UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

A practical model for load-unload-reload cycles on sand Dabeet, Antone E. 2008-02-02

You don't seem to have a PDF reader installed, try download the pdf

Item Metadata

Download

Media
24-ubc_2008_fall_dabeet_antone_e.pdf [ 3.89MB ]
Metadata
JSON: 24-1.0063096.json
JSON-LD: 24-1.0063096-ld.json
RDF/XML (Pretty): 24-1.0063096-rdf.xml
RDF/JSON: 24-1.0063096-rdf.json
Turtle: 24-1.0063096-turtle.txt
N-Triples: 24-1.0063096-rdf-ntriples.txt
Original Record: 24-1.0063096-source.json
Full Text
24-1.0063096-fulltext.txt
Citation
24-1.0063096.ris

Full Text

A PRACTICAL MODEL FOR LOAD-UNLOAD-RELOAD CYCLES ON SANDbyANTONE E. DABEETB.Sc., The American University in Cairo, 2005A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQIURMENTS FOR THE DEGREE OFMASTERS OF APPLIED SCIENCEinTHE FACULATY OF GRADUATE STUDEIES(CIVIL ENGINEERING)THE UNIVERSITY OF BRITISH COLUMBIA(VANCOUVER)October 2008© Antone Dabeet, 2008ABSTRACTThe behaviour of sands during loading has been studied in great detail. However, littlework has been devoted to understanding the response of sands in unloading. Drainedtriaxial tests indicate that, contrary to the expected elastic behaviour, sand often exhibitcontractive behaviour when unloaded. Undrained cyclic simple shear tests show that theincrease in pore water pressure generated during the unloading cycle often exceeds thatgenerated during loading. The tendency to contract upon unloading is important inengineering practice as an increase in pore water pressure during earthquake loadingcould result in liquefaction.This research contributes to filling the gap in our understanding of soil behaviour inunloading and subsequent reloading. The approach followed includes both theoreticalinvestigation and numerical implementation of experimental observations of stressdilatancy in unload-reload loops. The theoretical investigation is done at the micro-mechanical level. The numerical approach is developed from observations from drainedtriaxial compression tests. The numerical implementation of yield in unloading usesNorSand — a hardening plasticity model based on the critical state theory, and extendsupon previous understanding. The proposed model is calibrated to Erksak sand and thenused to predict the load-unload-reload behaviour of Fraser River sand. The trendspredicted from the theoretical and numerical approaches match the experimentalobservations 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 theprevious loading. It appears that major changes in particles arrangement occur once peakstress ratio is exceeded. The developed unload-reload model requires three additionalinput parameters, which were correlated to the monotonic parameters, to representhardening in unloading and reloading and the effect of induced fabric changes on stressdilatancy. The calibrated model gave accurate predictions for the results of triaxial testswith load-unload-reload cycles on Fraser River sand.11TABLE OF CONTENTSABSTRACT.iiTABLE OF CONTENTS iiiLIST OF TABLES viiLIST OF FIGURES ixLIST OF SYMBOLS xviACKNOWLEDGEMENTS xix1. INTRODUCTION I1.1. Research Objectives 41.2. Thesis Organization2. LITERATURE REVIEW 62.1. Experimental soil behaviour2.1.1. Typical stress-strain behaviour of sand 72.1.2. The Critical State 112.1.3. The state parameter 172.1.4. Yielding of sands 202.2. Triaxial testing2.3. Soil constitutive models 252.3.1. Elasto-plastic soil modelling 252.3.2. Simple soil models 292.3.3. Cam-Clay soil model 322.4. tress-iiiiaiancy 372.5. The NorSand soil model 452.5.1. Yield surface and flow rule 472.5.2. Hardening of the yield surface 502.5.3. Typical evolution of the yield surface 522.5.4. Elastic properties of NorSand 531112.5.5. Summary ofthe NorSand model .532.6. Soil behaviour in unloading 552.6.1. A Simple physical model 552.6.2. Thermo-mechanical approach 562.6.3. Unloading in NorSand 622.6.4. Summary 653. DILATANCY IN UNLOAD-RELOAD LOOPS: A THEORETICALINVESTIGATION 663.1. Micro-Mechanical perspective for dilatancy in unloading 663.2. Micro-Mechanical perspective for dilatancy in reloading 713.3. Summary 744. DILATANCY IN UNLOAD-RELOAD LOOPS: AN EXPERIMENTALINVESTIGATION 754.1. Sands Tested 754.1.1. Erksak Sand 754.1.2. Fraser River Sand 764.2. Testing program 774.2.1. Erksak Sand Testing Program 774.2.2. Fraser River Sand 794.3. Experimental observations4.4. Implications of experimental observations 935. A MODEL TO ACCOMMODATE UNLOAD-RELOAD LOOPS USINGNORSAND 965.1. Yield surface and internal cap5.2. Flow rule 1005.2.1. Flow rule in unloading 1005.2.2. Flow rule in reloading 1025.2.3. Potential surface in unloading 1065.3. Hardening in loading, unloading and reloading 1095.4. Comparison with other models 1145.5. Summary 1206. MODEL CALIBRATION 1216.1. Monotonic calibration for Erksak sand 1216.1.1. Critical state parameters 122iv6.1.2. Elasticityparameters.1286.1.3. Plasticity parameters1306.1.4. Summary of Erksak monotonic calibration 1326.2. Monotonic calibration for Fraser River sand 1356.2.1. Critical State parameters 1356.2.2. Elasticity parameters 1396.2.3. Plasticity parameters 1396.2.4. Summary of Fraser River Sand monotonic calibration 1426.3. Unload-reload calibration to Erksak sand 1426.3.1. Overview of Erksak Unload-Reload Calibration 1466.4. .ummary7. PREDICTIONS OF FRASER RIVER SAND UNLOAD-RELOADBEHAVIOUR 1517.1. Model parameters 1517.2. Model predictions 1527.3. Discussion of model predictions 1547.4. Summary 1568. SUMMARY AND CONCLUSIONS 1608.1. Context of Research 1608.2. Research Objectives 1618.3. Methodology 1618.4. Conclusions 1618.5. Suggestions for Future Work 163REFERENCES 165APPENDIX A: PREDICTION OF STRESS DILATANCY IN UNLOADING 170APPENDIX B: RESULTS OF THE UNLOAD-RELOAD CALIBRATION FORERKSAK SAND 176APPENDIX C: FRASER RIVER SAND MONOTONIC CALIBRATION RESULTS183VAPPENDIX D: STEPS TO IMPLEMENT THE LOAD-UNLOAD-RELOADMODELINACODE 189APPENDIX E: TRIAXIAL TESTING PROCEDURE 192viLIST OF TABLESTable 2.1. Summary of NorSand equations (modified after Jefferies and Shuttle, 2005).54Table 2.2. Summary ofNorSand parameters (after Jefferies and Shuttle, 2005) 55Table 4.1: Index properties of Fraser River and Erksak sands 76Table 4.2: Drained triaxial compression tests on Erksak Sand with load-unload-reloadcycles (data from www.golder.com/liq) 78Table 4.3: Undrained monotonic triaxial compression tests on Erksak sand (data fromBeen et. al., 1991) 78Table 4.4: Drained triaxial compression tests with load-unload-reload cycles on FraserRiver sand (data provided by Golder Associates) 79Table 4.5. Monotonic triaxial compression tests on Fraser River sand (data provided byGolder Associates) 80Table 4.6. Direction of volumetric changes in unloading for the load-unload-reload testsonES 81Table 5.1. Equations used in the triaxial compression version ofNorSand and their stepby step implementation in an Euler integration code 96Table 5.2. Summary of the unloading part of the model 115Table 5.3. Comparison between hardening in the proposed model and Drucker andSeereeram (1987) 119Table 6.1. Typical ranges for monotonic parameters (same as Table 2.2, modified afterJefferies and Shuttle, 2005) 122Table 6.2. using stress-dilatancy method for the unload-reload tests on Erksak sand.126Table 6.3. Summary of M, values for Erksak sand 127Table 6.4. Summary of monotonic calibration for Erksak sand 134Table 6.5. Summary ofNorSand monotonic calibration to Fraser River sand 141Table 6.6. Summary of the unload-reload calibration for Erksak sand 146viiTable 7.1. Parameters used for Fraser River sand unload-reload predictions 152viiiLIST OF FIGURESFigure 1.1. The behaviour of an elastic material in loading and unloading 2Figure 1.2. Results of a triaxial test on Erksak sand in volumetric strain vs. axial strain(reproduced after Golder, 1987) 2Figure 1.3. Drained simple shear tests on Fraser River sand (modified afterSriskandakumar, 2004) 3Figure 1.4. Cyclic direct simple shear test on Fraser River Sand (modified afterWijewickreme et al. , 2005) 4Figure 2.1. Schematic of typical results of a drained triaxial test on loose and dense sandsamples (a) deviator stress vs. axial or deviator strain (b) volumetric strain vs. axialor deviator strain 9Figure 2.2. Schematic of typical results of an undrained triaxial test on loose and densesand samples (a) deviator stress vs. axial or deviator strain (b) pore pressure vs. axialor deviator strain 10Figure 2.3. Schematic of stress strain curves for different mean effective stress values atconstant initial void ratio 11Figure 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) 12Figure 2.5. Results of simple shear tests on 1-mm diameter steel balls at constant normaleffective stress of 138 kPa (reproduced from Roscoe et. al., 1958) 13Figure 2.6. Drained triaxial compression tests on Chattahoochee River sand (reproducedafter Vesic and Clough, 1968) 14Figure 2.7. Critical State Line for Erksak 330/0.7 sand (reproduced from Been et al.,1991) 15Figure 2.8. The projection of the critical state line (a)p’- q (b) e-logp’ 18ixFigure 2.9. Stress paths for three undrained triaxial tests on Kogyuk 350/2 Sand(reproduced from Been & Jefferies, 1985) 19Figure 2.10. Peak friction angle as a function of state parameter for several sands(modified from Been & Jefferies, 1985) 19Figure 2.11. Projection of the yield surface inp’-q plane for Aoi Sand (reproduced fromYasufukuetal., 1991) 21Figure 2.12. Family of yield envelopes for Fuji River sand (reproduced from Ishihara andOkada, 1978) 21Figure 2.13. Schematic of the triaxial apparatus 24Figure 2.14. An example of a yield surface 27Figure 2.15. Definition of dilatancy (modified after Jefferies & Been, 2006) 27Figure 2.16. Definition of normality 28Figure 2.17. Example of the yield surface hardening 29Figure 2.18. Tresca yield criteria in 3-D stress space 30Figure 2.19. Normality to Tresca and Mohr-Coulomb surface 30Figure 2.20. Mohr-Coulomb yield criteria in 3-D stress space 32Figure 2.21. Parallel CSL and NCL in e-logp’ plot 36Figure 2.22. Original Cam-Clay yield surface 36Figure 2.23. Typical assembly of rigid rods. (a) stress conditions (b) deformationcharacteristics (reproduced from Rowe, 1962) 39Figure 2.24. Forces acting on a rigid block sliding on an inclined surface (reproducedfrom Rowe, 1962) 40Figure 2.25. Comparison between Rowe’s stress-dilatancy, Cam-Clay flow rule, andNova’s rule 42Figure 2.26. Dilatancy component of strength as a function of mean effective stress atfailure and relative density (reproduced from Bolton, 1986) 45Figure 2.27. Infinite number ofNCL’s (reproduced from Jefferies and Shuttle, 2002)... 47Figure 2.28. NorSand yield surface (modified after Jefferies and Shuttle, 2005) 51Figure 2.29. Minimum dilatancy as a function of state parameter at image for 13 sands(modified after Jefferies and Been, 2006) 52Figure 2.30. The Saw Tooth Model a) loading phase b) unloading phase 56xFigure 2.31. Energy balance as introduced by palmer (1967) 58Figure 2.32. Stress-dilatancy for Cam-Clay loading, Nova loading, and Jefferies (1997)unloading 60Figure 2.33. Schematic representation of work storage and dissipation according toCollins (2005) 61Figure 2.34. Movement of yield surface in NorSand: Case of unloading from a point onthe internal cap 64Figure 2.35 Movement of yield surface in NorSand: Case of unloading from a pointbefore reaching the internal cap 65Figure 3.1 Micro-mechanical representation of dilatancy for a uniform packing of rigidrods during both loading and unloading a) Minimum void ratio for ,8 = 60° b)Maximum void ratio forfi= 45° c) Minimum void ratio forfi= 30° 68Figure 3.2. Two different uniform assemblies of rigid rods; the dashed rectanglerepresents the basic unit volume (reproduced after Li and Dafalias, 2000) 71Figure 3.3 Theoretical expression based on grain to grain friction(q250)for theuniform packing in Figure 3.1 a) compared with a drained triaxial test on Erksak330/0.7 (p’= 100 kPa and e0 = 0.653) in stress ratio vs. dilatancy space, b) Anglebetween the vertical direction and the tangent at the interface between grains 73Figure 3 .4 Rowe’s stress-dilatancy relation based on grain to grain friction for the twopackings in Figure 3.2 74Figure 4.1. Data from ES_CID_867 (a) stress ratio vs. axial strain (b) volumetric vs. axialstrain (c) stress ratio vs. dilatancy 85Figure 4.2. Data from ES_CID_867 in shear stress vs. axial strain 86Figure 4.3. Results of FR_CID_02 in shear stress vs. axial strain 86Figure 4.4. Zoom on loops 1 and 2 for test ES_CID_867 87Figure 4.5. Zoom on the elastic zone in Figure 4. lc 88Figure 4.6. Data from ES_CID_868 (a) stress ratio vs. axial strain (b) volumetric vs. axialstrain (c) stress ratio vs. dilatancy 89Figure 4.7. Comparison of ES_CID_870 and ES_CID_872 with similar e0 and initialp’but different number of U-R loops (a) axial strain vs. stress ratio (b) axial strain vs.volumetric strain 90xiFigure 4.8. Comparison ofES_CID_861 and ES_CID_862 with similar e0 and initialp’but different number of U-R loops (a) axial strain vs. stress ratio (b) axial strain vs.volumetric strain 91Figure 4.9. Stress ratio vs. dilatancy for pre-peak and post-peak reloading loops(ES_CID_862) 92Figure 4.10. Stress ratio vs. dilatancy for different reload ioops (ES_CID_867) 92Figure 4.11. Dmin VS.iat Dmin for first and second loading of Erksak sand 93Figure 4.12. The saw tooth model (a) loading (b) unloading (Same as Figure 2.35) 94Figure 5.1. Yield surface and internal cap in NorSand, same as Figure 2.28 (modifiedafter Jefferies and Shuttle 2005) 99Figure 5.2. Demonstration of interpreted elastic and elasto-plastic zones on the results ofES_CID_682 in stress ratio vs. dilatancy plot 100Figure 5.3. Drained triaxial tests on Erksak sand with unload-reload loops plotted in thedilatancy vs. space 102Figure 5.4.‘?Land M for L3 and U3, respectively, for ES_CID_862 103Figure 5.5. Correlation between M andijfrom previous loading (drained triaxial testson Erksak sand) 103Figure 5.6. Predicted and measured stress-dilatancy for ES_CID_866 104Figure 5.7. Change of M for different reloading loops (ES_CID_862) 106Figure 5.8. The shape of the potential surface in unloading 109Figure 5.9. Expanded scale view of U2/L3 for ES_CID_868 in Figure 4.6a 111Figure 5.10. The direction of plastic strain increment ratios in unloading with thecorresponding yield surfaces and internal caps 113Figure 5.11. The direction of plastic strain increments ratios in unloading normal to thepotential surfaces 114Figure 5.12. Predicted and measured stress-dilatancy for ES_CID_866 117Figure 5.13. Drucker and Seereeram model (reproduced from Drucker and Seereeram,1987) 118Figure 5.14. Hardening according to Jefferies (1997) (same as Figure 2.35) 119Figure 6.1. M1 using Bishops method for Erksak sand 124Figure 6.2. using stress-dilatancy method (ES_CID_871) 125xiiFigure 6.3. Range ofM using the stress-dilatancy method from the last reloading loopsfor the 9 tests in Table 4.2 126Figure 6.4. CSL determination for Erksak sand from loose undrained tests 127Figure 6.5. Enlarged view of the elastic part in L3 for ES_CID_866 129Figure 6.6. The elastic bulk modulus from Equations 6.1 and 6.3 againstp’ for the elasticzone in L3 for ES_CID_866 130Figure 6.7. Trend lines through Dmjn vs. çti at Dmin for first and second peaks for Erksaksand 131Figure 6.8. Best fit to Hvs. çt, for Erksak sand 132Figure 6.9. Example fit to test ES_CID_867 133Figure 6.10. Recommended procedure for obtaining NorSand parameters 135Figure 6.11. using Bishop method for Fraser River sand 137Figure 6.12. Enlarged view of the dilatant zone for FR_CID_03 137Figure 6.13. using stress-dilatancy method for FR_CID_04 138Figure 6.14. CSL for Fraser River sand 138Figure 6.15. Peak dilatancy vs. çt’at peak for Fraser River sand 140Figure 6.16. Best fit for Hfor monotonic triaxial tests on Fraser River sand 140Figure 6.17. Example fit to test FR_CID_03 141Figure 6.18. Model fits using different H values compared to laboratory data (a) U2 forES_CID_867 (b) U3 for ES CID 867 144Figure 6.19. Model fits for different Hr values compared to L4 for ES_CID_867 145Figure 6.20. Model simulation for a changing and constant Hr values 145Figure 6.21. Model fits for constant and changingvaluescompared to ES_CID_867.146Figure 6.22. Comparison between calibrated NorSand model and ES_CID_867 148Figure 6.23. Zoom on the second loop of comparison between calibrated NorSand modelwith elasto-plastic unloading and ES_CID_867 149Figure 6.24. Zoom on the second loop of comparison between calibrated NorSand modeland ES_CID_867 with plastic unloading 149Figure 6.25. Zoom on the first loop for ES_CID_867 150Figure 7.1. Predictions for Test FR_CID_01 (a) q—i (b) i —‘j (c) s—8j 157xliiFigure 7.2. Predictions for Test FR_C1IJ_02 (a) q— (b),—&j (c) .,—&j 158Figure 7.3. Model simulation for Test FR_CID_02 in 6—ej with constant,‘of 4.34. ... 159Figure A. 1. Predicted and measured stress-dilatancy for ES_CID_860 171Figure A.2. Predicted and measured stress-dilatancy for ES_CID_86 1 171Figure A.3. Predicted and measured stress-dilatancy for ES_CID_862 172Figure A.4. Predicted and measured stress-dilatancy for ES_CID_866 172Figure A.5. Predicted and measured stress-dilatancy for ES_CID_867 173Figure A.6. Predicted and measured stress-dilatancy for ES_CID_868 173Figure A.7. Predicted and measured stress-dilatancy for ES_CID_870 174Figure A.8. Predicted and measured stress-dilatancy for ES_CID_871 174Figure A.9. Predicted and measured stress-dilatancy for ES_CID_872 175Figure A. 10. Predicted and measured stress-dilatancy for ES_CID_873 175Figure B. 1. Load-unload-reload calibration results compared to laboratory data forES_CID_860 176Figure B.2. Load-unload-reload calibration results compared to laboratory data forES CID 861 177Figure B.3. Load-unload-reload calibration results compared to laboratory data forES CID 862 178Figure B.4. Load-unload-reload calibration results compared to laboratory data forES CID 866 179Figure B.5. Load-unload-reload calibration results compared to laboratory data forES CID 867 180Figure B.6. Load-unload-reload calibration results compared to laboratory data forES_CID_868 181Figure B.7. Load-unload-reload calibration results compared to laboratory data forESCID_873 182Figure C. 1. Monotonic calibration results compared to tests data for FR_CID_03 183Figure C.2. Monotonic calibration results compared to tests data for FR_CID_04 184Figure C.3. Monotonic calibration results compared to tests data for FR_CID_05 185Figure C.4. Monotonic calibration results compared to tests data for FR_CID_06 186Figure C.5. Monotonic calibration results compared to tests data for FR_CU_0 1 187xivFigure C.6. Monotonic calibration results compared to tests data for FR_CU_02 188Figure D. 1. A diagram illustrating loading in NorSand 189Figure D.2. Description of unloading in the model 190Figure D.3. Description of reloading in the model 191xvLIST OF SYMBOLSc Mohr-Coulomb stress parameters representing cohesionCSL critical state lineD dilatancy(8v/Sq)Dr relative densitye void ratioE elastic young’s modulusG elastic shear modulusH hardening/softening modulus in loading, a NorSand model input parameterHr hardening/softening modulus in reloading, a NorSand model input parameterf1softening modulus in unloading, a NorSand model input parameterdimensionless shear rigidity parameter (G/p’), a NorSand model input parameterK elastic bulk modulusM critical state stress ratio (q/p’ at critical state), a NorSand model input parameterM stress ratio at image state (image is the boundary between contraction anddilation)M stress ratio at D”= 0 for the case of unloadingN volumetric coupling coefficient, a NorSand model input parameterNC normally consolidatedOCR over-consolidation ratiop mean stress, for triaxial conditions p= (j+2o)/3Pomean effective stress under initial conditionsPcapmean effective stress on the internal cappmean effective stress at first yield in unloadingPrefreference pressure equal to 100 kPa (often assumed equivalent to atmosphericpressure)xviq shear stress invariant, fortriaxial conditions q (1-o-3)v specific volume, 1+ eW total work doneF Altitude of CSL in e-log p’ space at 1 kPa, a NorSand model input parameter8j 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 = (61+ 263)6qshear strain invariant, for triaxial compression6q2(6i — 63)13xislope of the line relating Dmjn to çu at Dmin defined for the first peaks; is equivalentto usual usage of%2slope of the line relating Dmrn to çu at Dmin defined for the second peaksstress ratio, i(q/p’)1Lthe last value of stress ratio in a loading/reloading phaseK slope of the elastic swelling lines2joslope of CSL in e-logiop’ spaceslope of CSL in e-logep’ space, a NorSand model input parameterçt’ state parameter, ,u (e-e)6 angle of dilatationç4,,,constant volume friction angleqjRowe’s mobilised friction anglemaxpeak friction angleq grain to grain friction anglev Poisson’s ratiop soil densityo-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 failuret shear stress on the plane of failurexviiSubscripts• dot over a symbol denotes incrementc critical statedenotes image conditionsq shear invarianto initial,tc triaxial compressionu unloadingv volumetricSuperscriptseffective stresse elasticp plasticxviiiACKNOWLEDGEMENTSI would like to express my deepest gratitude to my supervisor Dr. Dawn Shuttle forherguidance, support and encouragement. Without her advice this work would not havebeen accomplished.I would like to thank my reviewer Dr. John Howie for his useful comments and myofficial supervisor Dr. Jim Atwater. The author would also like to acknowledge the helpof Mike Jefferies, Roberto Bonilla, and Golder Associates for providing access to thelaboratory testing on which this research is based. Thanks to my professors andcolleagues at the Geotechnical group at UBC for their encouragement and usefuldiscussions. The financial support provided by the University of British ColumbiaGraduate Fellowship and the Vancouver Geotechnical Society is highly appreciated.Finally, I owe an enormous debt to my family for their constant support during thepursuit of my Masters degree at UBC. This work is dedicated to my mother.xix1. INTRODUCTIONThe behaviour of sands during loading has been studied in great detail. However, littlework has been devoted to understanding the response of sands in unloading. This issurprising 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 conventionaltriaxial testas illustrated in Figure 1.1. The figure on the left hand side is a schematic illustratingtheexpected elastic trend of decreasing volume associated with increasing confining stress ina conventional triaxial test. The solid square represents the original element sizebeforeloading and the dashed square is the deformed element. According to elasticity, theelement is expected to recover its original size upon removing the confinement, asshownin the figure on the right hand side.Drained triaxial tests indicate that, contrary to the expected elastic behaviour ofincrease 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 load-unload-reload cycle. Positive volumetric strains denote contraction, i.e. decrease involume, 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 j =2.2%. 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 FraserRiver sand are plotted in Figure 1.3. A cyclic shear stress of 50 kPa is applied. It can be1noticed that unloading is associated with contraction, in some cycles more than that inloading. In drained simple shear tests, because the vertical effective stress remainsconstant, the expected elastic volumetric strains are zero. This is contrary to the observedbehaviour.JrIElastic loading4- 4-I‘IElastic unloadingBefore loading or after unloadingAfterloading or before unloadingFigure 1.1. The behaviour of an elastic material in loading and unloading.C-”>I0—1-2Figure 1.2. Results of a triaxial test on Erksak sand in volumetric strain vs. axial strain(reproduced after Golder, 1987).II-6:%2755025o-25-50-75Figure 1.3. Drained simple shear tests on Fraser River sand (modified afterSriskandakumar, 2004).The tendency to contract upon unloading during an earthquake is one contributoryfactor in soil liquefaction. The importance of contraction during unloading may beobserved in undrained cyclic simple shear tests. Figure 1.4 shows a cyclic simple sheartest on Fraser River Sand reported in Wijewickreme et al. (2005). Vertical effective stressis plotted on the x-axis and the applied shear stress is plotted on the y-axis. A decrease inthe vertical effective stress is associated with an increase in pore water pressure. It can beobserved that, apart from the first two cycles, the increase in pore water pressuregenerated during the unloading cycle often exceeds that generated during loading.330a-b:Loading20a’ kPa b-a:Unloading10Figure 1.4. Cyclic direct simple shear test on Fraser River Sand (modified afterWijewickreme et al. , 2005).Observed soil behaviour from both drained and undrained testing clearly indicates thatsoil behaviour in unloading is not wholly elastic. A constitutive model that yields inunloading is needed to predict this soil behaviour, and is the topic of this thesis. A basicrequirement of such a model is stress-dilatancy, i.e. the inter-relationship between stressratio ‘‘ and dilatancy ‘D’, where i qIp and D= ‘‘q’ (and are the incrementsof volumetric strain and shear strain invariant respectively).1.1. Research ObjectivesThe main objectives of this work are:1. Develop theoretical understanding of stress-dilatancy in unloading. Thisinvestigation includes the interaction between soil fabric and stress-dilatancy.2. Utilize the theoretical understanding to guide development of unload-reloadbehaviour, including yielding during unloading, into a constitutive model.This work will include developing an expression for stress-dilatancy in unloading basedon a discrete element approach, including the effect of fabric changes on dilatancy in4reloading, fabric represents “the arrangement of particles, particle groups and pore spacesin a soil” (Mitchell and Soga, 2005). Soil fabric is expected to change due to cyclicloading, consequently changing stress-dilatancy in reloading as compared to that for firstloading.A continuum model that yields in unloading is developed. The model uses the ideasfrom the theoretical investigation of stress-dilatancy in unloading and reloading. Thework will involve calibration of the model to experimental data and using the calibratedmodel to predict the results of drained load-unload-reload tests. The introduced modelutilizes the NorSand soil model, a critical state hardening plasticity model, as its startingpoint.1.2. Thesis OrganizationThe thesis is organized into 8 chapters. Chapter 2 provides an overview of literaturerelating to constitutive modelling for soils, with particular emphasis on soil behaviour inunloading. The theoretical investigation into stress-dilatancy in both unloading andreloading phases is investigated from a micro-mechanical point of view in Chapter3.Chapters 4 through 7 review experimental data to develop an improved constitutivemodel for yielding in unloading and reloading. Chapter 4 presents drained triaxial dataon 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 continuumconstitutive model, NorSand. Chapter 6 presents calibrations of the model. Monotoniccalibration of NorSand is done for both Erksak sand and Fraser River sand. Loadunload-reload calibration of the model is then undertaken on Erksak sand. The calibratedmodel predictions for load-unload-reload tests on Fraser River sand are presented inChapter 7. The conclusions from this work are summarized in Chapter 8.52. LITERATURE REVIEWThe behaviour of sands depends on many factors, including density and mean effectivestress. Constitutive models are necessary to capture the effect of these and other factorson soil behaviour, and to predict this behaviour for real engineering problems. Thischapter focuses on soil constitutive modelling with particular emphasis on soil behaviourin unloading. First, a brief description of the typical behaviour of sands as observed fromlaboratory data and the basics of triaxial testing is introduced. This is followed by adescription of the fundamentals of elasto-plastic constitutive models and some of thecommonly-used soil models are introduced, with emphasis on the critical state model,Cam-Clay. The interrelationship between stresses and dilatancy is then discussed. Thenthe NorSand soil model, used as the basis for the unloading/reloading development laterin this thesis, is introduced. Finally, a review of conceptual models for soils in unloadingis introduced.2.1. Experimental soil behaviourMuch of our understanding of soil behaviour comes from laboratory testing. The mainadvantage of laboratory testing is that the initial conditions and stress path can usually becontrolled. Typical soil behaviour is explained in this section by a review of laboratorytesting in the literature. The discussion includes selected factors which are observed fromlaboratory testing to affect stress-strain behaviour. The critical state theory is alsointroduced, together with a description of yield characteristics of sands.62.1.1. Typical stress-strain behaviour of sandTypical schematics of stress-strain curves for dense and loose sand in drained tests andwith the same applied stress conditions, starting from uniform all-around pressure, areshown in Figure 2.la. In Figure 2.la the deviator stress, q, is oj-o for triaxialconditions. The axial strain is 6j and the deviator strain,6qis 2(81 — 63)13 for triaxialcompression. Both sj and6qare commonly used to plot stress-strain curves in theliterature. They give similar trends. Typical behaviour for dense sand shows a peakvalue 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 ofstress 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. Volumetricstrain, 6, is defined as 6J+ 263 for triaxial conditions. In this thesis, positive strains arecompressive. Therefore positive volumetric strains denote contraction while negativevolumetric strains denote dilation. It can be seen that dense sand contracts initiallyduring shear and then dilates until a state is reached where volumetric strain remainsconstant. Loose sand contracts during shear until it reaches constant volume conditionsat large strains. Reynolds (1885) was first to show that dense sand dilates when shearedtowards failure while loose sand contracts.Typical undrained behaviour of sand is shown in Figure 2.2. As the undrainedcondition prevents volume change, the tendency to change in volume results in a porewater pressure change of opposite sign, which changes the effective stress conditions.Dense sand shows an increased strength with axial or deviator strain. This is associatedwith the development of negative (or decreasing) pore pressure.The strength of loose sand increases to a peak value. This is followed by a decrease instrength until reaching a constant value of strength which is independent of the strainlevel. The corresponding pore pressure increases with increasing strain level. The rate ofincrease decreases with strain, eventually reaching a constant pore pressure.7Soil strength is directly related to mean effective stress. For higher mean effectivestresses soil has a stiffer response and higher strength. Figure 2.3 is a schematicdemonstrating the effect of mean effective stress on stress-strain curve. The three plotshave identical initial void ratios.Although the behaviours described in Figure 2.1, Figure 2.2 and Figure 2.3 aregenerally applicable, differences in soil behaviour are observed for different soils, andalso for the same soil using different preparation methods. This occurs because differentsample preparation methods result in different initial fabric. Fabric refers to “thearrangement 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 tosub-rounded grains with sizes between 0.84 to 1.19mm (Figure 2.4). The two sampleshave a similar initial void ratio and mean effective stress. The only major differencebetween the two samples is the preparation method. One of the samples was prepared bytapping the sides of the mould. The other sample was prepared by tamping. The sampleprepared by tapping demonstrates a stiffer response, associated with a more dilativebehaviour, compared to that prepared by tamping.8(a)(b)II___________________Axial or deviator strainFigure 2.1. Schematic of typical results of a drained triaxial test on loose and dense sandsamples (a) deviator stress vs. axial or deviator strain (b) volumetric strain vs. axial ordeviator strain.Axial or deviator strainLoose sand9(a)Dense sandCLoose sandAxial or deviator strain(b)Loose sandt+ve-veDense sandAxial or deviator strainFigure 2.2. Schematic of typical results of an undrainedtriaxial test on loose and densesand samples (a) deviator stress vs. axial or deviatorstrain (b) pore pressure vs. axial ordeviator strain.10IFigure 2.3. Schematic of stress strain curves for different mean effective stress values atconstant initial void ratio.2.1.2. The Critical StateThe concept that soil will eventually reach a constant stress and void ratio state wasfirst introduced by Casagrande in 1936. He observed from shear box tests that both denseand loose sand, under same vertical effective stress, eventually reach a constant void ratioat which shear deformation continues at constant shear stress. These observations wereindependently confirmed over twenty years later by Roscoe et. al. (1958) who performedsimple shear tests on 1-mm diameter steel balls. All Roscoe et al.’s tests were done underconstant normal effective stress of 138 kPa. Regardless of the initial density, for thesame applied load of 138 kPa all samples reach similar specific volume at large sheardisplacements (see Figure 2.5). The specific volume is the volume occupied by a unitmass and is equal to (1+e).Axial or deviator strain11(a)200 —____________________________180Prepared by‘ 160tapping— — — — ———140120Prepared by• 10080• 6040200 I I I0 2 4 6 8 10 12Axial strain (%)(b)0.1Contractive0‘‘-0.1Prepared by8-0.2-0.3%%24Dilate12I:,,. -0.4-0.5tapping-0.6C-0.7-0.8-0.9Axial strain (%) VFigure 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 thedevelopment of what has become known as Critical State Soil Mechanics (CSSM). Thecritical state is defined as ‘the state at which a soil continues to deform at constant stressand constant void ratio” (Roscoe et. a!., 1958).121.65v0= 1.654— —- er:.+ v0= 1.638— —.—SI——%_+._•00— — — —E 1.63 — ... ..__v0— 1.625 —I— po* — — —p——p0v0 1.611 . *1.61— ...‘v0= 1.5981.590 5 10 1520Shear deformation (mm)Figure 2.5. Results of simple shear tests on 1-mm diametersteel balls at constant normaleffective stress of 138 kPa (reproduced from Roscoeet. al., 1958).However, this constant void ratio, usually known as the critical void ratio, has beenexperimentally shown to vary with stress level. The resultsof drained triaxial tests onChattahoochee River Sand are presented in Figure 2.6 (Vesic andClough, 1968). Thesedrained triaxial tests investigate the dependence of the critical void ratioon stress level.The two tests have identical void ratios but differentvalues of mean effective stress.Figure 2.6 shows that although all of the samplesare dense, the sample with the highermean effective stress contracts matching the behaviourof loose sand. Higher meaneffective stresses cause the particles to move aroundeach other, rather than over, andcrush, rather than simply override when sheared.This results in contractive behaviour.Therefore, the critical void ratio is a function of stresslevel.13(a)2Deviator strain (%)0 5 10 15Deviator strain (%)1.61.2C0.80.4020(b)10p’=34.3MPa&e0=0.69 - - —— — — — ——2-— IContractive-10 —• — -..————Figure 2.6. Drained triaxial compression tests on ChattahoocheeRiver sand (reproducedafter Vesic and Clough, 1968)The experimental observations described above led to the developmentof a theoreticalframework for soil behaviour, known as Critical StateSoil Mechanics (CSSM). CSSM isbased on two axioms:141. A unique critical state exists.2. The critical state is the final state to which all soils converge withincreasing shear strain.CSSM presents a fundamental framework for all soils. Because all soils eventually reachcritical state irrespective ofthecurrent void ratio and stress conditions, having a uniquecritical state is very useful. A unique critical state is an ideal framework around which toconstruct 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, beingboth independent of fabric, loading rate, stress path, and initial density. Figure 2.7 showsthat both moist compacted and pluviated samples in undrained tests finish at the samecritical state line. Drained tests were also observed to follow the same trend. The changein the slope of the critical state line at about 1000 kPa is thought to be due to grainscrushing at high mean effective stress levels.0.800.75 0 •C0.7.I-- 0.65 • Moist compacted - Undrained• Pluviated - Undrained0 60Moist compacted - DrainedCPluviated - Drained— Critical state line0.550.51 10 100 1000 10000Mean effective stress (kPa)Figure 2.7. Critical State Line for Erksak 330/0.7 sand (reproduced from Been et al.,1991).15The 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 deviatorstress, see Figure 2.1 a. Irrespective of the sample preparation method, the two samples inFigure 2.4a reached similar deviator stress values in the higher axial strain range(i.e.>6%).Hence soil behaviour can be understood within the frameworkof the critical state in thethree dimensional space ofp’,q and e. The slope of the projection of the critical state lineinp’-q is known as the critical friction ratio, M (Figure 2.8a). The projection of thecritical state line in e — logiop’ is given by:e=F—..Uog10p (2.1)Where e is the void ratio at the critical state, F ise atp’ = 1 kPa in e — log p’ plot, and2is the slope of the critical state line (see Figure 2.8b). Note that is definedin termsof logio and loge (and in this thesis are termedoand ?e respectively). Both are perfectlyacceptable. However care should be taken as 2 is often used in the literature withoutclarifying the base of the log used.The critical state is a very useful tool as both dense and loose sand are consideredtoend up at the critical state. At a particular mean effective stress level, soilwith e<e istermed dense while soil with e>e istermed loose. Drained dense tests dilate to reachthe critical state while drained loose tests contract to reach the critical state (Figure2.8b).As the critical state line is defmed to be unique, undrained tests reachthe same line as fordrained tests. This makes the critical state an extremely useful reference propertyforaccurate prediction of soil behaviour.162.1.3. The state parameterBeen & Jefferies (1985), using the results of seventy triaxial tests on Kogyuk sand,show that the “bulk characteristics of sands are not sufficient to characterize mechanicalbehaviour of granular materials”. Relative density or void ratio alone does not governsoil behaviour. Dense sand can behave similarly to loose sand at high confiningpressures as was previously shown for Chattahoochee River sand (Figure 2.6). The stateof 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 behaviourof sand is controlled by the state parameter, çt’. In order for the state parameter concept tobe useful, it needs to be defined relative to a reference condition that is unique and isindependent of initial conditions. The critical state is a proper framework as it satisfiesboth conditions. Equation 2.2 defines the state parameter.(2.2)The state parameter is dependent on mean effective stress as the critical void ratio isdependent on mean effective stress. It therefore represents soil behaviour better thanrelative density. This is for two reasons: First, relative density does not specify thecurrent state relative to critical state. Accordingly, relative density cannot be used topredict whether soil contracts or dilates before it reaches the critical state. Second, soilstrength depends on dilatancy, defined as the ratio between an increment of volumetricstrain and an increment of shear strain, and not void ratio, and dilatancy is inverselyproportional to mean effective stress. Figure 2.9 shows that tests with similar initial stateparameters have similar behaviour regardless of the difference in their relative densities,while tests with similar relative densities behaved very differently. The results of thethree tests are normalized to mean effective stress at the critical state,P’cs.Tests 103 and108 have similar initial state parameter. They demonstrate similar behaviour regardlessof 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, showdifferent behaviour.17The state parameter also influences some soil design parameters. A unique relationbetween the peak friction angle and the state parameter has been observed for a range ofdifferent sands (see Figure 2.10). Although there is scatter in the data, the trend ofdecreasing peak friction angle as state parameter increases is clear.(a)(b)00Figure 2.8. The projection of the critical state line (a)p’-q (b) e-logp’.Mean effective stress (kPa)Mean effective stress (kPa)180 0.5 1 1.5 2 2.5Figure 2.9. Stress paths for three undrained triaxial tests on Kogyuk 3 50/2 Sand(reproduced from Been & Jefferies, 1985).V(a-a‘S.0C,C(a(a.(a0S0,CC(a000CEs00C(a1..0- .1 - I________Figure 2.10. Peak friction angle as a function of state parameter for several sands(modified from Been & Jefferies, 1985).21.50.5Test 37:Pc= 350 kPaYb 003&r33000t<7/Testl03:pc = 5OkPa,tçvo.0.03&Dr=33%estlO8:p. =300kPa,— -0.03 & Dr = 50%)p’/p’cs41 -aDo.SD 0Upper boundwfloundS::.• Kogyok sand (0—10% tines)• cj32. z Beautortsand A (2—10% fines)W°aSoBeaufort sand B (5% fines)*Banding sand. f4auchipato sand (Castro, 1969)28 -‘ Vaigrinda sand (Bjerrurn eta!..1961)a Hokksund sand (NW)• Monterey no. 0 sand (Lade, 1972)Range of criticalfriction angle valuesS.—0-1State parametersU 01192.1.4. Yielding of sandsThe yielding point has been classically used to signify the end of recoverabledeformation, usually observed experimentally as a significant decrease in stiffness. Theyield surface may be intersected along any stress path and is composed of an infinitenumber of yielding points in the (e-p ‘-q) space. One of the earliest studies on yielding ofsoils is reported in Roscoe et al. (1958). Roscoe et al. derived an isometric yield curvefrom 39 drained simple shear tests done on 1 mm diameter steel balls. From a theoreticalviewpoint, it is more useful to plot the projection of the yield surface in the p ‘-q space, asshown for Aoi Sand in Figure 2.11 (Yasufuku et al., 1991). The data for all eight drainedtriaxial tests in Figure 2.11 started from the same stress state with OCR=2. The hollowcircles indicate yielding as evident from a sharp change in stress strain curves. The yieldsurface was drawn through these yield points. Note that the curve is not symmetricaround the p’ axis due to sand anisotropy. By repeating the same procedure for differentconsolidation stresses, a family of yield curves can be defined as shown in Figure 2.12for Fuji River sand (Ishihara and Okada, 1978). Experimental studies suggest that theyield surfaces typically have a similar shape, as can be seen for Aoi sand and Fuji Riversand.20200150100.s:-100-150-200— —— Yield surface--‘-- Stress path — . -- -— — — —— 14• Initial stress state .‘/s..S.. /S.I ///II200 40fr 6po 800fI%4%—%%8.4•’44..0—x*. — — —0Mean effective stress (kPa)Figure 2.12. Family of yield envelopes for Fuji River sand (reproduced from Ishihara andOkada, 1978).II8006004002000-200-400DOMean effective stress (kPa)Figure 2.11. Projection of the yield surface inp’-q plane for Aoi Sand (reproduced fromYasufuku et al., 1991).———8080’ — 84—4 SI0———4SSS4 S SI4*400 300 400 €00 6 0_._____..‘ •0I8 — — — —_• —I8___I004044___ —212.2. Triaxial testingAlthough no laboratory testing was undertaken as part of this work, existing triaxialtests form the basis for the constitutive model development.Triaxial testing is commonly used in both industry and research. This section describesconventional triaxial compression testing. The test involves consolidating a cylindricalspecimen under confining pressure, a-3 (for convenience it is assumed that theconsolidation is hydrostatic). A deviator stress of Ao- is then applied in the verticaldirection. The total stress in the vertical direction is o = 03.+ A oA typical arrangement of a conventional triaxial equipment is shown in Figure 2.13. Amulti-speed drive unit is used to apply the axial load. The triaxial cell is filled with deaired water. The soil sample has two porous discs (at the sample bottom and top) and issurrounded laterally by a rubber membrane. The top and bottom porous discs areattached to the upper and lower platens, respectively. The applied load is measured by aload 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 thepore pressure or volume changes and apply back pressure and cell pressure. The typicalsize 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 applyingdeviator stress in the consolidation phase. Water is allowed to drain out of the backpressure 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 pressureless the back pressure. During the shearing stage the sample is loaded by increasing theaxial load in increments for stress controlled testing or by applying displacementincrements for strain controlled testing. For undrained tests, water is not allowed to drain22during this stage and pore pressure is measured. For drained tests, water is allowed todrain 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,pandshear stress, q, invariants, where p’ = (a’i +2a ‘3)13 and q = (ai — a3). Volumetric strain,e, is defined as the sum of the principal strains (i.e. 6,, = 6j + 283). For stresses andstrains used to be work conjugate (meaning that the invariants, or the individual stressesand strains, can be used interchangeably), they must satisfy the following during aloading increment:q6+ p8,, = J;61 + J;82 + 0363 (2.3)Substituting the values ofp’,q, and 6,, in Equation 2.3 and rearranging gives thefollowing expression for the shear strain invariant, 6q:6q =(s —83) (2.4)The primary advantage of triaxial testing is that all of the principal stresses are knownand can be directly controlled. Hence, when used as part of constitutive modeldevelopment, no stresses or strains are left to be inferred. Having to assume stressconditions, introduces uncertainty into the appropriateness of any model. However, thetest is limited to applying only two independent principal stresses. This is a stress paththat rarely, if ever, corresponds to the nature of loading conditions in the field.23CD Cl) C) CD C) 0 CDW2oD•CD O) (DO) CoC 0’-1’CD-.0D3 CDC-) CD.C’) 0I- 0 00--ti 00C Cl)C) CD 0 0 CH -I Co CoCD0003 -‘Cl)CoCo-i.DCDCDDI- 0 Co 0.I HC, CDC, CD2.3. Soil constitutive modelsSoil constitutive modelling provides qualitative and quantitative understanding of soilbehaviour. ‘Proper’ models provide us with an understanding of soil constitutivebehaviour based on an appropriate framework that is derived from mechanics. The needfor ‘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. Becauseit is impractical to perform tests at every possible combination of stress level and voidratio, a useful constitutive model should be able to accurately predict changes in strengthand deformation characteristics for the full range of applicable combinations of stresslevel and void ratio.A brief description of elasto-plastic soil modelling is presented in this section. This isfollowed by an overview of some commonly-used soil models.2.3.1. Elasto-plastic soil modellingSoil is an elasto-plastic material (i.e. exhibits both elasticity and plasticity). Elasticityis associated with recoverable strains, and purely elastic behaviour is usually onlyobserved in soil at very small strains. Plasticity is associated with irrecoverabledeformations. A typical elasto-plastic continuum model comprises: elasticity, a yieldsurface, a flow rule, a hardening/softening rule.Elasticity: Elastic strains are recoverable. The direction of an elastic strain incrementfollows that of the stresses.25Yield 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 yieldsurface 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 strainincrements. As soil changes in volume due to shearing a flow rule is needed (also knownas a stress-dilatancy relation). There are two definitions in literature for dilatancy: theabsolute and the rate definition illustrated in Figure 2.15. The rate definition is morewidely used in constitutive model development, and in North American practicegenerally, and is used in this thesis. Accordingly, dilatancy is defined as the ratiobetween an increment of volumetric strain to an increment of shear strain (i.e.Associated flow was commonly used in the original soil constitutive models becausethese models do not violate Drucker’s postulate (Drucker, 1951). This means that theplastic strain increment ratio, ñ,’ / ñ’, is normal to the yield surface (Figure 2.16). Oncethe yield surface is defined, the flow rule is then automatically defined. This results insimpler and more stable models compared to non-associated flow models (i.e. plasticstrain increment ratio is not normal to the yield surface).26j)• —C12cI1ci)ci)C,)Figure 2.14. An example of a yield surface.Figure 2.15. Definition of dilatancy (modified after Jefferies & Been, 2006).Mean effective stress, p’627Q0. ..EcUUU C)ciFigure 2.16. Definition of normality.Hardening/softening rule: The hardening/softening rule specifies the movement of theyield surface due to an applied plastic strain increment. The yield surface size isincreased for the case of hardening while it decreases for the case of softening. Anexample of a hardening yield surface is shown in Figure 2.17. The stress point followsthe hardened yield surface according to the specified loading path. The requirement for astress point during loading to start and finish on the current yield surface is called theconsistency condition.Plastic strain increment directionnormal to yield surface-.Stress pointMean effective stress, p’plastic volumetric strain increment,6’28Hardened yield surface—_____after applying loadings increment(1) 4%4%/_4./—4..._Stresspointafter, ,/Yield surface before+44.loading incrementapplying loading4%)incrementInitial stress 4%4%I,point s 4%44Mean effective stress,p’Figure 2.17. Example of the yield surface hardening.2.3.2. Simple soil models1) The Tresca modelThe Tresca soil model is widely used for representing the undrained behaviour of clayin a total stress analysis. In the Tresca model yielding occurs when the maximum shearstress reaches a critical value, c (see Figure 2.19). For undrained conditions, Poisson’sratio, v, is 0.49999, implying a condition of no volume change. The Tresca modelrequires two parameters: the critical shear stress value, c, and the elastic Young’smodulus, E. This yield criterion results in the yield surface in 3-D stress space shown inFigure 2.18. Maximum shear stress is independent of mean stress. This makes theTresca model ideal for modelling the unconsolidated undrained (UU) behaviour of soilswhere the shear strength is not affected by an increase in confinement. Normality toTresca’s surface results in vertical plastic strain increments, i.e. zero plastic volumetricstrains with shearing (Figure 2.19).29aiFigure 2.18. Tresca yield criteria in 3-D stress space.ITresca failure criterionFigure 2.19. Normality to Tresca and Mohr-Coulomb surface.02 = 0303Mohr-Coulombfailure criterionIStrain incrementaccording tonormality4-Strain incrementaccording tonormalityUn, 8302) The Mohr-Coulomb modelThe 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 isa function of the mean stress. MC failure surface in the 3-D stress space is shown inFigure 2.20. Unlike Tresca, MC is applied as an effective stress model. MC requires twostrength parameters, c’ and qY’, where c’ represents the part of strength that is independentof normal stress and qY is the effective friction angle. It represents the part of strengththat is dependent on normal stress. Accordingly shear strength, z that causes yield isgiven by:r=c+cr,,tançz’ (2.5)Equation 2.5 is plotted in Figure 2.19 for c = 0. MC requires two additional elasticityparameters (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 dilationangle is equal to the friction angle. This results in unreasonably high volumetric strainsand hence MC is typically used as a non-associated flow model with a dilation angleclose to zero. MC gives reasonable predictions for strength in unconfined problems but itmodels both volume changes and pre-yield stresses badly.3102Figure 2.20. Mohr-Coulomb yield criteria in 3-D stress space.2.3.3. Cam-Clay soil modelCam-Clay is an associated flow constitutive model based on critical state soilmechanics, and one of the earliest advanced constitutive models for soil. There are twoversions of Cam Clay widely referenced in engineering practice. The original version ofCam-Clay was developed in the 1960’s by Schofield and Wroth (1968). Original CamClay (0CC) is not widely found in commercial software, although it is important toexplain the development of ideas used in the model, and as the basis of some later criticalstate 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 allcommercial codes used for geotechnical analysis. MCC is an extension of 0CC thatsought to address some of the deficiencies of the original model.I,0332The 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 flowrule and a hardening rule. 0CC accounts for elastic volumetric strains only (i.e. it is rigidin elastic shear). The slope of the elastic swelling line in e-log p space, shown in Figure2.21, is ,The rate of total work done on a unit volume of soil is given by:W=q&q+p’v (2.6)As only plastic strains are involved in the dissipated work (the elastic strains arerecoverable), Equation 2.6 may be rewritten in terms of plastic strain as:wP=wwe=q6+p8,” (2.7)Dividing byp’ and gives:wp(2.8)The term on the right hand side represents the dimensionless normalized plastic workdissipated. 0CC is based on the assumption that the rate of dissipation is constant and isequal to, the friction ratio at the critical state, M. This results in the 0CC flow rule as:D=M—i (2.9)All 0CC yield surfaces intersect the critical state line at the current critical state value ofmean effective stress, p’ (see Figure 2.22). The normal consolidation line, NCL, is33assumed to be parallel to the critical state line, CSL (see Figure 2.21). This assumptionpoorly represents observed sand behaviour. Jefferies and Been (2000) showed, forErksak sand, that there are an infinite number of normal consolidation lines that are notparallel 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 tothe yield surface as shown in Figure 2.22),(2.12)P 6qFrom 2.11 and 2.12,=0 (2.13)p D”+i7Substituting the value of D” from Equation 2.9 in Equation 2.13. Integrating andsubstituting ln(p )+ 1 for the integration constant at critical state conditions, i.e.p pgives the equation of the yield surface as:(2.14)Mp)34Under normally consolidated hydrostatic conditions,p’=p’0and i = 0. Substituting inEquation 2.14, givesP’c =p ‘0/ 2.718.The 0CC yield surface hardens for the case of i < Mand is associated with intersectingit at mean pressures greater than p’s. A hardening yield surface is associated withcontractive volumetric strains (see Figure 2.22). Hardening continues until i = Mwheresoil reaches the critical state and further shear strain increments do not cause any changein volume. If the stress point touches the yield surface at 17> M, softening occurs. Thisis associated with dilation until the stress point reaches critical state. The 0CC hardeningrule, given in Equation 2.15, is written in terms of the increment of plastic volumetricstrain. It is noteworthy that at critical state ‘ = 0 and therefore movement of the yieldsurface stops. Hence all stress paths will end at the critical state.p(1+e)6’(215)2-itRoscoe and Burland (1968) modified Original Cam-Clay in what became the ModifiedCam-Clay (MCC) model. The major difference between the two models is the shape ofthe yield surface. One of the problems with 0CC is that it predicts shear strains for thecase of hydrostatic loading. The elliptical yield surface of MCC predicts only volumetricstrains for the hydrostatic loading condition. 0CC overestimates the values of strainincrements at small strains. MCC accounts for elastic shear while 0CC is rigid in elasticshear.35eLogp’Figure 2.21. Parallel CSL and NCL in e-logp’ plot.in = 1pc)Figure 2.22. Original Cam-Clay yield surface.ElasticaP’cP?=p/2.718P’o=r-a logp’ — -362.4. Stress-DilatancyDilatancy was defined in Section 2.3.1 as the ratio between an increment of volumetricstrain and an increment of shear strain (i.e. D=This section discusses stressdilatancy (i.e. the inter-relationship between stress and dilatancy) in more detail. Anobjective of this thesis is to investigate stress-dilatancy in unloading and reloading.Reynolds (1885) showed that dense sand dilates when sheared towards failure whileloose sand contracts.The work of Taylor showed that soil strength is due to both the frictional resistancebetween the particles and the tendency of dense soil particles to override each other. Thedifference between the critical friction angle and the peak friction angle is caused bydilatancy.Rowe (1962) introduced a relation between stresses and dilatancy based on the study ofparticles in contact. Particles are assumed rigid, have circular cross-sections and areidentical. The forces at the contacts are assumed purely frictional. The importance ofRowe’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. Theangle of deviation of the tangent at the contacts between particles from direction 1 isdefined asft.L1 are L2 are the loads on each rod in directions 1 and 2 respectively. Atypical unit volume is ij 12 (it is assumed that the rods have unit length in the thirddirection). The volume of the assembly can be expressed by an integer number times thenumber of typical units of volume.The conditions at each contact between two particles in the assembly are similar tothose shown in Figure 2.24. This figure shows a rigid block sliding on an inclined37surface making an angle ,6 from the direction of L1. The component of the reaction forcenormal to the surface is N. The component of the reaction force parallel to the surface isNtan ,where is the particle to particle friction angle. Resolving the forces in the L1and L2 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) (2.18)c2 L21Where o =L1/12 and o =L2/11.8 and52are the deformations in directions 1 and 2respectively at an angle /1 relative to those at an angle /3 (see Figure 2.23b). From thegeometry, the following can be derived:211tanatanfi(2.19)681l2Where=8/11and82 = 82/12.Assuming that vertical compression, lateral expansionand volume increase are all positive gives:-= 1+-- (2.20)6i8138Where= 61+82.From Equations 2.18, 2.19 and 2.20:61__________= tan(q5 + 6)°2 2(1+ 6”tan 46(2.21)The term on the left hand side of Equation 2.21 represents the ratio between work done inthe direction of the major principal stress on the assembly to that done by the assemblyon the direction of the minor principal stress. This ratio is equal to one for the case wherethe 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 themajor principal stress is transferred to the minor principal stress.(a)-.(b)I6/2Figure 2.23. Typical assembly of rigid rods (a) stress conditions (b) deformationcharacteristics (reproduced from Rowe, 1962).I öiI239L2Figure 2.24. Forces acting on a rigid block sliding on an inclined surface (reproducedfrom Rowe, 1962).For a random mass of irregular particles, the value of /3’ changes withloading as theparticles orientations changes. It is assumed that this relocation happenssuch that “therate of internal work done is minimum” (Rowe,1962). This assumption changesEquation 2.21 to:=tan2(45+O.5q) (2.22)From experimental observations Rowe found it necessary to use qS(defined as thefunctional or mobilised friction angle) instead ofq,where q5j varies depending on densityand boundary conditions. The sign of the volumetric strain increment ischanged so thatvolume decrease is positive following the sign conventionused in soil mechanics. Rowe(1962) showed that Equation 2.22 is valid regardlessof the boundary conditions. Fortriaxial conditions, the minor principal stress is o3. Thisgives:= K(1— -) (2.23)J36iWhere,Li40K = tan2(45 + O.5b)(2.24)For triaxial conditions, Rowe(1969) showed that varies betweenthe inter-particlefriction angle and the critical statefriction angle. Under plane strain,ç5f is equal to thefriction angle at the critical statefor any packing up to peakstress ratio. In thep ‘-qspace, rearranging Equations2.23 and 2.24 and assumingq5results in Equation 2.25for triaxial compression.= 9(M—(2.25)9+3M—2Mi7Where,M=6sinq&,,(2.26)3—sinSchofield and Wroth (1968) introducedthe Cam-Clay dilatancy rulebased on plasticwork dissipation mechanismas in Equation 2.27 (Cam-Claywas described in detail inSection 2.3.2). Roscoe and Burland(1968) modified Original Cam-Clayin what becamethe Modified Cam-Clay (MCC)model. Equation 2.28 is the MCCflow rule.D”=M—ri(2.27)Dp=M(2.28)2iCam-Clay is widely used forsoft clay, but the dilatancy ruledoes not match sand datawell, particularly for dense sands.Nova addressed this issue in 1982and developed animproved stress-dilatancy rulebased on observations from laboratorydata (Equation412.29). Nova’s equation contains an additional volumetric coupling parameter (N) whichusually falls in the range of 0.2-0.4.D= (M—i7)(1-N)(2.29)Figure 2.25 plots the Rowe, Cam-Clay and Nova flow rules for M1 .27 and N0.25. Itis 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. = 31.6°).Figure 2.25. Comparison between Rowe’s stress-dilatancy, Cam-Clay flow rule, andNova’s rule.Bolton (1986) used a large database of both triaxial and plane strain tests to relate thecomponent of strength that is caused by dilatancy to initial density and mean effectivestress. The component of strength caused by dilatancy is represented by the differencebetween the peak friction angle, qY,,, and the friction angle at the critical state, qYL,,.Triaxial data show thatq —q5 is directly proportional to relative density and inversely-1.5 -1 -0.5 0 0.5 1 1.5 2D42proportional to mean effective stress at failure (see Figure 2.26). Bolton presentedEquation 2.30 from fits to triaxial laboratory data. Equation 2.30 is plotted in Figure 2.26for 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.— =3[Dr(1O—lnp’)—l](2.30)From plane strain data, Bolton found that the relation between the fraction of strengthcaused 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. Equation2.31, is very similar to Rowe’s relation in Equation 2.23.= 0.80 (2.31)0=sin1 —- =sin’ (2.32)6183Equation 2.31 is valid for plane strain boundary condition for the whole stress pathincluding at peak. Bolton’s work implies that the fraction of strength at peak caused bydilatancy,,ax —,for triaxial boundary conditions is:qi—Ø =0480m(2.33)The problem now is that, unlike for plane strain, the dilation angle does not have aphysical meaning for triaxial conditions. To derive Equation 2.33, it was assumed thatthe 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 stresspaths and initial densities. Assuming that the definition for the dilation angle, Equation2.32, is valid for triaxial conditions, they confirmed that at peak stress the friction angle is43uniquely related to 0maxregardless of the confining pressure and relative density. Theyalso found this relation between peak friction angle and peak dilatancy to be independentof stress path. They used different triaxial stress paths in the p ‘-q space which includedboth compression and extension tests. Accordingly, Vaid and Sasitharan proposed arelation between q5 —qand maximum dilation angle for triaxial conditions. Theymeasured q&1, using the Bishop method that involves plotting the data in peak dilation vs.peak friction angle (Bishop, 1971). A best fit linear trend line is plotted through the datapoints and the friction angle corresponding to zero peak dilatancy is çi. Their proposedrelation 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 onaverage. Therefore, it is not surprising that Equation 2.34, developed for Erksak sand, isdifferent 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 —, for triaxial conditions is around60% of that for plane strain conditions.44p’ at failure (kPa)Figure 2.26. Dilatancy component of strength as a function of mean effective stress atfailure and relative density (reproduced from Bolton, 1986).2.5. The NorSand soil modelThe constitutive model development in the following chapters is based on the generalframework of the NorSand soil model. Therefore, NorSand is described in some detail inthis section. The discussion is limited to triaxial compression boundary conditions.NorSand is an elasto-plastic critical state soil model developed by Jefferies (1993). Overthe last 15 years the NorSand model has been updated, primarily to incorporate varyingcritical 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 onthe monotonic version ofNorSand. The cyclic version will be described in section 2.6.3.NorSand was the first critical state model to realistically model sand in that, unlikeCam-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 the161412108- 642010 100 1000 1000045hardening of the yield surface which allows for more realistic prediction of dilatancy fordense soils. The model requires 8 input parameters that can be easily determined fromlaboratory data (three critical state parameters, three plasticity parameters, and twoelasticity 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 increasingshear strain.One of the main features of all versions of NorSand, which is a significant differencefrom Cam-Clay, is that NorSand has an infinity of normal consolidation lines (NCL) andnot every yield surface is required to pass through the critical state. This behaviour wasfirst reported by Tatsuoka and Ishihara (1974), from triaxial tests on Fuji River sand, whodemonstrated that the normal consolidation line (NCL) for sands is not unique, insteadbeing a function of density. They showed that looser samples yield at higher deviatorstress for a given mean effective stress. Jefferies and Been provided additional data toconfirm this finding in 2000 for Erksak sand (Jefferies and Been, 2000). The concept isillustrated in Figure 2.27. For every normal consolidation line there is a conjugate yieldsurface at each value of initial mean effective stress. The implications of having infiniteNCL locations are:• The yield surface could exist anywhere in the e-q-log(p’) space. It does notnecessarily 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 overconsolidation ratio must be used even if the sand was normally consolidated,46i.e. it did not experience higher mean effective stresses in its history. InNorSand, the “intrinsic state” of soil is separated from overconsolidation andthere is no need to assign an over-consolidation ratio to properly model densenormally consolidated sand (Jefferies, 1993). Instead, the concept of the stateparameter previously discussed is utilized to determine the current location ine-log p’ space relative to the critical state.2.5.1. Yield surface and flow ruleNorSand’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 verticalcap at alimiting dilatancy which occurs at a stress ratio coincident with peak stress conditions.InNorSand peak stress ratio,T7limit,is associated with peak dilatancy orDmin if the sign istaken in consideration (Figure 2.28). In the following discussion the curved portionofCSL0L)zNCLOver-consolidatedLogp’Figure 2.27. Infinite number ofNCL’s (reproduced from Jefferies and Shuttle, 2002).47the yield surface is called the outer yield surface and the vertical portion is called theinner cap or inner yield surface. A soil stress path may intersect the inner cap inunloading. This behaviour will be described in Section 2.6.3. Therefore the focus here ison the outer yield surface.NorSand defines the image condition as the boundary between the contractive anddilative behaviour in dense sands (see Figure 2.28). The image condition is differentiatedfrom the critical state in that it satisfies only one condition of the critical state. At theimage condition, D° = 0 but D1’ 0. The stress ratio (q/p’) at image, M, is a function ofM and iy. As soil reaches the critical state with shearing, the value of M approaches Muntil they are eventually equal at the critical state. The idea of changing M is verysimilar 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 thevariable 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).D—M,—i7 (2.35)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 2.13gives NorSand yield surface as:-7--=l—ln1Pr’(2.36)An expression for M is needed and Nova’s rule in Equation 2.29 is adopted here for peakconditions. Combining Nova’s rule at peak with equation 2.35 gives:M1=M+ND (2.37)48Been and Jefferies (1985) showed by plotting experimental data that there is a relationbetween Dmjn and state parameter, cv. There are three versions of this plot in the literaturedepending onp’ and e at whichylis 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 atDmin, it’ where = e — e1 (er, is the critical void ratio evaluated atp’,). A plot isshown in Figure 2.29.The three versions of the Dmjn vs. cv plot show a trend of increasing dilation rate withincreasing state parameter. The slope of the trend line through the data points is,aNorSand model parameter that is used to impose a limit on the minimum allowabledilation rate and is a function of soil fabric. The second version of theDmjn vs. cit plot isthe one adopted in this thesis. Accordingly,%= Dmin / cv. As elastic strains are negligibleat 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 plasticwork similar to dense sand, Equation 2.39 is changed to Equation 2.40, i.e. madesymmetric 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 (seeFigure 2.28). Evaluating Equation 2.36 at peak conditions and rearranging gives,—1M1),,‘ ,J— (2.41)maxSubstituting the value of D”mjn , as in Equation 2.38, in Equation 2.41 gives,I;— e’’— (2.42)maxThe relative position of the M, Mand77limitlines in Figure 2.28 is not constant.According to Equation 2.40, M1 tends to M as the critical state is approached until theyare eventually equal at the critical state where v 0.Tllimitalso decreases until it is equalto Mat critical state (see Equation 2.42).2.5.2. Hardening of the yield surfaceThe NorSand outer yield surface hardens until the point corresponding to Dmin Sreached. This is followed by a softening response until the yield surface stops changing insize at the critical state. As the NorSand yield surface size is controlled by thedimensionless ratio of (p /p),the hardening rule, representing the change in the size ofthe yield surface, is expressed by (pI p). The NorSand hardening rule takes the formof:50.= H[(J -(2.43)maxWhere H is the plastic hardening modulus, a model parameter. The hardening rule is afunction of 8’ because using s’ instead would result in a model that never gets pastimage as when i = M,, s = 0. The hardening rule gives better fit to data if it is give adependence on the shear stress level (Jefferies and Been, 2006). An exponential functionis 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).510.:.aa_Data from 13 sands•a:ii-0.2aII a1aa Ia —aaa — aI.4- II —a a•a aagao6 a aaaa: a —— aa a-0.8— 1-0.3 -0.2 -0.1 0State 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 surfaceThe hardening and softening ofNorSand yield surface is described as follow (seeJefferies 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 theyield surface during hardening controlled by the mean effective stress atimage,p’s.Soil remains contractive until the current mean effective stressequals p,.• Although the current stress ratio is equal to M, the movement of the yieldsurface does not stop because the image state only satisfies one condition ofthe critical state. The hardening continues with increasing shear strain in a52dilative manner until it reaches the surface corresponding to the limiting stressand maximum allowable dilation rate.• At this point, softening starts with a decreasing rate as it approaches thecritical state.• At the critical state M = M and the yield surface does not move any further.2.5.4. Elastic properties of NorSandIn Cam-Clay elasticity, the elastic shear strains are ignored. NorSand does not ignorethe elastic part of shear strains and variations on elasticity including standard linearelasticity and a range of stress dependent models have been implemented.2.5.5. Summary of the NorSand modelThe full set of equations that specify the NorSand model presented in the precedingsections are given in Table 2.1. Table 2.2 lists the parameters used in the model and theirtypical ranges. The parameter ranges were primarily obtained from calibrations to sand,so care should be exercised when applying to other soil types.53Table 2.1. Summary ofNorSand equations (modified after Jefferies and Shuttle, 2005).Aspect of NorSand EquationInternal model parameters,p- = e—e =F—2eln(p’)and,M.=M-x411Yield surface1—withI-E--l= e‘I M1)LL =M1PI} L1’JmaxFlowruleD’9 =M1—Hardening of outer yield(surface= He(1/M1)E(J’—[PjmaxElasticity= G /p54Table 2.2. Summary ofNorSand parameters (after Jefferies and Shuttle, 2005).ParameterITypical range DescriptionCritical stateF 0.9-1.4 The y-intercept of the elog(p curve at 1KPa2e0.01 — 0.07 The slope of CSL in elog(p’) space defined onbase eJvI 1.2-1.5 q/p’at critical statePlasticityH 50-500 Plastic hardening modulus%tc2.5-4.5 A parameter that limits thehardening of the yieldsurfaceN 0.2-0.4 The volumetric couplingparameter (used in Nova’srule)ElasticityJr100-800 Dimensionless shear rigidity(G/pv0.1-0.3 Poisson’s ratio2.6. Soil behaviour in unloadingWhile there have been relatively many studies addressing the overall cyclic behaviourof sand, little work has been done to study the behaviour of sand during the unloadingphase in detail. It is interesting that sand also shows contractive, in addition to theexpected dilative, behaviour when unloaded. The implications of this behaviour werediscussed in Chapter 1. This section discusses previous work on the topic.2.6.1. A Simple physical modelJefferies (1997) explains soil contraction in unloading in terms of stored potentialenergy during the loading phase. Assuming the saw tooth model represents how soildilates, when dense sand is loaded grains tend to climb over the slip surfaces (see Figure552.30a). This is associated with increase in volume as the voids between the teethareincreased. At the end of loading the potential energy of those particles has beenincreased by the virtue of their new location. When unloading, it is then easy to imaginethat those particles will tend to slide backwards (see Figure 2.30b). This is associatedwith decrease in volume as the voids between the saw teeth get smaller (d2<d1).2.6.2. Thermo-mechanical approachThe first law of thermodynamics states that “The increase in the internal energy of asystem is equal to the amount of energy added by heating the system, minus the amountlost as a result of the work done by the system on its surroundings”. Alternatively, plasticwork done on soil is either dissipated in the form of frictional energy or contributes to theincrease of internal energy.Cam-Clay assumes that all ‘plastic’ work done on soil is dissipated. This means thatplastic work does not contribute to changing internal energy. Part of the total workincrements is recoverable (termed ‘elastic’) and the other part is irrecoverable (termed(a)(b)Id1Id2‘IFigure 2.30. The Saw Tooth Model a) loading phase b) unloading phase.d1>d256‘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 changein internal energy is only due to an ‘elastic’ change of volumetric strain that can becalculated using the slope of the swelling line in a usual consolidation test (Schofield andWroth, 1968). The ‘plastic’ component of work is dissipated and the dissipation rate isassumed constant and equal to the critical friction ratio, M. The term on the right handside of Equation 2.45 represents plastic work dissipation and cannot be negative (as allwork dissipation is positive). Dividing Equation 2.45 through by ./i6q’ and rearrangingyields the Cam-Clay flow rule, Equation 2.46.(2.45)Where,is the ‘plastic’ work (unrecoverable according to Cam-Clay) done per unit volumeis the total work done per unit volumeis 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 inEquation 2.47. Nova (1982) derived his flow rule based on experimental observations.Dp=M7(2.47)1-NUpon substituting for D” and,jand rearranging,57+p’ =Mp + Np’ (2.48)If soils were not to violate the first law of thermodynamics, then work done on the soilsample 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. Thefirst term on the right hand side represents the dissipation mechanism as discussed earlierin this section. It is then reasonable to assume that the second term on the right hand sidecontributes 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 toassume that plastic work done on the sample is transferred into stored elastic energy. It isstored 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 onthe right hand side. Based on this assumption, any other term on the right hand siderepresents something other than dissipation of plastic work. Therefore, according tothermodynamics, it represents changed internal energy or stored ‘plastic’ energy.The idea of a change in internal energy due to change in plastic strains was firstproposed by Palmer (1967). Palmer’s approach is illustrated in Figure 2.31.Total Work Done[Dissipated frictional energy Change in internal energyLf()J L______________________rI\ rIDue to change in Due to change in ‘ (term1(rigid in elastic shear)J Lignored in Original Cam-Clay)Figure 2.31. Energy balance as introduced by palmer (1967).58To justify this, Palmer (1967) considers a hypothetical experiment where the state ofsoil moves along the critical state line in the e-p’ space. While moving along the CSLshear deformations -in this case all deformations are plastic as Palmer’s model as well asOriginal Cam-Clay are rigid in elastic shear- resisted by friction are not expected tocontribute to any change in volumetric strain and the Original Cam-Clay energy balanceequation reduces to:= (2.49)But because we are hypothetically moving on the CSL, and different pressures areassociated with different critical void ratios, then there must be a change in volumetricstrain. Most of this change is ‘plastic’ because the CSL is usually much steeper than theswelling lines. However, Equation 2.49 fails to predict this change. Therefore, anotherterm should be added to represent changes in internal energy due to change in plasticvolumetric strain. This term turns out to be the ‘N’ term on the right hand side ofEquation 2.48.Jefferies (1997) assumes that all the stored ‘plastic’ energy is recovered uponunloading. Solving Equation 2.48 for the case of unloading while changing the sign ofthe ‘N’ term, as it is energy recovered in unloading, gives the following:q8’+p8’ ——Mps’—Np6,’ (2.50)Upon rearranging and substituting,Dp=M(2.51)1+NEquations 2.46, 2.47 and 2.51 are plotted in Figure 2.32. It will be shown later thattriaxial laboratory data shows a different trend for stress-dilatancy in unloading from thatrepresented by Equation 2.51.59Stress-dilatancy in unloading\according to Jefferies (1997);\0.4D’1 =(-M- )i(1+N) forM =1.27andN=0.4-3 -2 -1DFigure 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 athermo-mechanical viewpoint, taking into consideration differences between the microscale where the particles interact, and the continuum scale where most soil constitutivemodels are defined. The model is summarized in Figure 2.33. The difference betweenCollins (2005) approach and that of Palmer (1967) and Jefferies (1997) is that the formerassumes that stored elastic energy is the cause for yield in unloading while this is not thecase for the latter. Collins model is illustrated in the following paragraph.Pure hydrostatic loading on the continuum scale is assumed and following the usualconvention 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 interms of elastic compression of soil particles. In subsequent unloading, part of the storedelastic work is released causing dilation while the other part can only be released ifNova’s rule (M =1.27 andN = 0.4)0.80 1 260associated with particle rearrangement. Particle rearrangement is not elastic and henceplasticity occurs during unloading. Hence it is implied that all plastic strains duringunloading are dilative. It is assumed that most of the total shear energy is dissipated asplastic work.Total work done in loadingHydrostatic compressioncomponentShear componentFigure 2.33. Schematic representation of work storage and dissipation according toCollins (2005).Stored (elasticcompression ofparticles on themicro scale)Dissipated(plastic particlerearrangement)Most of the total shearenergy is dissipated asplastic workUpon unloading, part of stored elasticenergy is recovered (elastic expansionwith no particles rearrangement)Stored elastic shear energy(Very little contribution tofrozen energy)Upon unloading, some of the storedelastic energy cannot be recoveredwithout particle rearrangement. Theenergy associated with particlerearrangement is termed ‘frozen energy’and is dissipated as plastic dilationduring unloading.612.6.3. Unloading in NorSandJefferies (1997) presented a framework for the NorSand model in unloading andsubsequent reloading. Because this model is extended as part of the current work, a moredetailed 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 loadingconditions. This section discusses in more details the unload-reload version ofNorSand.In unloading, soil yields at the inner cap. The inner yield surface (or inner cap) is thevertical part of the yield surface shown in Figure 2.28. Its location at the outer yieldsurface is chosen to fit within the framework of the NorSand model in loading and is avertical straight line for simplicity. This internal cap scales with the outer yield surfaceand is located at:Peap= e(_1)mjM1)(2.52)The NorSand flow rule in unloading was derived earlier in Section 2.6.2 as Equation2.51. Jefferies (1997) introduced a rule to govern the movement of the inner cap, i.e. ahardening rule, as:1___8v“lflII (2.53)Hp%P)Where,H is the hardening(softening) modulus in unloadingpjiis the mean effective stress at first yield in unloading (i.e. the mean stress of the capwhen first intersected)62So far the model definition is completed. The rest of this section presents twoexamples 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 yieldsurfaces corresponding to the load-unload-reload phases. The points of interest areannotated on the stress-strain curve, i.e. the plot at the left top side, and on the yieldsurfaces corresponding to loading, unloading and reloading, i.e. plots on the right top, leftbottom, right bottom sides respectively. The darker lines represent the surface wherecurrent yield is occurring. The thicker lines represent the stress path. The yield surfacehardens 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 ofthe yield surface. With continued shearing further strain causes softening and the stresspath softens to reach point 3.Another stress path is illustrated in Figure 2.35 which has a similar arrangement as forFigure 2.34. Unloading in this case occurs from a point before reaching the internal capthat represents peak conditions. Loading causes hardening of the yield surface along path1-2. The internal cap scales with the yield surface.In unloading, there are three possible cases for the stress point to move on or inside theyield surface:Case 1:The stress point touches the internal cap in loading, unloading then cause plasticsoftening of the yield surface. This is illustrated in Figure 2.34 where yieldingoccurs as the stress point moves on the internal cap from point 3 to point 4. Asthe 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 Figure2.35. Upon unloading, the yield surface does not move until the stress pointtouches the internal cap. Before this point, unloading is purely elastic (Figure632.35). After the stress point touches the cap yielding starts on the cap and theyield surfaces soften until the stress point reaches location 3.Case 3:The third case occurs for unloading early in the stress path. Under thesecircumstances the stress point does not touch the cap during unloading and thewhole unloading phase remains elastiö.Under all situations reloading is elastic as long as the stress point is inside the outeryield surface (see Figure 2.34 and Figure 2.35). Once the stress point touches the outeryield surface, plastic reloading continues as in the virgin loading phase.S-Loadingwhere0 200 200 200 400 000 660 001.201200.000.402Sreface where cuffent yield1• Cununtotresopoint1000 5 10 15 20 25er:%200 -3,.-‘Unloading200/,‘,surfaces where\,N20010005___________________ReloadingSurla swherecurrent y Id isI.0 urnng£Ittsti.reloodingp.100 200 300 400 600 6000 10 200 300 400 000 600 700Figure 2.34. Movement of yield surface in NorSand: Case of unloading from a point onthe internal cap.p.641.802200Loading°2Surface where cusT006 yield /7/::_____________________________________I:\pcciJmng0 5 16 65 20 0 570 200 300 400 600 800 7003003004200/V}Unloading230 A!Reloading/Suaces ere . It S ces ere‘:current yId is:ttcre1oad\\’”\0 100 200 300 400 570 800 700 0 100 200 300 400 500 600700p p’Figure 2.35 Movement of yield surface in NorSand: Case of unloading from a pointbefore reaching the internal cap.2.6.4. SummaryIt can be seen from this selective review of soil behaviour in unloading that soilbehaviour in unloading is still not well understOod. There is no agreement on the causeof yield in unloading, for example the Jefferies model implies that it is mainly caused byplastic shear deformation in loading while Collins attributes yield in unloading torearrangement caused by elastic dilation of particles associated with reduction in meaneffective stress. Clearly, this topic needs more investigation as it is important forearthquake engineering. A practical model that accounts for yield in unloading isrequired. Understanding stress-dilatancy in unloading is one of the requirements of sucha model and is discussed in the following chapter.653. DILATANCY IN UNLOAD-RELOADLOOPS: A THEORETICALINVESTIGATIONDilatancy in loading has been investigated by many researchers as discussed in Section2.4. Most elasto-plastic constitutive models have yield surfaces that were developed forstress paths involving increasing shear; a reduction of shear stress (i.e. unloading) withinthat surface is considered elastic. But contraction has been observed during unloading forthe standard triaxial stress path. Standard elasticity would predict expansion duringunloading. Hence, these observations suggest that the soil is yielding during unloading.Constitutive models that incorporate yield in unloading are rare. The topic is notwellcovered and is controversial as shown in Section 2.6. The objective of this chapter is todevelop theoretical understanding of dilatancy in unloading as well as in subsequentreloading phases. The investigation is done at the micro-mechanical level.3.1. Micro-Mechanical perspective for dilatancy in unloadingThe standard elasto-plastic approach assumes that soil is a continuum. However, inreality, soil is composed of discrete particles and plasticity is an abstraction used toexplain what really happens between the grains. It is of interest to develop anunderstanding of why soil contracts in unloading from a microscopic point of view.Rowe (1962) derived an expression for dilatancy in loading based on frictional forcesbetween rigid cylindrical rods (see Section 2.4). Rowe assumed identical rods that arerigid and have a circular cross-section. The forces at the contacts are assumedpurelyfrictional and the initial packing does not change during shearing. Packing represents thepattern at which particles are arranged relative to each other. For example, Figure 3.1shows one possible packing for the rods but three different particle assemblies. The three66different particle assemblies in the middle of Figures 3.1 a,b, and c have the samepacking. 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 itstays to the right ofparticle ‘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 theupper grain is pushed vertically downward the two side grains will move outwards. Thiswill be associated with an increase in volume if the angle between the tangent to grainsinterface and the vertical direction, f3>45° (see Figure 3.la). However, for f3<45° whenthe 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 workdone by the assembly on the minor principal stress for rigid rods, Rowe derived thefollowing equation (the complete derivation is given in Section 2.4):= tan(qS+,6)(3 1)2 ;(1+)tanfiWhere,a’j is the major principal effective stressa’2 is the minor principal effective stressis the rate of change in major principal strain2is the rate of change in minor principal strainis the rate of unit volume changeq5is grain to grain friction anglefiis the angle between the tangent to grains interface and the vertical directionAnd for the packing in Figure 3.1,67= tan(/1) tan(q5 +/3)(3.2)C)I2 4,8 12Figure 3.1 Micro-mechanical representation of dilatancy for a uniform packing of rigidrods during both loading and unloading a) Minimum void ratio forft= 600b) Maximumvoid ratio forfi= 450c) Minimum void ratio forfi = 30°LOADINGaI24,a)32’24-b)UNLOADINGI2f82! 2-12tW1124,I2oI 24-62124-a2o1I2o1282! 2-Equation 3.1 is valid for different packings of rigid rods but the stress ratio in Equation3.2 depends on the packing type (i.e. the pattern at which particles are arranged relative toeach other). For the packing in Figure 3.1, Rowe showed that stresses and strains in68direction 1 over those in direction 2 can be expressed as in Equations 3.2 and 3.3,respectively.1(3.3)82tan2/3Multiplying Equation 3.2 by 3.3 yields equation 3.1. Note that Equations 3.2 and 3.3 areidentical 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, thatEquations 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 basicunit 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(q5,2sin/5(34)1+2cos/1—— (1+2 cos/3)cos/3(3 5622sin2fl)Note that multiplying Equation 3.4 by 3.5 also yields Equation 3.1. Therefore, Equation3.1 is valid for different packings, while the ratio between stress in direction 1 to that indirection 2 is packing specific. Therefore, if the term 6v18i in Equation 3.1 is assumedto represent dilatancy, then there are different stress-dilatancy relations for differentpackings.69Unloading can be explained in the physical sense by reference to the illustrations on theright hand side of Figure 3.1. If the side grains are pushed inwards, the upper and lowergrains will move outwards. This is associated with decrease of volume iffi>450•As discussed above, Equation 3.1 is derived for loading based on an energy balancebetween the ratio of work done by a strain increment in direction 1 on the assembly tothat done by the assembly in direction 2. Part of work done in the form of a strainincrement on direction 1 is dissipated in the assembly by friction while the remainingwork is transferred to direction 2.Assuming that soil is an isotropic material and the packing does not change during theloading and unloading phases, work balance in unloading can be thought of as the ratiobetween the work done by a strain increment in direction 2 on the assembly to that doneby the assembly on direction 1. In other words, the proposed expression for dilatancy inunloading based on grain to grain friction is the same as the usual one of Rowe (1962)but with the assembly rotated by900,i.e.:8i ==tan(q5M+90—(3 6)U;62 J;(1+6v/6i)tan(90-fi)= tan(90— fi)tan(q + 90— ,6) (3.7)Figure 3.3a plots the proposed relation for dilatancy for unloading, i.e. Equations 3.6and 3.7, as compared to that for loading.fifor unloading is equal to 90°-fl of that forloading, as a consequence of rotating the assembly (Figure 3.3b). Note that Equation 3.7is 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 tograin friction angle,q,can be estimated for quartz as 25° (Rowe, 1962). Figure 3.3a70shows a comparison between the dilatancies for the loading phase and first unloading fora drained triaxial test on Erksak sand (p’ = 100 kPa and e0 = 0.653) to the theoreticalexpressions based on grain to grain friction for q = 25°. The fit is a very good one forboth loading and unloading considering the previously stated assumptions involved inderiving the theoretical expression. However, it is not as good for the reloading phaseand the second unloading ioop (not shown in the figure). This is to be expected as soilfabric changes with continued shearing, while the theoretical expression is only valid forthe packing in Figure 3.1.Packing A Packing BFigure 3.2. Two different uniform assemblies of rigid rods; the dashed rectanglerepresents the basic unit volume (reproduced after Li and Dafalias, 2000).3.2. Micro-Mechanical perspective for dilatancy in reloadingThe stress-dilatancy relation is expected to change for different reloading phases. Thereason is due to particle arrangement (i.e. fabric) changes during shearing. Let us assumethat we have two different uniform packings of rigid cylindrical rods similar to those inFigure 3.2. Packing B has a higher void ratio than packing A. For the sake of theargument, it is assumed that during an unload-reload phase the arrangement of the rodschanges from a packing similar to B to another one similar to A. This changecan be71thought of as being equivalent to change in fabric in real soils. As discussed in Section3.1, the stress-dilatancy relation is different for the two packings. Equation 3.1 is validfor the two packings. However, the stress ratio (i.e. oil a2) is different for packing Aand B as in Equations 3.2 and 3.4, respectively. The stress ratio in unloading for packingA is as in Equation 3.7. Similarly, the equation for stress ratio in unloading for packing Bis:-1-=tan(b +90—fl)2sin(90—fl)(3.8)M1+2sin(90—fl)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 stressratio for a given dilatancy is lower for the denser packing as expected. The trend fromtriaxial laboratory data agrees with the trends in Figure 3.4 as will be shown in the nextchapter.Rowe (1962) recognized that particle relocation occurs with shearing, and as a resultthe value of/I changes in a non-uniform manner because real soil particles are not of thesame size and shape. He assumed that this relocation would happen in a way such thatchanges in the values of/I would keep the rate of internal work done to a minimum. Thisassumption changes Equation 3.1 to Equation 3.9 which is independent offiand thereforeindependent of packing and density (the complete version of Rowe’s derivation is givenin Section 2.4). The assumption of minimum internal work predicts a single stressdilatancy relation to be valid for all packings. Li and Dafalias (2000) disagree withRowe’s use of the assumption of minimum internal work and therefore they predict thatthe 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 soilparticles. It contradicts the exact solution, i.e. Equations 3.1-3.8, that clearly shows thatstress-dilatancy is dependent on packing.72a)-056v/81Figure 3.3 Theoretical expression based on grain to grain friction (q=25°) for theuniform packing in Figure 3.1 a) compared with a drained triaxial test on Erksak 330/0.7(p’= 100 kPa and e0 = 0.653) in stress ratio vs. dilatancy space, b) Angle between thevertical direction and the tangent at the interface between grains.U; (1+ d V/V)= tan2(45+O.Sç!i) (3.9)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 asC,b)-1 -0.5 • 08 / s ion0.5‘JyLoading555035. 30—1 0 0.573shown in Figure 3.4. As the packing is expected to change in a reloading phasecompared to that for first loading, a change in stress-dilatancy is expected in reloading.3.3. SummaryThe study of two packings of rigid rods showed that yield in unloading occurs. In thischapter, a stress-dilatancy relation in unloading was derived based on a micro-mechanicalapproach. The derived relation turns out to be identical to Rowe’s stress-dilatancy inloading while rotating the packing of rods by900.The study of deformationcharacteristics in reloading using a micromechanical approach showed that stressdilatancy changes in reloading compared to first loading. Reloading is associated with amore dilative response than first loading.00.5Figure 3.4. Rowe’s stress-dilatancy relation based on grain to grain friction for the twopackings in Figure 3.2-1 -0.5 • 08 vi 6 1744. DILATANCY IN UNLOAD-RELOAD Loops: AN EXPERIMENTALINVESTIGATIONThe previous chapter addressed dilatancy in unloading and reloading from a micro-mechanical point of view. In order to compare the trends predicted from the micro-mechanical approach to the trends observed in real soils, and to apply these observedtrends to a general continuum model for future application to liquefaction modelling, thischapter investigates observed stress-dilatancy for two sands in drained triaxial tests thatinclude unloading and reloading cycles.4.1. Sands TestedErksak sand (ES) and Fraser River sand (FRS) were used in this study. ES was chosenbecause drained triaxial tests with load-unload-reload cycles were available (GolderAssociates, 1987; www.golder.com/liq). Note that the focus of this chapter is toinvestigate stress-dilatancy and therefore drained tests were used. FRS was chosenbecause of the availability of new monotonic triaxial tests and drained load-unload-reloadtriaxial tests undertaken by Golder Associates.4.1.1. Erksak SandErksak sand, a sand that was used in the construction of the Molikpak core in theCanadian Arctic, is a uniformly graded medium-grain sub-rounded sand, mainlycomposed 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 propertiesare presented in Table 4.1. Its specific gravity is 2.66. The index density measures, emin75and 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 SandFraser River sand originates from the alluvial deposits of Fraser River located in BritishColumbia, Canada. It is a uniformly graded medium-grain sand with angular to sub-rounded particles. It is mainly composed of Quartz, Feldspar and unaltered rockfragments with an average particles size of 260im(see Table 4.1). Its specific gravity,emin, and emaxare 2.75, 0.62, and 0.94, respectively.Table 4.1: Index properties of Fraser River and Erksak sandsFraser River sand, Erksak sand,Sriskandakumar (2004) Been et al. (1991)and Chillariage et. al. and Sasitharan(1997) (1989)Mineralogy 40% Quartz, 11% 73% Quartz, 22%feldspar, 45% Feldspar, and 5%unaltered rock other mineralsfragments and 4% ofother mineralsMedian grain size D50:pm 260 330Effective grain size D10:pm 170 190Uniformity coefficient 1.6 1.8Percentage passing no. 2000 0 7sieveSpecific gravity of particle 2.75 2.66Grain description Angular to sub- Sub-roundedroundedMaximum voids ratio em 0.94 0.775Minimum voids ratio emin 0.62 0.527764.2. Testing programAll tests reported in this section are conventional triaxial compression tests (i.e.confining stress is kept constant during the shearing phase). The testing procedure andsample preparation methods are described in Appendix E.The full Erksak testing program included 29 drained and 39 undrained triaxial tests. Asthis 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 thecritical state line (see Chapter 6). Of the available drained triaxial tests, the ten drainedtests that followed a conventional triaxial stress path, and also contained load-unload-reload cycles, were used for this work. The data for all of the Erksak tests are availablefor download from the web site: www.golder.com/liq.The FRS testing program included 6 drained and 2 undrained triaxial tests. Sixmonotonic tests on FRS, 4 drained and 2 undrained, are used for the monotoniccalibration of NorSand to FRS (see Chapter 6). The two drained unload-reload tests onFRS are used to validate the predictions of the calibrated unload-reload model in Chapter7.4.2.1. Erksak Sand Testing ProgramThe ES load-unload-reload tests are summarized in Table 4.2. The ten tests cover awide range of mean effective stresses (100-800 kPa). The range of void ratios is 0.603-0.723. All of the samples were water pluviated except for ES_CID_868 that was moisttamped. These ten tests were performed under drained conditions to enable review of thevolumetric change characteristics of sand. The number of unload-reload loops in the testsvaries between one and three. In some tests the unload-reload loops occur beforereaching peak strength, while other unload-reload loops are post-peak. This allows the77investigation of stress-dilatancy during unloading and reloading at different strain andstress levels.The ES undrained monotonic triaxial testing used to determine the critical state line issummarized in Table 4.3. All of the moist tamped tests are on loose samples with voidratios close to emax. Undrained conditions and loose samples were chosen so that thesamples can reach critical state within the limitation of the apparatus.Table 4.2: Drained triaxial compression tests on Erksak Sand with load-unload-reloadcycles (data from www.golder.com/liq)P’ e0 Preparation Number ofTest No. (kPa) method’ U-R loops2ES_CID_860 100 0.672 WP 1ES_CID_861 100 0.645 WP 2ES CID_862 100 0.645 WP 3ES_CID_866 400 0.698 WP 2ES_CID_867 400 0.680 WP 3ES_CID_868 400 0.723 MT 2ES_CID_870 800 0.653 WP 1ES_CID_871 800 0.637 WP 2ES_CID_872 800 0.652 WP 3ES CID_873 100 0.603 WP 3‘WP stands for water pluviated and MT stands for moist tamped.2U-R stands for unload-reloadTable 4.3: Undrained monotonic triaxial compression tests on Erksak sand (data fromBeenet.al., 1991)1MT stands for moist tamped.p’ e0 PreparationTest No. (kPa) method1ES_L_601 499 0.754 MTES L_604 699 0.768 MTES_L_605 500 0.766 MTES_L_606 701 0.759 MTES L607 701 0.748 MT784.2.2. Fraser River SandTable 4.4 presents the test conditions of two load-unload-reload tests undertakenrecently by Golder Associates on FRS. Both samples were moist tamped. The confiningpressure for each test is similar at 190 kPa and 198 kPa, but the corresponding void ratiosdiffer, being 0.89 and 0.72 respectively. Both tests had more unload-reload loops thanthe 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 tampedsamples were carried out (see Table 4.5 for a summary of the test details). The undrainedtests were used to obtain critical state parameters, while the drained tests were used forthe FRS monotonic calibration to the NorSand model. The consolidation stresses rangedfrom 50 kPa to 515 kPa ande0ranged from 0.63 to 0.91.Table 4.4: Drained triaxial compression tests with load-unload-reload cycles on FraserRiver sand (data provided by Golder Associates)p’ (kPa) e0 Preparation Number ofTest No. method’ UIR loopsFR_CID 01 190 0.89 MT 5FR_CID_02 198 0.72 MT 41MT stands for moist tamped.2UR stands for unload-reload.79P’ (kPa) PreparationTest No. e0 Test type method1FRCID 03 114 0.67 Drained MTFR_CID_04 410 0.63 Drained MTFR CID 05 515 0.69 Drained MTFR_CID_06 50 0.75 Drained MTFR_CU_01 388 0.91 Undrained MTFR_CU 02 196 0.82 Undrained MT‘MT stands for moist tamped.4.3. Experimental observationsThe main focus of the review of experimental results was stress-dilatancy in unloadingand subsequent reloading. To the knowledge of the author, an experimental review tospecifically 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 dilatein unloading and the amount of those volumetric changes. The effect of the loops onpeak strength and volumetric changes in reloading is also investigated.In the following discussion “U” refers to an unloading loop and “L” refers to a loadingor reloading loop. The number following the symbol denotes the order of a particularloop from the beginning of the test.A typical test on Erksak Sand is plotted in Figure 4.1. The strength of the sand, shownin Figure 4.la, does not seem to be highly affected by the unload-reload loops. The datashows 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 1occurs before the image state which marks the boundary between contractive and dilativebehaviour as annotated in Figure 4. lb. Loop Ui is not followed by a small peak and thestress 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 reloadingTable 4.5. Monotonic triaxial compression tests on Fraser River sand (data provided byGolder Associates)80appear larger in Figure 4.2 which plots shear stress vs. axial strain (rather than thestressratio vs. axial strain plotted in Figure 4.1 a). The other tests on Erksaksand (seeAppendix 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).Unloadingoccurs starting from pre-image conditions for Ui. Note thatvolumetric changesassociated with unloading are very small and are initially contractive followed by a smalldilative phase (see Figure 4.4 for a zoom on loop 1). However, for U2 and U3,volumetric changes associated with unloading are significant. They aredominantlycontractive (see Figure 4.4 for a zoom on loop 2). Note that unloading for those twoloops starts from post-image conditions.Table 4.6 summarizes the direction of volumetric changes in unloading for the load-unload-reload tests on ES. The information provided includes the stress ratio and axialstrain at image stage. For each unload-reload (U-R) loop the stress ratio at whichunloading starts is given, together with the axial strain, whether the ioop starts pre or postimage, and whether the volumetric strains are dilative or contractive. It can be observedthat whenever unloading happens from post-image conditions, volumetric strains areeither totally contractive or dominated by contraction. Conversely, if unloading occursfrom pre-image conditions, volumetric strains are either totally dilative or dominated bydilation.Table 4.6. Direction of volumetric changes in unloading for the load-unload-reload testsonESTest U-R loop’ Stress ratio Axial Pre/Post Dilative/ContractiveStrain (%) image during unloadingES_CID_860 Image 1.114 0.402 At -1 1.419 10.073 Post Con.ES_CID_861 Image 1.098 0.544 At -1 1.362 1.778 Post Mostly con.2 1.484 10.014 Post Con.81Test U-R loop’ Stress ratio Axial Pre/Post Dilative/ContractiveStrain (%) image during unloadingES_CID_862 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.ES_CID_866 Image 1.169 2.644 At -1 1.265 4.50 1 Post Mostly con.2 1.359 10.064 Post Mostly con.ES_CID_867 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.ES_CID_868 Image 1.209 3.68 At -1 1.2 12 4.047 Post Mostly con.2 1.269 10.146 Post Con.ES_CID_870 Image 1.177 1.807 At -1 1.352 10.056 Post Mostly con.ES_CID_871 Image 1.16 1.84 At -1 1.308 4.035 Post Mostly con.2 1.366 10.108 Post Mostly con.ES_CID_872 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.ES_CID_873 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.Image indicates the stressbehaviour in loading)ratio at image (image marks the boundary between contractive and dilative82Figure 4.1 c presents the data from the same example ES test used previously, testES CID 867, in stress-dilatancy space. The following equation was used for calculatingdilatancy from laboratory data:D8(+i) 8(—i)(4.1)6q(n+1) 6q(n—1)where 11 denotes the current step.In the case of unloading, positive dilatancy means volume increase while negativedilatancy indicates volume decrease. For unload phases U2 and U3 in Figure 4.lc, thesample contracts in unloading except for one point in U2. Unloading for those twophases starts from post-image stress ratios. Conversely, for Ui where the sample isunloaded from a pre-image stress ratio, the sample increases in volume at the beginningof the unloading phase. Then the sample contracts towards the end of the unloadingphase. This behaviour shows that soil does not unload in an elastic manner for U2 andU3. That the behaviour of U2 and U3 is not elastic is known for two reasons: 1) forconstant Poisson’s ratio elastic dilatancy should be constant 2) dilatancy has a negativesign which is not possible under the elastic framework for conventional triaxial stresspath unloading. For U 1, where the sample is unloaded from a pre-image stress ratio, thereis a small elastic part represented by the first three points in the dilatancy plot (see Figure4.5). However, there is some uncertainty in the interpretation of this part of the testbecause of the small number of data points. The elastic part is followed by plasticyielding.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 thesample preparation method.83Tests ES_C1Q870 and ES_CID_872 have similar e0 and initial p’. The onlysignificant difference between the tests is that the former has one U-R loop while thelatter has three loops. The difference in the number of loops does not cause a significanteffect on the stress ratio vs. axial strain curve (Figure 4.7a). The first loop inES_CID_872 causes only small change in volumetric strains while the second loopcauses significant contraction when compared to the results of ES_CID_870 (Figure4.7b). Note that the first loop in ES_CID_872 is pre-image while the second is post-image. In the third loop, both tests start from approximately similar points anddemonstrate similar behaviour. It can be observed that the volumetric strain curve forES_CID_872 after the second loop is steeper than that for ES_CID_870. This impliesthat 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 andvolumetric strain curves, are tests ES_CID_86 1 and ES_CID_862 which have identical e0and 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 anysignificant different between the results of the two tests.84(a)1.6LI L2 L3 L41.20 5 10 15:%Thlage‘I,tContraction5N1lat2-4•L1.ui.AL2$••U 2t “(L3tc020-1 -0.6 -0.2 0.2 0.6 1Contraction if unloading Contraction if loadingDilation if loading— DDilation if unloadingFigure 4.1. Data from ES_CID_867 (a) stress ratio vs. axial strain (b) volumetric vs. axialstrain (c) stress ratio vs. dilatancy:- 0.80.4010—1—2—3(b)0’(c)0851200 —_______________________________8006001000Hksand400e0 0.68p’ = 400 kPa2000• I I0 5 10 15 20 25c:%Figure 4.2. Data from ES_CID_867 in shear stress vs. axial strain1000800600400198 kPa20:0.00 5.00 10.00 15.00 20.00 25.00c: %1Contractive—1e-2Nlla-3-4-5Figure 4.3. Results of FR_CID_02 in shear stress vs. axial strain860.6Loop 1O552E1: %04L2 L3Loop20.0U245-0.46:%Figure 4.4. Zoom on loops 1 and 2 for test ES_CID_867.87-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4DFigure 4.5. Zoom on the elastic zone in Figure 4.lc.Volumetric strains in reloading phases are observed to be influenced by the unload-reload loops. This is investigated by plotting the data in stress-dilatancy space. A changein 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. Soilbecomes more dilatant and another peak dilatancy value (termed as second peak) isreached. The increase in peak dilatancy between the two peaks exceeds 50% in somecases (Figure 4.10).Peak dilatancy values for the available drained triaxial tests with unload-reload loopson 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 atthe critical state (see Section 2.1.3). Two different trends can be seen from laboratorydata for different reload loops. It is noteworthy that the slopes and intercepts of a trendline through the data points of the first peak dilatancy are different from those for thedilatancy of the second peaks.• Loading I• Unloading Ia Loading 2‘ Unloading 2xLoading 3• Unloading 3÷ Loading 41.4+4C.4 C• +•• x•• Points (solid squares)________:indicating an elastic zone0.8at the beginning of Uiwhile no such zone isobserved for U2 and U3•xCcCc.4+:÷•÷x0.688(a)1.61 .b•L1EI0.4>——-—----1 -0.6 -0.2 0.2 0.6Contraction if unloading Contraction if loadingDilation if loading— D— Dilation if unloadingFigure 4.6. Data from ES_CID_868 (a) stress ratio vs. axial strain (b) volumetric vs. axialstrain (c) stress ratio vs. dilatancy.LIL2 L320(b)1.55 10 15 Dilation0.50’0-0.5(c)89(a)2.000.800.400.0025(b)100”—1-2Figure 4.7. Comparison of ES_CID_870 and ES_CID_872 with similar e0 and initialp’but different number of U-R loops (a) axial strain vs. stress ratio (b) axial strain vs.volumetric strain.1.601.200 5 10 15 206:%90(a)2.001.601.20— ES CID 861O.80 — —ES CID 8620.40 — —0.00— I I I I(b)0 5%15 20 25Figure 4.8. Comparison ofES_CID_861 and ES_CID_862 with similar e0 and initialp’but different number of U-R loops (a) axial strain vs. stress ratio (b) axial strain vs.volumetric strain.91-0.6 -0.4 -0.2 0 0.2Figure 4.10. Stress ratio vs. dilatancy for different reload loops (ES_CID_867).-1.5 -1 -0.5 0 0.5DFigure 4.9. Stress ratio vs. dilatancy for pre-peak and post-peak reloading loops(ES_CID 862).I.2 peak_-_.I ÷1.2•L1ipeak ++. ++L4Ano++xD920• First peaksaA Second peaks-0.2aAaPA I AAAAAa-0.6 I I I-0.16 -0.12 -0.08 -0.04 0y atDmjnFigure 4.11. Dmin vs. iu at Dmjn for first and second loading of Erksak sand4.4. Implications of experimental observationsDeformation characteristics in unloading are seen to be highly dependent on the stressratio at the start of unloading (or the end of previous loading). If this stress ratio is lessthan that at image, unloading is dominated by a small amount of dilation. This behaviourmight be explained by elasticity. An elastic material expands in response to a decrease inmean effective stress (associated with unloading in conventional triaxial tests). However,once the image stress ratio is exceeded, unloading is associated with significant amountof contraction. This indicates non-elastic behaviour or yield in unloading. Therefore, theimage 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 aprerequisite for significant contraction in unloading. This contradicts many soil modelssuch as those presented in Section 2.3 where unloading happens inside the yield surfaceand it is elastic.93Contraction due to post-image unloading can be explained based on a simple physicalmodel 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 ofcontraction in unloading (Jefferies, 1997). If dilation can be thought of as soil particlessliding on top of each other, then a situation similar to that in Figure 4.12a develops inloading. 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 (Figure4. 12b). This is associated with contraction. Therefore, plastic dilation in loading isresponsible for the observed contraction in subsequent unloading. The model alsosuggests that the amount of contraction in unloading is related to the amount of dilationsin a previous loading. The more soil is allowed to dilate in loading, the more contractionis expected in subsequent unloading. This will be shown in Chapter 5.Id1I(b)Id2Id1>d2Figure 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. Thesaw-tooth model (Figure 4.12) is a friction based model. It can be thought of as a(a)94simplified 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 similarto those predicted by the model.It was observed that post-image U-R loops demonstrate a new peak in stress-straincurves (Figure 4.3). This is consistent with the behaviour that post-image unloading isassociated 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 changedonly if the previous loading phase exceeds peak stress ratio (refer to Section 4.3). Beenand Jefferies (1985) showed that there is a relation between peak dilatancy and stateparameter at peak dilatancy as previously discussed in Section 2.5. However, thisrelation is expected to change if fabric changes. Changes in fabric are induced due toshearing in unloading and reloading phases. The data suggests that once the peak stressratio is exceeded, soil goes through permanent changes in fabric.955. A MODEL TO ACCOMMODATE UNLOAD-RELOAD LOOPS USINGN0RSANDNorSand 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 somedetail in the literature review (see section 2.5). Jefferies (1997) also presented aframework for the behaviour of a NorSand material in unloading and reloading (seesection 2.6.3). This chapter expands on this framework to incorporate the observed soilbehaviour in unload-reload loops discussed in Chapter 4, supported by the theoreticalinvestigation 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 soilbehaviour. NorSand can be easily implemented in any programming language. Thesteps followed in coding the monotonic triaxial compression version of NorSand aresummarized in Table 5.1. The equations were derived and the parameters were definedin Section 2.5.Table 5.1. Equations used in the triaxial compression version ofNorSand and their stepby step implementation in an Euler integration code.Step description Equation1 Apply plastic shear strainincrement(8:)2 Obtain the value of stress ratio= M—• at image (M1)3 Calculate the current plasticD = M. —dilatation rate964 Get plastic volumetric strain=DPincrement(ct’)5 Get the current dilation limit D,=çi’ where,p’ =e—e,e=F—%lri(p) and—6 Apply the hardening rule to• 2 -‘change the size of the yieldP=14””1re(_x/M)Psurface due to the applied p1P1)LP]plastic shear strain increment7 Apply consistency conditionso that the stress state stays onthe yield surfaceWhere, L = -- (From Jefferies andpBeen, 2006)8 Update stresses, strains andstate parameter and add elasticstrains( M.i/Il+L(L—z)The objective of this chapter is to extend NorSand to include the new understanding ofyielding during unloading and subsequent reloading, introduced in this study. Theproposed model has been implemented in the Microsoft Excel Visual Basic Application(VBA) environment. Appendix D shows the main steps followed in coding the loadunload-reload model.97The four components of any elasto-plastic model, including NorSand, are elasticity, ayield surface, a plastic potential (i.e. a flow rule) and a hardening rule.5.1. Yield surface and internal capNorSand’s outer yield surface and the inner yield surface (or internal cap) wasdiscussed in some detail in Sections 2.5 (see Figure 5.1). Equation 2.52 (reproduced hereas Equation 5.1) specifies the location of the internal cap. This is the same location usedby Jefferies (1997). The current location of the cap fits the framework of the NorSandmodel in loading.p1Pcap— (—D,,../M)(5.1)The cap is taken as a vertical line for simplicity. It is assumed to intersect the outeryield surface at peak (i.e. minimum dilatancy if the sign is taken in consideration) whichoccurs at peak strength. Hence the location of the cap that defines yield in unloading iscoupled to the outer yield surface that was determined by the previous loading phase. Itwill be shown in the following sections that soil behaviour in unloading is related toprevious loading phases. It is therefore reasonable to have the location of the cap definedduring a previous loading phase. The adopted location of the internal cap is identical tothat for Jefferies (1997). This is to avoid inconsistencies with the NorSand model inloading. Consider a case where unloading starts from peak. For convenience assume thatan internal cap at the image state was chosen. Unloading would then start from a point tothe left of the internal cap and the consistency condition would not be satisfied as thestress point would not be on the internal cap.98Figure 5.1. Yield surface and internal cap in NorSand, same as Figure 2.28 (modifiedafter Jefferies and Shuttle 2005).The proposed location of the internal cap matches the observation from the datapresented in Chapter 4. It was shown that yield in unloading must occur to the left of theimage. This is consistent with the saw-tooth model where significant contraction inunloading (i.e. yield in unloading) occurs only for the case where soil was allowed todilate in a previous loading. By definition, dilation is only possible if stress ratio exceedsthat of image. Therefore, having the internal cap at peak is consistent with the NorSandmodel and matches observations from laboratory data.Soil unloaded after reaching peak stress yielded in unloading directly without goingthrough 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 unloadedfrom lower stress ratios showed a purely elastic phase before yielding. This behaviourcan be captured by placing the cap at the point representing peak stress in loading.99-1.5 -1 -0.5 0 1 1.5DFigure 5.2. Demonstration of interpreted elastic and elasto-plastic zones on the results ofES_CID_682 in stress ratio vs. dilatancy plot.5.2. Flow rule5.2.1. Flow rule in unloadingData 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 discussedin Chapter 4. Three observations may be made from the data in Figure 5.3. First, it canbe seen that almost all the plots for dilatancy in unloading are perpendicular to those inloading. Second, the position of stress-dilatancy curves in unloading is seen to depend onthe stress ratio at which previous loading stopped (see Figure 5.4). In the following thestress ratio at ]Y = 0 is defined as M (the subscript “u” denotes unloading). The higherthe stress ratio at which loading stops, the larger the value of M for a followingunloading phase. Lastly, the measured dilatancy in unloading stops changing at a valueof 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+k.4-.-..No sign of an elastic___________zone at the beginning ofU2 and U3x• Loading 1Unloading ILoading 2x Unloading 2z Loading 3• Unloading 3•+ Loading•••$••xX0.80.6x0.40.2xxxxxInterpreted as anelastic zone becauseof constant positivedilation values+ax•+a •x0.5100unloading loops within each test. Therefore, the location of the vertical part of the plot isdifferent for different tests and different loops within each test.It is assumed here that the Cam-Clay flow rule (D” = M - i) represents stress-dilatancyin loading (see Section 2.3 for more details). Because stress-dilatancy curves inunloading are almost perpendicular to those for loading, the proposed expression forstress-dilatancy in unloading is negative of that for Cam-Clay, and while replacing M byM. It takes the form:D=ii—M(5.2)The problem is how to get a representative value for M, as Figure 5.3 clearly showsthat for Equation 5.2 to be valid M needs to vary for different loops in the same test. Aspreviously discussed, the saw-tooth model implies that the amount of contraction inunloading is related to the amount of dilation in the previous loading phase. The value ofM for each unloadingphase 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 Mfor U3 (U3 follows L3) is higher than that for U2. It seems that the higher the last valueof i (denoted as‘lL)in a loading phase, the higher the value of M in subsequentunloading. To prove this point, values ofLfor different loading or reloading loops forall tests on ES are plotted against values of M for the corresponding subsequentunloading phases (see Figure 5.5). It can be seen thatLand M are directly related. Alinear trend line representing a best fit to the data points has the following equation:M=2q1 —1.5 (5.3)It is observed that there is a limit on the maximum dilatancy that can be reached inunloading at a value of = 0.4-0.5, as previously discussed. At this limiting valuedilatancy becomes constant for a particular unloading loop. Equation 5.4 is found to givea reasonable approximation for this maximum dilatancy in unloading (or minimumdilatancy if the sign is taken in consideration).101D=0.5 -M(5.4)Figure 5.6 shows an example comparison between the stress-dilatancy predicted by theproposed equations (Equations 5.2 to 5.4) and data from test ES_CID_866. Thepredictive ability of Equations 5.2 to 5.4 is shown for all ten Erksak tests in Appendix A.Figure 5.3. Drained triaxial tests on Erksak sand with unload-reload loops plotted in thedilatancy vs. space.5.2.2. Flow rule in reloadingThe usual NorSand flow rule for monotonic loading is D” =- i. Triaxial data onErksak sand plotted in Figure 5.3 show that stress-dilatancy is altered if the soil isunloaded and reloaded. It changes for different reloading loops (see Section 4.3). It isobserved that in most tests, the stress-dilatancy relation in reloading changes if theprevious loading phase reached peak dilatancy (see Figures 4.9, 4.10, and 4.11). It wasshown that peak dilatancy values increase for post-peak reloading phases.-1.5 -1 -0.5D0 0.5 1102-1.5 -1 -0.5 0 0.5 11 1.2 1.4 1.617LFigure 5.5. Correlation between M and iL from previous loading (drained triaxial testson Erksak sand).Value ofMfor thefollowing unloadingLast value of stressratio for L3(ii L)JLI—.—. UI—a-- L 2—— U 2—.—U3DFigure 5.4. ijL and M for L3 and U3, respectively, for ES_CID_862.1.81.4I0.6• Tests data—TrendlineM=2-1.5 •••• •103Figure 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 locationof the internal cap (the location and shape of the assumed internal cap in NorSand wasdiscussed in Section 5.1). Following the observed behaviour of change in peak dilatancyin post-peak reloading, it seems reasonable to introduce some changes to how soil dilatesin the code if the stress state reaches the cap in a previous loading/reloading. The triaxialdata for all tests on Erksak sand was plotted in Figure 4.11 in ,u vs. Dmjn space. It is clearthat 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 aspreviously discussed in Section 4.4 supported by the results of the theoreticalinvestigation in Section 3.2. It is noteworthy that if a trend line (based on a best fit to thedata points) is drawn through the points of the second peaks, it would have an interceptthat is far from zero. This is not in accordance with the critical state theory on whichNorSand is based.A major feature of NorSand is that it limits dilatancy based on a relation between çu andDmin. The slope of a linear trend line with a zero intercept in the çu and Dmjn plot is termedD-1.5 -1 -0.5 0 0.5 1104x(used as a parameter in NorSand). A zero intercept of the trend line is consistent withCritical State Soil Mechanics (CSSM). This is because if peak dilatancy happens at thecritical state (i.e. for a test on loose sand) then Dmjn=O and e = e (by definition t, mustequal zero as çti= e - es). Therefore, the trend line through the data points of the secondpeaks should also be drawn with an intercept of zero, as well as the trend for the firstpeaks.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 withthree or more loops does not provide sufficient information to determine whether otherpeaks exist or not. However, the stress-dilatancy relation is not expected to changeindefinitely. A change in stress-dilatancy relation is caused by changes in fabric. As thereported tests do not reach critical state (where major fabric changes occur), no furtherchanges in the stress-dilatancy relation are expected to occur.The proposed model assumes that only two peaks exist. Once the first peak is exceededin a loading/reloading phase, all subsequent reloading phases follow a different stressdilatancy 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 throughthe points of the first and second peaks. The slopes of the two lines are denoted asjandX2.The parameterxjis identical toxin standard NorSand. The code uses the secondpeaks value,X2,only if the stress state in the previous loading or reloading phase hits theinternal cap which represents peak conditions. This is consistent with observations intriaxial tests. The second peaks are attained only if peak stress was reached in a previousloading/reloading (see Section 4.3).The implication of a changingxon the NorSand model is twofold. Firstly, a change inxresults in a change in the location of the internal cap for a certain yield surface. Asxincreases 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, thecomputed values of M change (see Equation 5.5). A higherxvalue yields a smaller105This is consistent with the observed behaviour. Figure 5.7 shows that M for a secondpeak reloading (i.e. higher‘value) is higher than M for the first peak reloading with thesmallerx.M, =M-xNçuj (5.5)Where,=xjfor the case of first loading or previous loading/reloading does not touch the internalcap.X=X2for the case where previous loading/reloading touches the internal cap.D0 0.1 0.2Figure 5.7. Change of M for different reloading loops (ES CID 862).5.2.3. Potential surface in unloadingNorSand uses an associated flow rule, meaning that the plastic potential surface andyield surface are the same. In unloading, yield happens on the internal cap. As-0.6 -0.5 -0.4 -0.3 -0.2 -0.1106previously discussed in Section 5.1, the cap is a vertical line. Using an associated flowrule with a vertical cap yields zero dilatancy. However, a significant amount ofcontraction 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 tohave a potential surface that is different from the yield surface.An expression for the potential surface is derived as it will be necessary to implementthis model in any finite element formulation for future work. The derivation involvestwo 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 ratioq=ip’(5.6)Taking the differential of 5.6 gives:(5.7)And to satisfy normality,(5.8)P 8qFrom 5.7 and 5.8,=0 (5.9)p D+,7From Equations 5.2 and 5.9,=0 (5.10)p 2ii-MIntegrating Equation 5.10 gives:107f=C (5.11)pThe solution of the integral is:lnp+i—ln211—MI=C(5.12)When ij= M, the stress state would be at the imageand p= ‘iu, hence:c=lnp;+.-lnM(5.13)And the equation of the potential surface in unloading is:ln1—”+!1n_L_1=0(5.14)p)2MRearranging gives:— e_1_1(5.15)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. Theupper part is applicable for the case where (2ii/M -1) > 0 while the lower part is for(2ii/M -1) <0.108240400p’ (kPa)Figure 5.8. The shape of the potential surface in unloading5.3. Hardening in loading, unloading and reloadingHardening of the NorSand yield surface in loading, unloading and reloading isdescribed in Section 2.6.3. A similar framework is adopted here because it matches theway Erksak sand behaves. If unloading occurs from a low stress level, the behaviour iselastic until yield occurs when the stress path hit the internal cap. Figure 5.2 shows thatUi is dominated by elastic behaviour at the beginning of the unloading phase and yieldoccurs 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 inunloading for those loops without passing through an elastic phase.The outer yield surface softening during unloading is important for accuratepredictions. If the outer yield surface would not soften in unloading, reloading would beelastic until the stress level prior to unloading is exceeded. Figure 5.9 shows that200160804000 100 200 300109reloading is not entirely elastic: stiffness decreases before the stress level is as high as thestress level at the start of previous unloading.In unloading, the internal cap contracts. And because the internal cap intersects theouter yield surface, it softens as well. Jefferies (1997) introduced a rule for thecontraction of the internal cap, reproduced as Equation 5.16. The term ln(pjy/p) wasthought by Jefferies to introduce an effect similar to overconsolidation. The further thestress point is from first yield in unloading, the larger are the generated strains. This isconsistent with observations from laboratory data. Figure 5.9 shows an expanded view ofU2 for ES_CID_868. More axial strains are generated in U2 at lower stress ratios. Thelower the stress ratio, the further the stress point is from first yield in unloading..‘p1p’ (‘6v=———--TlnII(5.16)HpP)Where,H is the hardening (softening)modulus in unloadingp’,is the mean effective stress at first yield in unloading (i.e. the mean stress of the capwhen first intersected)1101.601.200.800.400.0011Figure 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 strainincrements. 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 computedaccording to Equation 5.16. This results in a division over zero as 8 /D. To getaround this problem, Equation 5.17 is used instead of Equation 5.16. Plastic shear strainincrements are first calculated according to Equation 5.17 then plastic volumetric strainincrements are recovered through stress-dilatancy (i.e. s’ = 8 D’). For the case of zeroD”, plastic volumetric strain increments become zero and the problem of having todivide over zero is solved. The sign of Equation 5.16 is changed as unloading isassociated with negative mean effective stress increments and negative plastic shearstrain increments.9 10L: %111•i‘ (P6q=—-—-1n1I (5.17)HpAs yield in unloading causes softening of the outer yield surface, it is important toderive an equation to quantitatively describe the amount of that softening. The size of theouter yield surface depends onp.As previously discussed in Section 2.6.3,Peapis relatedtop1as follow:p1Pcap— e(_i)u1/’M1)(5.18)From 5.18,.. IJ3Pcap—(5.19)Since the stress point remains on the internal cap in unloading, mean effective stress inEquation 5.17 is equal toPcapand,(5.20)From Equations 5.17 to 5.20,=sH/1fl,(5.21)P1 P112Equation 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 thechange in the size of the outer yield surface due to an applied plastic shear strainincrement relative to it original size before applying that increment. Figure 5.10 showsdifferent outer yield surfaces corresponding to different points on the unloading stresspath. The inner cap moves to the left with the stress path dragging the outer yield surfacewith it.As previously discussed in Section 5.2.3, the potential surface in unloading is differentfrom the yield surface (i.e. the internal cap). It can be noted from Figure 5.10 that plasticstrain 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 5.2.3for the derivation of the potential surface).Figure 5.10. The direction of plastic strain increment ratios in unloading with thecorresponding yield surfaces and internal caps.70030020010000 100 200 300 400 500 600113300250200o 150100500600Figure 5.11. The direction of plastic strain increments ratios in unloading normal to thepotential surfaces.5.4. Comparison with other modelsThis chapter presented an unload-reload model for sands that is based on the NorSandsoil 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 summaryof the unloadingpart of the model and the main assumptions are presented in Table 5.2. It was shown inthis chapter that those assumptions match the observations from laboratory resultspresented in Chapter 4.It was observed that soil becomes more dilatant in post-peak reloading loops. Thebehaviour was simulated in the model by a changing . The value of increases to ahigher value of2once first peak is exceeded.0 100 200 300 400 500pt114Table 5.2. Summary of the unloading part of the model.Model Equation AssumptionscomponentsYield surfacep.• Yield in unloading happens on a—P1vertical cap.cap—/ M1)• The inner cap intersects the outer yieldsurface at a point that corresponds toDmin in loading.Flow ruleDy” =—• Stress-dilatancy plots in unloading areperpendicular to those in loading.Where,• There is a direct relation between MM=—1.5and11Lof the previous loading.• There is a minimum value forAnd the minimumdilatancy in unloading.dilatancy in unloading is:D=O.5 -MHardening rule Movement of the internal• The further the stress point from firstcap: yield in unloading, the slower the rate, /of movement of the internal cap.— 1P1• The outer yield surface softens due to— H‘9Jyield in unloading.Softening of the outeryield surface in unloading:P±6PH/lfl(P4]Jefferies (1997) derived an equation for stress-dilatancy in unloading based on theassumption 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 (Equation5.22), and substituting for D’ and i (i.e. ‘ / and q/p’, respectively) and expandingyields Equation 5.23.D= (M—i)(5.22)(1-N)115q’+p’=Mpf:l÷Npt(5.23)The terms on the left hand side of Equation 5.23 represent plastic work done. The righthand side represents what soil does with that work. The first term on the right hand siderepresents energy dissipation (Schofield and Wroth, 1968). The second term on the righthand 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 increasedin loading as the particles assume new locations. This energy is released upon unloadingas the particles tend to recover their original locations before loading. This is associatedwith contractive response in unloading. For the unloading phase, ‘ <0 and the N termin 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(5.24)1+NEquation 5.24 predicts different trends, more contraction in unloading, compared tolaboratory data and the predictions of Equations 5.2 to 5.4 (Figure 5.12). The expressionassumes that all ‘plastic’ energy stored in a loading phase must be released in thesubsequent unloading phase which does not seem to be the case (i.e. only part of thisenergy is released in the subsequent unloading phase).The proposed model in this chapter, similar to measured laboratory data, shows thatcontraction in unloading depends on shear deformation in previous loading. This isconsistent with the saw tooth model where shear deformation is a major source of plasticwork stored in loading and recovered in unloading. Jefferies (1997) was the first to adoptthe saw tooth model to explain soil behaviour in unloading. Therefore, the proposedmodel and that for Jefferies (1997) are very similar conceptually.116• Loading I Unloading ILoading 2 x Unloading 2*Loading 3 —Model fit to UI and U2—Jefferies 1997 (M=I.27 and N=O.25)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 workstorage while isotropic compression is the major source. Collins model was described inSection 2.6.2. Pure isotropic loading on the continuum scale is assumed and following theusual convention the applied work may be separated into an elastic and plasticcomponent - 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 causingdilation while the other part can only be released if associated with particlesrearrangement (causes dilation as well). Particle rearrangement is not elastic and henceplasticity occurs during unloading. It is noteworthy that if soil particles were rigid,Collins model predicts no volumetric strains in unloading. However, it was shown inChapter 3 based on Rowe’s theoretical model that an assembly of rigid particles changesD-2 -1.5 -1 -0.5 0 0.5117in volume in unloading. Unloading according to Collins model is associated with dilationwhile laboratory data shows that significant contraction occurs in unloading.Dqy.s 2p,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 thestarting point. The yield surface moves to y.s. 2 during the loading path of A-B. Uponunloading (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 assumesthat 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 consolidatedconditions is elasto-plastic (Figure 5. 14b). Unloading is purely elastic only for the phasebefore the stress path hits the internal cap (Figure 5.14c). In this phase, the outer yieldsurface does not move. Otherwise, unloading is elasto-plastic and causes softening of theouter yield surface. Reloading is elastic until the stress path hits the outer yield surfaceBCy.sy.s I118(Figure 5. 14d). Clearly, this is different from Drucker and Seereeram model described inthe previous paragraph. The differences are summarized in Table 5.3.2.0C 300tee2Loading1202Surface where cuerent yield is/ .1Surfaces where(1occuering/current yield is— Stress path00/joccumng• Ci000 5 10 15 20 0 100 200 300 400 500 600 700cj:%300 ———————————1}iingUnloading200Reloading.Surfaces whereS ces ere1:’/174W—_-__-_ is10 100 200 300 400 000 600 700 0 100 200 200 400 500 600 700Figure 5.14. Hardening according to Jefferies (1997) (same as Figure 2.35).Table 5.3. Comparison between hardening in the proposed model and Drucker andSeereeram (1987).Phase/model Proposed hardening Drucker and Seereeram(identical to Jefferies, 1997) (1987)Unloading• Either elastic or elasto- • Purely elasticplastic• Yield surface always• Yield surface moves only moves with the stress pathif stress_path_hits_the_capReloading • Either elastic or elasto- • Elasto-plasticplastic• Yield surface always• Yield surface moves if moves with the stress paththe stress path hits theouter_yield_surface1195.5. SummaryThis chapter presented a practical continuum model for unload-reload cycles on sandsthat 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 inunloading, a behaviour that is consistent with observations from lab data presented inChapter 4 and the conclusions of the theoretical investigation of stress-dilatancypresented in Chapter 3. Like Jefferies model, yield in unloading is assumed to occur on avertical cap in the p ‘-q space. Unlike Jefferies model, stress-dilatancy in unload andreload 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 isdirectly proportional to the stress ratio at end of previous loading. The model accountsfor the observed increase in dilation for post-peak reloading.1206. MODEL CALIBRATIONThis chapter presents a load-unload-reload calibration of the model presented inChapter 5 to ten triaxial tests on Erksak sand (see Table 4.2). The calibrated model isthen used to predict two drained triaxial tests with unload-reload loops on Fraser Riversand later in Chapter 7. As the unload-reload model uses NorSand as its starting point, amonotonic 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 amonotonic calibration to Fraser River sand. Section 6.3 presents the unload-reloadcalibration to Erksak sand.6.1.Monotonic calibration for Erksak sandA calibration of NorSand for Erksak sand under monotonic loading in a triaxial test ispresented 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 2.1.2and Section 2.5. Table 2.2, reproduced here for convenience as Table 6.1, is a summaryof the required parameters and their typical ranges. The critical state parameters F and 2are 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 plastichardening parameter ‘H’ specifies the rate of the hardening of the yield surface. Theslope of a trend line with zero intercept through the data points in the Dmjnv’isdesignated as (e.g. Figure 2.29). It is used to control the maximum allowable absolutevalue of the dilation rate. The volumetric coupling parameter ‘N’ was introduced byNova (1982). It is based on fits to stress-dilatancy from the results of laboratory tests.121Finally, the elasticity parameters are the dimensionless shear rigidity parameter (G/pand Poisson’s ratio.Table 6.1. Typical ranges for monotonic parameters (same as Table 2.2, modified afterJefferies and Shuttle, 2005).ParameterITypical rangeIDescriptionCritical stateF 0.9-1.4 The y-intercept of the elog(p’) curve at 1KPa2e0.01 — 0.07 The slope of CSL in elog(p) space defined onbase e!vI 1.2-1.5 g/p’ at critical statePlasticityH 50-500 Plastic hardening modulusZIc2.5-4.5 A parameter that limits thehardening of the yieldsurfaceN 0.2-0.4 The volumetric couplingparameter (used in Nova’srule)ElasticityJr100-800 Dimensionless shear rigidity(G/p)v0.1-0.3 Poisson’s ratio6.1.1. Critical state parametersThere 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 stressratio from drained triaxial tests: the terminal value of stress ratio method, maximumcontraction method, Bishop method, and Stress-dilatancy method.1. Plotting the curves for each test in the stress ratio vs. strain space and simplypicking up the terminal value for the stress ratio. The problem with this method isthat the dense tests, and even most of the loose tests, do not go far enough toreach the critical state.1222. The stress ratio at maximum contraction is taken as Mk. This method assumesthat the stress ratio at maximum contraction is equivalent to that at the criticalstate. The point of maximum contraction is not the same as the critical state andhence the method would only be appropriate if the true stress dilatancy behaviourof soil was a unique locus (i.e. only one stress ratio corresponded to onedilatancy). Although this assumption has been used in flow rules such asCamClay, modified CamClay, and Nova, real soils do not show this behaviour.The real measured soil response shows a different stress ratio at maximumcontraction and at the critical state, consistent with Rowe’s idea of an evolving Mwith increasing strain. More recent soil models such as NorSand and Li andDafalias’s (2000) model also incorporate this evolving M. Therefore this methodof determining M, provides poor predictions.3. Bishop (1971) suggests plotting the data for all the available tests in Dmin - Tlmaxspace. The method is based on the idea that a very loose soil should reach thecritical state at peak stress ratio, i.e. Dmjn and i at peak should be zero and M1respectively. Hence, assuming the trend in the data is linear (which is consistentwith experimental measurements), the y-intercept of a trend line through the datapoints will correspond to Mk. The unload-reload Erksak tests have differentpeaks associated with different reloading phases as was previously discussed inSection 4.3. A peak is termed a ‘first peak’ if peak strength was never exceededin previous loading or reloading loops for a certain test. All other peaks aretermed ‘second peaks’. Two linear trend lines are plotted for first and secondpeaks data points in Figure 6.1 resulting in two M values. According to criticalstate, a single sand must have a single M, value. Therefore, the two M valuesmust be identical. However, Dmjn is affected by changes in fabric due to shearingin different reloading loops and therefore based on extrapolating data in Dmjn -i7,,,,is expected to change for different reloading loops. The trend line for thefirst peaks gives M, =1.15(q5= 28.85°) while that for the second peaks gives avery similar value of M1 .1(27.7°). Therefore, Bishops method gives M123in the range of 1.1-1.15. It can be noticed that the two best fit lines are parallel. Itseems that slope of the lines is very similar for different reloading phases.However, the y-intercept is slightly different.1.6‘ii = (N—1)D+M0(Bishop, 1971 and Nova, 1982)1=—0.8O5D,+ 1.15— 11O.8O4Dmm+1.1A1.41.2 ..* AAFirst peaks (FP)---•Trend line (SP)0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6Figure 6.1. using Bishops method for Erksak sand.4. The stress-dilatancy method for obtaining suggests linearly extrapolating theretUrning curves in the dilatant part of the stress-dilatancy plots to D = 0 (seeFigure 6.2). The extrapolated value of i at D 0 is Table 6.2 shows Ii/ttusing the stress-dilatancy method for all tests but for ES_CID_868 (post peakbehaviour for this test is questionable). Stress-dilatancy method gives valuesof 1.24-1.35 as shown in Figure 6.3 with an average of 1.286(= 31.97°).Ghafghazi & Shuttle (2006) recommended the use of the Bishop method or StressDilatancy approach. The Bishop method gave a lower M1 than the stress-dilatancymethod. Ghafghazi & Shuttle (2006) showed that “with only a small number of testsavailable, the Bishop method is sensitive to any outlying data points”. Table 6.3 showsthat the range of M obtained in this work using 9 tests with the Bishop method is lower1-N1124than the obtained by Ghafghazi & Shuttle (2006) for 34 tests using the same method.However, despite using many fewer tests than Ghafghazi and Shuttle, the M1 values ofboth sets of authors using the Stress-Dilatancy method are very similar. Therefore, thestress-dilatancy method seems to provide a repeatable value of even for a smallnumber of tests. Therefore, a value of = 1.286(q= 31 .97°).was adopted for Erksaksand.Figure 6.2. using stress-dilatancy method (ES_CID_87 1).-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4D125-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4DFigure 6.3. Range of M, using the stress-dilatancy method from the last reloading loopsfor the 9 tests in Table 4.2.Table 6.2. M1using stress-dilatancy method for the unload-reload tests on Erksak sand.Test name (extrapolated)1CID-G860 1.33CID-G861 1.35CID-G862 1.33CID-G866 1.27CID-G867 1.25CID-G868 Ignored2CID-G870 1.24CID-G871 1.24CID-G872 1.27CID-G873 1.3Average 1.286All values are extrapolated but for CID-872 that reached critical state.2Post-peak behaviour for this test is judged to be unrepresentative.1.551.51.451.4.•fExpected range for (1.24-1.35)1.21. 151.1050.95 * _0.9•0.85X:.. -126Table 6.3. Summary of M’ values for Erksak sandMethod/source Current work (9 tests) Ghafghazi & Shuttle,2006 (34 tests)Bishop method 1.1(lstpeak)1.261.15(2fldpeak)Stress-Dilatancy 1.286 1.28methodThe critical state parameters in e-log p’ space were determined using undrained triaxialtests from Jefferies and Been, 2006 (see Figure 6.4). The tests were described inSection4.2 and summarized in Table 4.3. The derived critical state parameters are F = 0.82 and= 0.03 1 (equivalent to Xe = 0.0135) which are very similar to F = 0.8 16and= 0.031 in Jefferies and Been (2006). These derived parameters are accurateforp’< 800 kPa. Been etal. (1991) showed that at higher mean effective stresses the linebecomes steeper, attributed to grain crushing, and therefore a single linear CSLwould notbe applicable. Hence the fits for tests ES_CID_870, ES_CID_87 1, and ES_C1Q872with high mean effective stress of 800 kPa are not expected to be accurate.Logp’ (kPa)Figure 6.4. CSL determination for Erksak sand from loose undrained tests.0.80.78. 0.760>0.740.720.7:.............................e = 0.82- O.01 logpAA End of test (undrthnedtriwd tesonosesamPIes;dáäfroi4Bi& — — —Jeffèries , 2006)—CSL10 100 10001276.1.2. Elasticity parametersIdeally, elastic shear modulus is measured using bender elements. However, tests withbender 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 unload-reload cycles. Equation 6.1 gives the elastic bulk modulus from best fit to theirexperimental data:/‘O.5K_C1(e_e5)p)Pref(6.1)where C is a material ‘compressibility’ constant equal to 260,Prefis 100 kPa and e, thevoid ratio at which the volumetric compressibility becomes zero is equal to 0.355. Theparameter e represents the void ratio where soil behaviour changes from predominantlyparticulate 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 modulusis given by:(6.2)p 4pHowever, there is scatter of as much as ± 50% in Jefferies and Been data. The elasticbulk modulus can be directly obtained from the unload-reload tests on Erksak sandpresented in Chapter 4. Figure 6.5 is an enlarged view of the elastic part at the beginningof L3 for ESCID_866. It can be seen that the plot in that zone is nearly linear and it isassumed that all deformations are elastic. The elastic bulk modulus was calculateddirectly from the data points for that elastic zone using a linear difference approximationas in the following equation:128Kpj÷i_Pj(i+ej+i+eje3—e1i 2(6.3)Wherej+% is the midpoint between consecutive measurements at which K is computed,jis the previous measurement, andj+1 is the next measurement. Figure 6.6 plots theresults of Equations 6.1 and 6.3 for the elastic part of L3. It is clear that Equation6.1underestimates the values of the bulk modulus compared to those directly computedfromlaboratory data (i.e. Equation 6.3). A higher value of C 750 in Equation 6.1 would givea better estimate of K. A better fit to the laboratory data was obtained usingequation 6.1using C 750, and e5 equal to 0.355 (i.e. the same e as in Jefferies and Been, 2000butdifferent C) as shown in Figure 6.6.0.70.20.3c1: %Figure 6.5. Enlarged view of the elastic part in L3 for ES_CID_866.129800700- • Lab data • C = 750 AC = 260:::400300200A AA AA A A1000 I I II400 420 440 460 480 500p’: kPaFigure 6.6. The elastic bulk modulus from Equations 6.1 and 6.3 against p’ for the elasticzone in L3 for ES CID 866.6.1.3. Plasticity parametersThe monotonic loading version of NorSand requires three plasticity parameter; N,,-andH.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 through77max Dminplot is (N-i) as shown in Figure 6.1. The two trend lines through both the firstand second peak points have very similar slopes of around 0.8. Therefore, the value of Nis 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 issmaller than that for second loading (i.e. after one unload-reload loop). The NorSand130plasticity parameter is defined as the slope of a linethrough the points in Figure 6.7,and correspondingly there are two values forxas discussed in Chapters 4 and 5. Notethat the best fit straight line through the second peaks does not have aY-intercept of zero.However, the trend line for the second peaks is required to passthrough the point (0,0) asif peak stress ratio (equivalent to minimum dilatancy) occurs atthe critical state then, bydefinition, Dmun must have a value of zero.The hardening modulus, H, was determined by iterative forwardmodelling of eachdrained pluviated triaxial test. The value of H was observed to be correlatedto .the initialvalue of state parameter (Figure 6.8). Appendix B showsthe fits for all of the drainedtests on Erksak sand in i — and—plots; an example fit for test ES_CID_867isgiven in Figure 6.9. The general trend linefor “I]” (i.e. H = -1727.3 çü, + 75.9) was usedto obtain the monotonic parts of thefits (i.e. before unload-reload cycles).-0.8v at DmiiiFigure 6.7. Trend lines through Dmin vs. çu at Dmjn for first and secondpeaks for Erksaksand0-0.2-0.4-0.6-0.16 -0.12 -0.08 -0.040131Although the hardening modulus is expected to also depend on initial fabric, for Erksaksand the moist tamped sample, ES_CID_868, also fitted well in the overall trend for thepluviated samples. Tests ES_CID_870, ES-CID_871, and ES-CID_872 showed adifferent trend for H values. This is likely due to the curved critical state line observedby Been et al. (1991), often associated with grain crushing at high mean effective stressvalues. If a steeper critical state line was used at higher mean effective stress, the valueof initial state, for tests ES-CID_870, ES-CID_87 1, and ES-CID_872 would be lessnegative and closer to the typical trend line. Hence these three tests were ignored in thecalibration.6.1.4. Summary of Erksak monotonic calibrationA summary of the Erksak monotonic calibration is presented in Table 6.4. Figure 6.10summarised the procedure followed for the monotonic calibration ofNorSand.H -1727.3V’0+75.9—‘I’,300Tests withp’ = 800 kPaignored in the calibrationES CID 868 withthe MT sample200100-0.20 -0.16 -0.12 -0.08 -0.04‘/‘0Figure 6.8. Best fit to Hvs. çte for Erksak sand.0.001322.00Model1.6011.200.800.400.00 I0 5 10 15 20 25s:%1.00.5Contraction0.0-0.5-1.0-1.5-2.0Lab results-2.5-3.0Li:%Figure 6.9. Example fit to test ES_CID_867.133Table 6.4. Summary of monotonic calibration for Erksak sandParameter Erksak sand RemarkCSLF0.82Altitude of CSL at 1 KPa, 0.0 135Slope of CSL, defined on base eM1.286Triaxial critical friction ratioPlasticityH-1727.3 + 759Monotonic plastic hardeningparameterZtc3.34 Slope of the line relating Dmin to t’atDmin defined for triaxial conditionsN 0.2 The volumetric coupling parameter(used in Nova’s rule)Elasticity134NorSand parametersObtain Critical StateparametersFigure 6.10. Recommended procedure for obtaining NorSand parameters.62.Monotonic calibration for Fraser River sandThe same general procedure was used for the monotonic calibration of Fraser Riversand as for Erksak sand in Section 6.1. Therefore, the calibration to Fraser River sand isonly briefly described in this section. The tests used in the calibration were described inSection 4.2 (see Table 4.5).6.2.1. Critical State parametersIt was shown in the previous section that the Bishop method can provide unrealisticparameter values, especially for small number of tests (four tests in this case). This alsoappeared to be the case for Fraser River Sand where the Bishop method gave a higher‘I,Bender elements or cyclic teststo get accurate values for theelastic parameters. Elasticproperties for commonly usedsands can be found in literatureTriaxial tests (both dense andloose samples; loose samplesreach critical state within thelimit of the apparatus)Jr4.JrDraw D,vs. ii to getJrUsing the previously obtainedparameters, run NorSand to get thevalues ofH that give accurate resultscompared to lab data. Plot H vs.135value of of 1.55(Ø= 38.02°), see Figure 6.11, which is greater than M from thestress-dilatancy approach. Therefore, the stress-dilatancy method was adopted. Thepost-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) forES_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 morereliable. Both tests gave of 1.42(4= 3 5.04°) using the stress-dilatancy method (seeFigure 6.13 for FR_CID_04). Therefore, of 1.42 was adopted for Fraser River sand.This is similar to 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 sixtests as in Figure 6.14. Chillarige et al. (1997) also performed triaxial tests on FraserRiver sand and got the critical state parameters. The line in Figure 6.14 has a larger yintercept value than that of Chillarige et al. (1997), but it is steeper. Most of theChillarige et al. data match the line in Figure 6.14 except for two outliers. It isnoteworthy that those two tests are drained and therefore might not have reached criticalstate. Drained tests that reach the critical state at large strains are often associated withlocalization.136Z0.68Dm1.55.1.4 I I0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8Figure 6.11. using Bishop method for Fraser River sand.1.90—— FR_CID_031.30I I I 1.20-0.90 -0.70 -0.50 -0.30 -0.10 0.10 0.30 0.50DFigure 6.12. Enlarged view of the dilatant zone for FRCID_03.137c=1.42o— FR_C ID_04-0.50 -0.40 -0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40 0.50DFigure 6.13. using stress-dilataney method for FR_CID_04.1.20 —__________________________________________________—• End of test — FR_CID_03e = 1.23+ 0.154 log(p’)c End oftest(afterChillarige ---—FR_CID_0410 100 1000 10000Log pt (kPa)Figure 6.14. CSL for Fraser River sand.1386.2.2. Elasticity parametersChillarige 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 defineelastic conditions.,‘,O.262G0—p(295—143e---1(6.4)PaJ6.2.3. Plasticity parametersThe value of the parameter was determined as 4.34 from the best fit to the data pointsin peak dilatancy vs. state parameter at peak (Figure 6.15). The slope of the Bishop linein Figure 6.11 is equal to N-i. Therefore the value of N is 0.32. Finally, the plastichardening modulus, H, which gives a good fit to stress strain and volumetric strain curvesis correlated to initial state parameter (see Figure 6.16). The calibrated model results ascompared to laboratory tests results are included in Appendix C, and an example fit totest FR_CID_03 is provided in Figure 6.17.139-0.1-0.2-0.3-0.5-0.6-0.7 I I-0.16 -0.14 -0.12 -0.1 -0.08 -0.06 -0.04v atDm,nFigure 6.15. Peak dilatancy vs. vat peak for Fraser River sand.120.H7+45.4I I I ‘.1-0.30 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10cL,0Figure 6.16. Best fit for H for monotonic triaxial tests on Fraser River sand.1406005004003002001000C020-2-4-6-10-12Figure 6.17. Example fit to test FR_CID_03.Table 6.5. Summary ofNorSand monotonic calibration to Fraser River sand.Parameter Fraser River Sand RemarkCSLF1.23Altitude of CSL at 1 KPa20.067Slope of CSL, defined on base e‘VIle1.42Triaxial critical friction ratioPlasticityH-305.7 + 45.4Monotonic plastic hardeningparameterZic4.34 Slope of the line relating Dmjn to çvatDmindefined for triaxial conditionsN 0.32 The volumçtric coupling parameter(used in Nova’s rule)ElasticityI0.262G=p (295_143e{—E--’11r =,Pref100 kPa andp is soilPj) PAfter Chillarige et al.density in ton/m3(1997)V 0.2 Poisson’s ratio (assumed)0 5 10 15 20 2581: %1416.2.4. Summary of Fraser River Sand monotonic calibrationTable 6.5 presents a summary ofNorSand monotonic calibration to Fraser River sand.6.3. Unload-reload calibration to Erksak sandThe unload-reload model requires three additional parameters: “Ha” and “Hr”, theplastic hardening modulus in unloading and reloading respectively andX2”,the slope ofa trend line through the second peaks.X2is used to capture the effect of induced fabricchanges when stress conditions exceed peak.Ideally it would be preferable to relate the three additional unload-reload parameters toparameters derived from a monotonic calibration. However, this is the first calibration ofthe unload-reload model and therefore it is being used to provide guidance on whetherH, Hr andX2show any relation to the monotonic calibration.The plastic hardening modulus in unloading “Ha” specifies the rate of movement of theinternal cap (i.e. inner yield surface) as previously discussed in Section 5.3. It is used inthe model as in Equation 5.17. Laboratory data on Erksak sand suggests that H is shearstrain level (at the start of unloading) dependent and that the response of soil in unloadingis softer for higher strain levels. Figure 6.1 8a shows the model fits for H = 30 and 40 tounload-reload loop U2 for test ES_CID_867 . An H of 40 is a better fit. The fits forunload-reload ioop U3 are presented in Figure 6.1 8b. An H of 20 is a better fit in thiscase. It is noteworthy that U3 is at a higher axial strain level than U2. Therefore, higherstrain levels appear to be associated with softer unloading. This trend of reducing Hwith increasing strain level was applicable for all the unload-reload tests. A constantvalue 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).142The plastic hardening modulus in reloading “Hr” specifies the rate of movement of theouter yield surface in reloading. Figure 6.19 shows that Hr = 4H fits laboratory data. It isalso clear that the results are not sensitive to Hr as it has a very high value (i.e. there is asmall difference in the model predictions for Hr = 4H a.nd Hr = 811). The first portion ofL4 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 yieldsurface prior to unload-reload is exceeded. Jefferies (1997) suggested that soil‘remembers’ its past yield surface before unload-reload. Current observations are inagreement with this observation. The plastic hardening modulus in reloading “Hr”calibrated well with a very high value of 4H until the yield surface prior to unload-reloadis exceeded. At this point Hr = H. Figure 6.20 shows model simulations for a test with aconstant Hr = 4H and one with a changing Hr. The solid dots are on the yield surfacebefore the unload-reload ioop. A constant Hr results in sharp peaks for the cases of L2and L3 that were not observed in laboratory results. A changing Hr is adopted as it isconsistent with laboratory data. Hr changes from a value of 4H to a value of H once theyield surface before unload-reload is exceeded.The slope of a trend line through the second peaks,12,is used to capture the effect ofinduced changes in particle arrangement on stress-dilatancy when stress conditionsexceed peak. A value of12= 4.71 was obtained in Section 6.1.3 (see Figure 6.7) (theoriginal value of for first loading was 3.34). The first loading y is used for pre-peakload or reload loops while12is used for post-peak reload loops. A higher value resultsin a more dilative behaviour. Figure 6.21 shows that a changing gives a better fit than aconstant.It is noteworthy that the two simulations (i.e. constant and changing ) areidentical for the first reloading loop, a pre-peak loop. However, they are different for theother two reloading loops as they are post-peak loops.Table 6.6 is a summary of the unload-reload calibration for Erksak sand presented inthis section.143(a)0.60.4ed \N::II0.2El: %(b)0.0 I I I7.5 8 8.5 9 9.5 10 10.5 11 11.5 1s: %Figure 6.18. Model fits using different H values compared to laboratory data (a) U2 forES_CID_867 (b) U3 for ES CID 867.1441.6H 8H—i 2H1.2 Elasto-plastic H4Hreloading0.8 reloadingModelresu20.40.09 10 11Li: %Figure 6.19. Model fits for different Hr values compared to L4 for ES_CID_867.1200H = 4H10008006004002000 2 4 6 8 10 12 14 16 18 20 22 24Li: %Figure 6.20. Model simulation for a changing and constant Hvalues.1451.00.50.0-0.5•:-1.0-1.5-2.0-2.5-3.0Figure 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 Erksak sand RemarkH 30Plastic hardening parameter inunloadingHr4H Plastic hardening parameter inreloadingX24.71 Slope of the line relatingDmin toj, atDmjn defined for the secondpeaks6.3.1. Overview of Erksak Unload-Reload CalibrationThe proposed model captures the main features of soil behaviour including the volumechanges observed during unloading and reloading accurately. Appendix B presents thecalibrations for the Erksak sand triaxial tests. Figure 6.22 shows a comparison betweenthe i61and6 61plots of the calibrated NorSand model and ES_CID_867.Unloading for Loop 2, shown at larger scale in Figure 6.23, does not go through anelastic phase. The response in unloading, phase a-b, is elasto-plastic. Unloading in theNorSand model calibration is elasto-plastic as the stress point touched the cap in the%146previous loading phase. A small amount of dilation is seen at the beginning of segmenta-b which is contrary to what is expected according to plasticity. The reason for that is atthe beginning of the unloading phase, the ratio of elastic strains that are dilative to totalstrains is large. This ratio decreases dramatically later in the unloading phase with plasticstrains taking over. Figure 6.24 shows the same simulation as in Figure 6.23 but withplastic unloading instead of elasto-plastic unloading. It is noteworthy that the smalldilative part in segment a-b disappeared. This proves that this dilative part is due to thelarge influence of elasticity at the beginning of the unloading phase. Similar behaviour isobserved in the simulations for all other tests. It was shown in Section 6.1.2 that thevalues of the elastic bulk modulus used in the model are lower than those directlycalculated form the unload-reload tests data (see Figure 6.6). This could be one of thereasons 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 thedecreasing p’ and this well known behaviour cannot be ignored. The elastic propertiesused for Erksak sand are not accurate in the absence of bender element testing (seeSection 6.1.2). Therefore, other data sets with bender elements tests are needed to verifythis point. Reloading for loop 2 is elastic (part b-c) until yield occurs once the stresspoint hits the outer yield surface.Unloading for the first loop, shown at large scale in Figure 6.25, starts with a linearelastic phase where soil expands in unloading (part a-b), followed by an elasto-plasticphase (part b-c). Note that the stress point does not touch the internal cap in the previousloading for this ioop. Reloading is elastic (part c-d) until the stress point touches theouter yield surface.1472.00Model1.601o.ooO1IO2’O 2561:%1.0 ——— ——________0.5Contractiont-2.5Lab results-3.0s:%Figure 6.22. Comparison between calibrated NorSand model and ES_CID_867.1481.0w0.50.0-0.5Figure 6.23. Zoom on the second ioop of comparison between calibrated NorSand modelwith elasto-plastic unloading and ES_CID_867.Figure 6.24. Zoom on the second loop of comparison between calibrated NorSand modeland ES_CID_867 with plastic unloading.8:%1.00.50.0-0.5e1: %1490.80.6• 0.400.20.081: %Figure 6.25. Zoom on the first ioop for ES_CID_867.6.4. SummaryThis chapter presented a triaxial monotonic calibration of the NorSand model to Erksaksand and Fraser River sand. NorSand requires 8 parameters: 3 critical state parameters, 3plasticity parameters, and 2 elasticity parameters. The unload-reload model presented inChapter 5 requires 3 additional parameters: H, Hr, and%2.An unload-reload calibrationwas performed on Erksak sand using cyclic triaxial tests. It was found that Hr and%2arerelated to the monotonic parameters Hand,j,respectively, such that Hr 4Hand12 =1.41i. Hr changes from a value of 4Hto a value ofHonce the yield surface beforeunload-reload is exceeded. A constantL[of 30 fits the unload plots on average.0 0.5 1 1.5 2 2.5 31507. PREDIcTIoNs OF FRASER RIVER SAND UI4LOAD-RELOADBEHAVIOURThe parameters/relations from the monotonic calibration for Fraser River sandsupplemented by the parameters/relations derived from the unload-reload calibration forErksak sand were used to predict the two load-unload-reload triaxial tests on Fraser Riversand using the modified NorSand model developed earlier in Chapter 5. The two triaxialtests modelled, FR_CID_0l and FR_CID_02 were described in Table 4.4. TestFR_CID_0l is on a loose sample (pl9OkPa & = 0.012) while test FR CID_02 is ona dense sample (p’=l98 kPa & yi = -0.156). Both samples were moist tamped.7.1. Model parametersAs discussed previously in Chapter 6, three additional material parameters are neededfor the proposed unload-reload calibration:“Ha” and ‘H;’, the plastic hardening modulusin unloading and reloading respectively andX2,the slope of a trend line through thesecond peaks of the Dmjn versus yat Dmin data. The same value forH as used for Erksaksand was assumed (i.e. H = 30). The relation Hr = 4H, derived for the Erksak Sandcalibration, was also used for the reloading portion of the stress path.Hr is used duringloading 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 ofX2for FraserRiver sand was obtained by assuming the same ratio between%iand%2for Fraser Riversand as obtained from the Erksak calibration. Table 7.1 is a summary of the unload-reload parameters used for the predictions. The monotonic parameters were presentedpreviously in Table 6.5.151Table 7.1. Parameters used for Fraser River sand unload-reload predictions.Parameter Erksak sand RemarkH 30 Plastic hardening parameter inunloadingH,. 4H Plastic hardening parameter inreloadingZ26.12 Slope of the line relatingDmjn toiatDmjn defined for the secondpeaks7.2. Model predictionsThe updated NorSand model was run using the Fraser River monotonic calibration andthe unload-reload properties/relations derived from the Erksak unload-reloadcalibration.No iteration to the input parameters to improve the fits wasattempted. The truemeasured and NorSand computed stress and strain responses aregiven in Figure 7.1 andFigure 7.2 for the loose (FR_CID_01) and dense (FR_CLD_02) tests respectively.The prediction for FR_CID_0 1 in bothq —es and i —j shows slightly lower curvesthan laboratory data in the range of 6j values greater than2% (Figures 7.1 a-b). Thedifference between the predictions and the laboratory data starts decreasingat higheraxial strains, i.e.> 15%. Although the predictions and laboratory datafor the monotonicparts 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 identicalto those for thelaboratory data. It can be noticed that the area inside the loops is largerat higher axialstrains. The predictions simulate this behaviour.The model generally predicts more contractive behaviour for FR_CID_01 comparedtolaboratory data (Figures 7. ic). Both laboratory data and the predictionsare very similarup to si of around 5%. At higher axial strains, the difference betweens from laboratory152data and predictions increases with a maximum difference of more than 1% at very highaxial strains.Similar to FR_CID_01, the monotonic parts of q —8j and —j for the predictions forFR_CID_02 are slightly lower compared to laboratory data (Figure 7.2a-b). Thedifference between the two plots, i.e. laboratory data and predictions, decreases at higheraxial strains close to critical state. The slopes of the unload-reload loops are almostidentical for both the predictions and laboratory data.The predictions are quite accurate in s, - s for FR CID 02. The maximum differencebetween i values for the predictions and laboratory data in-for FR_CID_02 isaround 0.5% at s = 25% (Figure 7.2c) However, this difference is much smaller atlower81values and the predictions are very accurate. The unload response for the loopat 8j = 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 forFR_CID_0 1. The model captures this behaviour accurately. FR_CID_02 has a higher Hvalue, and therefore a higher Hr. This yields a stiffer reloading response for FR_CID_02and the area inside the unload-reload loops is increased.The predictions for FR_CID_02 in the - plot better match laboratory data thanthose for FR_CID_01 (Figures 7.lc & 7.2c). This is because most of the tests used in themonotonic calibration for Fraser River sand were done on dense samples (see Section6.2). Therefore, the derived parameters are expected to better fit dense samples. Thepredictions 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 ofErksak sand provide a reasonably good calibration for Fraser River sand. In q —8j and17 —8j, the monotonic parts of the plots for the predictions for both tests are slightly lowerthan those for laboratory data. However, the slopes of the unload-reload parts are very153similar. In a,, — 6j, test FR_CID_02 with the dense sample had better fits than testFR_CIDO1 with the loose sample. This is expected as the calibration in Chapter 6 wasdone using dense samples. Using a constant H value fits the unloading parts on average.A constantI-Iaresults in a simpler calibration although it is evident that unloading issofter at higher strains.For future work, other sets of unload-reload tests are needed to better understand thefactors that H depends on. The predictions could be further improved by including moreunload-reload tests on loose samples in the calibration and tests with more unload-reloadloops.7.3. Discussion of model predictionsTo understand the unload-reload behaviour of sands, special laboratory testingprograms that are not commonly performed in industry are required. However,monotonic laboratory testing is part of the routine in industry. Hence, a model that cansimulate the unload-reload behaviour using unload-reload parameters that can becorrelated to monotonic parameters is of practical significance.A challenge for any constitutive model is obtaining good quality laboratory informationto calibrate the model. Typically, the simpler the model calibration, the greater the utilityof the model for real engineering problems. This section has investigated whether thestandard monotonic calibration for NorSand, supplemented by relations observedbetween the monotonic and three unload-reload parameters for Erksak sand, can be usedto 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“X2”,the slope of a trend line through the second peaks of the Dmjn versusçt’ atDmin data.Fromthe unload-reload calibration to Erksak sand presented in Chapter 6, it was shown that Hr154andX2are correlated to the monotonic H andXi, respectively (Hr 4H &X2 =l.4iXi).A constant value ofH equal to 30 was shown to fit the unloading curves on average.The predictions for Fraser River sand presented in this chapter suggest that usingcorrelations from the Erksak calibration gives quite good fits. The slopes of the predictedunload-reload loops in q—and i —6j were very similar to those for laboratory data. Thedense sample gave reasonably accurate fits in — 6j plot. The predictions for the densesample were better than those for the loose sample. However, the fits for the loose sampleare still quite good up to axial strain of around 5%.Similar to observation from Erksak data, Fraser River sand simulations show that thecode 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 betterpredictions. Figure 7.3 shows a model simulation for FR_CID_02 with constant equalto.The predictions with changing 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 ofaround 5%. This is because the reloading loops in that range are pre-peak (i.e. start frompre-peak conditions) and therefore both simulations are based on = 4.34. The laterreloading loops start from post-peak conditions and therefore % changes to 6.12 for thesimulation in Figure 7.2c. It is noteworthy that changing gives identical predictions asconstant,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 betweenmonotonic and unload-reload parameters was undertaken based on the triaxial tests for asingle sand, Erksak. It is likely that these correlations will be improved as the database oftriaxial 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 fewtriaxial tests with unload-reload loops should be done for more accurate results.1557.4. SummaryThe standard monotonic calibration for NorSand, supplemented by relations observedbetween the monotonic and the three unload-reload parameters for Erksak sand, wassuccessfully used to predict the cyclic behaviour of another sand: in this instance FraserRiver sand. The main finding of this chapter is that the unload-reload behaviour can besimulated using unload-reload parameters that can be correlated to monotonicparameters. It is likely that these correlations will be improved as the database of unloadreload tests expands.156(a)600Test data_______500 1400Predictioi300Fraser River sand200e0 =0.89100p’l9OkPa0 I0 5 10 15 20 256j ..%(b)2.0Test data1.61.20.80.40.00 5 10 15 20 258j %(c)4.Prediction2tatafcIDilative0- I IFigure 7.1. Predictions for Test FR_CID_0 1 (a) q—8j (b) i —si (c) s—8i.157(a)1000_____Test data800c 600400Fraser River sand= 0.72200= 198 kPa00 5 10 15 20 2561: %(b)2.0Test data1.6120.80.40.00 5 10 15 20 2561: %(c)IContractive°Prediction-4-5-6Figure 7.2. Predictions for Test FR_CID_02 (a) q—ei (b) i —6j (c)158NabthE1: %Figure 7.3. Model simulation for Test FR_CID_02 in 6—6i with constant,‘of 4.34.1598. SUMMARY AND CONCLUSIONS8.1. Context of ResearchThe behaviour of sands during loading has been studied in great detail. However, littlework has been devoted to understanding the response of sands in unloading. This issurprising as the behaviour of sands in unloading is of great practical importance forearthquake 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 mayexhibit contractive behaviour when unloaded. Drained cyclic simple shear tests showsimilar behaviour in unloading (Sriskandakumar, 2004). Therefore, it is clear that soilbehaviour in unloading is not wholly elastic. The drained behaviour of sands inunloading was investigated in this work as well as strength and deformationcharacteristics in reloading. A practical continuum model that accounts for inelasticunloading (i.e. yields in unloading) was introduced.The tendency to contract upon unloading during an earthquake could result inliquefaction. Undrained cyclic simple shear tests show that the increase in pore waterpressure generated during the unloading cycle often exceeds that generated duringloading. A model that yields in unloading is needed to predict this behaviour.1608.2. Research ObjectivesThe main objectives of this research were:1. Develop a theoretical understanding of stress-dilatancy in unloading. Thisinvestigation includes the interaction between soil fabric and stress-dilatancy.2. Utilize the theoretical understanding to guide development of unload-reloadbehaviour, including yielding during unloading, into a constitutive model.8.3. MethodologyThe 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 wasbased on observations from a series of triaxial tests with unload-reload loops on Erksaksand. Calibration of the model for monotonic and unload-reload stress path wereperformed for Erksak sand. To determine whether the monotonic to unload-reloadrelations observed from the Erksak unload-reload calibrations could be applied generally,a prediction for unload-reload on Fraser River Sand was undertaken. First a monotoniccalibration of NorSand to Fraser River sand was done. Then the monotonic calibration onFraser River sand and the unload-reload parameters for Erksak sand were used to predictthe results of triaxial tests with load-unload-reload cycles on Fraser River sand.8.4. ConclusionsThe theoretical study of deformation characteristic of an assembly of rigid rods showsthat the observed soil contraction in unloading is to be expected. The relation betweenstress and dilatancy during unloading depends on particle arrangement. As the161arrangement of particles is expected to change with cyclic loading, the stress-dilatancyrelation in reloading differs from that for first loading.The study of ten drained conventional triaxial tests on Erksak sand, including betweenone and three unload-reload loops, indicates that significant amounts of contraction canoccur during unloading. This is contrary to the dilatant elastic unloading response oftenassumed in constitutive models of soil. This observed behaviour is consistent with theresults of the theoretical study.Experimental observations indicate that unloading loops starting from pre-image stressratio are dominated by small amounts of dilation, while those starting from post-imagestress ratio are dominated by significant amounts of contraction. The effect of theunload-reload loops on peak strength is small. This observed contraction in unloadingcan be explained based on the saw-tooth model. The sawtooth model suggests that themore soil dilates in loading, the more potential energy the soil stores. This energy isavailable to be released as contraction in subsequent unloading, as observedexperimentally.The results of the series of tests on Erksak sand show that soil becomes more dilatant inpost-peak reloading phases (i.e. reloading loops occurring post-peak). However, thestress-dilatancy relation remains as in first loading for pre-peak reloading loops.The NorSand constitutive model was extended to represent the experimentally observedyielding 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 zerodilatancy in unloading, M. Soil is assumed to yield in unloading on a vertical cap atDmgn. This assumption fits the framework of monotonic NorSand and is consistent withobservations from laboratory data. The cap contracts in unloading dragging the outeryield surface with it. As a result the outer yield surface softens due to yield in unloading.162The 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 introducedin this work: “Ha” and “Hr”, the plastic hardening modulus in unloading and reloadingrespectively and“X2”,the slope of a trend line through the second peaks (i.e. for post-peak reloading phases) in Dmjn - çvplot.X2is used to capture the effect of inducedchanges 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 behaviourof sand under load-unload-reload cycles. H was significantly smaller than themonotonic loading hardening, H, and the constant value of 30 provided good fits to theexperimental unload-reload data on average. The code is not sensitive to changes in H,.because it has a high value of around 4H. %2was always observed to be higher than%and for Erksak sand%2exceededXiby 40%. The calibrated model predicted the resultsof triaxial tests with load-unload-reload cycles on Fraser River sand with good precision.Overall, this thesis introduced stress-dilatancy relations for unloading and reloadingbased on experimental observation, supported by the findings of an investigation done atthe micro-mechanical level. An elasto-plastic continuum model that yields in unloadingwas proposed. It was shown that the unload-reload behaviour can be simulated usingunload-reload parameters that can be correlated to monotonic parameters.8.5. Suggestions for Future WorkIn summary, this research presents a practical model for load-unload-reload cycles onsand that incorporate inelastic unloading. It accounts for induced changes in particlearrangement. The model gives accurate predictions for triaxial Laboratory data. Thisresearch is limited to a triaxial compression framework. For future research, theproposed model can be implemented in a finite elements code and therefore it needs to be163validated for general stress path conditions. An example of this wouldbe a stress pathwith decreasing mean effective stress at constant shear stress.164REFERENCESASTM 2006a, “Standardtest methods for maximumindex density of soils usingavibratory table (D4253-00-2006),In 2006 Annual Book of ASTMStandards, sect. 4,Vol. 4.08, ASTM, Philadelphia.ASTM 2006b, “Standardtest methods for minimumindex density of soils and calculationof relative density (D4254-00-2006),In 2006 Annual Bookof ASTM Standards, sect.4, Vol. 4.08. ASTM,Philadelphia.Been, K., and Jefferies,M., 1985, “State parameterfor 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 parametersfor undisturbed and remouldedsoilspecimens”, Proc. RoscoeMemorial Symp., Cambridge,3-58.Bolton, M. D., 1986,“Strength and dilatancyof sands”, Géotechnique,Vol. 36, No. 1,pp.65-78.Byrne, P., Cheung, H.,Yan, L., 1987, “Soil parametersfor deformationanalysis of sandmasses”, Canadian GeotechnicalJournal, Vol.24,pp.366-376.Casagrande, A., 1936, “Characteristicsof cohesionless soilsaffecting the stabilityofearth fill”, Journal ofBoston Society of Civil Engineers,Vol. 23,pp.257-276.165Chillarige, AnnajiRao, V., Robertson, P., Morgenstem, N., and Christian, H., 1997,“Evaluation of the in situ state of Fraser River sand”, Canadian GeotechnicalJournal, Vol. 34, No. 4, pp. 510-519.Collins, I. F., 2005, “The concept of stored plastic work or frozen elastic energy in soilmechnics”, Géotechnique, Vol. 55, No. 5,pp.373-382.Drucker, D. C., 1951, “A more fundamental approach to stress-strain relations”, Proc.First US National Congress of Applied Mechanics, ASME,pp.487-49 1.Drucker, D. C., and Seereeram, D., 1987, “Remaining at yield during unloading and otherunconventional 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.1629-1653.Golder Associates, 1987, “Report on unload testing to assess initial fabric of Erksaksand”, Golder Associates, Calgary, AlbertaGolder Associates. Erksak 330/0.7 triaxial tests, available at www.golder.com/liq.Ghafghazi, M., and Shuttle, D., 2006, “Accurate determination of the critical statefriction angle from triaxial tests”, Proc.59thCanadian Geotechnical Conference,Vancouver, BC., Canada.Ishihara, K., Okada, S., 1978, “Yielding of overconsolidated sand and liquefaction modelunder 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.91-103.166Jefferies, 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 isotropiccompression 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, JohnWiley& sons, New Jersey, USA.Nova, R., 1982, “A constitutive model for soil under monotonic and cyclic loading”,inSoil Mechanics Transient and Cyclic Loads, Wiley,pp. 343-373.Oda, M., 1972, “Initial fabrics and their relation to mechanical properties ofgranularmaterial”, 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 1.167Roscoe, 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 wetclay”, Engineering Plasticity, Cambridge University Press, Cambridge, pp. 535-609.Rowe, P. W., 1962, “The stress-dilatancy relation for static equilibrium of an assembly ofparticles in contact’, Proc. of the Royal Society, A269, 500-527.Rowe, P. W., 1969, “The relation between the shear strength of sands in triaxialcompression, 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 ofsand”, M.A.Sc Thesis, University of British Columbia, Vancouver, BC, Canada.Sriskandakumar, 5., 2004, “Cyclic loading response of Fraser River sand for validation ofnumerical models simulating centrifuge tests”, M.A. Sc Thesis, University of BritishColumbia, Vancouver, BC, Canada.Tatsuoka, F., and Ishihara, K., 1974, “Yielding of sand in triaxial compression”, Soilsand Foundations, Vol. 14,pp.63-76.Terzaghi, K., Peck, R., and Mesri, G., 1996, “Soil Mechanics in Engineering Practice”,3rdedition, John Wiley & sons, New York, USA.Vaid, Y. ,and Sasitharan, 5., 1992, “The strength and dilatancy of sand”, CanadianGeotechnical Journal, Vol. 29, No. 3,pp.522-526.Vesic, A. S., and Clough, G., W., 1968, “Behavior of granular materials under highstresses”, Journal of the Soil Mechanics and Foundations Division, Vol. 94, pp. 661-688.168Wijewickreme, D., Sriskandakumar, S., and Byrne, P., 2005, “Cyclic loading response ofloose air-pluviated Fraser River sand for validation of numerical models simulatingcentrifuge tests”, Canadian Geotechnical Journal, Vol. 42, No. 2,pp.550-561.Wood, D. M., 1990, “Soil behaviour and critical state soil mechanics”, CambridgeUniversity Press, Cambridge, UK.Yasufuku, N., Murata, H., and Hyodo, M., 1991, “Yield characteristics of anisotropicallyconsolidated sand under low and high stresses”, Soils and Foundations, Vol. 31, No.1,pp.95-109.169APPENDIX A: PREDICTION OF STRESS DILATANCY IN UNLOADINGIt is assumed that stress-dilatancy in unloading can be represented by EquationA. 1.This appendix presents a quantitative justification using the availabledata on Erksak sand(Table 4.2). Figures A.1 to A.1O show a comparison between Equations A.1to A.3 andlaboratory data.D°=ii—M (A.l)Where,= 2iL—1.5 (A.2)Where27Lis the last value of stress ratio from the previous loading phase. Plasticdilatancy 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 notpresented in thefollowing figures. Those loops are highly influenced by elasticity.Using plastic dilatancyrelation to fit those loops would give inaccurate results. It is assumedthat the effect ofelasticity is negligible for the unload/reload loops at higher stresslevels.1701 4• Loading I-s Unloading I-- • Loading2U -• •• : -- - - -- :-- Model fit to UIII-- 0.40.2-1.5 -1 -0.5D0 0.5Figure A. 1. Predicted and measured stress-dilatancy for ES_CID_860.• Loading I1.6a Unloading 1Loading 2 -x Unloading 2 - -)< -z Loading 3x - - -- aU0.8- -- Model fit to UI -- - IandU2•0.6U‘a 0.4K.;’I,0.2a’_,-1.5 -I -0.5 0 0.5DFigure A.2. Predicted and measured stress-dilatancy for ES_CID_86 1.171• Loading 11.6 -• Unloading 1•‘z’;’x Loading 2• -x Unloading 2• __ ‘—-- ILoading 3• -‘- x: - - - - -X0.8• Unloading 3- - 0.6+ Loading 4:-- :x4 - - - Model fit tot •0.4 •• U2 and U3-1.5L0.2 ‘a.1.5Figure A.3. Predicted and measured stress-dilatancy for ES_CID_862.• Loading 1Ia Unloading 1 -Loading2-Unloading2 xLoading 3 x0.8—— ,.4- -- ModelfittoUlj0.64.andU2C’ E. .•-1.5 -1 -0.5 0 0.5 1DFigure A.4. Predicted and measured stress-dilatancy for ES_CID_866.1724 t?DFigure A.5. Predicted and measured stress-dilatancy for ES_CID_867.DFigure A.6. Predicted and measured stress-dilatancy for ES_CID_868.• Loading I• Unloading 1Loading 2x Unloading 2Loading 3• Unloading 3+ Loading 4-- Model fit for U2 and U3...‘,•• - -0.80.6•I_ •a:04’— —— ..4;.,•I -- ••x41E •-2 -1.5 -1 -0.5 0 0.5 1• Loading 1• Unloading 1e Loading 2x Unloading 2xXx Loading 3 xX- -- Model fitto UI andU2X - -0.6;‘ 0.4a0.2x•xc-1.5 -1 -0.5 0 0.5 1$C A173__-__- - . Loading 1Unloading Ix Unloading 2I --+ZzLoading 3xX+. Unloading 3•‘-0.8+:,_ +Loading 4-- X - -- Model fit to U3-0.6x>0.4-ci’0.2-I.- .-1.5 -1 -0.5 0 0.5DFigure A.7. Predicted and measured stress-dilatancy for ES_CID_870.1.4 - . Loading 1-. Unloading ILoading 2•a‘1 - - - Model fit to Ui0.6 --1.5 -1 -0.5D::Figure A.8. Predicted and measured stress-dilatancy for ES_CID_87 1.174DFigure A.9. Predicted and measured stress-dilatancy for ES_CID_872.DA I1Figure A. 10. Predicted and measured stress-dilatancy for ES_CID_873.• Loading 14• Unloading ILoading 2x Unloading 2x Loading3- -- ModelfittoUlandU2 0.84,,0.60.40.2Es)-1.5 -1 -0.5 0 0.5 1• Loading I• Unloading Ii Loading 2x Unloading 2x Loading 3+ Unloading 3- Loading 4- -- Model fit to U2 and U3c_+1,xX:“)X•. 0.4+4*_ —I I - I-2 -1.5 -1 -0.5 0 0.5175APPENDIX B: RESULTS OF THE UNLOAD-RELOAD CALIBRATIONFOR ERKSAK SANDThis appendix presents a comparison between the calibrated model results for Erksaksand as compared to triaxial data. The load-unload-reload calibration used to produceFigures B.1 to B.7 was introduced in Section 6.1.2.001.601.200.800.400.00-2:%Figure B. 1. Load-unload-reload calibration results compared to laboratory data forES_CID_860.0 5 10 15 20 251762.00 — —_____________________________________________Model1.601’1.20Labres0.80 ,LIII0.400.000 5 10 15 20 25Ei: %1•IContractionoModel—1-2-3-4-5Figure B.2. Load-unload-reload calibration results compared to laboratory data forES_CID_86 1.1772.00Model1201sEfl0.400.00 — I0 5 10 1520 25e1: %tContractionFigure B.3. Load-unload-reload calibration results compared tolaboratory data forES_CID_862.1782.00Model1.601.200.800 00iai,resuits_f - I0.400 5 10 15 20 25ci: %tContractionMode.Coi1Di1aDilation—1Lab resultsFigure B.4. Load-unload-reload calibration results compared to laboratory data forES_CID_866.1792.00 ——-——— — ——___________________________Model1.6011.2010 15 20 25E: %1.00.5tContraction-25Labresults-3.0ei:%Figure B .5. Load-unload-reload calibration results, compared to laboratory data forES_CID_867.180ModelE1: %0 5 102.001.601.200.800.400.002I015 20 25CA)Figure B.6. Load-unload-reload calibration results compared to laboratory data forES_CID_868.1812.00Model1.601.200.800.400.000 5 10 15 20 25C1: %2tContraction0I lilation-203-4-6-8Figure B.7. Load-unload-reload calibration results compared to laboratory data forES_CID_873.182APPENDIX C: FRASER RIVER SAND MONOTONIC CALIBRATIONRESULTS6005004003002001000A monotonic calibration of NorSand for Fraser River sand was introduced in Section6.2. The tests used in the calibration are presented in Table 4.5. Figures C. 1 to C.6 showa comparison between the calibrated model results and laboratory data.0 5 10 15 20 258: %-2-6-8-10-121.00.90.80.70.6100 1000p’: kPaFigure C. 1. Monotonic calibration results compared to tests data for FR_CID_0318320000Sa0 5 10 15 20 256j. %20-2-4-6-8-101.00.90.80.70.6--— — — —100 1000p’: kPaFigure C.2. Monotonic calibration results compared to tests data for FR_CID_04.16001200800400FR_CID_041842000de=O.6910-1-2C)-3-4-5CC0 5 10 15 20 2581: %16001200800400— FR_CID_05 p’ = 515kPa---- ---_%%%%—— -0.90.80.70.6100 1000 10000p’kPaFigure C.3. Monotonic calibration results compared to tests data for FR_CID_05.185300200100020-2-6-8-100 5 10 15 20 258j. %.%%%SL%%%%%—--100p’: kPaNorSandFR_CID_06e0 = 0.75p’5OkPaI1.00.90.80.710 1000Figure C.4. Monotonic calibration results compared to tests data for FR_CID_06.186300250200150NorSande0 0.91100_____p’ 388 kPa50FR_CU_0100 5 10 15 20 25 3081: %400rM200100.1I0400300. 200Cs1000 100 200 300 400p’: kPaFigure C.5. Monotonic calibration results compared to tests data for FR_CU_0 1.187600500/orSd082200100FR_CU_02p’ = 196 kPa00 5 10 15 20 25 3081. %160120z8040-404003002001000 50 100 150 200 250 300 350 400p’: kPaFigure C.6. Monotonic calibration results compared to tests data for FR_CU_02188APPENDIX D: STEPS TO IMPLEMENT THE LOAD-UNLOAD-RELOAD MODEL IN A CODEFigures D.1 to D.3 show the steps that can be followed in coding the load-unload-reload model.8L) Addelastic strainsand move tostep 1U1L)Applyplastic shearstrainincrement2L) CalculateM and thecurrent plasticdilation rate6L) Applyconsistencycondition toget the newstress state5L) Harden!soften theyield surface4L) imposethe limit onthe maximumdilation rate3L) RecoverplasticvolumetricstrainsFigure D. 1. A diagram illustrating loading in NorSand.7L) Updatestrains andstate1895U) Soften theouter yield surface6U) Add elasticstrains. Then moveto Step 1R forsubsequentreloading4U) Recoverplastic volumetricstrain incrementsfrom stressdilatancy andupdate strains andstate1U) Get thecurrent location ofthe internal cap forthe last yieldsurface inloading/reloading.IfpPp’skipstep 2U.3U) Apply thehardening rule onthe inner yieldsurface (internalcap) and recoverthe new stress state2U) Recoverelastic strains.Yield surface doesnot change. If pbecomes= Poapgoto 3U. Otherwise,go directly to 6U.Figure D.2. Description of unloading in the model.1905R) Go to lUforsubsequentunloading1R) Apply shearstrain increment3R) Proceed as inloading startingfrom 2L. UseX2ifpeak is exceeded ina previous loading2R) Recover elasticstrains until yieldsurface defined in5U is exceededFigure D.3. Description of reloading in the model.4R) If largest pastyield surface isexceeded use Hinstead ofHr191APPENDIX E: TRIAXIAL TESTINGPROCEDURE1. IntroductionThis appendix briefly presents the triaxialtesting procedure and sample preparationmethods followed to produce the results ofthe triaxial tests presented in Chapter4. Amore comprehensive descriptionof the testing procedure can be found in Golder,1987.The triaxial apparatus used wasdescribed in Section 2.2 (Figure 2.13). Alltestsreported in Chapter 4 were displacementcontrolled. Load was applied bya 19mmdiameter stainless steel piston.The load was measured by a loadcell. Cell pressure wasmeasured using a pressure transducer. Volumechange was measured using a cylindricalpiston with a linear displacementtransducer that was calibrated to measurevolumechange.2. Sample preparation• Wet pluviation: A sampleof air-dried sand is placed ina long neck flask. The flaskis then filled with de-aired waterand placed under vacuum. The membranemouldis filled with de-aired water andthe flask is inverted with its neck25mm above thebottom of the mould. Sand thenflows out of the flask. The side ofthe mould istapped to reach the desiredvoid ratio.• Most tamping: The sample is tamped in6 layers. Distilled water is addedto airdried sand to yield water contentof 5-6%. 6 equal weights, prepared to givethedesired void ratio, are tamped insidethe mould using a tamper with adjustablestops.1923. Testing procedure• The lower platen is installed and the membrane (0.3mm thick) is attached usingthe 0-rings. The split mould is mounted and vacuum is applied to keep themembrane 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 andattached with 0-rings. Vacuum is released and a negative pore pressure of 10-2OkPa 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 topplaten and then locked. The LVDT is zeroed.• A cell pressure of 2OkPa is applied while the negative pore pressure is beingreleased. The change in volume is recorded.• For moist tamped samples, carbon dioxide is bubbled through the sample for 3hours. 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 tosaturation is recorded. The change in volume is also recorded.• The sample is consolidated hydrostatically by increasing the cell pressure inincrements. The change in height and volume are recorded.• All transducers are zeroed and the sample is sheared under either drained orundrained conditions.• The drainage line is shut and the sample is frozen. The water content and voidratio are determined using the frozen sample.193

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.24.1-0063096/manifest

Comment

Related Items