained  condition prevents volume change, the tendency to change in volume results in a pore water pressure change of opposite sign, which changes the effective stress conditions. Dense sand shows an increased strength with axial or deviator strain. This is associated with the development of negative (or decreasing) pore pressure. The strength of loose sand increases to a peak value. This is followed by a decrease in strength until reaching a constant value of strength which is independent of the strain level. The corresponding pore pressure increases with increasing strain level. The rate of increase decreases with strain, eventually reaching a constant pore pressure.  7  Soil strength is directly related to mean effective stress. For higher mean effective stresses soil has a stiffer response and higher strength.  Figure 2.3 is a schematic  demonstrating the effect of mean effective stress on stress-strain curve. The three plots have identical initial void ratios.  Although the behaviours described in Figure 2.1, Figure 2.2 and Figure 2.3 are generally applicable, differences in soil behaviour are observed for different soils, and also for the same soil using different preparation methods. This occurs because different sample preparation methods result in different initial fabric.  Fabric refers to “the  arrangement of particles, particle groups, and pore spaces in a soil” (Mitchell and Soga, 2005). Oda (1972) performed triaxial tests on a uniform sand composed of rounded to sub-rounded grains with sizes between 0.84 to 1.19mm (Figure 2.4). The two samples have a similar initial void ratio and mean effective stress. The only major difference between the two samples is the preparation method. One of the samples was prepared by tapping the sides of the mould. The other sample was prepared by tamping. The sample prepared by tapping demonstrates a stiffer response, associated with a more dilative behaviour, compared to that prepared by tamping.  8  (a)  Axial  (b)  or deviator strain  Loose sand  I I  Axial or deviator strain  Figure 2.1. Schematic of typical results of a drained triaxial test on loose and dense sand samples (a) deviator stress vs. axial or deviator strain (b) volumetric strain vs. axial or deviator strain.  9  (a) Dense sand  C  Loose sand Axial or deviator strain (b) Loose sand t+ve -ve Dense sand  Axial or deviator strain  Figure 2.2. Schematic of typical results of an undrained triaxial test on loose and dense sand samples (a) deviator stress vs. axial or deviator strain (b) pore pressure vs. axial or deviator strain.  10  I Axial or deviator strain Figure 2.3. Schematic of stress strain curves for different mean effective stress values at constant initial void ratio.  2.1.2. The Critical State The concept that soil will eventually reach a constant stress and void ratio state was first introduced by Casagrande in 1936. He observed from shear box tests that both dense and loose sand, under same vertical effective stress, eventually reach a constant void ratio at which shear deformation continues at constant shear stress. These observations were independently confirmed over twenty years later by Roscoe et. al. (1958) who performed simple shear tests on 1-mm diameter steel balls. All Roscoe et al.’s tests were done under constant normal effective stress of 138 kPa. Regardless of the initial density, for the same applied load of 138 kPa all samples reach similar specific volume at large shear displacements (see Figure 2.5). The specific volume is the volume occupied by a unit mass and is equal to (1  +  e).  11  (a) ‘  200  —________  180  Prepared by  160  tapping —  —  —  —  —  ——  140 120 •  Prepared by  100  80 •  60  40 20 0 0  2  I  I  I  4  6  8  Axial strain  10  12  (%)  (b) 0.1  Contractive  0 ‘‘  -0.1  %%24Dilate12  -0.2 Prepared by 8  -0.3 I:,, .  -0.4 -0.5  C  tapping  -0.6 -0.7 -0.8 -0.9 Axial strain  (%)  V  Figure 2.4. Effect of sample preparation method (a) deviator stress vs. axial strain (b) volumetric strain vs. axial strain. (modified after Mitchell and Soga, 2005).  This idea of a unique relation existing between stress level and void ratio led to the development of what has become known as Critical State Soil Mechanics (CS SM). The critical state is defined as ‘the state at which a soil continues to deform at constant stress and constant void ratio” (Roscoe et. a!.,  1958).  12  1.654  = 0 v  1.65  = 1.638 0 v  +  I +—  S %_  E  1.63  —  —  *  —  —  p 0 p  —  — 0  — ...  —  —  —  ._  — 1.625 0 v  o  —.  •0  —  I  —  —  —  er:.  —  ..  — p  —  0 v  1.611  .  *  1.61 —  ...‘  = 1.598 0 v 1.59 0  5  10  15  20  Shear deformation (mm) Figure 2.5. Results of simple shear tests on 1-mm diameter steel balls at constant normal effective stress of 138 kPa (reproduced from Roscoe et. al., 1958).  However, this constant void ratio, usually known as the critical void ratio, has been experimentally shown to vary with stress level. The results of drained triaxial tests on Chattahoochee River Sand are presented in Figure 2.6 (Vesic and Clough, 1968). These drained triaxial tests investigate the dependence of the critical void ratio on stress level. The two tests have identical void ratios but different values of mean effective stress. Figure 2.6 shows that although all of the samples are dense, the sample with the higher mean effective stress contracts matching the behaviour of loose sand.  Higher mean  effective stresses cause the particles to move around each other, rather than over, and crush, rather than simply override when sheared. This results in contractive behaviour. Therefore, the critical void ratio is a function of stress level.  13  (a)  2 1.6  1.2 C  0.8 0.4  0 0  5  10  15  20  Deviator strain (%)  (b) 10  0 =0.69 p’=34.3MPa&e —  —  —  —  —  -10  I  —  —•  —  -  —  —  2 -  -  Contractive  -..————  Deviator strain (%)  Figure 2.6. Drained triaxial compression tests on Chattahoochee River sand (reproduced after Vesic and Clough, 1968)  The experimental observations described above led to the development of a theoretical framework for soil behaviour, known as Critical State Soil Mechanics (CSSM). CSSM is based on two axioms:  14  1. A unique critical state exists. 2. The critical state is the final state to which all soils converge with increasing shear strain. CSSM presents a fundamental framework for all soils. Because all soils eventually reach critical state irrespective ofthecurrent void ratio and stress conditions, having a unique critical state is very useful. A unique critical state is an ideal framework around which to construct soil models around. The question of the uniqueness of the critical state was investigated by Been et al. (1991) who provided evidence to indicate that the critical state is likely unique, being both independent of fabric, loading rate, stress path, and initial density. Figure 2.7 shows that both moist compacted and pluviated samples in undrained tests finish at the same critical state line. Drained tests were also observed to follow the same trend. The change in the slope of the critical state line at about 1000 kPa is thought to be due to grains crushing at high mean effective stress levels.  0.8 0  0.75  0  •  C  0.7 . I-  •  0.65  Moist compacted Undrained • Pluviated Undrained 0 Moist compacted Drained C Pluviated Drained Critical state line -  -  -  06  -  —  0.55 0.5 1  100  10  1000  10000  Mean effective stress (kPa) Figure 2.7. Critical State Line for Erksak 330/0.7 sand (reproduced from Been et al., 1991). 15  The critical state is also unique in the p’-q space. Both loose sand and dense sand, under identical mean effective stress conditions, finished at the same value of deviator stress, see Figure 2.1 a. Irrespective of the sample preparation method, the two samples in Figure 2.4a reached similar deviator stress values in the higher axial strain range (i.e.  >  6%).  Hence soil behaviour can be understood within the framework of the critical state in the three dimensional space ofp’, q and e. The slope of the projection of the critical state line in p’- q is known as the critical friction ratio, M (Figure 2.8a). The projection of the critical state line in e logiop’ is given by: —  p 10 e =F—..Uog  (2.1)  Where e is the void ratio at the critical state, F is e atp’ = 1 kPa in e 2 is the slope of the critical state line (see Figure 2.8b). Note that of logio and loge (and in this thesis are termed o and  ?e  —  log p’ plot, and  is defined in terms  respectively). Both are perfectly  acceptable. However care should be taken as 2 is often used in the literature without clarifying the base of the log used. The critical state is a very useful tool as both dense and loose sand are considered to end up at the critical state. At a particular mean effective stress level, soil with e  <  e is  termed dense while soil with e> e is termed loose. Drained dense tests dilate to reach the critical state while drained loose tests contract to reach the critical state (Figure 2.8b). As the critical state line is defmed to be unique, undrained tests reach the same line as for drained tests. This makes the critical state an extremely useful reference property for accurate prediction of soil behaviour.  16  2.1.3. The state parameter  Been & Jefferies (1985), using the results of seventy triaxial tests on Kogyuk sand, show that the “bulk characteristics of sands are not sufficient to characterize mechanical behaviour of granular materials”. Relative density or void ratio alone does not govern soil behaviour.  Dense sand can behave similarly to loose sand at high confining  pressures as was previously shown for Chattahoochee River sand (Figure 2.6). The state of soil was described by Been & Jefferies as “a description of the physical conditions” which includes the influence of confinement and void ratio. In this sense, the behaviour of sand is controlled by the state parameter,  çt’.  In order for the state parameter concept to  be useful, it needs to be defined relative to a reference condition that is unique and is independent of initial conditions. The critical state is a proper framework as it satisfies both conditions. Equation 2.2 defines the state parameter.  (2.2)  The state parameter is dependent on mean effective stress as the critical void ratio is dependent on mean effective stress. It therefore represents soil behaviour better than relative density. This is for two reasons: First, relative density does not specify the current state relative to critical state. Accordingly, relative density cannot be used to predict whether soil contracts or dilates before it reaches the critical state. Second, soil strength depends on dilatancy, defined as the ratio between an increment of volumetric strain and an increment of shear strain, and not void ratio, and dilatancy is inversely proportional to mean effective stress. Figure 2.9 shows that tests with similar initial state parameters have similar behaviour regardless of the difference in their relative densities, while tests with similar relative densities behaved very differently. The results of the three tests are normalized to mean effective stress at the critical state, P’cs. Tests 103 and 108 have similar initial state parameter. They demonstrate similar behaviour regardless of the difference in relative density (33% for test 103 and 50% for test 108). However, tests 103 and 37 with identical Dr of 33%, and very different state parameters, show different behaviour. 17  The state parameter also influences some soil design parameters. A unique relation between the peak friction angle and the state parameter has been observed for a range of different sands (see Figure 2.10). Although there is scatter in the data, the trend of decreasing peak friction angle as state parameter increases is clear.  (a)  Mean effective stress (kPa)  (b)  0  0  Mean effective stress (kPa)  Figure 2.8. The projection of the critical state line (a)p’- q (b) e-logp’.  18  Test 37: Pc = 350 kPa 003& r33000t Yb  2  1.5  Test :pc 03 = 5OkPa, l t ç vo.0.03&Dr=33%  0.5  estlO8:p. =300kPa, -0.03 & Dr = 50%  <7/ —  0  0.5  1  1.5  2  p’/p’cs  )  2.5  Figure 2.9. Stress paths for three undrained triaxial tests on Kogyuk 3 50/2 Sand (reproduced from Been & Jefferies, 1985). 41  -a Do  .  V  SD  0  Upper bound  (a  -a ‘S. 0 C, C (a (a.  (a 0  S 0, C C  :  S  wflound  • cj • Kogyok sand (0—10% tines) W z Beautort sand A (2—10% fines) o Beaufort sand B (5% fines) * Banding sand. f4auchipato sand (Castro, 1969) ‘ Vaigrinda sand (Bjerrurn eta!.. 1961) a Hokksund sand (NW) • Monterey no. 0 sand (Lade, 1972)  (a 0 0 0 C  32  0 0 C (a 1..  28  .  Es  -  0  -  .1  Range of critical friction angle values  . °  aS  S.  I  -  —0-1 State parameters  U  Figure 2.10. Peak friction angle as a function of state parameter for several sands (modified from Been & Jefferies, 1985).  19  01  2.1.4. Yielding of sands The yielding point has been classically used to signify the end of recoverable deformation, usually observed experimentally as a significant decrease in stiffness. The yield surface may be intersected along any stress path and is composed of an infinite number of yielding points in the (e-p ‘-q) space. One of the earliest studies on yielding of soils is reported in Roscoe et al. (1958). Roscoe et al. derived an isometric yield curve from 39 drained simple shear tests done on 1 mm diameter steel balls. From a theoretical viewpoint, it is more useful to plot the projection of the yield surface in the p ‘-q space, as shown for Aoi Sand in Figure 2.11 (Yasufuku et al., 1991). The data for all eight drained triaxial tests in Figure 2.11 started from the same stress state with OCR=2. The hollow circles indicate yielding as evident from a sharp change in stress strain curves. The yield surface was drawn through these yield points. Note that the curve is not symmetric around the p’ axis due to sand anisotropy. By repeating the same procedure for different consolidation stresses, a family of yield curves can be defined as shown in Figure 2.12 for Fuji River sand (Ishihara and Okada, 1978). Experimental studies suggest that the yield surfaces typically have a similar shape, as can be seen for Aoi sand and Fuji River sand.  20  800  --‘--  —  600  —  •  I I  —  Yield surface Stress path Initial stress state ..  —  —  .  --  -—  —  —  /  .‘  ..  400  —  1  /  I  4 s  S  /  S.  //  200  0  I  200  %4%  I  6po  40fr  800  f I DO  —  %%  -200  8.4•’  4  x*.  —  0 —  —  0  4..  —  -400  Mean effective stress (kPa) Figure 2.11. Projection of the yield surface inp’-q plane for Aoi Sand (reproduced from Yasufuku et al., 1991).  200 ——  150  —  8  0  8  0’  8  —  4  —  100  0  S  4  I —  4  ——  S  S  *  300  400  400  €00  6 0  •  _._____..‘  0 I 8  -100  S I  4  .s:  S  S  4  —  —  —  —  _•  —  I  8___  I 0  4  0 44___  0  —  -150 -200  Mean effective stress (kPa) Figure 2.12. Family of yield envelopes for Fuji River sand (reproduced from Ishihara and Okada, 1978). 21  2.2.  Triaxial testing  Although no laboratory testing was undertaken as part of this work, existing triaxial tests form the basis for the constitutive model development. Triaxial testing is commonly used in both industry and research. This section describes conventional triaxial compression testing. The test involves consolidating a cylindrical specimen under confining pressure, a3 (for convenience it is assumed that the consolidation is hydrostatic). A deviator stress of Ao- is then applied in the vertical direction. The total stress in the vertical direction is o =  03.+  Ao  A typical arrangement of a conventional triaxial equipment is shown in Figure 2.13. A multi-speed drive unit is used to apply the axial load. The triaxial cell is filled with de aired water. The soil sample has two porous discs (at the sample bottom and top) and is surrounded laterally by a rubber membrane.  The top and bottom porous discs are  attached to the upper and lower platens, respectively. The applied load is measured by a load cell. The axial displacement is measured using a linear displacement transducer (LVDT). There are three pressure connections to the system that are used to measure the pore pressure or volume changes and apply back pressure and cell pressure. The typical size of the cylindrical soil specimen is 36mm in diameter and 76mm in length.  Typically, specimens are hydrostatically consolidated by increasing the cell pressure. Non-hydrostatic consolidation could be done as well, though less common, by applying deviator stress in the consolidation phase. Water is allowed to drain out of the back pressure line until the pore pressure is equal to the back pressure. During consolidation, the sample contracts and the effective stress increases to a value equal to the cell pressure less the back pressure. During the shearing stage the sample is loaded by increasing the axial load in increments for stress controlled testing or by applying displacement increments for strain controlled testing. For undrained tests, water is not allowed to drain  22  during this stage and pore pressure is measured. For drained tests, water is allowed to drain and volumetric strains are measured usually using a differential pressure transducer. The axial displacement is measured using the LVDT.  Triaxial data is presented in this thesis in terms of the mean effective stress, p and shear stress, q, invariants, where p’  =  (a’i  +2a ‘3)13 and q = (ai  e, is defined as the sum of the principal strains (i.e.  6,,  = 6j  —  +  ). Volumetric strain, 3 a 283). For stresses and  strains used to be work conjugate (meaning that the invariants, or the individual stresses and strains, can be used interchangeably), they must satisfy the following during a loading increment:  6 + p8,, q  =  (2.3)  1 + J;8 J;6 2 + 0363  Substituting the values of p’, q, and  6,,  in Equation 2.3 and rearranging gives the  following expression for the shear strain invariant,  6q =(s  q: 6  (2.4)  —83)  The primary advantage of triaxial testing is that all of the principal stresses are known and can be directly controlled.  Hence, when used as part of constitutive model  development, no stresses or strains are left to be inferred.  Having to assume stress  conditions, introduces uncertainty into the appropriateness of any model. However, the test is limited to applying only two independent principal stresses. This is a stress path that rarely, if ever, corresponds to the nature of loading conditions in the field.  23  CD  0  C)  CD  Cl) C)  CD  W  -.  C  0 0  CD  C)  D• CD O) (DO) CoC 0’’CD 1  2o  C Cl)  0--ti 00  0  C, CD  Co  -‘  D CD  Co  3 Co  CD  H -I  CD  3  Cl)  D  Co -i. CD  000  D  CD  H  I  C’) 0  .  C-)  C, CD  0.  I0 Co  0 0  I-  2.3. Soil constitutive models Soil constitutive modelling provides qualitative and quantitative understanding of soil behaviour.  ‘Proper’ models provide us with an understanding of soil constitutive  behaviour based on an appropriate framework that is derived from mechanics. The need for ‘good’ constitutive models is ever increasing because, with the advance in computers, more complex numerical analyses are becoming a routine practice.  Soil behaviour depends on many factors including stress level and void ratio. Because it is impractical to perform tests at every possible combination of stress level and void ratio, a useful constitutive model should be able to accurately predict changes in strength and deformation characteristics for the full range of applicable combinations of stress level and void ratio.  A brief description of elasto-plastic soil modelling is presented in this section. This is followed by an overview of some commonly-used soil models.  2.3.1. Elasto-plastic soil modelling Soil is an elasto-plastic material (i.e. exhibits both elasticity and plasticity). Elasticity is associated with recoverable strains, and purely elastic behaviour is usually only observed in soil at very small strains.  Plasticity is associated with irrecoverable  deformations. A typical elasto-plastic continuum model comprises: elasticity, a yield surface, a flow rule, a hardening/softening rule. Elasticity: Elastic strains are recoverable. The direction of an elastic strain increment follows that of the stresses.  25  Yield surface: The yield surface is the boundary between elastic and plastic strains.  Figure 2.14 is an example of a typical yield surface. A stress probe inside the yield surface causes elastic strains while a probe outside the surface causes plastic strains. Flow rule: A flow rule controls the direction and relative magnitude of the plastic strain  increments. As soil changes in volume due to shearing a flow rule is needed (also known as a stress-dilatancy relation). There are two definitions in literature for dilatancy: the absolute and the rate definition illustrated in Figure 2.15. The rate definition is more widely used in constitutive model development, and in North American practice generally, and is used in this thesis.  Accordingly, dilatancy is defined as the ratio  between an increment of volumetric strain to an increment of shear strain (i.e.  Associated flow was commonly used in the original soil constitutive models because these models do not violate Drucker’s postulate (Drucker, 1951). This means that the plastic strain increment ratio, ñ,’ / ñ’, is normal to the yield surface (Figure 2.16). Once the yield surface is defined, the flow rule is then automatically defined. This results in simpler and more stable models compared to non-associated flow models (i.e. plastic strain increment ratio is not normal to the yield surface).  26  j)  •— C12 cI1  ci)  ci) C,)  Mean effective stress, p’ Figure 2.14. An example of a yield surface.  6  Figure 2.15. Definition of dilatancy (modified after Jefferies & Been, 2006).  27  Plastic strain increment direction normal to yield surface  -.  Q 0 .  .  .E  c  Stress point  U U U  C)  ci  Mean effective stress, p’ plastic volumetric strain increment, 6’ Figure 2.16. Definition of normality.  Hardening/softening rule: The hardening/softening rule specifies the movement of the yield surface due to an applied plastic strain increment.  The yield surface size is  increased for the case of hardening while it decreases for the case of softening. An example of a hardening yield surface is shown in Figure 2.17. The stress point follows the hardened yield surface according to the specified loading path. The requirement for a stress point during loading to start and finish on the current yield surface is called the consistency condition.  28  Hardened yield surface after applying loading increment  —  s  (1)  4% 4%  / /  —  4.  _  ._Stresspointafter loading increment  4..  ,  ,  /  )  Yield surface before + applying loading increment Initial stress point  I,  44.  4% 4% 4%  s  4% 44  Mean effective stress,p’ Figure 2.17. Example of the yield surface hardening.  2.3.2. Simple soil models 1) The Tresca model The Tresca soil model is widely used for representing the undrained behaviour of clay in a total stress analysis. In the Tresca model yielding occurs when the maximum shear stress reaches a critical value, c (see Figure 2.19). For undrained conditions, Poisson’s ratio, v, is 0.49999, implying a condition of no volume change.  The Tresca model  requires two parameters: the critical shear stress value, c, and the elastic Young’s modulus, E. This yield criterion results in the yield surface in 3-D stress space shown in Figure 2.18. Maximum shear stress is independent of mean stress. This makes the Tresca model ideal for modelling the unconsolidated undrained (UU) behaviour of soils where the shear strength is not affected by an increase in confinement. Normality to  Tresca’s surface results in vertical plastic strain increments, i.e. zero plastic volumetric strains with shearing (Figure 2.19).  29  ai  02 = 03  03  Figure 2.18. Tresca yield criteria in 3-D stress space.  Mohr-Coulomb failure criterion  I Strain increment  according to normality Strain increment 4-  I  according to normality  Tresca failure criterion Un,  8  Figure 2.19. Normality to Tresca and Mohr-Coulomb surface.  30  2) The Mohr-Coulomb model  The Mohr-Coulomb (MC) model is a very simple elastic perfectly plastic soil model (i.e. the yield surface does not harden with increasing shear). Like the Tresca model, elasticity is assumed linear elastic, but now the shear strength is no longer constant, but is a function of the mean stress. MC failure surface in the 3-D stress space is shown in Figure 2.20. Unlike Tresca, MC is applied as an effective stress model. MC requires two strength parameters, c’ and qY’, where c’ represents the part of strength that is independent of normal stress and qY is the effective friction angle. It represents the part of strength that is dependent on normal stress. Accordingly shear strength, z that causes yield is given by:  (2.5)  r=c+cr,,tançz’  Equation 2.5 is plotted in Figure 2.19 for c  =  0. MC requires two additional elasticity  parameters (Young’s modulus, E, and Poisson’s ratio, v) and the dilation angle. Applying normality to MC surface, i.e. using associated flow, implies that the dilation angle is equal to the friction angle. This results in unreasonably high volumetric strains and hence MC is typically used as a non-associated flow model with a dilation angle close to zero. MC gives reasonable predictions for strength in unconfined problems but it models both volume changes and pre-yield stresses badly.  31  I,  03  02  Figure 2.20. Mohr-Coulomb yield criteria in 3-D stress space.  2.3.3. Cam-Clay soil model Cam-Clay is an associated flow constitutive model based on critical state soil mechanics, and one of the earliest advanced constitutive models for soil. There are two versions of Cam Clay widely referenced in engineering practice. The original version of Cam-Clay was developed in the 1960’s by Schofield and Wroth (1968). Original Cam Clay (0CC) is not widely found in commercial software, although it is important to explain the development of ideas used in the model, and as the basis of some later critical state models, including the NorSand model used as the starting point for the current work. Conversely, Modified Cam Clay (MCC) is found as an inbuilt model in almost all commercial codes used for geotechnical analysis. MCC is an extension of 0CC that sought to address some of the deficiencies of the original model.  32  The Original Cam-Clay model is a work dissipation model (Schofield and Wroth, 1968). As for any elasto-plastic model, it is composed of elasticity, yield surface, a flow rule and a hardening rule. 0CC accounts for elastic volumetric strains only (i.e. it is rigid in elastic shear). The slope of the elastic swelling line in e-log p space, shown in Figure 2.21, is  ,  The rate of total work done on a unit volume of soil is given by:  (2.6)  W=q&q+p’v  As only plastic strains are involved in the dissipated work (the elastic strains are recoverable), Equation 2.6 may be rewritten in terms of plastic strain as:  (2.7)  wP =wwe =q6+p8,”  Dividing byp’ and  gives:  wp  (2.8)  The term on the right hand side represents the dimensionless normalized plastic work dissipated. 0CC is based on the assumption that the rate of dissipation is constant and is equal to, the friction ratio at the critical state, M. This results in the 0CC flow rule as:  (2.9)  D=M—i  All 0CC yield surfaces intersect the critical state line at the current critical state value of mean effective stress, p’ (see Figure 2.22). The normal consolidation line, NCL, is  33  assumed to be parallel to the critical state line, CSL (see Figure 2.21). This assumption poorly represents observed sand behaviour. Jefferies and Been (2000) showed, for Erksak sand, that there are an infinite number of normal consolidation lines that are not parallel to CSL. The 0CC yield surface may be derived as follows. By definition:  q=ip  (2.10)  Taking the differential of 2.10 gives:  (2.11)  As 0CC uses associated flow, to satisf’ normality (i.e. plastic strain increments normal to the yield surface as shown in Figure 2.22),  (2.12) P  q 6  From 2.11 and 2.12,  p  7 D”+i  (2.13)  =0  Substituting the value of D” from Equation 2.9 in Equation 2.13. Integrating and substituting ln(p )+ 1 for the integration constant at critical state conditions, i.e. p  p  gives the equation of the yield surface as:  M  (2.14)  p)  34  0 and Under normally consolidated hydrostatic conditions,p’=p’  i  =  0. Substituting in  Equation 2.14, gives P’c = p ‘0/ 2.718. The 0CC yield surface hardens for the case of i < Mand is associated with intersecting it at mean pressures greater than p’s.  A hardening yield surface is associated with  contractive volumetric strains (see Figure 2.22). Hardening continues until  i  =  Mwhere  soil reaches the critical state and further shear strain increments do not cause any change in volume. If the stress point touches the yield surface at 17> M, softening occurs. This is associated with dilation until the stress point reaches critical state. The 0CC hardening rule, given in Equation 2.15, is written in terms of the increment of plastic volumetric strain. It is noteworthy that at critical state  ‘  =  0 and therefore movement of the yield  surface stops. Hence all stress paths will end at the critical state.  p(1+e)6’ 2-it  (215)  Roscoe and Burland (1968) modified Original Cam-Clay in what became the Modified Cam-Clay (MCC) model. The major difference between the two models is the shape of the yield surface. One of the problems with 0CC is that it predicts shear strains for the case of hydrostatic loading. The elliptical yield surface of MCC predicts only volumetric strains for the hydrostatic loading condition. 0CC overestimates the values of strain increments at small strains. MCC accounts for elastic shear while 0CC is rigid in elastic shear.  35  in  =  1  pc) Elastic  a e  P? =p/2.718  P’o  P’c  =r-a  logp’  —  -  Logp’ Figure 2.21. Parallel CSL and NCL in e-logp’ plot.  Figure 2.22. Original Cam-Clay yield surface. 36  2.4. Stress-Dilatancy Dilatancy was defined in Section 2.3.1 as the ratio between an increment of volumetric strain and an increment of shear strain (i.e. D =  This section discusses stress  dilatancy (i.e. the inter-relationship between stress and dilatancy) in more detail. An objective of this thesis is to investigate stress-dilatancy in unloading and reloading. Reynolds (1885) showed that dense sand dilates when sheared towards failure while loose sand contracts. The work of Taylor showed that soil strength is due to both the frictional resistance between the particles and the tendency of dense soil particles to override each other. The difference between the critical friction angle and the peak friction angle is caused by dilatancy. Rowe (1962) introduced a relation between stresses and dilatancy based on the study of particles in contact. Particles are assumed rigid, have circular cross-sections and are identical. The forces at the contacts are assumed purely frictional. The importance of Rowe’s work is that it relates stresses to dilatancy throughout deformation to failure. To explain Rowe’s model a typical assembly of rods is shown in Figure 2.23a. The angle of deviation of the tangent at the contacts between particles from direction 1 is defined as  1 are L 2 are the loads on each rod in directions 1 and 2 respectively. A ft. L  typical unit volume is ij 12 (it is assumed that the rods have unit length in the third direction). The volume of the assembly can be expressed by an integer number times the number of typical units of volume.  The conditions at each contact between two particles in the assembly are similar to those shown in Figure 2.24. This figure shows a rigid block sliding on an inclined  37  . The component of the reaction force 1 surface making an angle ,6 from the direction of L normal to the surface is N. The component of the reaction force parallel to the surface is Ntan  ,  where  1 is the particle to particle friction angle. Resolving the forces in the L  and L 2 directions gives:  Ltan(ç +/J)  (2.16)  From Figure 2.23a: tana=!L  (2.17)  From Equations 2.16 and 2.17:  =2L=tanatan(b+fl) 2 L c 1 2  (2.18)  1 8 and 52 are the deformations in directions 1 and 2 1 / 2 =L 1 and o . 2 / 1 Where o =L respectively at an angle /1 relative to those at an angle /3 (see Figure 2.23b). From the geometry, the following can be derived:  (2.19)  tanatanfi 2 1 1  6  81l2  Where  =  8/ 11 and  82  =  82/12. Assuming  that vertical compression, lateral expansion  and volume increase are all positive gives:  -=  i 6  (2.20)  1+-81  38  Where  = 61+82.  =  61 °2 2  From Equations 2.18, 2.19 and 2.20:  tan(q5  +  6)  (2.21)  tan 46  (1 + 6”  The term on the left hand side of Equation 2.21 represents the ratio between work done in the direction of the major principal stress on the assembly to that done by the assembly on the direction of the minor principal stress. This ratio is equal to one for the case where the particle to particle friction angle,  is equal to zero, i.e. in the absence of inter-  particle friction the dissipated work is equal to zero and therefore all work done on the major principal stress is transferred to the minor principal stress. (a)  -.  (b)  I 6/2  I  öiI2  Figure 2.23. Typical assembly of rigid rods (a) stress conditions (b) deformation characteristics (reproduced from Rowe, 1962).  39  L2  Li  Figure 2.24. Forces acting on a rigid block sliding on an inclined surface (reproduced from Rowe, 1962). For a random mass of irregular particles, the value of /3’ changes with loading as the particles orientations changes. It is assumed that this relocation happens such that “the rate of internal work done is minimum” (Rowe, 1962).  This assumption changes  Equation 2.21 to:  =tan ( 2 45+O.5q)  (2.22)  From experimental observations Rowe found it necessary to use qS (defined as the functional or mobilised friction angle) instead of q, where q5j varies depending on density and boundary conditions. The sign of the volumetric strain increment is changed so that volume decrease is positive following the sign convention used in soil mechanics. Rowe (1962) showed that Equation 2.22 is valid regardless of the boundary conditions. For triaxial conditions, the minor principal stress is o . This gives: 3  =  3 J  K(1 —  -)  (2.23)  6i  Where,  40  K = tan (45 + O.5b) 2  (2.24)  For triaxial conditions, Rowe (1969) showed that  varies between the inter-particle  friction angle and the critical state friction angle. Under plane strain, 5 ç is equal to the f friction angle at the critical state for any packing up to peak stress ratio. In the p ‘-q space, rearranging Equations 2.23 and 2.24 and assuming  q results in Equation 2.25 5  for triaxial compression.  =  9(M 7 9+3M—2Mi —  (2.25)  Where,  M=  6sinq&,, 3—sin  (2.26)  Schofield and Wroth (1968) introduced the Cam-Clay dilatancy rule based on plastic work dissipation mechanism as in Equation 2.27 (Cam-Clay was described in detail in Section 2.3.2). Roscoe and Burland (1968) modified Original Cam-Clay in what became the Modified Cam-Clay (MCC) model. Equation 2.28 is the MCC flow rule.  D”=M—ri  (2.27)  Dp=M 2i  (2.28)  Cam-Clay is widely used for soft clay, but the dilatancy rule does not match sand data well, particularly for dense sands. Nova addressed this issue in 1982 and developed an improved stress-dilatancy rule based on observations from laboratory data (Equation  41  2.29). Nova’s equation contains an additional volumetric coupling parameter (N) which usually falls in the range of 0.2-0.4.  D  =  (M—i ) 7 (1-N)  (2.29)  Figure 2.25 plots the Rowe, Cam-Clay and Nova flow rules for M1 .27 and N0.25. It is noteworthy that the trends are fairly similar in the dilatant range (i.e. for negative D”) for a typical critical friction ratio of 1.27 (i.e.  -1.5  -1  -0.5  0  =  D  31.6°).  0.5  1  1.5  2  Figure 2.25. Comparison between Rowe’s stress-dilatancy, Cam-Clay flow rule, and Nova’s rule.  Bolton (1986) used a large database of both triaxial and plane strain tests to relate the component of strength that is caused by dilatancy to initial density and mean effective stress. The component of strength caused by dilatancy is represented by the difference between the peak friction angle, qY,,, and the friction angle at the critical state, qYL,,. Triaxial data show that q  —  q5 is directly proportional to relative density and inversely  42  proportional to mean effective stress at failure (see Figure 2.26).  Bolton presented  Equation 2.30 from fits to triaxial laboratory data. Equation 2.30 is plotted in Figure 2.26 for different Dr values. This relation is very useful as knowing effective stress conditions, relative density and critical friction angle, peak friction angle could be computed.  —  (2.30)  =3[Dr(1O—lnp’)—l]  From plane strain data, Bolton found that the relation between the fraction of strength caused by dilatancy, i.e.  ‘  —  Ø,  and the angle of dilation, 0, is as in Equation 2.31,  where 0 is defined as in Equation 2.32. Bolton showed that his Equation, i.e. Equation 2.31, is very similar to Rowe’s relation in Equation 2.23.  (2.31)  = 0.80  1 0=sin  (2.32)  =sin’  —-  6183  Equation 2.31 is valid for plane strain boundary condition for the whole stress path including at peak. Bolton’s work implies that the fraction of strength at peak caused by dilatancy, ,ax  qi  —Ø  , —  for triaxial boundary conditions is:  (2.33)  =0480m  The problem now is that, unlike for plane strain, the dilation angle does not have a physical meaning for triaxial conditions. To derive Equation 2.33, it was assumed that the definition of the dilation angle in Equation 2.32 is valid for triaxial conditions.  Vaid and Sasitharan (1992) performed triaxial tests on Erksak sand with different stress paths and initial densities. Assuming that the definition for the dilation angle, Equation 2.32, is valid for triaxial conditions, they confirmed that at peak stress the friction angle is 43  uniquely related to 0 max regardless of the confining pressure and relative density. They also found this relation between peak friction angle and peak dilatancy to be independent of stress path. They used different triaxial stress paths in the p ‘-q space which included both compression and extension tests. Accordingly, Vaid and Sasitharan proposed a relation between q5  —  q and maximum dilation angle for triaxial conditions. They  measured q& , using the Bishop method that involves plotting the data in peak dilation vs. 1 peak friction angle (Bishop, 1971). A best fit linear trend line is plotted through the data points and the friction angle corresponding to zero peak dilatancy is çi. Their proposed relation is given by:  —  =  0•330  (2.34)  The factor on the right hand side of Equation 2.34 is lower than that in Equation 2.33, i.e. 0.33 is lower than 0.48. Equations 2.33 and 2.34 were developed for triaxial conditions. It should be noted that Equation 2.33 was developed to fit the data for 11 sands on average. Therefore, it is not surprising that Equation 2.34, developed for Erksak sand, is different from Equation 2.33.  Overall, according to Bolton, from Equations 2.31 evaluated at peak and Equation 2.33, the fraction of strength caused by dilatancy, qS 60% of that for plane strain conditions.  44  —  ,  for triaxial conditions is around  16 14 12 10 8 -  6 4 2  0 10  1000  100  10000  p’ at failure (kPa) Figure 2.26. Dilatancy component of strength as a function of mean effective stress at failure and relative density (reproduced from Bolton, 1986).  2.5.  The NorSand soil model  The constitutive model development in the following chapters is based on the general framework of the NorSand soil model. Therefore, NorSand is described in some detail in this section.  The discussion is limited to triaxial compression boundary conditions.  NorSand is an elasto-plastic critical state soil model developed by Jefferies (1993). Over the last 15 years the NorSand model has been updated, primarily to incorporate varying critical image stress ratio, M, and to provide improved predictions under plane strain. The version of Jefferies and Shuttle (2005) is described below. This section focuses on the monotonic version of NorSand. The cyclic version will be described in section 2.6.3. NorSand was the first critical state model to realistically model sand in that, unlike Cam-Clay, it predicts realistic dilatancy for dense soils (Jefferies and Shuttle, 2005). Like Cam-Clay, NorSand assumes normality, but NorSand also imposes a limit on the  45  hardening of the yield surface which allows for more realistic prediction of dilatancy for dense soils. The model requires 8 input parameters that can be easily determined from laboratory data (three critical state parameters, three plasticity parameters, and two elasticity parameters). NorSand, like other critical state models, is based on two basic axioms:  •  A unique critical state exists  •  The critical state is the final state to which all soils converge with increasing shear strain.  One of the main features of all versions of NorSand, which is a significant difference from Cam-Clay, is that NorSand has an infinity of normal consolidation lines (NCL) and not every yield surface is required to pass through the critical state. This behaviour was first reported by Tatsuoka and Ishihara (1974), from triaxial tests on Fuji River sand, who demonstrated that the normal consolidation line (NCL) for sands is not unique, instead being a function of density. They showed that looser samples yield at higher deviator stress for a given mean effective stress. Jefferies and Been provided additional data to confirm this finding in 2000 for Erksak sand (Jefferies and Been, 2000). The concept is illustrated in Figure 2.27. For every normal consolidation line there is a conjugate yield surface at each value of initial mean effective stress. The implications of having infinite NCL locations are:  •  The yield surface could exist anywhere in the e-q-log(p’) space. It does not necessarily need to intersect the critical state line as in Cam-Clay. Therefore, the hardening of the yield surface cannot be uniquely controlled by void ratio, and the slopes of NCL and the swelling line as for the OCCIMCC model. Hardening in NorSand is controlled by the plastic hardening parameter, H, that is a function of the state parameter and soil fabric.  •  To get representative predictions for dense sand in OCC/MCC, a high over consolidation ratio must be used even if the sand was normally consolidated,  46  i.e. it did not experience higher mean effective stresses in its history. In NorSand, the “intrinsic state” of soil is separated from overconsolidation and there is no need to assign an over-consolidation ratio to properly model dense normally consolidated sand (Jefferies, 1993). Instead, the concept of the state parameter previously discussed is utilized to determine the current location in e-log p’ space relative to the critical state.  CSL 0  L)  z  NCL Over-consolidated  Logp’ Figure 2.27. Infinite number of NCL’s (reproduced from Jefferies and Shuttle, 2002).  2.5.1. Yield surface and flow rule NorSand’s outer yield surface has an identical shape to the Original Cam-Clay surface (see Figure 2.28). In addition NorSand’s yield surface also has a straight vertical cap at a limiting dilatancy which occurs at a stress ratio coincident with peak stress conditions. In NorSand peak stress ratio,  7limit, T  is associated with peak dilatancy or Dmin if the sign is  taken in consideration (Figure 2.28). In the following discussion the curved portion of  47  the yield surface is called the outer yield surface and the vertical portion is called the inner cap or inner yield surface.  A soil stress path may intersect the inner cap in  unloading. This behaviour will be described in Section 2.6.3. Therefore the focus here is on the outer yield surface. NorSand defines the image condition as the boundary between the contractive and dilative behaviour in dense sands (see Figure 2.28). The image condition is differentiated from the critical state in that it satisfies only one condition of the critical state. At the image condition, D° M and  iy.  =  0 but D ’ 1  0. The stress ratio (q/p’) at image, M, is a function of  As soil reaches the critical state with shearing, the value of M approaches M  until they are eventually equal at the critical state. The idea of changing M is very similar to Rowe’s mobilised stress ratio, or mobilised friction angle  qc,  in Equation 2.24.  NorSand’s flow rule is very similar to the Original Cam-Clay flow rule except the variable M is used instead of M, as in Equation 2.35. The model uses associated flow (i.e. plastic strain ratio increments are normal to the yield surface).  (2.35)  7 D—M,—i  The derivation of NorSand yield surface follows the same steps as that for Cam-Clay (Equations 2.10-2.14). Substituting the value of D”, i.e. Equation 2.35, in Equation 2.13 gives NorSand yield surface as:  (2.36)  -7--=l—ln1Pr’  An expression for M is needed and Nova’s rule in Equation 2.29 is adopted here for peak conditions. Combining Nova’s rule at peak with equation 2.35 gives:  (2.37)  =M+ND 1 M  48  Been and Jefferies (1985) showed by plotting experimental data that there is a relation between Dmjn and state parameter, cv. There are three versions of this plot in the literature depending onp’ and e at which  yl  is evaluated:  1. Data is plotted in Dmin vs. the state parameter at initial conditions, 2. Data is plotted in Dmjn vs. the state parameter evaluated at Dmjn. 3. Data is plotted in Dmjn vs. the state parameter for image conditions evaluated at Dmin,  it’  where  =  e  —  1 (er, is the critical void ratio evaluated atp’,). A plot is e  shown in Figure 2.29.  The three versions of the Dmjn vs.  cv plot  show a trend of increasing dilation rate with  increasing state parameter. The slope of the trend line through the data points is  ,  a  NorSand model parameter that is used to impose a limit on the minimum allowable dilation rate and is a function of soil fabric. The second version of the Dmjn vs.  cit  plot is  the one adopted in this thesis. Accordingly, % = Dmin / cv. As elastic strains are negligible at peak conditions,  ,  =  / cv can alternatively be written as:  D  (2.38)  Combining Equations 2.37 and 2.38 gives an expression for M as:  (2.39)  The derivation considered dense sand only. As loose sand is expected to dissipate plastic work similar to dense sand, Equation 2.39 is changed to Equation 2.40, i.e. made symmetric about the critical state (Jefferies and Been, 2006).  49  =  M  —  xNlct’I  (2.40)  For a given outer yield surface, the location of the point at Dmin needs to be defined (see Figure 2.28). Evaluating Equation 2.36 at peak conditions and rearranging gives,  ,,‘  ,J  ) 1 1M —  (2.41)  —  max  Substituting the value of D”mjn  I;  —  ,  as in Equation 2.38, in Equation 2.41 gives,  e’’  (2.42)  —  max  The relative position of the M, Mand  7limit 7  lines in Figure 2.28 is not constant.  According to Equation 2.40, M 1 tends to M as the critical state is approached until they are eventually equal at the critical state where v  0. lT limit also decreases until it is equal  to Mat critical state (see Equation 2.42).  2.5.2. Hardening of the yield surface  The NorSand outer yield surface hardens until the point corresponding to  Dmin S  reached. This is followed by a softening response until the yield surface stops changing in size at the critical state. As the NorSand yield surface size is controlled by the dimensionless ratio of (p /p  ), the hardening rule, representing the change in the size of  the yield surface, is expressed by (pI p). The NorSand hardening rule takes the form of:  50  .  = H  [(J  (2.43) max  -  Where H is the plastic hardening modulus, a model parameter. The hardening rule is a function of  8’  image as when  because using s’ instead would result in a model that never gets past i  =  M,, s  =  0. The hardening rule gives better fit to data if it is give a  dependence on the shear stress level (Jefferies and Been, 2006). An exponential function is used to introduce this dependence. Hence, equation 2.43 is changed to:  []  =  He  (1/Mi)[Jmax  (2.44)  -  Figure 2.28. NorSand yield surface (modified after Jefferies and Shuttle, 2005).  51  a_  0 :.a . •  Data from 13 sands  a  :ii  a  -0.2  I  1  a a  a  o6  a a ag  II  — a  a  •a a  a  I  a a  — a —  a I  .4-  I  a a  a  a  :  a  ——  a a  -0.8 —  -0.3  a  1 -0.2 -0.1 State parameter at image,  Figure 2.29. Minimum dilatancy as a function of state parameter at image for 13 sands (modified after Jefferies and Been, 2006).  2.5.3. Typical evolution of the yield surface  The hardening and softening ofNorSand yield surface is described as follow (see Jefferies 1993):  •  It is assumed that we are starting with a soil denser than the critical state.  •  With increasing shear strain the yield surface hardens, with the size of the yield surface during hardening controlled by the mean effective stress at image, p’s. equals p,.  •  Soil remains contractive until the current mean effective stress  Although the current stress ratio is equal to M, the movement of the yield surface does not stop because the image state only satisfies one condition of the critical state. The hardening continues with increasing shear strain in a  52  0  dilative manner until it reaches the surface corresponding to the limiting stress and maximum allowable dilation rate. •  At this point, softening starts with a decreasing rate as it approaches the critical state.  •  At the critical state M  =  M and the yield surface does not move any further.  2.5.4. Elastic properties of NorSand In Cam-Clay elasticity, the elastic shear strains are ignored. NorSand does not ignore the elastic part of shear strains and variations on elasticity including standard linear elasticity and a range of stress dependent models have been implemented.  2.5.5. Summary of the NorSand model  The full set of equations that specify the NorSand model presented in the preceding sections are given in Table 2.1. Table 2.2 lists the parameters used in the model and their typical ranges. The parameter ranges were primarily obtained from calibrations to sand, so care should be exercised when applying to other soil types.  53  Table 2.1. Summary of NorSand equations (modified after Jefferies and Shuttle, 2005). Aspect of NorSand  Equation  Internal model parameters  ,p-  =  e  —  e =F—2eln(p’) and,  M. =M-x411 Yield surface  LL = 1 1 M  with  —  PI}  Flowrule  9 =M D’ — 1  Hardening of outer yield  (  surface  =  He  I -E--l L1’  =  G /p  54  e  Jmax  )E(J’ 1 (1/M  [ Elasticity  =  Pjmax  —  ‘  IM ) 1  Table 2.2. Summary of NorSand parameters (after Jefferies and Shuttle, 2005). Parameter Critical state F  I  Typical range 0.9-1.4 0.01  2e  —  0.07  1.2-1.5  JvI Plasticity H %tc  50-500 2.5-4.5  N  0.2-0.4  Elasticity Jr  100-800  v  0.1-0.3  Description The y-intercept of the e log(p curve at 1KPa The slope of CSL in e log(p’) space defined on base e q/p’at critical state Plastic hardening modulus A parameter that limits the hardening of the yield surface The volumetric coupling parameter (used in Nova’s rule) Dimensionless shear rigidity (G/p Poisson’s ratio  2.6. Soil behaviour in unloading  While there have been relatively many studies addressing the overall cyclic behaviour of sand, little work has been done to study the behaviour of sand during the unloading phase in detail. It is interesting that sand also shows contractive, in addition to the expected dilative, behaviour when unloaded. The implications of this behaviour were discussed in Chapter 1. This section discusses previous work on the topic.  2.6.1. A Simple physical model Jefferies (1997) explains soil contraction in unloading in terms of stored potential energy during the loading phase. Assuming the saw tooth model represents how soil dilates, when dense sand is loaded grains tend to climb over the slip surfaces (see Figure 55  2.30a). This is associated with increase in volume as the voids between the teeth are increased.  At the end of loading the potential energy of those particles has been  increased by the virtue of their new location. When unloading, it is then easy to imagine that those particles will tend to slide backwards (see Figure 2.30b). This is associated 2<d ). 1 with decrease in volume as the voids between the saw teeth get smaller (d  I  (a)  1 d  (b)  I 2 d  ‘I >d 1 d 2 Figure 2.30. The Saw Tooth Model a) loading phase b) unloading phase.  2.6.2. Thermo-mechanical approach The first law of thermodynamics states that “The increase in the internal energy of a system is equal to the amount of energy added by heating the system, minus the amount lost as a result of the work done by the system on its surroundings”. Alternatively, plastic work done on soil is either dissipated in the form of frictional energy or contributes to the increase of internal energy. Cam-Clay assumes that all ‘plastic’ work done on soil is dissipated. This means that plastic work does not contribute to changing internal energy. Part of the total work increments is recoverable (termed ‘elastic’) and the other part is irrecoverable (termed  56  ‘plastic’) as in Equation 2.45. Cam-Clay is rigid in ‘elastic’ shear and only recovers ‘elastic’ volumetric strain. Therefore, the Cam-Clay approach assumes that any change in internal energy is only due to an ‘elastic’ change of volumetric strain that can be calculated using the slope of the swelling line in a usual consolidation test (Schofield and Wroth, 1968). The ‘plastic’ component of work is dissipated and the dissipation rate is assumed constant and equal to the critical friction ratio, M. The term on the right hand side of Equation 2.45 represents plastic work dissipation and cannot be negative (as all work dissipation is positive). Dividing Equation 2.45 through by ./i6q’ and rearranging yields the Cam-Clay flow rule, Equation 2.46.  (2.45)  Where, is the ‘plastic’ work (unrecoverable according to Cam-Clay) done per unit volume  is the total work done per unit volume  is the ‘elastic’ work (or recoverable) per unit volume  (2.46)  However, the Cam-Clay flow rule does not fit sand data as well as Nova’s rule in Equation 2.47. Nova (1982) derived his flow rule based on experimental observations.  Dp=M7 1-N  Upon substituting for D” and  (2.47)  ,j  and rearranging,  57  + p’  =  (2.48)  Mp + Np’  If soils were not to violate the first law of thermodynamics, then work done on the soil sample is either dissipated or contributes to a change in the internal energy of the sample. The two terms on the right hand side of Equation 2.48 represent plastic work done. The first term on the right hand side represents the dissipation mechanism as discussed earlier in this section. It is then reasonable to assume that the second term on the right hand side contributes to a change in internal energy. In other words it represents a stored energy. Jefferies (1997),calls it ‘plastic’ stored energy. It is not elastic as it is not reasonable to assume that plastic work done on the sample is transferred into stored elastic energy. It is stored energy, i.e. not dissipated, because the term can take negative sign. Cam-Clay assumes that all plastic work dissipation is represented by the first term on the right hand side. Based on this assumption, any other term on the right hand side represents something other than dissipation of plastic work.  Therefore, according to  thermodynamics, it represents changed internal energy or stored ‘plastic’ energy.  The idea of a change in internal energy due to change in plastic strains was first proposed by Palmer (1967). Palmer’s approach is illustrated in Figure 2.31.  Total Work Done  frictional energy [ Dissipated f() L J r  1  Change in internal energy  L______________________ I  \  Due to change in (rigid in elastic shear)  J  I  r  L  Due to change in ‘ (term ignored in Original Cam-Clay)  Figure 2.31. Energy balance as introduced by palmer (1967).  58  To justify this, Palmer (1967) considers a hypothetical experiment where the state of soil moves along the critical state line in the e-p’ space. While moving along the CSL shear deformations -in this case all deformations are plastic as Palmer’s model as well as Original Cam-Clay are rigid in elastic shear- resisted by friction are not expected to contribute to any change in volumetric strain and the Original Cam-Clay energy balance equation reduces to:  (2.49)  =  But because we are hypothetically moving on the CSL, and different pressures are associated with different critical void ratios, then there must be a change in volumetric strain. Most of this change is ‘plastic’ because the CSL is usually much steeper than the swelling lines. However, Equation 2.49 fails to predict this change. Therefore, another term should be added to represent changes in internal energy due to change in plastic volumetric strain. This term turns out to be the ‘N’ term on the right hand side of Equation 2.48.  Jefferies (1997) assumes that all the stored ‘plastic’ energy is recovered upon unloading. Solving Equation 2.48 for the case of unloading while changing the sign of the ‘N’ term, as it is energy recovered in unloading, gives the following:  (2.50)  q8’+p8’ ——Mps’—Np6,’  Upon rearranging and substituting,  Dp=M 1+N  (2.51)  Equations 2.46, 2.47 and 2.51 are plotted in Figure 2.32. It will be shown later that triaxial laboratory data shows a different trend for stress-dilatancy in unloading from that represented by Equation 2.51.  59  Nova’s rule (M = 1.27 andN = 0.4) 0.8 Stress-dilatancy in unloading\ according to Jefferies (1997); 1 =(-M- )i(1+N) forM = D’ 1.27andN=0.4  \  -3  -2  0.4  0  -1  1  2  D Figure 2.32. Stress-dilatancy for Cam-Clay loading, Nova loading, and Jefferies (1997) unloading.  Collins (2005) discusses a different conceptual model for yield in unloading from a thermo-mechanical viewpoint, taking into consideration differences between the micro scale where the particles interact, and the continuum scale where most soil constitutive models are defined. The model is summarized in Figure 2.33. The difference between Collins (2005) approach and that of Palmer (1967) and Jefferies (1997) is that the former assumes that stored elastic energy is the cause for yield in unloading while this is not the case for the latter. Collins model is illustrated in the following paragraph. Pure hydrostatic loading on the continuum scale is assumed and following the usual convention the applied work may be separated into an elastic and plastic component. During loading part of the applied work is dissipated while the remainder is stored in terms of elastic compression of soil particles. In subsequent unloading, part of the stored elastic work is released causing dilation while the other part can only be released if  60  associated with particle rearrangement. Particle rearrangement is not elastic and hence plasticity occurs during unloading. Hence it is implied that all plastic strains during unloading are dilative. It is assumed that most of the total shear energy is dissipated as plastic work.  Total work done in loading  Hydrostatic compression component  Stored (elastic compression of particles on the micro scale)  Shear component  Dissipated (plastic particle rearrangement)  Most of the total shear energy is dissipated as plastic work  Upon unloading, part of stored elastic energy is recovered (elastic expansion with no particles rearrangement)  Stored elastic shear energy (Very little contribution to frozen energy)  Upon unloading, some of the stored elastic energy cannot be recovered without particle rearrangement. The energy associated with particle rearrangement is termed ‘frozen energy’ and is dissipated as plastic dilation during unloading. Figure 2.33. Schematic representation of work storage and dissipation according to Collins (2005).  61  2.6.3. Unloading in NorSand Jefferies (1997) presented a framework for the NorSand model in unloading and subsequent reloading. Because this model is extended as part of the current work, a more detailed discussion of the Jefferies (1997) unloading model is provided in this section. The NorSand model was described in Section 2.5 with emphasis on monotonic loading conditions. This section discusses in more details the unload-reload version of NorSand. In unloading, soil yields at the inner cap. The inner yield surface (or inner cap) is the vertical part of the yield surface shown in Figure 2.28. Its location at the outer yield surface is chosen to fit within the framework of the NorSand model in loading and is a vertical straight line for simplicity. This internal cap scales with the outer yield surface and is located at:  Peap =  (2.52)  e(_1)mjM1)  The NorSand flow rule in unloading was derived earlier in Section 2.6.2 as Equation 2.51. Jefferies (1997) introduced a rule to govern the movement of the inner cap, i.e. a hardening rule, as:  8v  1  “lflI Hp  I  (2.53)  %P)  Where, H is the hardening (softening) modulus in unloading pji is the mean effective stress at first yield in unloading (i.e. the mean stress of the cap when first intersected)  62  So far the model definition is completed.  The rest of this section presents two  examples of stress paths (Figure 2.34 and Figure 2.35) to illustrate the model behaviour. Figure 2.34 shows a stress-strain curve with a single unload-reload loop and the yield surfaces corresponding to the load-unload-reload phases.  The points of interest are  annotated on the stress-strain curve, i.e. the plot at the left top side, and on the yield surfaces corresponding to loading, unloading and reloading, i.e. plots on the right top, left bottom, right bottom sides respectively. The darker lines represent the surface where current yield is occurring. The thicker lines represent the stress path. The yield surface hardens with loading until the internal cap is reached as in Figure 2.34, i.e. path 1-2. Point 2 in the figure represents peak strength and is associated with the maximum size of the yield surface. With continued shearing further strain causes softening and the stress path softens to reach point 3. Another stress path is illustrated in Figure 2.35 which has a similar arrangement as for Figure 2.34. Unloading in this case occurs from a point before reaching the internal cap that represents peak conditions. Loading causes hardening of the yield surface along path 1-2. The internal cap scales with the yield surface. In unloading, there are three possible cases for the stress point to move on or inside the yield surface:  Case 1: The stress point touches the internal cap in loading, unloading then cause plastic softening of the yield surface. This is illustrated in Figure 2.34 where yielding occurs as the stress point moves on the internal cap from point 3 to point 4. As the cap moves with the stress point, the outer yield surface also softens. Case 2: The stress point does not touch the internal cap in loading, as shown in Figure 2.35. Upon unloading, the yield surface does not move until the stress point touches the internal cap. Before this point, unloading is purely elastic (Figure  63  2.35). After the stress point touches the cap yielding starts on the cap and the yield surfaces soften until the stress point reaches location 3. Case 3: Under these  The third case occurs for unloading early in the stress path.  circumstances the stress point does not touch the cap during unloading and the whole unloading phase remains elastiö. Under all situations reloading is elastic as long as the stress point is inside the outer yield surface (see Figure 2.34 and Figure 2.35). Once the stress point touches the outer yield surface, plastic reloading continues as in the virgin loading phase.  S  -  Loading  2  1.20  where  120  Sreface where cuffent yield 0.00  100  0.40  •  Cununtotresopoint  1  0  5  15  10  20  0  25  200  200  200  400  000  660  00  er:% 5  200  -  3  Reloading  Unloading  ,.-‘  200  /  200  surfaces where  ,‘,  Surla swhere current y Id is 0 urnng  100  I. £Ittsti.relooding  0  100  200  300  400  600  600  0  p.  10  200  400  300  000  600  700  p.  Figure 2.34. Movement of yield surface in NorSand: Case of unloading from a point on the internal cap.  64  2  1.80  :  Loading  200  °  2  /7  Surface where cusT006 yield  /  \pcciJmng  I:  0  16  5  65  20  570  0  300  300  200  /  /  Unloading  V}  200  400  300  600  800  700  4  A!  230  Suaces ere current yId is  .  Reloading  It  S  ces ere  ttcre1oad\\’”\ :  ‘: 0  100  200  400  300  570  800  700  0  p  100  200  300  400  500  600  700  p’  2.35 Movement of yield surface in NorSand: Case of unloading from a point before reaching the internal cap.  Figure  2.6.4. Summary It can be seen from this selective review of soil behaviour in unloading that soil behaviour in unloading is still not well understOod. There is no agreement on the cause of yield in unloading, for example the Jefferies model implies that it is mainly caused by plastic shear deformation in loading while Collins attributes yield in unloading to rearrangement caused by elastic dilation of particles associated with reduction in mean effective stress.  Clearly, this topic needs more investigation as it is important for  earthquake engineering.  A practical model that accounts for yield in unloading is  required. Understanding stress-dilatancy in unloading is one of the requirements of such a model and is discussed in the following chapter.  65  3. DILATANCY IN UNLOAD-RELOAD LOOPS: A THEORETICAL INVESTIGATION  Dilatancy in loading has been investigated by many researchers as discussed in Section 2.4. Most elasto-plastic constitutive models have yield surfaces that were developed for stress paths involving increasing shear; a reduction of shear stress (i.e. unloading) within that surface is considered elastic. But contraction has been observed during unloading for the standard triaxial stress path.  Standard elasticity would predict expansion during  unloading. Hence, these observations suggest that the soil is yielding during unloading. Constitutive models that incorporate yield in unloading are rare. The topic is not well covered and is controversial as shown in Section 2.6. The objective of this chapter is to develop theoretical understanding of dilatancy in unloading as well as in subsequent reloading phases. The investigation is done at the micro-mechanical level.  3.1. Micro-Mechanical perspective for dilatancy in unloading The standard elasto-plastic approach assumes that soil is a continuum. However, in reality, soil is composed of discrete particles and plasticity is an abstraction used to explain what really happens between the grains.  It is of interest to develop an  understanding of why soil contracts in unloading from a microscopic point of view. Rowe (1962) derived an expression for dilatancy in loading based on frictional forces between rigid cylindrical rods (see Section 2.4). Rowe assumed identical rods that are rigid and have a circular cross-section. The forces at the contacts are assumed purely frictional and the initial packing does not change during shearing. Packing represents the pattern at which particles are arranged relative to each other. For example, Figure 3.1 shows one possible packing for the rods but three different particle assemblies. The three  66  different particle assemblies in the middle of Figures 3.1 a,b, and c have the same packing. The relative arrangement of particles in the three assemblies does not change, i.e. if particle ‘x’ happens to be to the right of particle ‘y’ in the first assembly, then it stays to the right of particle ‘y’ in the other two assemblies. A change in the volume of the packing can be explained by taking four particles aside. In loading, as illustrated using the four particles on the left hand side of Figure 3.1, if the upper grain is pushed vertically downward the  two  side grains will move outwards. This  will be associated with an increase in volume if the angle between the tangent to grains interface and the vertical direction, f3  >  45° (see Figure 3.la). However, for f3  <  45° when  the upper particle is pushed down the assembly decreases in volume (see Figure 3.1 c).  By computing the work done by the major principal stress on the assembly to the work done by the assembly on the minor principal stress for rigid rods, Rowe derived the following equation (the complete derivation is given in Section 2.4):  =  2  ; (1+  )  tan(qS + ,6) tan fi  (3 1)  Where, a’j  is the major principal effective stress  a’2  is the minor principal effective stress is the rate of change in major principal strain  2  is the rate of change in minor principal strain is the rate of unit volume change  q is grain to grain friction angle 5 fi is the angle between the tangent to grains interface and the vertical direction And for the packing in Figure 3.1,  67  =  tan(/1) tan(q5  +  (3.2)  /3)  a  LOADING  UNLOADING  a) I2  32’  I2  4,  2  82!  4-  2  oI 2 4-  -  12  f  t  b)  12 4, 1 W  I2 1 o  2 a  82! 2  I2  -  6212 4-  o12  C)  I2 4,  8 12  Figure 3.1 Micro-mechanical representation of dilatancy for a uniform packing of rigid rods during both loading and unloading a) Minimum void ratio for ft = 600 b) Maximum void ratio forfi = 450 c) Minimum void ratio forfi = 30°  Equation 3.1 is valid for different packings of rigid rods but the stress ratio in Equation 3.2 depends on the packing type (i.e. the pattern at which particles are arranged relative to each other). For the packing in Figure 3.1, Rowe showed that stresses and strains in  68  direction 1 over those in direction 2 can be expressed as in Equations 3.2 and 3.3, respectively.  1 2 tan  82  (3.3)  /3  Multiplying Equation 3.2 by 3.3 yields equation 3.1. Note that Equations 3.2 and 3.3 are identical to Equations 2.20 and 2.21 for a = /3, which is the case for this packing. For the packing on the right hand side of Figure 3.2 Li and Dafalias (2000) showed, following a similar derivation as for Equations 2.20 and 2.21 in Section 2.4, that Equations 3.4 and 3.5 below should be used instead of equations 3.2 and 3.3, respectively. The reason for having different equations is that the volume of the basic unit defined by the dashed rectangle in Figure 3.2 for each of the packings is different. The complete derivation is given in Li and Dafalias (2000).  --=tan(q5,  ——  62  2sin/5 1+2cos/1  (34)  (1+2 cos /3) cos /3 fl 2 2sin  (3 5 )  Note that multiplying Equation 3.4 by 3.5 also yields Equation 3.1. Therefore, Equation 3.1 is valid for different packings, while the ratio between stress in direction 1 to that in direction 2 is packing specific. Therefore, if the term  6v18i  in Equation 3.1 is assumed  to represent dilatancy, then there are different stress-dilatancy relations for different packings.  69  Unloading can be explained in the physical sense by reference to the illustrations on the right hand side of Figure 3.1. If the side grains are pushed inwards, the upper and lower grains will move outwards. This is associated with decrease of volume if fi>  450•  As discussed above, Equation 3.1 is derived for loading based on an energy balance between the ratio of work done by a strain increment in direction 1 on the assembly to that done by the assembly in direction 2. Part of work done in the form of a strain increment on direction 1 is dissipated in the assembly by friction while the remaining work is transferred to direction 2. Assuming that soil is an isotropic material and the packing does not change during the loading and unloading phases, work balance in unloading can be thought of as the ratio between the work done by a strain increment in direction 2 on the assembly to that done by the assembly on direction 1. In other words, the proposed expression for dilatancy in unloading based on grain to grain friction is the same as the usual one of Rowe (1962) but with the assembly rotated by  8i U;62  =  =  =  J;(1+6v/6i)  tan(90  —  900,  i.e.:  tan(q5M +90— tan(90-fi)  (3 6)  fi) tan(q + 90— ,6)  (3.7)  Figure 3.3a plots the proposed relation for dilatancy for unloading, i.e. Equations 3.6 and 3.7, as compared to that for loading.  fi  for unloading is equal to 90°-fl of that for  loading, as a consequence of rotating the assembly (Figure 3.3b). Note that Equation 3.7 is only valid for the packing in Figure 3.1.  Erksak 330/0.7 sand is a quartz sand with an average grain size of 330.tm. The grain to grain friction angle, q, can be estimated for quartz as 25° (Rowe, 1962). Figure 3.3a  70  shows a comparison between the dilatancies for the loading phase and first unloading for 0 a drained triaxial test on Erksak sand (p’ = 100 kPa and e  expressions based on grain to grain friction for q  =  =  0.653) to the theoretical  25°. The fit is a very good one for  both loading and unloading considering the previously stated assumptions involved in deriving the theoretical expression. However, it is not as good for the reloading phase and the second unloading ioop (not shown in the figure). This is to be expected as soil fabric changes with continued shearing, while the theoretical expression is only valid for the packing in Figure 3.1.  Packing B  Packing A  Figure 3.2. Two different uniform assemblies of rigid rods; the dashed rectangle represents the basic unit volume (reproduced after Li and Dafalias, 2000).  3.2. Micro-Mechanical perspective for dilatancy in reloading The stress-dilatancy relation is expected to change for different reloading phases. The reason is due to particle arrangement (i.e. fabric) changes during shearing. Let us assume that we have two different uniform packings of rigid cylindrical rods similar to those in Figure 3.2. Packing B has a higher void ratio than packing A. For the sake of the argument, it is assumed that during an unload-reload phase the arrangement of the rods changes from a packing similar to B to another one similar to A. This change can be  71  thought of as being equivalent to change in fabric in real soils. As discussed in Section 3.1, the stress-dilatancy relation is different for the two packings. Equation 3.1 is valid ) is different for packing A 2 for the two packings. However, the stress ratio (i.e. oil a and B as in Equations 3.2 and 3.4, respectively. The stress ratio in unloading for packing A is as in Equation 3.7. Similarly, the equation for stress ratio in unloading for packing B is:  -1-=tan(bM +90—fl)  2sin(90—fl) 1+2sin(90—fl)  (3.8)  Equations 3.1, 3.2, and 3.4 were used to plot the loading curves in Figure 3.4. Equations 3.6, 3.7 and 3.8 were used to plot the unloading curves. The predicted stress ratio for a given dilatancy is lower for the denser packing as expected. The trend from triaxial laboratory data agrees with the trends in Figure 3.4 as will be shown in the next chapter.  Rowe (1962) recognized that particle relocation occurs with shearing, and as a result the value of/I changes in a non-uniform manner because real soil particles are not of the same size and shape. He assumed that this relocation would happen in a way such that changes in the values of/I would keep the rate of internal work done to a minimum. This assumption changes Equation 3.1 to Equation 3.9 which is independent of fi and therefore independent of packing and density (the complete version of Rowe’s derivation is given in Section 2.4). The assumption of minimum internal work predicts a single stress dilatancy relation to be valid for all packings. Li and Dafalias (2000) disagree with Rowe’s use of the assumption of minimum internal work and therefore they predict that the stress-dilatancy relation is different for different packings. Rowe’s stress-dilatancy, Equation 3.9, is an idealization that is applicable for a random mass of irregular soil particles. It contradicts the exact solution, i.e. Equations 3.1-3.8, that clearly shows that stress-dilatancy is dependent on packing.  72  a)  C,  -0.5  -1  0  •  8  b)  / s  0.5  i on ‘Jy  Loading  55 50  35 30  .  -05  —1  0  0.5  6v/81  Figure 3.3 Theoretical expression based on grain to grain friction (q=25°) for the uniform packing in Figure 3.1 a) compared with a drained triaxial test on Erksak 330/0.7 0 = 0.653) in stress ratio vs. dilatancy space, b) Angle between the (p’= 100 kPa and e vertical direction and the tangent at the interface between grains.  = U;  5+O.Sç!i) 4 2( tan  (3.9)  (1+ d V/V )  Overall, this section showed that the stress-dilatancy relation is dependent on packing. For example, packings A and B in Figure 3.2 have different stress-dilatancy relations as  73  shown in Figure 3.4. As the packing is expected to change in a reloading phase compared to that for first loading, a change in stress-dilatancy is expected in reloading.  3.3. Summary The study of two packings of rigid rods showed that yield in unloading occurs. In this chapter, a stress-dilatancy relation in unloading was derived based on a micro-mechanical approach. The derived relation turns out to be identical to Rowe’s stress-dilatancy in loading while rotating the packing of rods by  900.  The study of deformation  characteristics in reloading using a micromechanical approach showed that stress dilatancy changes in reloading compared to first loading. Reloading is associated with a more dilative response than first loading.  0  -1  -0.5  0  •  8  vi  6  0.5  1  Figure 3.4. Rowe’s stress-dilatancy relation based on grain to grain friction for the two packings in Figure 3.2  74  4. DILATANCY IN UNLOAD-RELOAD Loops: AN EXPERIMENTAL INVESTIGATION  The previous chapter addressed dilatancy in unloading and reloading from a micromechanical point of view. In order to compare the trends predicted from the micromechanical approach to the trends observed in real soils, and to apply these observed trends to a general continuum model for future application to liquefaction modelling, this chapter investigates observed stress-dilatancy for two sands in drained triaxial tests that include unloading and reloading cycles.  4.1. Sands Tested Erksak sand (ES) and Fraser River sand (FRS) were used in this study. ES was chosen because drained triaxial tests with load-unload-reload cycles were available (Golder Associates, 1987; www.golder.com/liq).  Note that the focus of this chapter is to  investigate stress-dilatancy and therefore drained tests were used.  FRS was chosen  because of the availability of new monotonic triaxial tests and drained load-unload-reload triaxial tests undertaken by Golder Associates.  4.1.1. Erksak Sand Erksak sand, a sand that was used in the construction of the Molikpak core in the Canadian Arctic, is a uniformly graded medium-grain sub-rounded sand, mainly composed of Quartz and Feldspar. The gradation of Erksak sand used, Erksak 330/0.7, had an average particles size of 330 jim and fines content of 0.7%. The Index properties are presented in Table 4.1. Its specific gravity is 2.66. The index density measures, emin 75  and emax, according to ASTM test methods D4253-00 and D4254-00 are 0.525 and 0.775, respectively (ASTM 2006a; ASTM 2006b; after Sasitharan, 1989).  4.1.2. Fraser River Sand Fraser River sand originates from the alluvial deposits of Fraser River located in British Columbia, Canada. It is a uniformly graded medium-grain sand with angular to subrounded particles.  It is mainly composed of Quartz, Feldspar and unaltered rock  fragments with an average particles size of 260 im (see Table 4.1). Its specific gravity, emin, and emax are 2.75, 0.62, and 0.94, respectively.  Table 4.1: Index properties of Fraser River and Erksak sands  Mineralogy  Median grain size : 50 pm D Effective grain size : 10 pm D Uniformity coefficient Percentage passing no. 200 sieve Specific gravity of particle Grain description Maximum voids ratio em Minimum voids ratio emin  Fraser River sand,  Erksak sand,  Sriskandakumar (2004) and Chillariage et. al. (1997)  Been et al. (1991) and Sasitharan (1989)  40% Quartz, 11% feldspar, 45% unaltered rock fragments and 4% of other minerals 260 170 1.6  73% Quartz, 22% Feldspar, and 5% other minerals  0  07  2.75  2.66  Angular to subrounded 0.94 0.62  Sub-rounded  76  330 190 1.8  0.775 0.527  4.2. Testing program  All tests reported in this section are conventional triaxial compression tests (i.e. confining stress is kept constant during the shearing phase). The testing procedure and sample preparation methods are described in Appendix E. The full Erksak testing program included 29 drained and 39 undrained triaxial tests. As this investigation focuses on volumetric changes drained tests were of primary interest, although five of the undrained tests on ES were used to better define the location of the critical state line (see Chapter 6). Of the available drained triaxial tests, the ten drained tests that followed a conventional triaxial stress path, and also contained load-unloadreload cycles, were used for this work. The data for all of the Erksak tests are available for download from the web site: www.golder.com/liq.  The FRS testing program included 6 drained and 2 undrained triaxial tests.  Six  monotonic tests on FRS, 4 drained and 2 undrained, are used for the monotonic calibration of NorSand to FRS (see Chapter 6). The two drained unload-reload tests on FRS are used to validate the predictions of the calibrated unload-reload model in Chapter 7.  4.2.1. Erksak Sand Testing Program The ES load-unload-reload tests are summarized in Table 4.2. The ten tests cover a wide range of mean effective stresses (100-800 kPa). The range of void ratios is 0.6030.723. All of the samples were water pluviated except for ES_CID_868 that was moist tamped. These ten tests were performed under drained conditions to enable review of the volumetric change characteristics of sand. The number of unload-reload loops in the tests varies between one and three.  In some tests the unload-reload loops occur before  reaching peak strength, while other unload-reload loops are post-peak. This allows the  77  investigation of stress-dilatancy during unloading and reloading at different strain and stress levels. The ES undrained monotonic triaxial testing used to determine the critical state line is summarized in Table 4.3. All of the moist tamped tests are on loose samples with void ratios close to emax. Undrained conditions and loose samples were chosen so that the samples can reach critical state within the limitation of the apparatus.  Table 4.2: Drained triaxial compression tests on Erksak Sand with load-unload-reload cycles (data from www.golder.com/liq) P’ Test No. (kPa) ES_CID_860 100 ES_CID_861 100 ES CID_862 100 ES_CID_866 400 ES_CID_867 400 ES_CID_868 400 ES_CID_870 800 ES_CID_871 800 ES_CID_872 800 ES CID_873 100  0 e  0.672 0.645 0.645 0.698 0.680 0.723 0.653 0.637 0.652 0.603  Preparation method’ WP WP WP WP WP MT WP WP WP WP  Number of 2 U-R loops 1 2 3 2 3 2 1 2 3 3  ‘WP stands for water pluviated and MT stands for moist tamped. 2 U -R stands for unload-reload  Table 4.3: Undrained monotonic triaxial compression tests on Erksak sand (data from Beenet.al., 1991)  Test No. ES_L_601 ES L_604 ES_L_605 ES_L_606 ES L607  0 e  p’ (kPa) 499 699 500 701 701  0.754 0.768 0.766 0.759 0.748  MT stands for moist tamped. 1  78  Preparation 1 method MT MT MT MT MT  4.2.2. Fraser River Sand  Table 4.4 presents the test conditions of two load-unload-reload tests undertaken recently by Golder Associates on FRS. Both samples were moist tamped. The confining pressure for each test is similar at 190 kPa and 198 kPa, but the corresponding void ratios differ, being 0.89 and 0.72 respectively. Both tests had more unload-reload loops than the ES tests (four and five loops, while the maximum number of loops for ES was three).  Four drained and two undrained monotonic triaxial compression tests on moist tamped samples were carried out (see Table 4.5 for a summary of the test details). The undrained tests were used to obtain critical state parameters, while the drained tests were used for the FRS monotonic calibration to the NorSand model. The consolidation stresses ranged from 50 kPa to 515 kPa and e ranged from 0.63 to 0.91. 0  Table 4.4: Drained triaxial compression tests with load-unload-reload cycles on Fraser River sand (data provided by Golder Associates)  p’ (kPa) Test No. FR_CID 01 FR_CID_02  190 198  Preparation method’ 0.89 MT 0.72 MT 0 e  MT stands for moist tamped. 1 2 U R stands for unload-reload.  79  Number of UIR loops 5 4  Table 4.5. Monotonic triaxial compression tests on Fraser River sand (data provided by Golder Associates) P’ (kPa) Test No. FRCID 03 FR_CID_04 FR CID 05 FR_CID_06 FR_CU_01 FR_CU 02  114 410 515 50 388 196  Test type Drained Drained Drained Drained Undrained Undrained  0 e 0.67 0.63 0.69 0.75 0.91 0.82  Preparation 1 method MT MT MT MT MT MT  ‘MT stands for moist tamped.  4.3. Experimental observations The main focus of the review of experimental results was stress-dilatancy in unloading and subsequent reloading. To the knowledge of the author, an experimental review to specifically investigate stress-dilatancy trends in unloading has not been done before. The issue of concern .is what factors determine whether soil is likely to contract or dilate in unloading and the amount of those volumetric changes. The effect of the loops on peak strength and volumetric changes in reloading is also investigated.  In the following discussion “U” refers to an unloading loop and “L” refers to a loading or reloading loop. The number following the symbol denotes the order of a particular loop from the beginning of the test.  A typical test on Erksak Sand is plotted in Figure 4.1. The strength of the sand, shown in Figure 4.la, does not seem to be highly affected by the unload-reload loops. The data shows that loop U 1 does not introduce a local peak in the stress-strain curve. However, loops U2 and U3 show small peaks slightly affecting the stress-strain curve. Loop U 1 occurs before the image state which marks the boundary between contractive and dilative behaviour as annotated in Figure 4. lb. Loop Ui is not followed by a small peak and the stress strain curve seems to continue as if the unload-reload loop did not exist. However, loops U2 and U3, post-image loops, are followed by small peaks. The peaks on reloading 80  appear larger in Figure 4.2 which plots shear stress vs. axial strain (rather than the stress ratio vs. axial strain plotted in Figure 4.1 a).  The other tests on Erksak sand (see  Appendix B) and Fraser River sand (Figure 4.3) demonstrate similar behaviour. Conversely, volumetric strains are dramatically influenced by the unload-reload loops. Both the absolute values and the rates of change are affected (Figure 4.lb). Unloading occurs starting from pre-image conditions for Ui.  Note that volumetric changes  associated with unloading are very small and are initially contractive followed by a small However, for U2 and U3,  dilative phase (see Figure 4.4 for a zoom on loop 1).  volumetric changes associated with unloading are significant.  They are dominantly  contractive (see Figure 4.4 for a zoom on loop 2). Note that unloading for those two loops starts from post-image conditions. Table 4.6 summarizes the direction of volumetric changes in unloading for the loadunload-reload tests on ES. The information provided includes the stress ratio and axial strain at image stage.  For each unload-reload (U-R) loop the stress ratio at which  unloading starts is given, together with the axial strain, whether the ioop starts pre or post image, and whether the volumetric strains are dilative or contractive. It can be observed that whenever unloading happens from post-image conditions, volumetric strains are either totally contractive or dominated by contraction. Conversely, if unloading occurs from pre-image conditions, volumetric strains are either totally dilative or dominated by dilation. Table 4.6. Direction of volumetric changes in unloading for the load-unload-reload tests onES Test  ES_CID_860  ES_CID_861  U-R loop’  Stress ratio  Axial  Pre/Post  Dilative/Contractive  Strain (%)  image  during unloading  Image  1.114  0.402  At  1  1.419  10.073  Post  Image  1.098  0.544  At  1  1.362  1.778  Post  Mostly con.  2  1.484  10.014  Post  Con.  81  -  Con. -  Test  U-R loop’  Strain ES_CID_862  ES_CID_866  ES_CID_867  ES_CID_868  ES_CID_870  ES_CID_871  ES_CID_872  ES_CID_873  Pre/Post  Stress ratio Axial  (%) image  Dilative/Contractive during unloading  Image  1.163  0.667  At  1  0.353  0.9  Pre  Dil.  2  1.377  1.768  Post  Mostly con.  3  1.481  10.122  Post  Con.  Image  1.169  2.644  At  1  1.265  4.50 1  Post  Mostly con.  2  1.359  10.064  Post  Mostly con.  Image  1.107  1.082  At  1  1.102  1.033  Pre  Mostly dil.  2  1.338  4.05 1  Post  Mostly con.  3  1.393  10.101  Post  Con.  Image  1.209  3.68  At  1  1.2 12  4.047  Post  Mostly con.  2  1.269  10.146  Post  Con.  Image  1.177  1.807  At  1  1.352  10.056  Post  Image  1.16  1.84  At  1  1.308  4.035  Post  Mostly con.  2  1.366  10.108  Post  Mostly con.  Image  1.209  1.814  At  1  1  1.022  Pre  Dil.  2  1.348  4.0 18  Post  Mostly con.  3  1.405  10.107  Post  Con.  Image  1.194  0.3 19  At  1  1.079  0.234  Pre  Dii.  2  1.5 13  1.53 1  Post  Mostly con.  3  1.5  10.073  Post  Con.  -  -  -  -  -  Mostly con. -  -  -  Image indicates the stress ratio at image (image marks the boundary between contractive and dilative behaviour in loading)  82  Figure 4.1 c presents the data from the same example ES test used previously, test ES CID 867, in stress-dilatancy space. The following equation was used for calculating dilatancy from laboratory data:  D  where  8(+i)  8(—i)  q(n+1) 6  q(n—1) 6  11  (4.1)  denotes the current step.  In the case of unloading, positive dilatancy means volume increase while negative dilatancy indicates volume decrease. For unload phases U2 and U3 in Figure 4.lc, the sample contracts in unloading except for one point in U2. Unloading for those two phases starts from post-image stress ratios.  Conversely, for Ui where the sample is  unloaded from a pre-image stress ratio, the sample increases in volume at the beginning of the unloading phase. Then the sample contracts towards the end of the unloading phase. This behaviour shows that soil does not unload in an elastic manner for U2 and U3. That the behaviour of U2 and U3 is not elastic is known for two reasons: 1) for constant Poisson’s ratio elastic dilatancy should be constant 2) dilatancy has a negative sign which is not possible under the elastic framework for conventional triaxial stress path unloading. For U 1, where the sample is unloaded from a pre-image stress ratio, there is a small elastic part represented by the first three points in the dilatancy plot (see Figure 4.5). However, there is some uncertainty in the interpretation of this part of the test  because of the small number of data points. The elastic part is followed by plastic yielding. Similar behaviour is observed for test ES_CID_868 with the moist tamped sample (Figure 4.6). The previously described behaviour of sand seems to be independent of the sample preparation method.  83  Tests ES_C1Q870 and ES_CID_872 have similar e 0 and initial p’.  The only  significant difference between the tests is that the former has one U-R loop while the latter has three loops. The difference in the number of loops does not cause a significant effect on the stress ratio vs. axial strain curve (Figure 4.7a).  The first loop in  ES_CID_872 causes only small change in volumetric strains while the second loop causes significant contraction when compared to the results of ES_CID_870 (Figure 4.7b). Note that the first loop in ES_CID_872 is pre-image while the second is postimage.  In the third loop, both tests start from approximately similar points and  demonstrate similar behaviour. It can be observed that the volumetric strain curve for ES_CID_872 after the second loop is steeper than that for ES_CID_870. This implies that the unloading ioop influences volumetric changes patterns in subsequent reloading. This point will be discussed in detail later in this section.  Another two tests with very similar initial conditions, and very similar stress-strain and volumetric strain curves, are tests ES_CID_86 1 and ES_CID_862 which have identical e 0 and initial p’. The first test has two U-R loops while the second has three U-R loops. The additional loop in ES_CID_862 is pre-image and therefore does not cause any significant different between the results of the two tests.  84  (a)  1.6  L2  LI  L3  L4  1.2  :-  0.8  0.4  0 0  10  5  : %  (b) 1  15  20  Thlage  ‘I,  t  0  Contraction  5N1lat2 0 0’  —1  —2  —3  (c)  -4 •  t  •L1  .ui AL2 U 2 (L3  .  $• “  tc0 -1  -0.6  -0.2  Contraction if unloading Dilation if loading  —  0.2  D  0.6  1  Contraction if loading Dilation if unloading  Figure 4.1. Data from ES_CID_867 (a) stress ratio vs. axial strain (b) volumetric vs. axial strain (c) stress ratio vs. dilatancy  85  1200  —_______________________________  1000  800 600 400  0 0.68 e Hksand p’ = 400 kPa  200 0•  0  5  I  I  10  15 c:  25  20  %  Figure 4.2. Data from ES_CID_867 in shear stress vs. axial strain  1000 800 600 400  198 kPa  20:  0.00  5.00  10.00  15.00  c:  20.00  25.00  %  1  —1  Contractive  e  -2  Nlla  -3 -4 -5  Figure 4.3. Results of FR_CID_02 in shear stress vs. axial strain  86  0.6  Loop 1  O552 : 1 E  L2  %  L3  Loop2  04  0.0  U2  45  -0.4 6:  %  Figure 4.4. Zoom on loops 1 and 2 for test ES_CID_867.  87  • Loading I • Unloading I a Loading 2 Unloading 2 x Loading 3 • Unloading 3 ÷ Loading 4  1.4  ‘  +4C  Cc  .4  C +  •  Cc  .4  • •  :  •  Points (solid squares) indicating an elastic zone at the beginning of Ui while no such zone is observed for U2 and U3  •  +  x  :  0.8 ÷•  0.6  x  •  ÷  x  -1  -0.8  -0.6  -0.2  -0.4  0  0.2  0.4  D Figure 4.5. Zoom on the elastic zone in Figure 4.lc.  Volumetric strains in reloading phases are observed to be influenced by the unloadreload loops. This is investigated by plotting the data in stress-dilatancy space. A change in stress-dilatancy relation implies a change in the slope of the volumetric strain curve. Figure 4.9 shows that the stress-dilatancy relation is almost the same for phases Li -L3. Once peak stress ratio is exceeded in L3, stress-dilatancy relation changes for L4. Soil becomes more dilatant and another peak dilatancy value (termed as second peak) is reached. The increase in peak dilatancy between the two peaks exceeds 50% in some cases (Figure 4.10). Peak dilatancy values for the available drained triaxial tests with unload-reload loops on Erksak Sand are plotted against the state parameter at peak dilatancy (Figure 4.11). The state parameter is equal to the difference between the current void ratio and that at the critical state (see Section 2.1.3). Two different trends can be seen from laboratory data for different reload loops. It is noteworthy that the slopes and intercepts of a trend line through the data points of the first peak dilatancy are different from those for the dilatancy of the second peaks.  88  (a)  1.6  L3  L2  LI  20  (b) 1.5  0.5 0’ 0 10  5  15  Dilation  -0.5  (c) 1 .b  •L1  E  I  0.4  >——-—---1  -0.6  Contraction if unloading Dilation if loading  -0.2 —  D  0.2  —  0.6  Contraction if loading Dilation if unloading  Figure 4.6. Data from ES_CID_868 (a) stress ratio vs. axial strain (b) volumetric vs. axial strain (c) stress ratio vs. dilatancy.  89  (a) 2.00  1.60  1.20  0.80  0.40  0.00 0  5  10  15  6:  (b)  20  25  %  1  0 0”  —1  -2  Figure 4.7. Comparison of ES_CID_870 and ES_CID_872 with similar e 0 and initialp’ but different number of U-R loops (a) axial strain vs. stress ratio (b) axial strain vs. volumetric strain.  90  (a)  2.00 1.60 1.20 —  O.80  ES CID 861 —  —  ES CID 862  0.40  —  0.00— 0  I  I  5  %  —  I  I  15  20  25  (b)  0 and initialp’ Figure 4.8. Comparison ofES_CID_861 and ES_CID_862 with similar e but different number of U-R loops (a) axial strain vs. stress ratio (b) axial strain vs. volumetric strain.  91  -1.5  0  -0.5  -1  0.5  D  Figure 4.9. Stress ratio vs. dilatancy for pre-peak and post-peak reloading loops (ES_CID 862).  I.  2 peak _-_.  •L1  I  ÷  1.2  ipeak  + +  +.  A  +L4  ++x no  -0.6  -0.4  -0.2  0  D Figure 4.10. Stress ratio vs. dilatancy for different reload loops (ES_CID_867).  92  0.2  0 • First peaks a  -0.2  A  Second peaks  a  A  a  P  I  A  A A  A  A  A  a -0.6 -0.16  I  I  I  -0.12  -0.08  -0.04  0  y atDmjn Figure 4.11. Dmin vs.  iu  at Dmjn for first and second loading of Erksak sand  4.4. Implications of experimental observations Deformation characteristics in unloading are seen to be highly dependent on the stress ratio at the start of unloading (or the end of previous loading). If this stress ratio is less than that at image, unloading is dominated by a small amount of dilation. This behaviour might be explained by elasticity. An elastic material expands in response to a decrease in mean effective stress (associated with unloading in conventional triaxial tests). However, once the image stress ratio is exceeded, unloading is associated with significant amount of contraction. This indicates non-elastic behaviour or yield in unloading. Therefore, the image condition defines the first possible location where yield in unloading can occur. Yield in unloading must occur at a post-image location as dilation in loading is a prerequisite for significant contraction in unloading. This contradicts many soil models such as those presented in Section 2.3 where unloading happens inside the yield surface and it is elastic.  93  Contraction due to post-image unloading can be explained based on a simple physical model as previously discussed in Section 2.6.1. When stress ratio exceeds that for image, the sample starts to dilate. It stores potential energy that can be recovered in the form of contraction in unloading (Jefferies, 1997). If dilation can be thought of as soil particles sliding on top of each other, then a situation similar to that in Figure 4.12a develops in loading. Upon unloading which can be thought of as pushing the upper part to the left, the particles would want to slide back to their original location prior to loading (Figure 4. 12b). This is associated with contraction. Therefore, plastic dilation in loading is responsible for the observed contraction in subsequent unloading.  The model also  suggests that the amount of contraction in unloading is related to the amount of dilations in a previous loading. The more soil is allowed to dilate in loading, the more contraction is expected in subsequent unloading. This will be shown in Chapter 5.  I  (a)  1 d  I  (b)  I 2 d  I >d 1 d 2 Figure 4.12. The saw tooth model (a) loading (b) unloading (Same as Figure 2.35).  It seems that this simple model can explain the observed behaviour in unloading. The saw-tooth model (Figure 4.12) is a friction based model.  94  It can be thought of as a  simplified version, or an abstraction, of Rowe’s micro-mechanical model. In Chapter 3, Rowe’s model was extended to unloading. The trends observed in Section 4.3 are similar to those predicted by the model.  It was observed that post-image U-R loops demonstrate a new peak in stress-strain curves (Figure 4.3). This is consistent with the behaviour that post-image unloading is associated with contraction and a denser soil is expected to have higher peak strength. Triaxial tests on Erksak sand show that dilatancy in reloading is significantly changed only if the previous loading phase exceeds peak stress ratio (refer to Section 4.3). Been and Jefferies (1985) showed that there is a relation between peak dilatancy and state parameter at peak dilatancy as previously discussed in Section 2.5.  However, this  relation is expected to change if fabric changes. Changes in fabric are induced due to shearing in unloading and reloading phases. The data suggests that once the peak stress ratio is exceeded, soil goes through permanent changes in fabric.  95  5. A MODEL TO ACCOMMODATE UNLOAD-RELOAD LOOPS USING N0RSAND  NorSand is a strain hardening/softening plasticity model based on critical state theory. The most widely used version of the code that only yields in loading is described in some Jefferies (1997) also presented a  detail in the literature review (see section 2.5).  framework for the behaviour of a NorSand material in unloading and reloading (see section 2.6.3). This chapter expands on this framework to incorporate the observed soil behaviour in unload-reload loops discussed in Chapter 4, supported by the theoretical investigation in Chapter 3. NorSand is chosen in this study because of its simplicity, small number of parameters and accurate representation of the major aspects of soil behaviour. NorSand can be easily implemented in any programming language. The steps followed in coding the monotonic triaxial compression version of NorSand are summarized in Table 5.1. The equations were derived and the parameters were defined in Section 2.5.  Table 5.1. Equations used in the triaxial compression version of NorSand and their step by step implementation in an Euler integration code.  Step description 1  Equation  Apply plastic shear strain increment (8:)  2  Obtain the value of stress ratio  =  M  =  M.  —  • at image (M ) 1 3  Calculate the current plastic  D  dilatation rate  96  —  4  Get plastic volumetric strain  = DP  increment ( ct’) 5  Get the current dilation limit  D, =  çi’  where,  p’ =e—e, e =F—%lri(p) and  —  6  Apply the hardening rule to  •  change the size of the yield  P 1 p  surface due to the applied  -‘  2  =  4””1 1 ) L 1 P  re(_x/M)  P P  ]  plastic shear strain increment 7  Apply consistency condition  ( L  i/Il+  so that the stress state stays on the yield surface Where, L =  --  M. (L—z)  (From Jefferies and  p  Been, 2006) 8  Update stresses, strains and state parameter and add elastic strains  The objective of this chapter is to extend NorSand to include the new understanding of yielding during unloading and subsequent reloading, introduced in this study.  The  proposed model has been implemented in the Microsoft Excel Visual Basic Application (VBA) environment. Appendix D shows the main steps followed in coding the load unload-reload model.  97  The four components of any elasto-plastic model, including NorSand, are elasticity, a yield surface, a plastic potential (i.e. a flow rule) and a hardening rule.  5.1. Yield surface and internal cap NorSand’s outer yield surface and the inner yield surface (or internal cap) was discussed in some detail in Sections 2.5 (see Figure 5.1). Equation 2.52 (reproduced here as Equation 5.1) specifies the location of the internal cap. This is the same location used by Jefferies (1997). The current location of the cap fits the framework of the NorSand model in loading.  Pcap  1 p —  (5.1)  ) 1 (—D,,../M  The cap is taken as a vertical line for simplicity. It is assumed to intersect the outer yield surface at peak (i.e. minimum dilatancy if the sign is taken in consideration) which occurs at peak strength. Hence the location of the cap that defines yield in unloading is coupled to the outer yield surface that was determined by the previous loading phase. It will be shown in the following sections that soil behaviour in unloading is related to previous loading phases. It is therefore reasonable to have the location of the cap defined during a previous loading phase. The adopted location of the internal cap is identical to that for Jefferies (1997). This is to avoid inconsistencies with the NorSand model in loading. Consider a case where unloading starts from peak. For convenience assume that an internal cap at the image state was chosen. Unloading would then start from a point to the left of the internal cap and the consistency condition would not be satisfied as the stress point would not be on the internal cap.  98  Figure 5.1. Yield surface and internal cap in NorSand, same as Figure 2.28 (modified after Jefferies and Shuttle 2005). The proposed location of the internal cap matches the observation from the data presented in Chapter 4. It was shown that yield in unloading must occur to the left of the image. This is consistent with the saw-tooth model where significant contraction in unloading (i.e. yield in unloading) occurs only for the case where soil was allowed to dilate in a previous loading. By definition, dilation is only possible if stress ratio exceeds that of image. Therefore, having the internal cap at peak is consistent with the NorSand model and matches observations from laboratory data.  Soil unloaded after reaching peak stress yielded in unloading directly without going through an elastic phase (Figure 5.2). It is not certain whether peak was reached for L2. However, it is clear that the stress ratio is very close to reaching peak. Samples unloaded from lower stress ratios showed a purely elastic phase before yielding. This behaviour can be captured by placing the cap at the point representing peak stress in loading.  99  No sign of an elastic zone at the beginning of U2 and U3  +  k  • Loading 1 Unloading I  x  x Unloading 2  x  z Loading 3  X  • Unloading 3• Loading  •• x  ••  x  xx x  •  $ -1.5  -1  .-  . .  Loading 2  +  .4-  Interpreted as an elastic zone because of constant positive dilation values  0.8 0.6 0.4  a a  0.2  x • +  +  •  x  -0.5  0  0.5  x 1  1.5  D  Figure 5.2. Demonstration of interpreted elastic and elasto-plastic zones on the results of ES_CID_682 in stress ratio vs. dilatancy plot.  5.2. Flow rule  5.2.1. Flow rule in unloading Data indicate that dilatancy in unloading depends on the previous loading phase. Figure 5.3 shows the stress-dilatancy plots for the triaxial tests on Erksak sand discussed in Chapter 4. Three observations may be made from the data in Figure 5.3. First, it can be seen that almost all the plots for dilatancy in unloading are perpendicular to those in loading. Second, the position of stress-dilatancy curves in unloading is seen to depend on the stress ratio at which previous loading stopped (see Figure 5.4). In the following the stress ratio at ]Y  =  0 is defined as M (the subscript “u” denotes unloading). The higher  the stress ratio at which loading stops, the larger the value of M for a following unloading phase. Lastly, the measured dilatancy in unloading stops changing at a value of =  j  0.4-0.5, and plots vertically in the stress-dilatancy plot. Note that the value of  0.4-0.5 corresponds to different dilatancy values for different tests and different  100  unloading loops within each test. Therefore, the location of the vertical part of the plot is different for different tests and different loops within each test.  It is assumed here that the Cam-Clay flow rule (D” in loading (see Section 2.3 for more details).  =  M  -  i)  represents stress-dilatancy  Because stress-dilatancy curves in  unloading are almost perpendicular to those for loading, the proposed expression for stress-dilatancy in unloading is negative of that for Cam-Clay, and while replacing M by M. It takes the form:  (5.2)  D=ii—M  The problem is how to get a representative value for M, as Figure 5.3 clearly shows that for Equation 5.2 to be valid M needs to vary for different loops in the same test. As previously discussed, the saw-tooth model implies that the amount of contraction in unloading is related to the amount of dilation in the previous loading phase. The value of M for each unloading phase is then expected to depend on the previous loading phase.  The last value of  i  for L3 in Figure 5.4 is higher than that for L2. Note that value of M  for U3 (U3 follows L3) is higher than that for U2. It seems that the higher the last value of  i  (denoted as  ‘lL)  in a loading phase, the higher the value of M in subsequent  unloading. To prove this point, values of L for different loading or reloading loops for all tests on ES are plotted against values of M for the corresponding subsequent unloading phases (see Figure 5.5). It can be seen that  L  and M are directly related. A  linear trend line representing a best fit to the data points has the following equation:  1 —1.5 M =2q  (5.3)  It is observed that there is a limit on the maximum dilatancy that can be reached in unloading at a value of  =  0.4-0.5, as previously discussed.  At this limiting value  dilatancy becomes constant for a particular unloading loop. Equation 5.4 is found to give a reasonable approximation for this maximum dilatancy in unloading (or minimum dilatancy if the sign is taken in consideration).  101  (5.4)  D=0.5 -M  Figure 5.6 shows an example comparison between the stress-dilatancy predicted by the proposed equations (Equations 5.2 to 5.4) and data from test ES_CID_866.  The  predictive ability of Equations 5.2 to 5.4 is shown for all ten Erksak tests in Appendix A.  -1.5  -1  -0.5  D  0  1  0.5  Figure 5.3. Drained triaxial tests on Erksak sand with unload-reload loops plotted in the dilatancy vs. space.  5.2.2. Flow rule in reloading The usual NorSand flow rule for monotonic loading is D”  =  -  i.  Triaxial data on  Erksak sand plotted in Figure 5.3 show that stress-dilatancy is altered if the soil is unloaded and reloaded. It changes for different reloading loops (see Section 4.3). It is observed that in most tests, the stress-dilatancy relation in reloading changes if the previous loading phase reached peak dilatancy (see Figures 4.9, 4.10, and 4.11). It was shown that peak dilatancy values increase for post-peak reloading phases.  102  Last value of stress ratio for L3 (ii L)  Value of M for the following unloading  —.—.  UI  —a-- L 2 ——  U2  —.—U3  -1.5  -0.5  -1  0  JLI 0.5  D  Figure 5.4.  ijL  and M for L3 and U3, respectively, for ES_CID_862.  1.8 • Tests data —Trendline 1.4  M=2-1.5  I  ••  •  •  •  0.6 1  1.2  1.4  1.6  7L 1  Figure 5.5. Correlation between M and on Erksak sand).  iL  from previous loading (drained triaxial tests  103  1  -1.5  -1  -0.5  0  0.5  1  D Figure 5.6. Predicted and measured stress-dilatancy for ES_CID_866.  In NorSand the peak stress is coincident with peak dilatancy, which is also the location of the internal cap (the location and shape of the assumed internal cap in NorSand was discussed in Section 5.1). Following the observed behaviour of change in peak dilatancy in post-peak reloading, it seems reasonable to introduce some changes to how soil dilates in the code if the stress state reaches the cap in a previous loading/reloading. The triaxial data for all tests on Erksak sand was plotted in Figure 4.11 in ,u vs. Dmjn space. It is clear that there are two different trends for the different peaks within each test and for all tests. The points for seconds peaks are lower on the plot compared to the points of first peaks. Change in particle arrangement due to the cyclic load is responsible for this change as previously discussed in Section 4.4 supported by the results of the theoretical investigation in Section 3.2. It is noteworthy that if a trend line (based on a best fit to the data points) is drawn through the points of the second peaks, it would have an intercept that is far from zero. This is not in accordance with the critical state theory on which NorSand is based. A major feature of NorSand is that it limits dilatancy based on a relation between çu and Dmin. The slope of a linear trend line with a zero intercept in the çu and Dmjn plot is termed  104  x (used as a parameter in NorSand).  A zero intercept of the trend line is consistent with  Critical State Soil Mechanics (CSSM). This is because if peak dilatancy happens at the critical state (i.e. for a test on loose sand) then Dmjn=O and e equal zero as  çti=  e  -  =  e (by definition  t,  must  es). Therefore, the trend line through the data points of the second  peaks should also be drawn with an intercept of zero, as well as the trend for the first peaks.  It could be argued that soil reaches other peaks for subsequent unload-reload loops (i.e. a third peak may exist). The limited number of tests in the currently available data with three or more loops does not provide sufficient information to determine whether other peaks exist or not. However, the stress-dilatancy relation is not expected to change indefinitely. A change in stress-dilatancy relation is caused by changes in fabric. As the reported tests do not reach critical state (where major fabric changes occur), no further changes in the stress-dilatancy relation are expected to occur.  The proposed model assumes that only two peaks exist. Once the first peak is exceeded in a loading/reloading phase, all subsequent reloading phases follow a different stress dilatancy relation with a different peak (i.e. second peak) associated with more dilation. Those two peaks can be represented in NorSand by the slopes of two trend lines through the points of the first and second peaks. The slopes of the two lines are denoted as j and X2.  The parameter  xj  is identical to x in standard NorSand. The code uses the second  peaks value, X2, only if the stress state in the previous loading or reloading phase hits the internal cap which represents peak conditions. This is consistent with observations in triaxial tests. The second peaks are attained only if peak stress was reached in a previous loading/reloading (see Section 4.3). The implication of a changing x on the NorSand model is twofold. Firstly, a change in  x results in a change  in the location of the internal cap for a certain yield surface. As x  increases from an initial value ofj to a larger value of %2, the location of the internal cap (Figure 5.1) is shifted to the left. This allows for higher dilatancy values. Secondly, the computed values of M change (see Equation 5.5). A higher x value yields a smaller  105  This is consistent with the observed behaviour. Figure 5.7 shows that M for a second peak reloading (i.e. higher  ‘  value) is higher than M for the first peak reloading with the  smaller x. M,  =M-xNçuj  (5.5)  Where,  =xj for the case of first loading or previous loading/reloading does not touch the internal cap. X=X2  for the case where previous loading/reloading touches the internal cap.  -0.6  -0.5  -0.4  -0.3  -0.2  -0.1  0  0.1  0.2  D Figure 5.7. Change of M for different reloading loops (ES CID 862).  5.2.3. Potential surface in unloading NorSand uses an associated flow rule, meaning that the plastic potential surface and yield surface are the same.  In unloading, yield happens on the internal cap.  106  As  previously discussed in Section 5.1, the cap is a vertical line. Using an associated flow rule with a vertical cap yields zero dilatancy.  However, a significant amount of  contraction was observed in unloading. Therefore, a non-associated flow rule is used (Equation 5.2). Having a non-associated flow rule in unloading makes it necessary to have a potential surface that is different from the yield surface. An expression for the potential surface is derived as it will be necessary to implement this model in any finite element formulation for future work. The derivation involves two assumptions: normality (i.e. plastic strain increments ratio is normal to the surface) and the stress-dilatancy relation (Equation 5.2). Starting with the definition of stress ratio  (5.6)  q=ip’ Taking the differential of 5.6 gives:  (5.7) And to satisfy normality, (5.8) P  q 8  From 5.7 and 5.8,  p  7 D+,  (5.9)  =0  From Equations 5.2 and 5.9,  p  2ii-M  (5.10)  =0  Integrating Equation 5.10 gives:  107  f  (5.11)  =C p  The solution of the integral is: (5.12)  lnp+i—ln211—MI=C  When ij  =  M, the stress state would be at the image and p  = ‘iu,  hence:  (5.13)  c=lnp;+.-lnM  And the equation of the potential surface in unloading is:  ln1—”+!1n_L_1=0 p) 2 M  (5.14)  Rearranging gives:  (5.15) — e_1_1  Equations 5.14 and 5.15 were used to plot the potential surface in Figure 5.8 for =  1.2. The potential surface has two parts that eventually meet at a high p’ value. The  upper part is applicable for the case where (2ii/M -1) (2ii/M -1) <0.  108  >  0 while the lower part is for  240  200  160  80  40  0 0  100  200  300  400  p’ (kPa)  Figure 5.8. The shape of the potential surface in unloading  5.3.  Hardening in loading, unloading and reloading  Hardening of the NorSand yield surface in loading, unloading and reloading is described in Section 2.6.3. A similar framework is adopted here because it matches the way Erksak sand behaves. If unloading occurs from a low stress level, the behaviour is elastic until yield occurs when the stress path hit the internal cap. Figure 5.2 shows that Ui is dominated by elastic behaviour at the beginning of the unloading phase and yield occurs only later on. Note that unloading in Ui starts from a low stress ratio. However, for the other loops, unloading is dominated by plasticity.  NorSand would yield in  unloading for those loops without passing through an elastic phase. The outer yield surface softening during unloading is important for accurate predictions. If the outer yield surface would not soften in unloading, reloading would be elastic until the stress level prior to unloading is exceeded.  109  Figure 5.9 shows that  reloading is not entirely elastic: stiffness decreases before the stress level is as high as the stress level at the start of previous unloading. In unloading, the internal cap contracts. And because the internal cap intersects the outer yield surface, it softens as well.  Jefferies (1997) introduced a rule for the  contraction of the internal cap, reproduced as Equation 5.16. The term ln(pjy/p) was thought by Jefferies to introduce an effect similar to overconsolidation. The further the stress point is from first yield in unloading, the larger are the generated strains. This is consistent with observations from laboratory data. Figure 5.9 shows an expanded view of U2 for ES_CID_868. More axial strains are generated in U2 at lower stress ratios. The lower the stress ratio, the further the stress point is from first yield in unloading.  .  ‘p  6v  1 p’ (‘ =———--TlnI I Hp P)  (5.16)  Where, H is the hardening (softening) modulus in unloading p’, is the mean effective stress at first yield in unloading (i.e. the mean stress of the cap when first intersected)  110  1.60 1.20 0.80 0.40 0.00 11  10  9  L:  %  Figure 5.9. Expanded scale view of U2/L3 for ES_CID_868 in Figure 4.6a.  The use of Equation 5.16 in the code can result in infinite plastic shear strain increments. Consider the case of  i  =  M. According to Equation 5.2 Ef becomes zero.  And for the sake of the argument, assume that volumetric strain increments are computed according to Equation 5.16. This results in a division over zero as  8  / D. To get  around this problem, Equation 5.17 is used instead of Equation 5.16. Plastic shear strain increments are first calculated according to Equation 5.17 then plastic volumetric strain increments are recovered through stress-dilatancy (i.e. s’  =  8  D’). For the case of zero  D”, plastic volumetric strain increments become zero and the problem of having to divide over zero is solved.  The sign of Equation 5.16 is changed as unloading is  associated with negative mean effective stress increments and negative plastic shear strain increments.  111  •  i  ‘ (P =—-—-1n q 1 6 I  (5.17)  Hp  As yield in unloading causes softening of the outer yield surface, it is important to derive an equation to quantitatively describe the amount of that softening. The size of the outer yield surface depends on p. As previously discussed in Section 2.6.3, Peap is related 1 as follow: top  p1  Pcap —  (5.18)  e(_i)u1/’M1)  From 5.18, .  .  I  J3  Pcap  (5.19) —  Since the stress point remains on the internal cap in unloading, mean effective stress in Equation 5.17 is equal to Pcap and,  (5.20)  From Equations 5.17 to 5.20,  1 P  = s H /1fl,  (5.21)  P  112  Equation 5.21 describes the softening of the outer yield surface due to yield in unloading. As the size of NorSand yield surface is controlled by p, the term p /p’ describes the change in the size of the outer yield surface due to an applied plastic shear strain increment relative to it original size before applying that increment. Figure 5.10 shows different outer yield surfaces corresponding to different points on the unloading stress path. The inner cap moves to the left with the stress path dragging the outer yield surface with it.  As previously discussed in Section 5.2.3, the potential surface in unloading is different from the yield surface (i.e. the internal cap). It can be noted from Figure 5.10 that plastic strain ratios (i.e.  6’/  st’) represented by the arrows are  not normal to the internal caps.  However, the arrows are normal to the potential surfaces in Figure 5.11 (see Section 5.2.3 for the derivation of the potential surface).  300  200  100  0 0  100  200  300  400  500  Figure 5.10. The direction of plastic strain increment ratios in unloading with the corresponding yield surfaces and internal caps.  113  600  700  300 250 200 o 150 100 50 0 0  100  400  300  200  500  t p Figure 5.11. The direction of plastic strain increments ratios in unloading normal to the potential surfaces.  5.4. Comparison with other models This chapter presented an unload-reload model for sands that is based on the NorSand soil model. One of the main features of the proposed model is that it yields in unloading. The model uses a non-associated flow rule in unloading. A summary of the unloading part of the model and the main assumptions are presented in Table 5.2. It was shown in this chapter that those assumptions match the observations from laboratory results presented in Chapter 4. It was observed that soil becomes more dilatant in post-peak reloading loops. The behaviour was simulated in the model by a changing higher value of 2 once first peak is exceeded.  114  .  The value of  increases to a  600  Table 5.2. Summary of the unloading part of the model. Model components Yield surface  Equation  Assumptions p.  cap  Dy”  =  •  •  —  Where, M  =  Yield in unloading happens on a  vertical cap.  ) 1 /M  —  Flow rule  •  P1  —  •  —1.5 •  And the minimum dilatancy in unloading is:  The inner cap intersects the outer yield surface at a point that corresponds to Dmin in loading. Stress-dilatancy plots in unloading are perpendicular to those in loading. There is a direct relation between M and 1 1L of the previous loading. There is a minimum value for dilatancy in unloading.  D=O.5 -M Hardening rule  Movement of the internal cap: /  ,  —  —  1 P 1 H ‘  •  •  9 J  The further the stress point from first yield in unloading, the slower the rate of movement of the internal cap. The outer yield surface softens due to yield in unloading.  Softening of the outer yield surface in unloading:  P±6PH/lfl(P4]  Jefferies (1997) derived an equation for stress-dilatancy in unloading based on the assumption that soil stores ‘plastic’ energy in loading that is recovered upon unloading. The model was described in Section 2.6.3. Starting from Nova’s flow rule (Equation 5.22), and substituting for D’ and  i  (i.e.  ‘  /  and q/p’, respectively) and expanding  yields Equation 5.23.  D  =  (M—i) (1-N)  (5.22)  115  (5.23)  q’+p’ =Mpf:l÷Npt  The terms on the left hand side of Equation 5.23 represent plastic work done. The right hand side represents what soil does with that work. The first term on the right hand side represents energy dissipation (Schofield and Wroth, 1968). The second term on the right hand side represents ‘plastic’ energy stored in loading and recovered in unloading (Jefferies, 1997). The saw tooth model gives a simple physical explanation of ‘plastic’ energy storage. Accordingly, the potential energy of individual soil particles is increased in loading as the particles assume new locations. This energy is released upon unloading as the particles tend to recover their original locations before loading. This is associated with contractive response in unloading. For the unloading phase,  ‘  <0 and the N term  in Equation 5.23 takes a negative sign as it represents ‘plastic’ energy recovered. Substituting and rearranging gives Equation 5.24 for stress-dilatancy in unloading. Equation 5.24 is plotted in Figure 5.12.  Dp=  M77 1+N  (5.24)  Equation 5.24 predicts different trends, more contraction in unloading, compared to laboratory data and the predictions of Equations 5.2 to 5.4 (Figure 5.12). The expression assumes that all ‘plastic’ energy stored in a loading phase must be released in the subsequent unloading phase which does not seem to be the case (i.e. only part of this energy is released in the subsequent unloading phase). The proposed model in this chapter, similar to measured laboratory data, shows that contraction in unloading depends on shear deformation in previous loading.  This is  consistent with the saw tooth model where shear deformation is a major source of plastic work stored in loading and recovered in unloading. Jefferies (1997) was the first to adopt the saw tooth model to explain soil behaviour in unloading. Therefore, the proposed model and that for Jefferies (1997) are very similar conceptually. 116  -2  -1.5  -1  -0.5  0  0.5  D • Loading I Loading 2 * Loading 3 —Jefferies 1997 (M=I.27 and N=O.25)  Unloading I x Unloading 2 —Model fit to UI and U2  Figure 5.12. Predicted and measured stress-dilatancy for ES_CID_866.  The proposed model and that for Jefferies (1997) are different from Collins (2005) conceptual model that assumes plastic shear is not a significant source of plastic work storage while isotropic compression is the major source. Collins model was described in Section 2.6.2. Pure isotropic loading on the continuum scale is assumed and following the usual convention the applied work may be separated into an elastic and plastic component  -  or + or.  During loading, part of the applied work is dissipated  (Or) while the remainder is stored in terms of elastic compression of soil particles (or).  In subsequent unloading, part of the stored elastic work is released causing  dilation while the other part can only be released if associated with particles rearrangement (causes dilation as well). Particle rearrangement is not elastic and hence plasticity occurs during unloading. It is noteworthy that if soil particles were rigid, Collins model predicts no volumetric strains in unloading. However, it was shown in Chapter 3 based on Rowe’s theoretical model that an assembly of rigid particles changes  117  in volume in unloading. Unloading according to Collins model is associated with dilation while laboratory data shows that significant contraction occurs in unloading.  D B q C  y.s  y.s 2 y.s I  p, Figure 5.13. Drucker and Seereeram model (reproduced from Drucker and Seereeram, 1987). Drucker and Seereeram (1987) proposed a hypothetical model for yield in unloading (Figure 5.13). It is assumed that point A, located on yield surface (y.s.) number 1, is the starting point. The yield surface moves to y.s. 2 during the loading path of A-B. Upon unloading (i.e. path B-C), the yield surface moves with the stress path reaching y.s. 3. During subsequent reloading (i.e. path C-D), the yield surface returns to y.s. 2. Accordingly, the yield surface always moves with the stress path. The model assumes that B-C is purely elastic while A-B and C-D are elasto-plastic.  Hardening in the proposed model which is identical to hardening proposed by Jefferies (1997) was discussed in Section 2.6.3 (Figure 5.14). Loading for normally consolidated conditions is elasto-plastic (Figure 5. 14b). Unloading is purely elastic only for the phase before the stress path hits the internal cap (Figure 5.14c). In this phase, the outer yield surface does not move. Otherwise, unloading is elasto-plastic and causes softening of the outer yield surface. Reloading is elastic until the stress path hits the outer yield surface  118  (Figure 5. 14d). Clearly, this is different from Drucker and Seereeram model described in the previous paragraph. The differences are summarized in Table 5.3.  2.0C  300  2  tee 120  2  / .1  Surface where cuerent yield is  (1  /  occuering Stress  —  •  Loading  00  path  /  Ci0  Surfaces where  current yield is occumng  j  0 0  10  5  15  20  100  0  200  300  400  500  600  700  cj:%  300  1} .  _-__-_  1:’ /  Unloading  iing  ———————————  Reloading  200  Surfaces where is  ces ere  S 1  174W—  0  100  200  300  400  000  600  700  100  0  200  200  400  500  600  Figure 5.14. Hardening according to Jefferies (1997) (same as Figure 2.35).  Table 5.3. Comparison between hardening in the proposed model and Drucker and Seereeram (1987). Phase/model Unloading  Reloading  Proposed hardening (identical to Jefferies, 1997) • Either elastic or elastoplastic • Yield surface moves only if stress_path_hits_the_cap • Either elastic or elastoplastic • Yield surface moves if the stress path hits the outer_yield_surface  119  Drucker and Seereeram (1987) • Purely elastic • Yield surface always moves with the stress path • •  Elasto-plastic Yield surface always moves with the stress path  700  5.5. Summary This chapter presented a practical continuum model for unload-reload cycles on sands that takes the NorSand soil model as its starting point (Jefferies, 1993; Jefferies, 1997, Jefferies and Shuttle, 2005). One of the main features of the model is that it yields in unloading, a behaviour that is consistent with observations from lab data presented in Chapter 4 and the conclusions of the theoretical investigation of stress-dilatancy presented in Chapter 3. Like Jefferies model, yield in unloading is assumed to occur on a vertical cap in the p ‘-q space. Unlike Jefferies model, stress-dilatancy in unload and reload phases in the proposed model is consistent with the observations from lab data. Unloading is linked to previous loading such that the amount of dilation in unloading is directly proportional to the stress ratio at end of previous loading. The model accounts for the observed increase in dilation for post-peak reloading.  120  6. MODEL CALIBRATION  This chapter presents a load-unload-reload calibration of the model presented in Chapter 5 to ten triaxial tests on Erksak sand (see Table 4.2). The calibrated model is then used to predict two drained triaxial tests with unload-reload loops on Fraser River sand later in Chapter 7. As the unload-reload model uses NorSand as its starting point, a monotonic calibration of NorSand for both sands is performed first. The monotonic calibration to Erksak sand is presented in Section 6.1. Section 6.2 is a monotonic calibration to Fraser River sand.  Section 6.3 presents the unload-reload  calibration to Erksak sand.  6.1.Monotonic calibration for Erksak sand A calibration of NorSand for Erksak sand under monotonic loading in a triaxial test is presented in this section. The calibration uses the tests described earlier in Chapter 4 (Tables 4.2 & 4.3). The required parameters were previously described in Section 2.1.2 and Section 2.5. Table 2.2, reproduced here for convenience as Table 6.1, is a summary of the required parameters and their typical ranges. The critical state parameters F and 2 are the slope and the y-intercept of the critical state line in e-log(p plot, respectively. The critical stress ratio for triaxial compression,  is q/p’ at critical state. The plastic  hardening parameter ‘H’ specifies the rate of the hardening of the yield surface. The slope of a trend line with zero intercept through the data points in the Dmjn v’ is designated as  (e.g. Figure 2.29). It is used to control the maximum allowable absolute  value of the dilation rate. The volumetric coupling parameter ‘N’ was introduced by Nova (1982). It is based on fits to stress-dilatancy from the results of laboratory tests.  121  Finally, the elasticity parameters are the dimensionless shear rigidity parameter (G/p and Poisson’s ratio.  Table 6.1. Typical ranges for monotonic parameters (same as Table 2.2, modified after Jefferies and Shuttle, 2005). Parameter Critical state F  I  Typical range  0.9-1.4 0.01  2e  !vI Plasticity H  —  0.07  1.2-1.5  ZIc  50-500 2.5-4.5  N  0.2-0.4  I  Description  The y-intercept of the e log(p’) curve at 1KPa The slope of CSL in e log(p) space defined on base e g/p’ at critical state Plastic hardening modulus A parameter that limits the hardening of the yield surface The volumetric coupling parameter (used in Nova’s rule)  Elasticity Jr  100-800  v  0.1-0.3  Dimensionless shear rigidity (G/p) Poisson’s ratio  6.1.1. Critical state parameters There is more than one way to obtain M from triaxial data. Ghafghazi & Shuttle (2006) reviewed four methods reported in the literature to obtain the critical state stress ratio from drained triaxial tests: the terminal value of stress ratio method, maximum contraction method, Bishop method, and Stress-dilatancy method. 1. Plotting the curves for each test in the stress ratio vs. strain space and simply picking up the terminal value for the stress ratio. The problem with this method is that the dense tests, and even most of the loose tests, do not go far enough to reach the critical state. 122  2. The stress ratio at maximum contraction is taken as Mk. This method assumes that the stress ratio at maximum contraction is equivalent to that at the critical state. The point of maximum contraction is not the same as the critical state and hence the method would only be appropriate if the true stress dilatancy behaviour of soil was a unique locus (i.e. only one stress ratio corresponded to one dilatancy).  Although this assumption has been used in flow rules such as  CamClay, modified CamClay, and Nova, real soils do not show this behaviour. The real measured soil response shows a different stress ratio at maximum contraction and at the critical state, consistent with Rowe’s idea of an evolving M with increasing strain. More recent soil models such as NorSand and Li and Dafalias’s (2000) model also incorporate this evolving M. Therefore this method of determining M, provides poor predictions.  3. Bishop (1971) suggests plotting the data for all the available tests in Dmin  -  Tlmax  space. The method is based on the idea that a very loose soil should reach the critical state at peak stress ratio, i.e. Dmjn and  i  at peak should be zero and M 1  respectively. Hence, assuming the trend in the data is linear (which is consistent with experimental measurements), the y-intercept of a trend line through the data The unload-reload Erksak tests have different  points will correspond to Mk.  peaks associated with different reloading phases as was previously discussed in Section 4.3. A peak is termed a ‘first peak’ if peak strength was never exceeded in previous loading or reloading loops for a certain test. All other peaks are termed ‘second peaks’. Two linear trend lines are plotted for first and second peaks data points in Figure 6.1 resulting in two M values. According to critical state, a single sand must have a single M, value. Therefore, the two M values must be identical. However, Dmjn is affected by changes in fabric due to shearing in different reloading loops and therefore  based on extrapolating data in Dmjn  -  i7,,,, is expected to change for different reloading loops. The trend line for the  first peaks gives M, =1.15 (q 5 very similar value of M1 .1  =  28.85°) while that for the second peaks gives a  ( 27.7°). Therefore, Bishops method gives M 123  in the range of 1.1-1.15. It can be noticed that the two best fit lines are parallel. It seems that slope of the lines is very similar for different reloading phases. However, the y-intercept is slightly different.  1.6 ‘ii  =  (Bishop, 1971 and Nova, 1982) 0 (N—1)D+M  1=—0.8O5D,+ 1.15 —  1-N  11O.8O4Dmm+1.1A  1.4 1  1.2  A  A  .. *  First peaks (FP)  ---•  Trend line (SP)  0  Figure 6.1.  -0.1  -0.2  -0.4  -0.3  -0.6  -0.5  using Bishops method for Erksak sand.  4. The stress-dilatancy method for obtaining  suggests linearly extrapolating the  retUrning curves in the dilatant part of the stress-dilatancy plots to D Figure 6.2). The extrapolated value of  i  at D  0 is  =  0 (see  Table 6.2 shows Ii/tt  using the stress-dilatancy method for all tests but for ES_CID_868 (post peak behaviour for this test is questionable). Stress-dilatancy method gives of 1.24-1.35 as shown in Figure 6.3 with an average of 1.286  (  =  values  31.97°).  Ghafghazi & Shuttle (2006) recommended the use of the Bishop method or Stress Dilatancy approach.  The Bishop method gave a lower M 1 than the stress-dilatancy  method. Ghafghazi & Shuttle (2006) showed that “with only a small number of tests available, the Bishop method is sensitive to any outlying data points”. Table 6.3 shows that the range of M obtained in this work using 9 tests with the Bishop method is lower 124  than the  obtained by Ghafghazi & Shuttle (2006) for 34 tests using the same method.  However, despite using many fewer tests than Ghafghazi and Shuttle, the M 1 values of both sets of authors using the Stress-Dilatancy method are very similar. Therefore, the even for a small  stress-dilatancy method seems to provide a repeatable value of number of tests. Therefore, a value of  =  1.286 (q  =  31 .97°).was adopted for Erksak  sand.  -0.5  -0.4  -0.3  -0.2  -0.1  0  0.1  D Figure 6.2.  using stress-dilatancy method (ES_CID_87 1).  125  0.2  0.3  0.4  1.55 1.5 1.45  1.4  Expected range for .  (1.24-1.35)  1.2 1. 15  •  1.1 05 f 0.95  *  _  •  0.9 X  0.85  -0.5  -0.4  -0.3  -0.2  -0.1  0.1  0  :..  -  0.2  0.3  D Figure 6.3. Range of M, using the stress-dilatancy method from the last reloading loops for the 9 tests in Table 4.2.  Table 6.2. M 1 using stress-dilatancy method for the unload-reload tests on Erksak sand. (extrapolated  Test name CID-G860 CID-G861 CID-G862 CID-G866 CID-G867 CID-G868 CID-G870 CID-G871 CID-G872 CID-G873 Average 2  1.33 1.35 1.33 1.27 1.25 2 Ignored 1.24 1.24 1.27 1.3 1.286  All values are extrapolated but for CID-872 that reached critical state. Post-peak behaviour for this test is judged to be unrepresentative.  126  )1  0.4  Table 6.3. Summary of M’ values for Erksak sand Ghafghazi & Shuttle,  Current work (9 tests)  Method/source  2006 (34 tests) peak) t 1.1 (ls fld 2 peak) 1.15 ( 1.286  Bishop method Stress-Dilatancy method  1.26 1.28  The critical state parameters in e-log p’ space were determined using undrained triaxial tests from Jefferies and Been, 2006 (see Figure 6.4). The tests were described in Section 4.2 and summarized in Table 4.3. The derived critical state parameters are F = 0.82 and 0.0135) which are very similar to F = 0.8 16 and  =  0.03 1 (equivalent to  =  0.031 in Jefferies and Been (2006). These derived parameters are accurate for  Xe  =  p’< 800 kPa. Been et al. (1991) showed that at higher mean effective stresses the line becomes steeper, attributed to grain crushing, and therefore a single linear CSL would not be applicable. Hence the fits for tests ES_CID_870, ES_CID_87 1, and ES_C1Q872 with high mean effective stress of 800 kPa are not expected to be accurate.  0.8  0.78  :.............................  e .  0  >  0.76  =  0.82- O.01 logp  A  A  0.74  End of test (undrthnedtriwd tes onosesamPIes;dáäfroi4Bi& Jeffèries 2006)  —  —  —  ,  0.72  —CSL  0.7 100  10  Logp’ (kPa) Figure 6.4. CSL determination for Erksak sand from loose undrained tests.  127  1000  6.1.2. Elasticity parameters  Ideally, elastic shear modulus is measured using bender elements. However, tests with bender elements on Erksak sand were not available. In the absence of bender tests, elastic properties may be estimated from the elastic portion of unload-reload loops. Jefferies and Been (2000) presented hydrostatic compression triaxial tests with unloadreload cycles.  Equation 6.1 gives the elastic bulk modulus from best fit to their  experimental data: /  ‘O.5  K_C1 Pref )p) 5 (e_e  (6.1)  where C is a material ‘compressibility’ constant equal to 260, Pref is 100 kPa and e, the void ratio at which the volumetric compressibility becomes zero is equal to 0.355. The parameter e represents the void ratio where soil behaviour changes from predominantly particulate to that of a solid, and is significantly less than the typically defined emm (e.g. ASTM D-4254-00). Assuming a Poisson’s ratio of 0.2, the dimensionless shear modulus is given by:  (6.2) p  p 4  However, there is scatter of as much as ± 50% in Jefferies and Been data. The elastic bulk modulus can be directly obtained from the unload-reload tests on Erksak sand presented in Chapter 4. Figure 6.5 is an enlarged view of the elastic part at the beginning of L3 for ESCID_866. It can be seen that the plot in that zone is nearly linear and it is assumed that all deformations are elastic.  The elastic bulk modulus was calculated  directly from the data points for that elastic zone using a linear difference approximation as in the following equation:  128  K  pj÷i_Pj(i+ej+i+ej 2 —e 3 e i 1  (6.3)  Wherej+% is the midpoint between consecutive measurements at which K is computed,j is the previous measurement, and j+1 is the next measurement. Figure 6.6 plots the results of Equations 6.1 and 6.3 for the elastic part of L3. It is clear that Equation 6.1 underestimates the values of the bulk modulus compared to those directly computed from laboratory data (i.e. Equation 6.3). A higher value of C  750 in Equation 6.1 would give  a better estimate of K. A better fit to the laboratory data was obtained using equation 6.1 using C  750, and e 5 equal to 0.355 (i.e. the same e as in Jefferies and Been, 2000 but  different C) as shown in Figure 6.6.  0.7  0.2  0.3  : % 1 c  Figure 6.5. Enlarged view of the elastic part in L3 for ES_CID_866.  129  800 700  -  • Lab data • C = 750  AC  =  260  : :  :  A  A  400 300 200  A  A  A  A  A  100 0  400  I  I  I  I  420  440  460  480  500  p’: kPa Figure 6.6. The elastic bulk modulus from Equations 6.1 and 6.3 against p’ for the elastic zone in L3 for ES CID 866.  6.1.3. Plasticity parameters The monotonic loading version of NorSand requires three plasticity parameter; N,  ,-  and  H. The N parameter in NorSand is defined in the same way as the N used by Nova (1982). It is derived from experimental stress-dilatancy data. The slope of the trend line through 77max  Dmin plot is (N-i) as shown in Figure 6.1. The two trend lines through both the first  and second peak points have very similar slopes of around 0.8. Therefore, the value of N is 0.2. Figure 6.7 plots the dilatancy at peak versus the corresponding state parameter at peak. The figure clearly shows that the slope of the trend for the dilatancy at first peak is smaller than that for second loading (i.e. after one unload-reload loop). The NorSand  130  plasticity parameter  is defined as the slope of a line through the points in Figure 6.7,  and correspondingly there are two values for x as discussed in Chapters 4 and 5. Note that the best fit straight line through the second peaks does not have a Y-intercept of zero. However, the trend line for the second peaks is required to pass through the point (0,0) as if peak stress ratio (equivalent to minimum dilatancy) occurs at the critical state then, by definition, Dmun must have a value of zero. The hardening modulus, H, was determined by iterative forward modelling of each drained pluviated triaxial test. The value of H was observed to be correlated to .the initial value of state parameter (Figure 6.8). Appendix B shows the fits for all of the drained tests on Erksak sand in  i  —  and  —  plots; an example fit for test ES_CID_867 is  given in Figure 6.9. The general trend line for “I]” (i.e. H = -1727.3  çü, +  75.9) was used  to obtain the monotonic parts of the fits (i.e. before unload-reload cycles).  0  -0.2  -0.4  -0.6  -0.8 -0.16  -0.08  -0.12  -0.04  0  v at D miii  Figure 6.7. Trend lines through Dmin vs. çu at Dmjn for first and second peaks for Erksak sand  131  Although the hardening modulus is expected to also depend on initial fabric, for Erksak sand the moist tamped sample, ES_CID_868, also fitted well in the overall trend for the pluviated samples.  Tests ES_CID_870, ES-CID_871, and ES-CID_872 showed a  different trend for H values. This is likely due to the curved critical state line observed by Been et al. (1991), often associated with grain crushing at high mean effective stress values. If a steeper critical state line was used at higher mean effective stress, the value of initial state,  for tests ES-CID_870, ES-CID_87 1, and ES-CID_872 would be less  negative and closer to the typical trend line. Hence these three tests were ignored in the calibration.  6.1.4. Summary of Erksak monotonic calibration A summary of the Erksak monotonic calibration is presented in Table 6.4. Figure 6.10 summarised the procedure followed for the monotonic calibration of NorSand.  —‘I’,  H  -1727.3 V’ 0 +75.9 300  ES CID 868 with the MT sample 200  Tests withp’ = 800 kPa ignored in the calibration  -0.20  -0.16  100  -0.08  -0.12 ‘/‘0  Figure 6.8. Best fit to Hvs.  çte  for Erksak sand.  132  -0.04  0.00  2.00 Model  1  1.60 1.20 0.80 0.40 0.00  I  0  5  15  10  20  s: % 1.0 0.5 Contraction 0.0 -0.5 -1.0 -1.5 -2.0 Lab results  -2.5 -3.0 Li:  Figure 6.9. Example fit to test ES_CID_867.  133  %  25  Table 6.4. Summary of monotonic calibration for Erksak sand  Parameter CSL F ,  M Plasticity H  Erksak sand 0.82  Altitude of CSL at 1 KPa  0.0 135  Slope of CSL, defined on base e  1.286  -1727.3  Ztc  3.34  N  0.2  Remark  +  Triaxial critical friction ratio  759  Monotonic plastic hardening parameter  Slope of the line relating Dmin to t’at Dmin defined for triaxial conditions The volumetric coupling parameter  ( used in Nova’s rule) Elasticity  134  NorSand parameters  ‘I,  Triaxial tests (both dense and loose samples; loose samples reach critical state within the limit of the apparatus)  Bender elements or cyclic tests to get accurate values for the elastic parameters. Elastic properties for commonly used sands can be found in literature  Jr Obtain Critical State parameters  4. Draw D , vs.  ii  to get  Jr  Jr  Using the previously obtained parameters, run NorSand to get the values of H that give accurate results compared to lab data. Plot H vs. Figure 6.10. Recommended procedure for obtaining NorSand parameters.  62.Monotonic calibration for Fraser River sand The same general procedure was used for the monotonic calibration of Fraser River sand as for Erksak sand in Section 6.1. Therefore, the calibration to Fraser River sand is only briefly described in this section. The tests used in the calibration were described in Section 4.2 (see Table 4.5).  6.2.1. Critical State parameters It was shown in the previous section that the Bishop method can provide unrealistic parameter values, especially for small number of tests (four tests in this case). This also appeared to be the case for Fraser River Sand where the Bishop method gave a higher  135  value of  of 1.55  (Ø  =  stress-dilatancy approach.  38.02°), see Figure 6.11, which is greater than M from the Therefore, the stress-dilatancy method was adopted.  The  post-peak behaviour for two of the tests (FR_CID-03 & FR_CID_06) was questionable. Figure 6.12 shows that the post-peak data points (i.e. the returning curves) for ES_CID_03 in stress-dilatancy plot are scattered and do not follow a consistent trend. However, post-peak behaviour for tests FR_CID_04 & FR_CID_05 seems to be more reliable. Both tests gave  of 1.42  (4 3 5.04°) using the stress-dilatancy method (see =  Figure 6.13 for FR_CID_04). Therefore, This is similar to  of 1.42 was adopted for Fraser River sand.  of 1.4 (q = 34.58°) obtained by Chillarige et al. (1997).  The critical state parameters in the e-log p ‘space are deduced from the data for the six tests as in Figure 6.14. Chillarige et al. (1997) also performed triaxial tests on Fraser River sand and got the critical state parameters. The line in Figure 6.14 has a larger y intercept value than that of Chillarige et al. (1997), but it is steeper.  Most of the  Chillarige et al. data match the line in Figure 6.14 except for two outliers.  It is  noteworthy that those two tests are drained and therefore might not have reached critical state. Drained tests that reach the critical state at large strains are often associated with localization.  136  Z0.68Dm1.55.  1.4 0  Figure 6.11.  I  I  -0.1  -0.2  -0.4  -0.3  -0.5  -0.6  -0.7  -0.8  using Bishop method for Fraser River sand.  1.90  ——  FR_CID_03  1.30  -0.90  I  I  -0.70  -0.50  I  1.20  -0.10  -0.30  0.10  D  Figure 6.12. Enlarged view of the dilatant zone for FRCID_03.  137  0.30  0.50  —  FR_C ID_04  c=1.42o  -0.50  Figure 6.13.  -0.40  -0.30  -0.20  -0.10  0.00 D  0.10  0.20  0.30  0.40  0.50  using stress-dilataney method for FR_CID_04.  1.20  —_____________________________________  • End of test  e  10  =  1.23 + 0.154 log(p’)  100  1000 Log p t (kPa)  Figure 6.14. CSL for Fraser River sand.  138  FR_CID_03 ---—FR_CID_04  —  c End oftest(afterChillarige  10000  6.2.2. Elasticity parameters Chillarige et al. (1997) performed tests with bender elements on Fraser River sand. They introduced the expression in Equation 6.4. It was adopted in the code to define elastic conditions.  ,‘  —p 0 G  ,O.26  2  (6.4)  (295—143e---1 Pa J  6.2.3. Plasticity parameters The value of the  parameter was determined as 4.34 from the best fit to the data points  in peak dilatancy vs. state parameter at peak (Figure 6.15). The slope of the Bishop line in Figure 6.11 is equal to N-i. Therefore the value of N is 0.32. Finally, the plastic hardening modulus, H, which gives a good fit to stress strain and volumetric strain curves is correlated to initial state parameter (see Figure 6.16). The calibrated model results as compared to laboratory tests results are included in Appendix C, and an example fit to test FR_CID_03 is provided in Figure 6.17.  139  -0.1 -0.2 -0.3  -0.5 -0.6 -0.7 -0.16  I  I  -0.14  -0.12  -0.08  -0.1  -0.06  -0.04  v atDm,n  Figure 6.15. Peak dilatancy vs. vat peak for Fraser River sand.  .  120  H7+45.4  -0.30  -0.25  -0.20  I  I  I  -0.15  -0.10  -0.05  ‘.1  0.00  0.05  0 cL,  Figure 6.16. Best fit for H for monotonic triaxial tests on Fraser River sand.  140  0.10  600 500 400 300 200 100 0 0  5  15  10 81:  20  25  %  2 0 -2 C 0  -4  -6 -10 -12  Figure 6.17. Example fit to test FR_CID_03.  Table 6.5. Summary of NorSand monotonic calibration to Fraser River sand.  Parameter  Fraser River Sand  CSL F  Remark  1.23  Altitude of CSL at 1 KPa  0.067  2  Slope of CSL, defined on base e  1.42  ‘VIle  Triaxial critical friction ratio  Plasticity  H  -305.7  Monotonic plastic hardening parameter Slope of the line relating Dmjn to çvat Dmin defined for triaxial conditions The volumçtric coupling parameter (used in Nova’s rule)  + 45.4  4.34  Zic  N  0.32  Elasticity  I  0.26  G  =  2  1r =  p (295_143e{—E--’1 Pj)  P  ,Pref  100 kPa andp is soil  density in ton/m 3  After Chillarige et al.  (1997) V  0.2  Poisson’s ratio (assumed)  141  6.2.4. Summary of Fraser River Sand monotonic calibration Table 6.5 presents a summary of NorSand monotonic calibration to Fraser River sand.  6.3. Unload-reload calibration to Erksak sand The unload-reload model requires three additional parameters: “Ha” and “Hr”, the plastic hardening modulus in unloading and reloading respectively and  X2”,  the slope of  a trend line through the second peaks. X2 is used to capture the effect of induced fabric changes when stress conditions exceed peak. Ideally it would be preferable to relate the three additional unload-reload parameters to parameters derived from a monotonic calibration. However, this is the first calibration of the unload-reload model and therefore it is being used to provide guidance on whether H, Hr and X2 show any relation to the monotonic calibration.  The plastic hardening modulus in unloading “Ha” specifies the rate of movement of the internal cap (i.e. inner yield surface) as previously discussed in Section 5.3. It is used in the model as in Equation 5.17. Laboratory data on Erksak sand suggests that H is shear strain level (at the start of unloading) dependent and that the response of soil in unloading is softer for higher strain levels. Figure 6.1 8a shows the model fits for H unload-reload loop U2 for test ES_CID_867  .  =  30 and 40 to  An H of 40 is a better fit. The fits for  unload-reload ioop U3 are presented in Figure 6.1 8b. An H of 20 is a better fit in this case. It is noteworthy that U3 is at a higher axial strain level than U2. Therefore, higher strain levels appear to be associated with softer unloading. This trend of reducing H with increasing strain level was applicable for all the unload-reload tests. A constant value of H  =  30 was adopted for simplicity. This value fits the unload plots on average  (see Appendix B for the fits of Erksak sand tests).  142  The plastic hardening modulus in reloading “Hr” specifies the rate of movement of the outer yield surface in reloading. Figure 6.19 shows that Hr = 4H fits laboratory data. It is also clear that the results are not sensitive to Hr as it has a very high value (i.e. there is a small difference in the model predictions for Hr  =  4H a.nd Hr  =  811). The first portion of  L4 is elastic and therefore the model gives similar results regardless of the value of Hr. The reload loops in stress strain curves are very steep until a point where the yield surface prior to unload-reload is exceeded.  Jefferies (1997) suggested that soil  ‘remembers’ its past yield surface before unload-reload. agreement with this observation.  Current observations are in  The plastic hardening modulus in reloading “Hr”  calibrated well with a very high value of 4H until the yield surface prior to unload-reload is exceeded. At this point Hr constant Hr  =  =  H. Figure 6.20 shows model simulations for a test with a  4H and one with a changing Hr. The solid dots are on the yield surface  before the unload-reload ioop. A constant Hr results in sharp peaks for the cases of L2 and L3 that were not observed in laboratory results. A changing Hr is adopted as it is consistent with laboratory data. Hr changes from a value of 4H to a value of H once the yield surface before unload-reload is exceeded.  The slope of a trend line through the second peaks, 12, is used to capture the effect of induced changes in particle arrangement on stress-dilatancy when stress conditions exceed peak. A value of 12  =  4.71 was obtained in Section 6.1.3 (see Figure 6.7) (the  original value of for first loading was 3.34). The first loading y is used for pre-peak load or reload loops while 12 is used for post-peak reload loops. A higher in a more dilative behaviour. Figure 6.21 shows that a changing constant  .  value results  gives a better fit than a  It is noteworthy that the two simulations (i.e. constant and changing ) are  identical for the first reloading loop, a pre-peak loop. However, they are different for the other two reloading loops as they are post-peak loops.  Table 6.6 is a summary of the unload-reload calibration for Erksak sand presented in this section.  143  (a) 0.6  e  0.4  d  :  \N I  I  0.2 %  El:  (b) 0.0  I  7.5  8  8.5  I  I  9  9.5  10  10.5  11  11.5  1  s: % Figure 6.18. Model fits using different H values compared to laboratory data (a) U2 for ES_CID_867 (b) U3 for ES CID 867.  144  1.6  1.2  Elasto-plastic reloading  0.8  reloading  H H  8H 4H  —  i 2H  Model resu 2 0.4  0.0 11  10  9  Li:  %  Figure 6.19. Model fits for different Hr values compared to L4 for ES_CID_867.  1200  H = 4H  1000 800 600 400 200  0  2  4  6  8  10  14  12 Li:  16  18  20  %  Figure 6.20. Model simulation for a changing and constant Hvalues.  145  22  24  1.0 0.5 0.0 -0.5 •: -1.0 -1.5 -2.0 -2.5 -3.0  % Figure 6.21. Model fits for constant and changing values compared to ES_CID_867.  Table 6.6. Summary of the unload-reload calibration for Erksak sand.  Parameter  Remark  Erksak sand  H  30  Hr  4H  X2  4.71  Plastic hardening parameter in unloading Plastic hardening parameter in reloading Slope of the line relating Dmin to j, at Dmjn defined for the second peaks  6.3.1. Overview of Erksak Unload-Reload Calibration The proposed model captures the main features of soil behaviour including the volume changes observed during unloading and reloading accurately. Appendix B presents the calibrations for the Erksak sand triaxial tests. Figure 6.22 shows a comparison between the  i 61  and  6 61  plots of the calibrated NorSand model and ES_C ID_867.  Unloading for Loop 2, shown at larger scale in Figure 6.23, does not go through an elastic phase. The response in unloading, phase a-b, is elasto-plastic. Unloading in the NorSand model calibration is elasto-plastic as the stress point touched the cap in the  146  previous loading phase. A small amount of dilation is seen at the beginning of segment a-b which is contrary to what is expected according to plasticity. The reason for that is at the beginning of the unloading phase, the ratio of elastic strains that are dilative to total strains is large. This ratio decreases dramatically later in the unloading phase with plastic strains taking over. Figure 6.24 shows the same simulation as in Figure 6.23 but with plastic unloading instead of elasto-plastic unloading.  It is noteworthy that the small  dilative part in segment a-b disappeared. This proves that this dilative part is due to the large influence of elasticity at the beginning of the unloading phase. Similar behaviour is observed in the simulations for all other tests. It was shown in Section 6.1.2 that the values of the elastic bulk modulus used in the model are lower than those directly calculated form the unload-reload tests data (see Figure 6.6). This could be one of the reasons for the bigger loops at the beginning of a-b compared to those for laboratory data. Although plastic unloading fits the current data set better than elasto-plastic unloading, the latter was adopted in the model. Particles are expected to expand elastically with the decreasing p’ and this well known behaviour cannot be ignored. The elastic properties used for Erksak sand are not accurate in the absence of bender element testing (see Section 6.1.2). Therefore, other data sets with bender elements tests are needed to verify this point. Reloading for loop 2 is elastic (part b-c) until yield occurs once the stress point hits the outer yield surface.  Unloading for the first loop, shown at large scale in Figure 6.25, starts with a linear elastic phase where soil expands in unloading (part a-b), followed by an elasto-plastic phase (part b-c). Note that the stress point does not touch the internal cap in the previous loading for this ioop. Reloading is elastic (part c-d) until the stress point touches the outer yield surface.  147  2.00 Model 1.60  1  o.ooO1IO  2’O %  61: 1.0  25  ———  ——________  0.5 Contraction  t Lab results  -2.5 -3.0  s: %  Figure 6.22. Comparison between calibrated NorSand model and ES_CID_867.  148  1.0  0.5  w 0.0  -0.5 8:  %  Figure 6.23. Zoom on the second ioop of comparison between calibrated NorSand model with elasto-plastic unloading and ES_CID_867.  1.0  0.5  0.0  -0.5  : % 1 e  Figure 6.24. Zoom on the second loop of comparison between calibrated NorSand model and ES_CID_867 with plastic unloading.  149  0.8  0.6  •  0  0.4  0.2  0.0 0  0.5  1.5  1  81:  2  2.5  3  %  Figure 6.25. Zoom on the first ioop for ES_CID_867.  6.4. Summary This chapter presented a triaxial monotonic calibration of the NorSand model to Erksak sand and Fraser River sand. NorSand requires 8 parameters: 3 critical state parameters, 3 plasticity parameters, and 2 elasticity parameters. The unload-reload model presented in Chapter 5 requires 3 additional parameters: H, Hr, and %2. An unload-reload calibration was performed on Erksak sand using cyclic triaxial tests. It was found that Hr and %2 are related to the monotonic parameters Hand ,j, respectively, such that Hr  4Hand 12  1.41i. Hr changes from a value of 4Hto a value ofHonce the yield surface before unload-reload is exceeded. A constant L[ of 30 fits the unload plots on average.  150  =  7. PREDIcTIoNs OF FRASER RIVER SAND UI4LOAD-RELOAD BEHAVIOUR  The parameters/relations from the monotonic calibration for Fraser River sand supplemented by the parameters/relations derived from the unload-reload calibration for Erksak sand were used to predict the two load-unload-reload triaxial tests on Fraser River sand using the modified NorSand model developed earlier in Chapter 5. The two triaxial tests modelled, FR_CID_0l and FR_CID_02 were described in Table 4.4. FR_CID_0l is on a loose sample (pl O kPa & 9 a dense sample (p’=l 98 kPa &  yi  =  =  Test  0.012) while test FR CID_02 is on  -0.156). Both samples were moist tamped.  7.1. Model parameters As discussed previously in Chapter 6, three additional material parameters are needed for the proposed unload-reload calibration: “Ha” and ‘H;’, the plastic hardening modulus in unloading and reloading respectively and X2, the slope of a trend line through the second peaks of the Dmjn versus sand was assumed (i.e. H  =  y at Dmin data. The same value for H as used for Erksak  30). The relation Hr  =  4H, derived for the Erksak Sand  calibration, was also used for the reloading portion of the stress path. Hr is used during loading until the yield surface in existence prior to the current unload-reload is exceeded. At this point Hr returns to the monotonic loading value of H. The value of X2 for Fraser River sand was obtained by assuming the same ratio between  %i  and %2 for Fraser River  sand as obtained from the Erksak calibration. Table 7.1 is a summary of the unloadreload parameters used for the predictions. The monotonic parameters were presented previously in Table 6.5.  151  Table 7.1. Parameters used for Fraser River sand unload-reload predictions. Parameter H  7.2.  Erksak sand 30  H,.  4H  Z2  6.12  Remark Plastic hardening parameter in unloading Plastic hardening parameter in reloading Slope of the line relating Dmjn to i at Dmjn defined for the second peaks  Model predictions  The updated NorSand model was run using the Fraser River monotonic calibration and the unload-reload properties/relations derived from the Erksak unload-reload calibration. No iteration to the input parameters to improve the fits was attempted.  The true  measured and NorSand computed stress and strain responses are given in Figure 7.1 and Figure 7.2 for the loose (FR_CID_01) and dense (FR_CLD_02) tests respectively.  The prediction for FR_CID_0 1 in both q —es and than laboratory data in the range of  6j values  i —j  shows slightly lower curves  greater than 2% (Figures 7.1 a-b). The  difference between the predictions and the laboratory data starts decreasing at higher axial strains, i.e.> 15%. Although the predictions and laboratory data for the monotonic parts of the curves are slightly different, they are very similar in the unload-reload parts. The slopes of the predicted unload-reload loops are almost identical to those for the laboratory data. It can be noticed that the area inside the loops is larger at higher axial strains. The predictions simulate this behaviour. The model generally predicts more contractive behaviour for FR_CID_01 compared to laboratory data (Figures 7. ic). Both laboratory data and the predictions are very similar up to  si of  around 5%. At higher axial strains, the difference between  152  s  from laboratory  data and predictions increases with a maximum difference of more than 1% at very high axial strains.  Similar to FR_CID_01, the monotonic parts of q —8j and  —j  for the predictions for  FR_CID_02 are slightly lower compared to laboratory data (Figure 7.2a-b).  The  difference between the two plots, i.e. laboratory data and predictions, decreases at higher axial strains close to critical state. The slopes of the unload-reload loops are almost identical for both the predictions and laboratory data.  The predictions are quite accurate in s, between  i  at  8j  81  =  s for FR CID 02. The maximum difference  values for the predictions and laboratory data in  around 0.5% at s lower  -  =  -  for FR_CID_02 is  25% (Figure 7.2c) However, this difference is much smaller at  values and the predictions are very accurate. The unload response for the loop  5% is stiffer than the predictions while it is softer for the unload loop at  j  =  10%.  This is similar to observations for Erksak sand described in Section 6.3. The area inside the unload-reload loops is larger for FR_CID_02 than that for FR_CID_0 1. The model captures this behaviour accurately. FR_CID_02 has a higher H value, and therefore a higher Hr. This yields a stiffer reloading response for FR_CID_02 and the area inside the unload-reload loops is increased.  The predictions for FR_CID_02 in the  -  plot better match laboratory data than  those for FR_CID_01 (Figures 7.lc & 7.2c). This is because most of the tests used in the monotonic calibration for Fraser River sand were done on dense samples (see Section 6.2). Therefore, the derived parameters are expected to better fit dense samples. The predictions are still accurate even for test FR_CIDO1 up to strain levels of around 5%. In summary, it seems that the unload-reload parameters obtained from the calibration of Erksak sand provide a reasonably good calibration for Fraser River sand. In q 17 —8j,  —8j  and  the monotonic parts of the plots for the predictions for both tests are slightly lower  than those for laboratory data. However, the slopes of the unload-reload parts are very 153  similar.  In a,,  —  6j,  test FR_CID_02 with the dense sample had better fits than test  FR_CIDO1 with the loose sample. This is expected as the calibration in Chapter 6 was done using dense samples. Using a constant H value fits the unloading parts on average. A constant I-Ia results in a simpler calibration although it is evident that unloading is softer at higher strains. For future work, other sets of unload-reload tests are needed to better understand the factors that H depends on. The predictions could be further improved by including more unload-reload tests on loose samples in the calibration and tests with more unload-reload loops.  7.3. Discussion of model predictions To understand the unload-reload behaviour of sands, special laboratory testing programs that are not commonly performed in industry are required.  However,  monotonic laboratory testing is part of the routine in industry. Hence, a model that can simulate the unload-reload behaviour using unload-reload parameters that can be correlated to monotonic parameters is of practical significance. A challenge for any constitutive model is obtaining good quality laboratory information to calibrate the model. Typically, the simpler the model calibration, the greater the utility of the model for real engineering problems. This section has investigated whether the standard monotonic calibration for NorSand, supplemented by relations observed between the monotonic and three unload-reload parameters for Erksak sand, can be used to accurately predict the behaviour of a different sand: in this instance Fraser River sand. Three additional material parameters are needed for the unload-reload calibration: “He” and H;’, the plastic hardening modulus in unloading and reloading respectively and the slope of a trend line through the second peaks of the Dmjn versus  çt’ at  “X2”,  Dmin data. From  the unload-reload calibration to Erksak sand presented in Chapter 6, it was shown that Hr 154  and X2 are correlated to the monotonic H and Xi , respectively (Hr  4H & X2  =  iXi). 4 l.  A constant value of H equal to 30 was shown to fit the unloading curves on average. The predictions for Fraser River sand presented in this chapter suggest that using correlations from the Erksak calibration gives quite good fits. The slopes of the predicted unload-reload loops in q  —  and  i —6j  were very similar to those for laboratory data. The  dense sample gave reasonably accurate fits in  —  6j  plot. The predictions for the dense  sample were better than those for the loose sample. However, the fits for the loose sample are still quite good up to axial strain of around 5%.  Similar to observation from Erksak data, Fraser River sand simulations show that the code is not sensitive to changes in H,. because it has a high value of 4H (see Figure 6.19). As in the Erksak calibration, H of 30 fits the data on average. Changing  gives better  predictions. Figure 7.3 shows a model simulation for FR_CID_02 with constant to  .  The predictions with changing  equal  shown in Figure 7.2b better fit the laboratory data.  It can be seen both simulations (i.e. changing and constant  ) are identical up to e of  around 5%. This is because the reloading loops in that range are pre-peak (i.e. start from pre-peak conditions) and therefore both simulations are based on  =  4.34. The later  reloading loops start from post-peak conditions and therefore % changes to 6.12 for the simulation in Figure 7.2c. It is noteworthy that changing constant  ,  gives identical predictions as  for FR_CID_0 1 as it is a loose test that never reaches peak.  Overall, the results of the prediction are promising. The current correlation between monotonic and unload-reload parameters was undertaken based on the triaxial tests for a single sand, Erksak. It is likely that these correlations will be improved as the database of triaxial tests with high resolution measurements of the unload reload behaviour expands. However, where possible it is best to calibrate the unload-reload response directly. A few triaxial tests with unload-reload loops should be done for more accurate results.  155  7.4.  Summary  The standard monotonic calibration for NorSand, supplemented by relations observed between the monotonic and the three unload-reload parameters for Erksak sand, was successfully used to predict the cyclic behaviour of another sand: in this instance Fraser River sand. The main finding of this chapter is that the unload-reload behaviour can be simulated using unload-reload parameters that can be correlated to monotonic parameters. It is likely that these correlations will be improved as the database of unload reload tests expands.  156  (a)  600 Test data  1  500 400  Predictioi 300 Fraser River sand 0 =0.89 e p’l9OkPa  200 100 I  0  0  5  15  10 j 6  (b)  20  25  20  25  ..%  2.0  Test data 1.6  1.2  0.8  0.4  0.0 0  5  10  15 8j  (c)  %  4. Prediction  2 I  tatafc Dilative  0-  I  Figure 7.1. Predictions for Test FR_CID_0 1 (a)  157  I  q—8j  (b)  i —si  (c)  s—8i.  (a) 1000  Test data 800 c 600 400  Fraser River sand 200  = =  0.72 198 kPa  0 0  5  10  (b)  61: %  2.0  15  20  25  20  25  Test data  1.6 12 0.8 0.4 0.0 0  5  10  15 61: %  (c) I Contractive  ° Prediction  -4  -5 -6  Figure 7.2. Predictions for Test FR_CID_02 (a) q—ei (b)  158  i —6j  (c)  Nabth  : 1 E  %  Figure 7.3. Model simulation for Test FR_CID_02 in  159  6—6i  with constant  ,‘  of 4.34.  8. SUMMARY AND CONCLUSIONS  8.1. Context of Research The behaviour of sands during loading has been studied in great detail. However, little work has been devoted to understanding the response of sands in unloading. This is surprising as the behaviour of sands in unloading is of great practical importance for earthquake engineering.  An elastic material is expected to expand upon unloading in a conventional triaxial test. Drained triaxial tests indicate that, contrary to the expected elastic behaviour, sand may exhibit contractive behaviour when unloaded. Drained cyclic simple shear tests show similar behaviour in unloading (Sriskandakumar, 2004). Therefore, it is clear that soil behaviour in unloading is not wholly elastic.  The drained behaviour of sands in  unloading was investigated in this work as well as strength and deformation characteristics in reloading.  A practical continuum model that accounts for inelastic  unloading (i.e. yields in unloading) was introduced. The tendency to contract upon unloading during an earthquake could result in liquefaction. Undrained cyclic simple shear tests show that the increase in pore water pressure generated during the unloading cycle often exceeds that generated during loading. A model that yields in unloading is needed to predict this behaviour.  160  8.2. Research Objectives The main objectives of this research were: 1. Develop a theoretical understanding of stress-dilatancy in unloading. This investigation includes the interaction between soil fabric and stress-dilatancy. 2. Utilize the theoretical understanding to guide development of unload-reload behaviour, including yielding during unloading, into a constitutive model.  8.3. Methodology The theoretical approach followed is based on Rowe’s stress-dilatancy (Rowe, 1962). The introduced unload-reload model adopted the NorSand soil model (Jefferies, 1993; Jefferies and Shuttle, 2005) as its starting point. The flow rule used in the model was based on observations from a series of triaxial tests with unload-reload loops on Erksak sand.  Calibration of the model for monotonic and unload-reload stress path were  performed for Erksak sand.  To determine whether the monotonic to unload-reload  relations observed from the Erksak unload-reload calibrations could be applied generally, a prediction for unload-reload on Fraser River Sand was undertaken. First a monotonic calibration of NorSand to Fraser River sand was done. Then the monotonic calibration on Fraser River sand and the unload-reload parameters for Erksak sand were used to predict the results of triaxial tests with load-unload-reload cycles on Fraser River sand.  8.4. Conclusions The theoretical study of deformation characteristic of an assembly of rigid rods shows that the observed soil contraction in unloading is to be expected. The relation between stress and dilatancy during unloading depends on particle arrangement.  161  As the  arrangement of particles is expected to change with cyclic loading, the stress-dilatancy relation in reloading differs from that for first loading. The study of ten drained conventional triaxial tests on Erksak sand, including between one and three unload-reload loops, indicates that significant amounts of contraction can occur during unloading. This is contrary to the dilatant elastic unloading response often assumed in constitutive models of soil. This observed behaviour is consistent with the results of the theoretical study. Experimental observations indicate that unloading loops starting from pre-image stress ratio are dominated by small amounts of dilation, while those starting from post-image stress ratio are dominated by significant amounts of contraction.  The effect of the  unload-reload loops on peak strength is small. This observed contraction in unloading can be explained based on the saw-tooth model. The sawtooth model suggests that the more soil dilates in loading, the more potential energy the soil stores. This energy is available to be released as contraction in subsequent unloading, as observed experimentally.  The results of the series of tests on Erksak sand show that soil becomes more dilatant in post-peak reloading phases (i.e. reloading loops occurring post-peak).  However, the  stress-dilatancy relation remains as in first loading for pre-peak reloading loops. The NorSand constitutive model was extended to represent the experimentally observed yielding during unloading. The introduced model uses non-associated flow in unloading. Dilatancy in unloading is a function of s1ress ratio,  ,  and the stress ratio (q/p at zero  dilatancy in unloading, M. Soil is assumed to yield in unloading on a vertical cap at Dmgn. This assumption fits the framework of monotonic NorSand and is consistent with  observations from laboratory data. The cap contracts in unloading dragging the outer yield surface with it. As a result the outer yield surface softens due to yield in unloading.  162  The resulting model requires the standard 8 NorSand monotonic parameters (T, 2, M, H, Zic,, N, ‘r,and v). It also requires three additional unload-reload parameters introduced in this work: “Ha” and “Hr”, the plastic hardening modulus in unloading and reloading respectively and  “X2”,  the slope of a trend line through the second peaks (i.e. for post-  peak reloading phases) in Dmjn  -  çvplot. X2 is used to capture the effect of induced  changes in particles arrangement on stress-dilatancy when stress conditions exceed peak. The results of the calibration show that the model captures the details of the behaviour of sand under load-unload-reload cycles.  H was significantly smaller than the  monotonic loading hardening, H, and the constant value of 30 provided good fits to the experimental unload-reload data on average. The code is not sensitive to changes in H,. was always observed to be higher than % 2 because it has a high value of around 4H. % exceeded Xi by 40%. The calibrated model predicted the results 2 and for Erksak sand % of triaxial tests with load-unload-reload cycles on Fraser River sand with good precision. Overall, this thesis introduced stress-dilatancy relations for unloading and reloading based on experimental observation, supported by the findings of an investigation done at the micro-mechanical level. An elasto-plastic continuum model that yields in unloading was proposed. It was shown that the unload-reload behaviour can be simulated using unload-reload parameters that can be correlated to monotonic parameters.  8.5. Suggestions for Future Work In summary, this research presents a practical model for load-unload-reload cycles on sand that incorporate inelastic unloading. It accounts for induced changes in particle arrangement. The model gives accurate predictions for triaxial Laboratory data. This research is limited to a triaxial compression framework.  For future research, the  proposed model can be implemented in a finite elements code and therefore it needs to be  163  validated for general stress path conditions. An example of this would be a stress path with decreasing mean effective stress at constant shear stress.  164  REFERENCES  ASTM 2006a, “Standard test methods for maximum index density of soils using a vibratory table (D4253-00-2006), In 2006 Annual Book of ASTM Standards, sect. 4, Vol. 4.08, ASTM, Philadelphia. ASTM 2006b, “Standard test methods for minimum index density of soils and calculation of relative density (D4254-00-2006), In 2006 Annual Book of ASTM Standards, sect. 4, Vol. 4.08. ASTM, Philadelphia. Been, K., and Jefferies, M., 1985, “State parameter for sands”, Geotechnique, Vol. 35, No. 2, pp. 99-112. Been, K., Jefferies, M., and Hachey, J., 1991, “Critical state of sands”, Geotechnique, Vol. 41,No. 3,pp. 365-381. Bishop, A. W., 1971, “Shear strength parameters for undisturbed and remoulded soil specimens”, Proc. Roscoe Memorial Symp., Cambridge, 3-58. Bolton, M. D., 1986, “Strength and dilatancy of sands”, Géotechnique, Vol. 36, No. 1, pp. 65-78. 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C., and Seereeram, D., 1987, “Remaining at yield during unloading and other unconventional elasto-plastic response”, Journal of Applied Mechanics, Vol. 54, No. 1, pp. 22-26. Duncan, J. M., and Chang, C., 1970, “Nonlinear analysis of stress and strain in soils”, Journal of the Soil Mechanics and Foundations Division, Vol. 96, No. 5, pp. 16291653. Golder Associates, 1987, “Report on unload testing to assess initial fabric of Erksak sand”, Golder Associates, Calgary, Alberta Golder Associates. Erksak 330/0.7 triaxial tests, available at www.golder.com/liq. Ghafghazi, M., and Shuttle, D., 2006, “Accurate determination of the critical state friction angle from triaxial tests”, Proc.  th 59  Canadian Geotechnical Conference,  Vancouver, BC., Canada. Ishihara, K., Okada, S., 1978, “Yielding of overconsolidated sand and liquefaction model under cyclic stresses”, Soils and Foundations, Vol. 18, No. 1, pp. 57-72. Jefferies, M., 1993, ‘NorSand; a simple critical state model for sand” Géotechnique, Vol. 43,No. “pp. 91-103.  166  Jefferies, M., 1997, “Plastic work and isotropic softening in unloading” Geotechnique, Vol. 47, No. 5, pp. 1037-1042. Jefferies, M., and Been, K., 2000, “Implications for critical state theory from isotropic compression of sand” Geotechnique, Vol. 50, No. 4, pp. 419-429. Jefferies, M., and Been, K., 2006, “Soil liquefaction: A critical state approach”, Taylor & Francis, London and New York. Jefferies, M., and Shuttle, D., 2002, “Dilatancy in general Cambridge-type models” Géotechnique, Vol. 52, No. 9, pp. 625-638. Jefferies, M., and Shuttle, D., 2005, “NorSand: Features, calibration and use”, Geotechnical special publication No. 128, American Society of Civil Engineers, United States, pp. 204-236.  Li, X., and Dafalias, Y., 2000, “Dilatancy for cohesionless soils” Geotechnique, Vol. 50, No. 4, pp. 449-460. Mitchell, J., and Soga, K., 2005, “Fundamentals of soil behavior”, 3’ edition, John Wiley & sons, New Jersey, USA. Nova, R., 1982, “A constitutive model for soil under monotonic and cyclic loading”, in Soil Mechanics Transient and Cyclic Loads, Wiley, pp. 343-373. Oda, M., 1972, “Initial fabrics and their relation to mechanical properties of granular material”, Soils and Foundations, Vol. 12, No. 1, pp. 17-37. Palmer, A. C., 1967, “Stress-strain relations for clays: Energy theory”, Géotechnique, Vol. 17, No. 4, pp. 348-358. Reynolds, 0., 1885, “On the dilatancy of media composed of rigid particles in contact, with experimental illustrations”, Philosophical Magazine, Vol. 20, pp. 469-48 1.  167  Roscoe, K., Schofield, A., and Wroth, C., 1958, “On yielding of soils”, Géotechnique, Vol. 8, No. 1, PP. 22-53. Roscoe, K., and Burland, J., 1968, “On the generalised stress-strain behaviour of wet clay”, Engineering Plasticity, Cambridge University Press, Cambridge, pp. 535-609. Rowe, P. W., 1962, “The stress-dilatancy relation for static equilibrium of an assembly of particles in contact’, Proc. of the Royal Society, A269, 500-527. Rowe, P. W., 1969, “The relation between the shear strength of sands in triaxial compression, plane strain and direct shear”, Géotechnique, Vol. 19, No. 1, pp. 75-86. Schofield, A. N., and Wroth, C. P., 1968, “Critical state soil mechanics”, McGraw-Hill, London. Sasitharan, 5., 1989, “Stress path dependency of dilatancy and stress-strain response of sand”, M.A.Sc Thesis, University of British Columbia, Vancouver, BC, Canada. Sriskandakumar, 5., 2004, “Cyclic loading response of Fraser River sand for validation of numerical models simulating centrifuge tests”, M.A. Sc Thesis, University of British Columbia, Vancouver, BC, Canada. Tatsuoka, F., and Ishihara, K., 1974, “Yielding of sand in triaxial compression”, Soils and Foundations, Vol. 14, pp. 63-76. Terzaghi, K., Peck, R., and Mesri, G., 1996, “Soil Mechanics in Engineering Practice”, rd 3  edition, John Wiley & sons, New York, USA.  Vaid, Y. ,and Sasitharan, 5., 1992, “The strength and dilatancy of sand”, Canadian Geotechnical Journal, Vol. 29, No. 3, pp. 522-526. Vesic, A. S., and Clough, G., W., 1968, “Behavior of granular materials under high stresses”, Journal of the Soil Mechanics and Foundations Division, Vol. 94, pp. 661688.  168  Wijewickreme, D., Sriskandakumar, S., and Byrne, P., 2005, “Cyclic loading response of loose air-pluviated Fraser River sand for validation of numerical models simulating centrifuge tests”, Canadian Geotechnical Journal, Vol. 42, No. 2, pp. 550-561. Wood, D. M., 1990, “Soil behaviour and critical state soil mechanics”, Cambridge University Press, Cambridge, UK. Yasufuku, N., Murata, H., and Hyodo, M., 1991, “Yield characteristics of anisotropically consolidated sand under low and high stresses”, Soils and Foundations, Vol. 31, No. 1, pp. 95-109.  169  APPENDIX A: PREDICTION OF STRESS DILATANCY IN UNLOADING  It is assumed that stress-dilatancy in unloading can be represented by Equation A. 1. This appendix presents a quantitative justification using the available data on Erksak sand (Table 4.2). Figures A.1 to A.1O show a comparison between Equations A.1 to A.3 and laboratory data.  (A.l)  D°=ii—M  Where, =  2iL —1.5  (A.2)  Where 72L is the last value of stress ratio from the previous loading phase. Plastic dilatancy in unloading becomes constant at:  D =0.5—Ma  (A.3)  Note that the fits for the unload/reload loops at low stress levels are not presented in the following figures. Those loops are highly influenced by elasticity. Using plastic dilatancy relation to fit those loops would give inaccurate results. It is assumed that the effect of elasticity is negligible for the unload/reload loops at higher stress levels.  170  14  •  Loading I  s  Unloading I  U  -  --  Loading2  •  •  • -  -  •: II  -  - -  :  -  0.4  --  0.2  -1.5  -0.5  -1  0.5  0  D  Figure A. 1. Predicted and measured stress-dilatancy for ES_CID_860.  •  Loading I  a  Unloading 1  1.6 Loading 2 x  Unloading 2  z  Loading 3  -  - -  )<  x - -  -  a  U  -  -  -  Model fit to UI andU2 K.;’  - -  -  0.6  • ‘a  0.8  I U  0.4  I,  0.2  a’ _,  -1.5  -I  -0.5  0  0.5  D  Figure A.2. Predicted and measured stress-dilatancy for ES_CID_86 1.  171  Model fit to UI  1.6  •  •  : -  :-  -  -  -  X -  -  Loading 3  0.6  :  x •  L0.2  ‘a.  -  •  Unloading 3  +  Loading 4  -  -  •  Unloading 2  I  x 0.8  0.4  t  Unloading 1  x ‘—  ‘-  -  •  Loading 2  -  __  Loading 1  x  •‘z’;’  •  • -  4  -  -  Model fit to U2 and U3  -  •  1.5  -1.5  Figure A.3. Predicted and measured stress-dilatancy for ES_CID_862.  I  •  Loading 1  a  Unloading 1  -  Loading2 x  Unloading2  0.8  x  Loading 3  ——  -  -1.5  -  -  ModelfittoUl andU2  -1  j  44.  0.6  ,.  C’ E.  -0.5  • . 0  0.5  D  Figure A.4. Predicted and measured stress-dilatancy for ES_CID_866.  172  1  4  •  Loading I  •  Unloading 1  t?  •  Loading 3 •  Unloading 3  +  Loading 4  -  •- ...‘, -  Unloading 2  -  0.8  —  •  •I_  0.6  •  a  :  Model fit for U2 and U3  -1.5  -2  ..  4;.,  Loading 2 x  —  —  -1  I  --  •  04’  41E  -0.5  •  x  0  •  0.5  1  D Figure A.5. Predicted and measured stress-dilatancy for ES_CID_867.  • • e x x -  -  -  Loading 1 Unloading 1 Loading 2 Unloading 2 Loading 3 Model fitto UI andU2  x X  X  x  -  X  x  -  • x  0.6 ;‘  c  0.4  a  0.2  $C  -1.5  -1  -0.5  0  0.5  D Figure A.6. Predicted and measured stress-dilatancy for ES_CID_868.  173  A  1  -_  - -  I  -  X  -I  -1.5  -1  x  Unloading 2 Loading 3 Unloading 3 Loading 4 Model fit to U3  .  +  0.8  -  +:,_  +  X  -  -  -  0.6 x>  ci’  Loading 1 Unloading I  + Zz  x •‘ - -  .  0.4 0.2 -  .  .-  -0.5  0  0.5  D  Figure A.7. Predicted and measured stress-dilatancy for ES_CID_870.  1.4  .  -  .  •a ‘1 0.6  -  -  -  :: -1.5  -1  -0.5 D  Figure A.8. Predicted and measured stress-dilatancy for ES_CID_87 1.  174  -  Loading 1 Unloading I Loading 2 Model fit to Ui  • • x x -  -  -  Loading 1 Unloading I Loading 2 Unloading 2 Loading3 ModelfittoUlandU2  4  0.8  4,,  0.6  0.4  Es)  0.2 -1.5  -1  0  -0.5  1  0.5  D Figure A.9. Predicted and measured stress-dilatancy for ES_CID_872.  A  •  Loading I  •  Unloading I  i  Loading 2  x  Unloading 2  x  Loading 3  +  Unloading 3  -  -  c _+1,  -  _  —  xX  )X  Loading 4  -  I  :“  •.  0.4  Model fit to U2 and U3 +4*  -2  I  I  -1.5  -1  -  I  1  -0.5  0  0.5  D Figure A. 10. Predicted and measured stress-dilatancy for ES_CID_873.  175  APPENDIX B: RESULTS OF THE UNLOAD-RELOAD CALIBRATION FOR ERKSAK SAND  This appendix presents a comparison between the calibrated model results for Erksak sand as compared to triaxial data. The load-unload-reload calibration used to produce Figures B.1 to B.7 was introduced in Section 6.1.  2.00 1.60 1.20 0.80 0.40 0.00 0  5  10  15  20  25  :%  -2  Figure B. 1. Load-unload-reload calibration results compared to laboratory data for ES_CID_860.  176  2.00  —  —_____________________________________________  Model 1.60  1’  1.20  Labres  0.80  ,  0.40 0.00  LIII  0  10  5  15  20  Ei: %  1•  I  Contraction  o Model —1 -2 -3 -4 -5  Figure B.2. Load-unload-reload calibration results compared to laboratory data for ES_CID_86 1.  177  25  2.00 Model  120 1sEfl  0.40 0.00  I  —  0  5  20  15  10  : % 1 e  t  Contraction  Figure B.3. Load-unload-reload calibration results compared to laboratory data for ES_CID_862.  178  25  2.00 Model 1.60 1.20 iai,resuits_f  I  -  0.80 0.40 0 00  0  5  10  15  ci:  20  25  %  t  Contraction  i1Di1aDilation Mode .  Co  —1  Lab  results  Figure B.4. Load-unload-reload calibration results compared to laboratory data for ES_CID_866.  179  2.00  ——-———  —  ——___________________________  Model 1.60  1  1.20  10  20  15 E:  %  1.0 0.5  t  Labresults  -25  -3.0  Contraction  ei: %  Figure B .5. Load-unload-reload calibration results, compared to laboratory data for ES_CID_867.  180  25  2.00 Model  1.60 1.20 0.80 0.40 0.00 0  5  15  10 : 1 E  20  25  %  2  I  CA)  0  Figure B.6. Load-unload-reload calibration results compared to laboratory data for ES_CID_868.  181  2.00  Model  1.60 1.20 0.80 0.40 0.00 0  5  15  10 : 1 C  20  %  2  t  Contraction  I  lilation  0 -2 03  -4  -6 -8  Figure B.7. Load-unload-reload calibration results compared to laboratory data for ES_CID_873.  182  25  APPENDIX RESULTS  C:  FRASER RIVER SAND MONOTONIC CALIBRATION  A monotonic calibration of NorSand for Fraser River sand was introduced in Section 6.2. The tests used in the calibration are presented in Table 4.5. Figures C. 1 to C.6 show a comparison between the calibrated model results and laboratory data. 600 500 400 300 200 100 0 0  5  15  10 8:  20  25  %  -2  -6 -8 -10 -12 1.0 0.9 0.8 0.7 0.6 1000  100  p’: kPa Figure C. 1. Monotonic calibration results compared to tests data for FR_CID_03  183  2000 1600 1200  Sa  800 400  FR_CID_04  0 0  5  15  10  6j.  25  20  %  2 0 -2  -4 -6 -8 -10 1.0 0.9 0.8 0.7  --  —  —  —  —  0.6 1000  100  p’: kPa Figure C.2. Monotonic calibration results compared to tests data for FR_CID_04.  184  2000 1600 1200 800  de=O.69 FR_CID_05 p’ = 515 kPa  400  —  0  15  10  5  81:  25  20  %  1 0 -1 -2 C)  -3 -4 -5  0.9 ----  %%%%  C  0.8  C  0.7  ---_  —  —  -  0.6 1000  100  10000  p’ kPa Figure C.3. Monotonic calibration results compared to tests data for FR_CID_05.  185  300  200  100  0 e  NorSand FR_CID_06  p’  =  0.75 5OkPa  0 0  5  15  10  20  25  %  j. 8  2 0 -2  -6 -8 -10  1.0  .%  %  %  SL  0.9  %%  I  %% %  0.8  —--  0.7 10  100  1000  p’: kPa Figure C.4. Monotonic calibration results compared to tests data for FR_CID_06.  186  300 250 200 150  NorSand  100 50  e 0 p’  FR_CU_01  0.91 388 kPa  0 0  10  5  15 81:  20  25  30  %  400 rM  200  100.1 I  0  0 400 300 .  200  Cs  100  0  100  200  300  400  p’: kPa Figure C.5. Monotonic calibration results compared to tests data for FR_CU_0 1.  187  600 500  /orSd  200 100  FR_CU_02  p’  =  082 196 kPa  0 5  0  15  10  81.  20  25  30  %  160 120 z  80 40  -40  400 300 200 100  0  50  100  150  200  250  300  350  400  p’: kPa Figure C.6. Monotonic calibration results compared to tests data for FR_CU_02  188  APPENDIX D: STEPS TO IMPLEMENT THE LOAD-UNLOADRELOAD MODEL IN A CODE Figures D.1 to D.3 show the steps that can be followed in coding the load-unloadreload model.  1L)Apply plastic shear strain increment  8L) Add elastic strains and move to step 1U  7L) Update strains and state  2L) Calculate M and the current plastic dilation rate  6L) Apply consistency condition to get the new stress state  3L) Recover plastic volumetric strains  5L) Harden! soften the yield surface  4L) impose the limit on the maximum dilation rate  Figure D. 1. A diagram illustrating loading in NorSand.  189  6U) Add elastic strains. Then move to Step 1R for subsequent reloading  1U) Get the current location of the internal cap for the last yield surface in loading/reloading. Ifp Pp’ skip step 2U.  5U) Soften the outer yield surface  2U) Recover elastic strains. Yield surface does not change. If p becomes = Poap go to 3U. Otherwise, go directly to 6U.  4U) Recover plastic volumetric strain increments from stress dilatancy and update strains and state  3U) Apply the hardening rule on the inner yield surface (internal cap) and recover the new stress state  Figure D.2. Description of unloading in the model.  190  5R) Go to lUfor subsequent unloading  1R) Apply shear strain increment  4R) If largest past yield surface is exceeded use H instead of Hr  2R) Recover elastic strains until yield surface defined in 5U is exceeded  3R) Proceed as in loading starting from 2L. Use X2 if peak is exceeded in a previous loading  Figure D.3. Description of reloading in the model.  191  APPENDIX E: TRIAXIAL TESTING PROCEDURE  1. Introduction This appendix briefly presents the triaxial testing procedure and sample preparation methods followed to produce the results of the triaxial tests presented in Chapter 4. A more comprehensive description of the testing procedure can be found in Golder, 1987. The triaxial apparatus used was described in Section 2.2 (Figure 2.13). All tests reported in Chapter 4 were displacement controlled. Load was applied by a 19mm diameter stainless steel piston. The load was measured by a load cell. Cell pressure was measured using a pressure transducer. Volume change was measured using a cylindrical piston with a linear displacement transducer that was calibrated to measure volume change.  2. Sample preparation  •  Wet pluviation: A sample of air-dried sand is placed in a long neck flask. The flask is then filled with de-aired water and placed under vacuum. The membrane mould is filled with de-aired water and the flask is inverted with its neck 25mm above the bottom of the mould. Sand then flows out of the flask. The side of the mould is tapped to reach the desired void ratio.  •  Most tamping: The sample is tamped in 6 layers. Distilled water is added to air dried sand to yield water content of 5-6%. 6 equal weights, prepared to give the desired void ratio, are tamped inside the mould using a tamper with adjustable stops.  192  3. Testing procedure •  The lower platen is installed and the membrane (0.3mm thick) is attached using the 0-rings. The split mould is mounted and vacuum is applied to keep the membrane stretched.  •  The sample is prepared using one of the previously described methods.  •  The top platen is placed and the membrane is rolled around the platen and attached with 0-rings. Vacuum is released and a negative pore pressure of 102OkPa is applied. The mould is then removed.  •  The sample diameter is measured at 5 locations and height is determined.  •  The cell is filled with water. The loading piston is set in contact with the top platen and then locked. The LVDT is zeroed.  •  A cell pressure of 2OkPa is applied while the negative pore pressure is being released. The change in volume is recorded.  •  For moist tamped samples, carbon dioxide is bubbled through the sample for 3 hours. The sample is then flooded with de-aired water from bottom.  •  The cell pressure and back pressure are increased gradually by 100 kPa and the “B” value is measured. The piston is unlocked and the change in height due to saturation is recorded. The change in volume is also recorded.  •  The sample is consolidated hydrostatically by increasing the cell pressure in increments. The change in height and volume are recorded.  •  All transducers are zeroed and the sample is sheared under either drained or undrained conditions.  •  The drainage line is shut and the sample is frozen. The water content and void ratio are determined using the frozen sample.  193  

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